{"id":15661,"date":"2019-09-05T12:08:15","date_gmt":"2019-09-05T16:08:15","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/chapter\/solve-systems-of-nonlinear-equations-2\/"},"modified":"2019-09-05T12:08:15","modified_gmt":"2019-09-05T16:08:15","slug":"solve-systems-of-nonlinear-equations-2","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/chapter\/solve-systems-of-nonlinear-equations-2\/","title":{"raw":"Solve Systems of Nonlinear Equations","rendered":"Solve Systems of Nonlinear Equations"},"content":{"raw":"[latexpage]<div class=\"textbox textbox--learning-objectives\"><h3 itemprop=\"educationalUse\">Learning Objectives<\/h3>By the end of this section, you will be able to: <ul><li>Solve a system of nonlinear equations using graphing<\/li><li>Solve a system of nonlinear equations using substitution<\/li><li>Solve a system of nonlinear equations using elimination<\/li><li>Use a system of nonlinear equations to solve applications<\/li><\/ul><\/div><div data-type=\"note\" id=\"fs-id1163873512681\" class=\"be-prepared\"><ol id=\"fs-id1163873600190\" type=\"1\"><li>Solve the system by graphing: \\(\\left\\{\\begin{array}{c}x-3y=-3\\hfill \\\\ x+y=5\\hfill \\end{array}.\\)<span data-type=\"newline\"><br \/><\/span> If you missed this problem, review <a href=\"\/contents\/a236420d-b79a-43e3-8f7c-1b8e18c5045e#fs-id1167834279490\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li><li>Solve the system by substitution: \\(\\left\\{\\begin{array}{c}x-4y=-4\\hfill \\\\ -3x+4y=0\\hfill \\end{array}.\\)<span data-type=\"newline\"><br \/><\/span> If you missed this problem, review <a href=\"\/contents\/a236420d-b79a-43e3-8f7c-1b8e18c5045e#fs-id1167835328722\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li><li>Solve the system by elimination: \\(\\left\\{\\begin{array}{c}3x-4y=-9\\hfill \\\\ 5x+3y=14\\hfill \\end{array}.\\)<span data-type=\"newline\"><br \/><\/span> If you missed this problem, review <a href=\"\/contents\/a236420d-b79a-43e3-8f7c-1b8e18c5045e#fs-id1167834121139\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li><\/ol><\/div><div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1163873610411\"><h3 data-type=\"title\">Solve a System of Nonlinear Equations Using Graphing<\/h3><p id=\"fs-id1163873783473\">We learned how to solve systems of linear equations with two variables by graphing, substitution and elimination. We will be using these same methods as we look at nonlinear systems of equations with two equations and two variables. A <strong data-effect=\"bold\">system of nonlinear equations<\/strong> is a system where at least one of the equations is not linear.<\/p><p id=\"fs-id1163873731492\">For example each of the following systems is a <span data-type=\"term\">system of nonlinear equations<\/span>.<\/p><div data-type=\"equation\" id=\"fs-id1163873644550\" class=\"unnumbered\" data-label=\"\">\\(\\begin{array}{ccccccc}\\hfill \\left\\{\\begin{array}{c}{x}^{2}+{y}^{2}=9\\hfill \\\\ {x}^{2}-y=9\\hfill \\end{array}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{5em}{0ex}}\\left\\{\\begin{array}{c}9{x}^{2}+{y}^{2}=9\\hfill \\\\ y=3x-3\\hfill \\end{array}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{5em}{0ex}}\\left\\{\\begin{array}{c}x+y=4\\hfill \\\\ y={x}^{2}+2\\hfill \\end{array}\\hfill \\end{array}\\)<\/div><div data-type=\"note\" id=\"fs-id1163873820382\"><div data-type=\"title\">System of Nonlinear Equations<\/div><p id=\"fs-id1163873946484\">A <strong data-effect=\"bold\">system of nonlinear equations<\/strong> is a system where at least one of the equations is not linear.<\/p><\/div><p id=\"fs-id1163873850941\">Just as with systems of linear equations, a solution of a nonlinear system is an ordered pair that makes both equations true. In a nonlinear system, there may be more than one solution. We will see this as we solve a system of nonlinear equations by graphing.<\/p><p id=\"fs-id1163873940693\">When we solved systems of linear equations, the solution of the system was the point of intersection of the two lines. With systems of nonlinear equations, the graphs may be circles, parabolas or hyperbolas and there may be several points of intersection, and so several solutions. Once you identify the graphs, visualize the different ways the graphs could intersect and so how many solutions there might be.<\/p><p id=\"fs-id1163873892611\">To solve systems of nonlinear equations by graphing, we use basically the same steps as with systems of linear equations modified slightly for nonlinear equations. The steps are listed below for reference.<\/p><div data-type=\"note\" id=\"fs-id1163873798518\" class=\"howto\"><div data-type=\"title\">Solve a system of nonlinear equations by graphing.<\/div><ol id=\"fs-id1163870411020\" type=\"1\" class=\"stepwise\"><li>Identify the graph of each equation. Sketch the possible options for intersection.<\/li><li>Graph the first equation.<\/li><li>Graph the second equation on the same rectangular coordinate system.<\/li><li>Determine whether the graphs intersect.<\/li><li>Identify the points of intersection.<\/li><li>Check that each ordered pair is a solution to both original equations.<\/li><\/ol><\/div><div data-type=\"example\" id=\"fs-id1163870369820\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1163873620675\"><div data-type=\"problem\" id=\"fs-id1163873632057\"><p id=\"fs-id1163873866469\">Solve the system by graphing: \\(\\left\\{\\begin{array}{c}x-y=-2\\hfill \\\\ y={x}^{2}\\hfill \\end{array}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163870381445\"><table id=\"fs-id1163873912472\" class=\"unnumbered unstyled can-break\" summary=\"Identify each graph of the system x minus y is equal to negative 2, which is a line, and y is equal to x squared, which is a parabola. Sketch the possible options for intersection of a parabola and a line. When a parabola and a line do not intersect, there the system has 0 solutions. When a parabola and line intersect at a single point, the system has one solution. When a parabola and line intersect at two points, the system has two solutions. Graph the line x minus y is equal to negative two. The slope intercept form of the line is y is equal to x plus 2. Graph the parabola, x squared. On a coordinate plane, the line has a slope of 1 and a y-intercept of 2 and the parabola has a vertex at (0, 0) and opens upward. They appear to be (2, 4) and (negative 1, 1). Check to make sure each solution makes both equations true. For (2, 4), is 2 minus 4 equal to negative 2? Negative is equal to negative 2. For (2, 4), is 4 equal to 2 squared? 4 is equal to 4. For (negative 1, 1), is negative 1 minus 1 equal to negative 2? Negative 2 is equal to negative 2. For (negative 1, 1), is 1 equal to the square of negative 1. 1 is equal to 1. The solutions are (2, 4) and (negative 1, 1).\" data-label=\"\"><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Identify each graph.<\/td><td data-valign=\"top\" data-align=\"left\">\\(\\left\\{\\begin{array}{ccc}x-y=-2\\hfill &amp; &amp; \\text{line}\\hfill \\\\ y={x}^{2}\\hfill &amp; &amp; \\text{parabola}\\hfill \\end{array}\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Sketch the possible options for<span data-type=\"newline\"><br \/><\/span>intersection of a parabola and a line.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163873819964\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_001a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Graph the line, \\(x-y=-2.\\)<span data-type=\"newline\"><br \/><\/span>Slope-intercept form \\(y=x+2.\\)<span data-type=\"newline\"><br \/><\/span>Graph the parabola, \\(y={x}^{2}.\\)<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163874012484\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_001b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Identify the points of intersection.<\/td><td data-valign=\"top\" data-align=\"left\">The points of intersection appear to be \\(\\left(2,4\\right)\\) and \\(\\left(-1,1\\right).\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Check to make sure each solution makes<span data-type=\"newline\"><br \/><\/span>both equations true.<span data-type=\"newline\"><br \/><\/span>\\(\\left(2,4\\right)\\)<span data-type=\"newline\"><br \/><\/span> \\(\\phantom{\\rule{1em}{0ex}}\\begin{array}{cccccccc}\\hfill x-y&amp; =\\hfill &amp; -2\\hfill &amp; &amp; &amp; \\hfill y&amp; =\\hfill &amp; {x}^{2}\\hfill \\\\ \\hfill 2-4&amp; \\stackrel{?}{=}\\hfill &amp; -2\\hfill &amp; &amp; &amp; \\hfill 4&amp; \\stackrel{?}{=}\\hfill &amp; {2}^{2}\\hfill \\\\ \\hfill -2&amp; =\\hfill &amp; -2\u2713\\hfill &amp; &amp; &amp; \\hfill 4&amp; =\\hfill &amp; 4\u2713\\hfill \\end{array}\\)<span data-type=\"newline\"><br \/><\/span><span data-type=\"newline\"><br \/><\/span> \\(\\left(-1,1\\right)\\)<span data-type=\"newline\"><br \/><\/span> \\(\\phantom{\\rule{0.3em}{0ex}}\\begin{array}{cccccccc}\\hfill x-y&amp; =\\hfill &amp; -2\\hfill &amp; &amp; &amp; \\hfill y&amp; =\\hfill &amp; {x}^{2}\\hfill \\\\ \\hfill -1-1&amp; \\stackrel{?}{=}\\hfill &amp; -2\\hfill &amp; &amp; &amp; \\hfill 1&amp; \\stackrel{?}{=}\\hfill &amp; {\\left(-1\\right)}^{2}\\hfill \\\\ \\hfill -2&amp; =\\hfill &amp; -2\u2713\\hfill &amp; &amp; &amp; \\hfill 1&amp; =\\hfill &amp; 1\u2713\\hfill \\end{array}\\)<\/td><td data-valign=\"top\" data-align=\"left\"><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\">The solutions are \\(\\left(2,4\\right)\\) and \\(\\left(-1,1\\right).\\)<\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1163873625836\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1163873812945\"><div data-type=\"problem\" id=\"fs-id1163870547413\"><p id=\"fs-id1163873506131\">Solve the system by graphing: \\(\\left\\{\\begin{array}{c}x+y=4\\hfill \\\\ y={x}^{2}+2\\hfill \\end{array}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163870335137\"><span data-type=\"media\" id=\"fs-id1163873507545\" data-alt=\"This graph shows the equations of a system, x plus y is equal to 4 and y is equal x squared plus 2, and the x y-coordinate plane. The line has a slope of negative 1 and a y-intercept at 4. The vertex of the parabola is (0, 2) and opens upward. The line and parabola intersect at the points (negative 2, 6) and (1, 3), which are labeled.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_301_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows the equations of a system, x plus y is equal to 4 and y is equal x squared plus 2, and the x y-coordinate plane. The line has a slope of negative 1 and a y-intercept at 4. The vertex of the parabola is (0, 2) and opens upward. The line and parabola intersect at the points (negative 2, 6) and (1, 3), which are labeled.\" \/><\/span><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1163873507073\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1163873860966\"><div data-type=\"problem\" id=\"fs-id1163870619657\"><p id=\"fs-id1163870644518\">Solve the system by graphing: \\(\\left\\{\\begin{array}{c}x-y=-1\\hfill \\\\ y=\\text{\u2212}{x}^{2}+3\\hfill \\end{array}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163873534978\"><span data-type=\"media\" id=\"fs-id1163873864859\" data-alt=\"This graph shows the equations of a system, x minus y is equal to negative 1 and y is equal to negative x squared plus three, and the x y-coordinate plane. The line has a slope of 1 and a y-intercept at 1. The vertex of the parabola is (0, negative 3) and opens upward. The line and parabola intersect at the points (negative 2, negative 1) and (1, 2), which are labeled.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_302_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows the equations of a system, x minus y is equal to negative 1 and y is equal to negative x squared plus three, and the x y-coordinate plane. The line has a slope of 1 and a y-intercept at 1. The vertex of the parabola is (0, negative 3) and opens upward. The line and parabola intersect at the points (negative 2, negative 1) and (1, 2), which are labeled.\" \/><\/span><\/div><\/div><\/div><p id=\"fs-id1165926714193\">To identify the graph of each equation, keep in mind the characteristics of the \\({x}^{2}\\) and \\({y}^{2}\\) terms of each conic.<\/p><div data-type=\"example\" id=\"fs-id1163873789993\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1163873807796\"><div data-type=\"problem\" id=\"fs-id1163873635154\"><p id=\"fs-id1163873530732\">Solve the system by graphing: \\(\\left\\{\\begin{array}{c}y=-1\\hfill \\\\ {\\left(x-2\\right)}^{2}+{\\left(y+3\\right)}^{2}=4\\hfill \\end{array}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163873679329\"><table id=\"fs-id1163873668699\" class=\"unnumbered unstyled can-break\" summary=\"Identify each graph of the system y is equal to negative 1, which is a line, and the quantity x minus 2 squared plus the quantity y plus 3 squared is equal to 4, which is a circle. Sketch the possible options for the intersection of a circle and a line. When a circle and a line do not intersect, there the system has 0 solutions. When a circle and line intersect at a single point, the system has one solution. When a circle and line intersect at two points, the system has two solutions. Graph the circle, the quantity x minus 2 squared plus the quantity y plus 3 squared is equal to 4. Its center is (2, negative 3) and it has a radius 2 units. Graph the line, y is equal to negative 1. It is a horizontal line. Identify the points of intersection. The point of intersection appears to be (2, negative 1). Check to make sure the solution makes both equations true. Substitute the coordinates from (2, negative 1) into the quantity x minus 2 squared plus the quantity y plus 3 squared is equal to 4. Is the quantity 2 minus 2 squared plus the quantity negative 1 plus 3 squared equal to 4? Is 0 squared plus 2 squared equal to 4? 4 is equal to 4. Substitute the coordinates from (2, negative 1) into y is equal to negative 1. Negative 1 is equal to negative 1. The solution is (2, negative 1).\" data-label=\"\"><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Identify each graph.<\/td><td data-valign=\"top\" data-align=\"left\">\\(\\left\\{\\begin{array}{ccc}y=-1\\hfill &amp; &amp; \\text{line}\\hfill \\\\ {\\left(x-2\\right)}^{2}+{\\left(y+3\\right)}^{2}=4\\hfill &amp; &amp; \\text{circle}\\hfill \\end{array}\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Sketch the possible options for the<span data-type=\"newline\"><br \/><\/span>intersection of a circle and a line.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163873865333\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_002a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Graph the circle, \\({\\left(x-2\\right)}^{2}+{\\left(y+3\\right)}^{2}=4\\)<span data-type=\"newline\"><br \/><\/span>Center: \\(\\left(2,-3\\right)\\) radius: 2<span data-type=\"newline\"><br \/><\/span>Graph the line, \\(y=-1.\\)<span data-type=\"newline\"><br \/><\/span>It is a horizontal line.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163873810952\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_002b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Identify the points of intersection.<\/td><td data-valign=\"top\" data-align=\"left\">The point of intersection appears to be \\(\\left(2,-1\\right).\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Check to make sure the solution makes<span data-type=\"newline\"><br \/><\/span>both equations true.<span data-type=\"newline\"><br \/><\/span> \\(\\left(2,-1\\right)\\)<span data-type=\"newline\"><br \/><\/span> \\(\\begin{array}{cccccccc}\\hfill {\\left(x-2\\right)}^{2}+{\\left(y+3\\right)}^{2}&amp; =\\hfill &amp; 4\\hfill &amp; &amp; &amp; \\hfill y&amp; =\\hfill &amp; -1\\hfill \\\\ \\hfill {\\left(2-2\\right)}^{2}+{\\left(-1+3\\right)}^{2}&amp; \\stackrel{?}{=}\\hfill &amp; 4\\hfill &amp; &amp; &amp; \\hfill -1&amp; =\\hfill &amp; -1\u2713\\hfill \\\\ \\hfill {\\left(0\\right)}^{2}+{\\left(2\\right)}^{2}&amp; \\stackrel{?}{=}\\hfill &amp; 4\\hfill &amp; &amp; &amp; &amp; &amp; \\\\ \\hfill 4&amp; =\\hfill &amp; 4\u2713\\hfill &amp; &amp; &amp; &amp; &amp; \\end{array}\\)<\/td><td data-valign=\"top\" data-align=\"left\"><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\">The solution is \\(\\left(2,-1\\right).\\)<\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1163873514495\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1163873549503\"><div data-type=\"problem\" id=\"fs-id1163873782736\"><p id=\"fs-id1163873539054\">Solve the system by graphing: \\(\\left\\{\\begin{array}{c}x=-6\\hfill \\\\ {\\left(x+3\\right)}^{2}+{\\left(y-1\\right)}^{2}=9\\hfill \\end{array}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163873798472\"><span data-type=\"media\" id=\"fs-id1163873857093\" data-alt=\"This graph shows the equations of a system, x is equal to negative 6 and the quantity x plus 3 squared plus the quantity y minus 1 squared is equal to 9, which is a circle, on the x y-coordinate plane. The line is a vertical line. The center of the circle is (negative 3, 1) and it has a radius of 3 units. The point of intersection between the line and circle is (negative 6, 1).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_303_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows the equations of a system, x is equal to negative 6 and the quantity x plus 3 squared plus the quantity y minus 1 squared is equal to 9, which is a circle, on the x y-coordinate plane. The line is a vertical line. The center of the circle is (negative 3, 1) and it has a radius of 3 units. The point of intersection between the line and circle is (negative 6, 1).\" \/><\/span><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1163870590841\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1163873770551\"><div data-type=\"problem\" id=\"fs-id1163870661129\"><p id=\"fs-id1163873672805\">Solve the system by graphing: \\(\\left\\{\\begin{array}{c}y=4\\hfill \\\\ {\\left(x-2\\right)}^{2}+{\\left(y+3\\right)}^{2}=4\\hfill \\end{array}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163870548959\"><span data-type=\"media\" id=\"fs-id1163873760122\" data-alt=\"This graph shows the equations of a system, y is equal to negative 4 and the quantity x minus 2 squared plus the quantity y plus 3 squared is equal to 4, which is a circle, on the x y-coordinate plane. The line is a horizontal line. The center of the circle is (2, negative 3) and it has a radius of 2 units. There is no point of intersection between the line and circle, so the system has no solution.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_304_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows the equations of a system, y is equal to negative 4 and the quantity x minus 2 squared plus the quantity y plus 3 squared is equal to 4, which is a circle, on the x y-coordinate plane. The line is a horizontal line. The center of the circle is (2, negative 3) and it has a radius of 2 units. There is no point of intersection between the line and circle, so the system has no solution.\" \/><\/span><\/div><\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1163873506935\"><h3 data-type=\"title\">Solve a System of Nonlinear Equations Using Substitution<\/h3><p id=\"fs-id1163873621055\">The graphing method works well when the points of intersection are integers and so easy to read off the graph. But more often it is difficult to read the coordinates of the points of intersection. The substitution method is an algebraic method that will work well in many situations. It works especially well when it is easy to solve one of the equations for one of the variables.<\/p><p id=\"fs-id1163870411088\">The substitution method is very similar to the substitution method that we used for systems of linear equations. The steps are listed below for reference.<\/p><div data-type=\"note\" id=\"fs-id1163873858056\" class=\"howto\"><div data-type=\"title\">Solve a system of nonlinear equations by substitution.<\/div><ol id=\"fs-id1163873621537\" type=\"1\" class=\"stepwise\"><li>Identify the graph of each equation. Sketch the possible options for intersection.<\/li><li>Solve one of the equations for either variable.<\/li><li>Substitute the expression from Step 2 into the other equation.<\/li><li>Solve the resulting equation.<\/li><li>Substitute each solution in Step 4 into one of the original equations to find the other variable.<\/li><li>Write each solution as an ordered pair.<\/li><li>Check that each ordered pair is a solution to <strong data-effect=\"bold\">both<\/strong> original equations.<\/li><\/ol><\/div><div data-type=\"example\" id=\"fs-id1163873865634\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1163873506444\"><div data-type=\"problem\" id=\"fs-id1163869585875\"><p id=\"fs-id1163873660517\">Solve the system by using substitution: \\(\\left\\{\\begin{array}{c}9{x}^{2}+{y}^{2}=9\\hfill \\\\ y=3x-3\\hfill \\end{array}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163873587705\"><table id=\"fs-id1163873798427\" class=\"unnumbered unstyled can-break\" summary=\"Identify each graph of the system, 9 x squared plus y squared is equal to 9, which is an ellipse, and y is equal to 3 x minus 3, which is a line. Sketch the possible options for intersection of an ellipse and a line. When an ellipse and a line do not intersect, the system has 0 solutions. When an ellipse and line intersect at a single point, the system has one solution. When an ellipse and line intersect at two points, the system has two solutions. The equation y is equal to 3 x minus 3 is solved for y already. Substitute 3 x minus 3 for y in the equation, 9 x squared plus y squared is equal to 9. Solve the equation for x. 9 x squared plus the quantity 3 x minus 3 end quantity squared is equal to 9. 9 x squared plus 9 x squared minus 18 x plus 9 is equal to 9. 18 x squared minus 18 x is equal to 0. 18 x times the quantity x minus 1) is equal to 0. So, x is equal to 0 or x is equal to 1. Substitute x is equal to 0 and x is equal to 1 into y is equal to 3 x minus 3 to find y. For x is equal to 0, the result is y is equal to 3 times 0 minus 3, which simplifies to y is equal to 3. For x is equal to 1, the result is y is equal to 3 times 1 minus 3, which simplifies to is equal to 0. The ordered pairs are (0, negative 3) and (1, 0). Check both ordered pairs in both equations. Substitute the coordinates in (0, negative 3) in 9 x squared plus y squared is equal to 9. Is 9 times 0 squared plus negative 3 squared equal to 9? Is 0 plus 9 equal to 9? 9 is equal to 9. Substitute the coordinates in (0, negative 3) in y is equal to 3 x minus 3. Is negative 3 equal to 3 times 0 minus 3? Is negative 3 equal to 0 minus 3? Negative 3 is equal to negative 3. Substitute the coordinates in (1, 0) in 9 x squared plus y squared is equal to 9. Is 9 times 1 squared plus 0 squared equal to 9? Is 9 plus 0 equal to 9? 9 is equal to 9. Substitute the coordinates in (1, 0) in y is equal to 3 x minus 3. Is 0 equal to 3 times 1 minus 3? Is 0 equal to 3 minus 3? 0 is equal to 3. The solutions are (0, negative 3) and (1, 0).\" data-label=\"\"><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Identify each graph.<\/td><td data-valign=\"top\" data-align=\"left\">\\(\\left\\{\\begin{array}{ccc}9{x}^{2}+{y}^{2}=9\\hfill &amp; &amp; \\text{ellipse}\\hfill \\\\ y=3x-3\\hfill &amp; &amp; \\text{line}\\hfill \\end{array}\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Sketch the possible options for intersection of an<span data-type=\"newline\"><br \/><\/span>ellipse and a line.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163869070525\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_003a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">The equation \\(y=3x-3\\) is solved for <em data-effect=\"italics\">y<\/em>.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_003b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163873889838\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_003c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Substitute \\(3x-3\\) for <em data-effect=\"italics\">y<\/em> in the first equation.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163873635449\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_003d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Solve the equation for <em data-effect=\"italics\">x<\/em>.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163873676950\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_003e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163870638800\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_003f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Substitute \\(x=0\\) and \\(x=1\\) into \\(y=3x-3\\) to find <em data-effect=\"italics\">y<\/em>.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163870660800\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_003g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163870359253\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_003h_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\">The ordered pairs are \\(\\left(0,-3\\right),\\) \\(\\left(1,0\\right).\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Check <strong data-effect=\"bold\">both<\/strong> ordered pairs in <strong data-effect=\"bold\">both<\/strong> equations.<span data-type=\"newline\"><br \/><\/span> \\(\\left(0,-3\\right)\\)<span data-type=\"newline\"><br \/><\/span> \\(\\begin{array}{}\\\\ \\hfill 9{x}^{2}+{y}^{2}&amp; =\\hfill &amp; 9\\hfill &amp; &amp; &amp; &amp; &amp; \\hfill y&amp; =\\hfill &amp; 3x-3\\hfill \\\\ \\hfill 9\u00b7{0}^{2}+{\\left(-3\\right)}^{2}&amp; \\stackrel{?}{=}\\hfill &amp; 9\\hfill &amp; &amp; &amp; &amp; &amp; \\hfill -3&amp; \\stackrel{?}{=}\\hfill &amp; 3\u00b70-3\\hfill \\\\ \\hfill 0+9&amp; \\stackrel{?}{=}\\hfill &amp; 9\\hfill &amp; &amp; &amp; &amp; &amp; \\hfill -3&amp; \\stackrel{?}{=}\\hfill &amp; 0-3\\hfill \\\\ \\hfill 9&amp; =\\hfill &amp; 9\u2713\\hfill &amp; &amp; &amp; &amp; &amp; \\hfill -3&amp; =\\hfill &amp; -3\u2713\\hfill \\end{array}\\)<span data-type=\"newline\"><br \/><\/span> \\(\\left(1,0\\right)\\)<span data-type=\"newline\"><br \/><\/span> \\(\\phantom{\\rule{1.3em}{0ex}}\\begin{array}{cccccccccc}\\hfill 9{x}^{2}+{y}^{2}&amp; =\\hfill &amp; 9\\hfill &amp; &amp; &amp; &amp; &amp; \\hfill \\phantom{\\rule{0.8em}{0ex}}y&amp; =\\hfill &amp; 3x-3\\hfill \\\\ \\hfill 9\u00b7{1}^{2}+{0}^{2}&amp; \\stackrel{?}{=}\\hfill &amp; 9\\hfill &amp; &amp; &amp; &amp; &amp; \\hfill \\phantom{\\rule{0.8em}{0ex}}0&amp; \\stackrel{?}{=}\\hfill &amp; 3\u00b71-3\\hfill \\\\ \\hfill 9+0&amp; \\stackrel{?}{=}\\hfill &amp; 9\\hfill &amp; &amp; &amp; &amp; &amp; \\hfill \\phantom{\\rule{0.8em}{0ex}}0&amp; \\stackrel{?}{=}\\hfill &amp; 3-3\\hfill \\\\ \\hfill 9&amp; =\\hfill &amp; 9\u2713\\hfill &amp; &amp; &amp; &amp; &amp; \\hfill \\phantom{\\rule{0.8em}{0ex}}0&amp; =\\hfill &amp; 0\u2713\\hfill \\end{array}\\)<\/td><td data-valign=\"top\" data-align=\"left\"><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\">The solutions are \\(\\left(0,-3\\right),\\left(1,0\\right).\\)<\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1163873808977\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1163873599263\"><div data-type=\"problem\" id=\"fs-id1163870152141\"><p id=\"fs-id1163873860353\">Solve the system by using substitution: \\(\\left\\{\\begin{array}{c}{x}^{2}+9{y}^{2}=9\\hfill \\\\ y=\\frac{1}{3}x-3\\hfill \\end{array}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163873535063\"><p id=\"fs-id1163869190295\">No solution<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1163873808862\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1163873526545\"><div data-type=\"problem\" id=\"fs-id1163873791536\"><p id=\"fs-id1163873679187\">Solve the system by using substitution: \\(\\left\\{\\begin{array}{c}4{x}^{2}+{y}^{2}=4\\hfill \\\\ y=x+2\\hfill \\end{array}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163870359066\"><p id=\"fs-id1163873999144\">\\(\\left(-\\frac{4}{5},\\frac{6}{5}\\right),\\left(0,2\\right)\\)<\/p><\/div><\/div><\/div><p id=\"fs-id1165926668831\">So far, each system of nonlinear equations has had at least one solution. The next example will show another option.<\/p><div data-type=\"example\" id=\"fs-id1163873668205\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1163873821502\"><div data-type=\"problem\" id=\"fs-id1163873940603\"><p id=\"fs-id1163870550662\">Solve the system by using substitution: \\(\\left\\{\\begin{array}{c}{x}^{2}-y=0\\hfill \\\\ y=x-2\\hfill \\end{array}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163873679567\"><table id=\"fs-id1163870358764\" class=\"unnumbered unstyled can-break\" summary=\"Identify each graph of the system, x squared minus y is equal to 0, which is a parabola, and y is equal to x minus 2, which is a line. Sketch the possible options for intersection of a parabola and a line. When a parabola and a line do not intersect, the system has 0 solutions. When a parabola and line intersect at a single point, the system has one solution. When a parabola and line intersect at two points, the system has two solutions. The equation y is equal to x minus 2 is solved for y already. Substitute the expression, x minus 2, for y into the equation x squared minus y is equal to 0. The result is x squared minus the quantity x minus 2 is equal to 0, which simplifies to x squared minus x plus 2 is equal to 0. Solve the equation for x. It doesn&#x2019;t factor easily, so we can check the discriminant, which is given by b squared minus 4 a c. Negative 1 squared minus 4 times 1 times 2 simplifies to negative 7. The discriminant is negative, so there is no real solution. The system has no solution.\" data-label=\"\"><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Identify each graph.<\/td><td data-valign=\"top\" data-align=\"center\">\\(\\left\\{\\begin{array}{ccc}{x}^{2}-y=0\\hfill &amp; &amp; \\text{parabola}\\hfill \\\\ y=x-2\\hfill &amp; &amp; \\text{line}\\hfill \\end{array}\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Sketch the possible options for<span data-type=\"newline\"><br \/><\/span>intersection of a parabola and a line<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163873602664\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_004a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">The equation \\(y=x-2\\) is solved for <em data-effect=\"italics\">y<\/em>.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163873783295\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_004b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163873606631\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_004c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Substitute \\(x-2\\) for <em data-effect=\"italics\">y<\/em> in the first equation.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163873862652\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_004d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Solve the equation for <em data-effect=\"italics\">x<\/em>.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163873753879\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_004e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">This doesn\u2019t factor easily, so we can<span data-type=\"newline\"><br \/><\/span>check the discriminant.<\/td><td data-valign=\"top\" data-align=\"center\"><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">\\(\\begin{array}{c}\\hfill {b}^{2}-4ac\\hfill \\\\ \\hfill {\\left(-1\\right)}^{2}-4\u00b71\u00b72\\hfill \\\\ \\\\ \\hfill \\phantom{\\rule{0.05em}{0ex}}-7\\hfill \\end{array}\\)<\/td><td data-valign=\"top\" data-align=\"left\">The discriminant is negative, so there is no real solution.<span data-type=\"newline\"><br \/><\/span>The system has no solution.<\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1163873514214\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1163873676484\"><div data-type=\"problem\" id=\"fs-id1163866890428\"><p id=\"fs-id1163873791504\">Solve the system by using substitution: \\(\\left\\{\\begin{array}{c}{x}^{2}-y=0\\hfill \\\\ y=2x-3\\hfill \\end{array}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163873796772\"><p id=\"fs-id1163873525731\">No solution<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1163870170606\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1163870491247\"><div data-type=\"problem\" id=\"fs-id1163873769859\"><p id=\"fs-id1163870386337\">Solve the system by using substitution: \\(\\left\\{\\begin{array}{c}{y}^{2}-x=0\\hfill \\\\ y=3x-2\\hfill \\end{array}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163873781771\"><p id=\"fs-id1163873521469\">\\(\\left(\\frac{4}{9},-\\frac{2}{3}\\right),\\left(1,1\\right)\\)<\/p><\/div><\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1163873642794\"><h3 data-type=\"title\">Solve a System of Nonlinear Equations Using Elimination<\/h3><p id=\"fs-id1163873782962\">When we studied systems of linear equations, we used the method of elimination to solve the system. We can also use elimination to solve systems of nonlinear equations. It works well when the equations have both variables squared. When using elimination, we try to make the coefficients of one variable to be opposites, so when we add the equations together, that variable is eliminated.<\/p><p id=\"fs-id1163870547923\">The elimination method is very similar to the elimination method that we used for systems of linear equations. The steps are listed for reference.<\/p><div data-type=\"note\" id=\"fs-id1163873659109\" class=\"howto\"><div data-type=\"title\">Solve a system of equations by elimination.<\/div><ol id=\"fs-id1163873782036\" type=\"1\" class=\"stepwise\"><li>Identify the graph of each equation. Sketch the possible options for intersection.<\/li><li>Write both equations in standard form.<\/li><li>Make the coefficients of one variable opposites.<span data-type=\"newline\"><br \/><\/span>Decide which variable you will eliminate.<span data-type=\"newline\"><br \/><\/span>Multiply one or both equations so that the coefficients of that variable are opposites.<\/li><li>Add the equations resulting from Step 3 to eliminate one variable.<\/li><li>Solve for the remaining variable.<\/li><li>Substitute each solution from Step 5 into one of the original equations. Then solve for the other variable.<\/li><li>Write each solution as an ordered pair.<\/li><li>Check that each ordered pair is a solution to <strong data-effect=\"bold\">both<\/strong> original equations.<\/li><\/ol><\/div><div data-type=\"example\" id=\"fs-id1163873603178\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1163873534849\"><div data-type=\"problem\" id=\"fs-id1163873544878\"><p id=\"fs-id1163873757998\">Solve the system by elimination: \\(\\left\\{\\begin{array}{c}{x}^{2}+{y}^{2}=4\\hfill \\\\ {x}^{2}-y=4\\hfill \\end{array}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163873788806\"><table id=\"fs-id1163873892760\" class=\"unnumbered unstyled can-break\" summary=\"Identify each graph of the system, x squared plus y squared is equal to 4, which is a circle, and x squared minus y is equal to 4, which is a parabola. Sketch the possible options for intersection of a circle and a parabola. When a circle and a parabola do not intersect, the system has 0 solutions. When circle and a parabola intersect at a single point, the system has one solution. When a circle and a parabola intersect at two points, the system has two solutions. When a circle and a parabola intersect at three points, the system has three solutions. When a circle and a parabola intersect at four points, the system has four solutions. Both equations are already in standard form. To get opposite coefficients of x squared, we will multiply the second equation by negative 1. The system is now x squared plus y squared is equal to 4 and negative 1 times the quantity x squared minus y is equal to negative 1 times 4. Simplify. The system is now x squared plus y squared is equal to 4 and negative x squared plus y is equal to negative 4. Add the two equations to eliminate x squared. The result is y squared plus y is equal to 0. Solve for y. Write the equation as y times the quantity y plus 1 is equal to 0. The result is y is equal to 0 and y plus 1 is equal to 0, which simplifies to y is equal to negative 1. Substitute y is equal to 0 and y is equal to negative 1 into one of the original equations. For y is equal to 0, x squared minus y is equal to 4 becomes x squared minus 0 is equal to 4. It simplifies to x squared equals 4, and then x is equal to plus or minus 2. For y is equal to negative 1, x squared minus y is equal to 4 becomes x squared minus negative 1 is equal to 4, which simplifies to x squared is equal to 3. The result is x is equal to plus or minus square root of 3. Write each solution as an ordered pair. The ordered pairs are (negative 2, 0), (2, 0), (square root of 3, negative 1), and (negative square root of 3, negative 1). Check that each ordered pair is a solution to both original equations. We will leave the checks for each of the four solutions to you. The solutions are (negative 2, 0), (2, 0), (square root of 3, negative 1), and (negative square root of 3, negative 1).\" data-label=\"\"><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Identify each graph.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163873812129\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_005a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Sketch the possible options for<span data-type=\"newline\"><br \/><\/span>intersection of a circle and a parabola.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163873925043\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_005b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Both equations are in standard form.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163873632086\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_005c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">To get opposite coefficients of \\({x}^{2},\\)<span data-type=\"newline\"><br \/><\/span>we will multiply the second equation by \\(-1.\\)<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163873606298\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_005d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Simplify.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163873661413\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_005e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Add the two equations to eliminate \\({x}^{2}.\\)<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163873667924\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_005f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Solve for <em data-effect=\"italics\">y<\/em>.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163873664895\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_005g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163873625115\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_005h_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Substitute \\(y=0\\) and \\(y=-1\\) into one of<span data-type=\"newline\"><br \/><\/span>the original equations. Then solve for <em data-effect=\"italics\">x<\/em>.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163873866321\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_005i_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163873854943\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_005j_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Write each solution as an ordered pair.<\/td><td data-valign=\"top\" data-align=\"center\">The ordered pairs are<span data-type=\"newline\"><br \/><\/span>\\(\\left(-2,0\\right)\\) \\(\\left(2,0\\right).\\)<span data-type=\"newline\"><br \/><\/span>\\(\\left(\\sqrt{3},-1\\right)\\left(\\text{\u2212}\\sqrt{3},-1\\right)\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Check that each ordered pair is a<span data-type=\"newline\"><br \/><\/span>solution to <strong data-effect=\"bold\">both<\/strong> original equations.<\/td><td data-valign=\"top\" data-align=\"center\"><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">We will leave the checks for each of<span data-type=\"newline\"><br \/><\/span>the four solutions to you.<\/td><td data-valign=\"top\" data-align=\"center\">The solutions are \\(\\left(-2,0\\right),\\) \\(\\left(2,0\\right),\\) \\(\\left(\\sqrt{3},-1\\right),\\) and<span data-type=\"newline\"><br \/><\/span>\\(\\left(\\text{\u2212}\\sqrt{3},-1\\right).\\)<\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1163870291935\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1163873809154\"><div data-type=\"problem\" id=\"fs-id1163873809156\"><p id=\"fs-id1163873533770\">Solve the system by elimination: \\(\\left\\{\\begin{array}{c}{x}^{2}+{y}^{2}=9\\hfill \\\\ {x}^{2}-y=9\\hfill \\end{array}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163873741414\"><p id=\"fs-id1163873741416\">\\(\\left(-3,0\\right),\\left(3,0\\right),\\left(-2\\sqrt{2},-1\\right),\\left(2\\sqrt{2},-1\\right)\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1163873666234\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1163873998114\"><div data-type=\"problem\" id=\"fs-id1163873998116\"><p id=\"fs-id1163870346533\">Solve the system by elimination: \\(\\left\\{\\begin{array}{c}{x}^{2}+{y}^{2}=1\\hfill \\\\ -x+{y}^{2}=1\\hfill \\end{array}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163873751619\"><p id=\"fs-id1163873658452\">\\(\\left(-1,0\\right),\\left(0,1\\right),\\left(0,-1\\right)\\)<\/p><\/div><\/div><\/div><p id=\"fs-id1165926616304\">There are also four options when we consider a circle and a hyperbola.<\/p><div data-type=\"example\" id=\"fs-id1163873606122\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1163873635082\"><div data-type=\"problem\" id=\"fs-id1163873635084\"><p id=\"fs-id1163873791386\">Solve the system by elimination: \\(\\left\\{\\begin{array}{c}{x}^{2}+{y}^{2}=7\\hfill \\\\ {x}^{2}-{y}^{2}=1\\hfill \\end{array}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163869407987\"><table id=\"fs-id1163873660150\" class=\"unnumbered unstyled can-break\" summary=\"Identify each graph of the system, x squared plus y squared is equal to 7, which is a circle, and x squared minus y squared is equal to 1, which is a hyperbola. Sketch the possible options for intersection of a circle and a hyperbola. When a circle and a hyperbola do not intersect, the system has 0 solutions. When circle and a hyperbola intersect at a single point, the system has one solution. When a circle and a hyperbola intersect at two points, the system has two solutions. When a circle and a hyperbola intersect at three points, the system has three solutions. When a circle and a hyperbola intersect at four points, the system has four solutions. Both equations are already in standard form. The coefficients of y squared are opposite, so we will add the equations. The result is 2 x squared is equal to 8. Simplify. The result is x squared is equal to 4, which further simplifies to x is equal to plus or minus 2. Substitute x is equal to 2 and x is equal to negative 2 into one of the original equations. Then solve for y. For x is equal 2, x squared plus y squared is equal to 7 becomes 2 squared plus y squared is equal to 7. 4 plus y squared is equal to 7, which simplifies to y squared is equal to 3. So, the result is y is equal to plus or minus square root of 3. For x is equal to negative 2, x squared plus y squared is equal to 7 becomes negative 2 squared plus y squared is equal to 7. 4 plus y squared is equal to 7, which simplifies to y squared is equal to 3. The result is y is equal to plus or minus square root of 3. Write each solution as an ordered pair. The ordered pairs are (negative 2, square root of 3), (negative 2, negative square root of 3), (2, square root of 3), and (2, negative square root of 3). Check that the ordered pair is a solution to both original equations. We will leave the checks for each of the four solutions to you. The solutions are (negative 2, square root of 3), (negative 2, negative square root of 3), (2, square root of 3), and (2, negative square root of 3).\" data-label=\"\"><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Identify each graph.<\/td><td data-valign=\"top\" data-align=\"center\">\\(\\left\\{\\begin{array}{ccc}{x}^{2}+{y}^{2}=7\\hfill &amp; &amp; \\text{circle}\\hfill \\\\ {x}^{2}-{y}^{2}=1\\hfill &amp; &amp; \\text{hyperbola}\\hfill \\end{array}\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Sketch the possible options for intersection<span data-type=\"newline\"><br \/><\/span>of a circle and hyperbola.<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1163873807546\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_006_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Both equations are in standard form.<\/td><td data-valign=\"top\" data-align=\"center\">\\(\\left\\{\\begin{array}{c}{x}^{2}+{y}^{2}=7\\hfill \\\\ {x}^{2}-{y}^{2}=1\\hfill \\end{array}\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">The coefficients of \\({y}^{2}\\) are opposite, so we<span data-type=\"newline\"><br \/><\/span>will add the equations.<\/td><td data-valign=\"top\" data-align=\"center\">\\(\\begin{array}{c}\\underset{__________}{\\left\\{\\begin{array}{c}{x}^{2}+{y}^{2}=7\\hfill \\\\ {x}^{2}-{y}^{2}=1\\hfill \\end{array}}\\hfill \\\\ \\\\ 2{x}^{2}\\phantom{\\rule{2em}{0ex}}=8\\hfill \\end{array}\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Simplify.<\/td><td data-valign=\"top\" data-align=\"center\">\\(\\begin{array}{ccc}\\hfill {x}^{2}&amp; =\\hfill &amp; 4\\hfill \\\\ \\hfill x&amp; =\\hfill &amp; \\text{\u00b1}2\\hfill \\end{array}\\)<span data-type=\"newline\"><br \/><\/span>\\(x=2\\phantom{\\rule{1.5em}{0ex}}x=-2\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Substitute \\(x=2\\) and \\(x=-2\\) into one of the<span data-type=\"newline\"><br \/><\/span>original equations. Then solve for <em data-effect=\"italics\">y<\/em>.<\/td><td data-valign=\"top\" data-align=\"center\">\\(\\begin{array}{ccccccccc}\\hfill {x}^{2}+{y}^{2}&amp; =\\hfill &amp; 7\\hfill &amp; &amp; &amp; &amp; \\hfill {x}^{2}+{y}^{2}&amp; =\\hfill &amp; 7\\hfill \\\\ \\hfill {2}^{2}+{y}^{2}&amp; =\\hfill &amp; 7\\hfill &amp; &amp; &amp; &amp; \\hfill {\\left(-2\\right)}^{2}+{y}^{2}&amp; =\\hfill &amp; 7\\hfill \\\\ \\hfill 4+{y}^{2}&amp; =\\hfill &amp; 7\\hfill &amp; &amp; &amp; &amp; \\hfill 4+{y}^{2}&amp; =\\hfill &amp; 7\\hfill \\\\ \\hfill {y}^{2}&amp; =\\hfill &amp; 3\\hfill &amp; &amp; &amp; &amp; \\hfill {y}^{2}&amp; =\\hfill &amp; 3\\hfill \\\\ \\hfill y&amp; =\\hfill &amp; \\text{\u00b1}\\sqrt{3}\\hfill &amp; &amp; &amp; &amp; \\hfill y&amp; =\\hfill &amp; \\text{\u00b1}\\sqrt{3}\\hfill \\end{array}\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Write each solution as an ordered pair.<\/td><td data-valign=\"top\" data-align=\"center\">The ordered pairs are \\(\\left(-2,\\sqrt{3}\\right),\\) \\(\\left(-2,\\text{\u2212}\\sqrt{3}\\right),\\)<span data-type=\"newline\"><br \/><\/span>\\(\\left(2,\\sqrt{3}\\right),\\) and \\(\\left(2,\\text{\u2212}\\sqrt{3}\\right).\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Check that the ordered pair is a solution to<span data-type=\"newline\"><br \/><\/span><strong data-effect=\"bold\">both<\/strong> original equations.<\/td><td data-valign=\"top\" data-align=\"left\"><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">We will leave the checks for each of the four<span data-type=\"newline\"><br \/><\/span>solutions to you.<\/td><td data-valign=\"top\" data-align=\"center\">The solutions are \\(\\left(-2,\\sqrt{3}\\right),\\) \\(\\left(-2,\\text{\u2212}\\sqrt{3}\\right),\\) \\(\\left(2,\\sqrt{3}\\right),\\)<span data-type=\"newline\"><br \/><\/span>and \\(\\left(2,\\text{\u2212}\\sqrt{3}\\right).\\)<\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1163864794326\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1163873670070\"><div data-type=\"problem\" id=\"fs-id1163869152940\"><p id=\"fs-id1163869152942\">Solve the system by elimination: \\(\\left\\{\\begin{array}{c}{x}^{2}+{y}^{2}=25\\hfill \\\\ {y}^{2}-{x}^{2}=7\\hfill \\end{array}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163873621488\"><p id=\"fs-id1163870228725\">\\(\\left(-3,-4\\right),\\left(-3,4\\right),\\left(3,-4\\right),\\left(3,4\\right)\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1163873633652\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1163873633655\"><div data-type=\"problem\" id=\"fs-id1163873633657\"><p id=\"fs-id1163873665020\">Solve the system by elimination: \\(\\left\\{\\begin{array}{c}{x}^{2}+{y}^{2}=4\\hfill \\\\ {x}^{2}-{y}^{2}=4\\hfill \\end{array}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163873632586\"><p id=\"fs-id1163873632588\">\\(\\left(-2,0\\right),\\left(2,0\\right)\\)<\/p><\/div><\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1163873764789\"><h3 data-type=\"title\">Use a System of Nonlinear Equations to Solve Applications<\/h3><p id=\"fs-id1163870170091\">Systems of nonlinear equations can be used to model and solve many applications. We will look at an everyday geometric situation as our example.<\/p><div data-type=\"example\" id=\"fs-id1163873625652\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1163873625654\"><div data-type=\"problem\" id=\"fs-id1163873625656\"><p id=\"fs-id1163873919072\">The difference of the squares of two numbers is 15. The sum of the numbers is 5. Find the numbers.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163873919076\"><table id=\"fs-id1163873651790\" class=\"unnumbered unstyled can-break\" summary=\"Identify what we are looking for. We are looking for two different numbers. Define the variables. Let x be equal to the first number. Let y be equal to the second number. Translate the information into a system of equations. The first sentence is &#x2018;The difference of the squares of two numbers is 15.&#x2019; Represent it with x squared minus y squared is equal to 15. The second sentence is &#x2018;The sum of the numbers is 5.&#x2019; Represent it with x plus y is equal to 5. The equations x squared minus y squared is equal to 15 and x plus y is equal to 5 form the system. Solve the system by substitution. Solve the second equation for x. The result is x is equal to 5 minus y. Substitute x into the first equation, x squared minus y squared is equal to 15. It becomes the quantity 5 minus y squared minus y squared is equal to 15. Expand and simplify. The result is the quantity 25 minus 10 y plus y squared end quantity minus y squared is equal to 15. 25 minus 10 y plus y squared minus y squared is equal to 15. 25 minus 10 y is equal to 15. Solve for y. Negative y is equal to negative 10. The result is y is equal to 1. Substitute back into the second equation, x plus y is equal to 5. It becomes x plus 1 is equal to 5, which simplifies to x is equal to 4. The numbers are 1 and 4.\" data-label=\"\"><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Identify what we are looking for.<\/td><td data-valign=\"top\" data-align=\"left\">Two different numbers.<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Define the variables.<\/td><td data-valign=\"top\" data-align=\"left\">\\(x=\\) first number<span data-type=\"newline\"><br \/><\/span>\\(y=\\) second number<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Translate the information into a system of<span data-type=\"newline\"><br \/><\/span>equations.<\/td><td data-valign=\"top\" data-align=\"left\"><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">First sentence.<\/td><td data-valign=\"top\" data-align=\"justify\">The difference of the squares of two numbers is 15.<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163873595897\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_007a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Second sentence.<\/td><td data-valign=\"top\" data-align=\"left\">The sum of the numbers is 5.<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163873632234\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_007b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Solve the system by substitution<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163873864424\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_007c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Solve the second equation for <em data-effect=\"italics\">x<\/em>.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163873652600\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_007d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Substitute <em data-effect=\"italics\">x<\/em> into the first equation.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163870645034\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_007e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163873752067\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_007f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Expand and simplify.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163873639497\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_007g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163873679776\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_007h_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Solve for <em data-effect=\"italics\">y<\/em>.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163873668569\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_007i_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163873660235\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_007j_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Substitute back into the second equation.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163869405960\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_007k_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163873869881\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_007l_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"center\">The numbers are 1 and 4.<\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1163873850715\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1163873605130\"><div data-type=\"problem\" id=\"fs-id1163873605132\"><p id=\"fs-id1163873605134\">The difference of the squares of two numbers is \\(-20.\\) The sum of the numbers is 10. Find the numbers.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163873674089\"><p id=\"fs-id1163873674091\">4 and 6<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1163873645610\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1163873817518\"><div data-type=\"problem\" id=\"fs-id1163873817521\"><p id=\"fs-id1163873892044\">The difference of the squares of two numbers is 35. The sum of the numbers is \\(-1.\\) Find the numbers.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163870406471\"><p id=\"fs-id1163870406473\">\\(-18\\) and 17<\/p><\/div><\/div><\/div><div data-type=\"example\" id=\"fs-id1163870487923\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1163870487925\"><div data-type=\"problem\" id=\"fs-id1163870487927\"><p id=\"fs-id1163870271495\">Myra purchased a small 25\u201d TV for her kitchen. The size of a TV is measured on the diagonal of the screen. The screen also has an area of 300 square inches. What are the length and width of the TV screen?<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163870593254\"><table id=\"fs-id1163870593257\" class=\"unnumbered unstyled can-break\" summary=\"Identify what we are looking for. We are looking for the length and width of the rectangle. Define the variables. Let x be equal to the width of the rectangle. Let y be equal to the length of the rectangle. Draw a diagram to help visualize the situation. The diagram is a rectangle with the width labeled x, the length labeled y, and its diagonal labeled 25 inches. Translate the information into a system of equations. &#x2018;The diagonal of the right triangle is 25 is represented by the equation x squared plus y squared is equal to 25 squared, which is simplified to x squared plus y squared is equal to 625. &#x2018;The area of the rectangle is 300&#x2019; is represented by the equation x y is equal to 300. The equations form the system x squared plus y squared is equal to 625 and x y is equal to 300. Solve the system using substitution. Solve x y is equal to 300 for x. The result is x is equal to 300 divided by y. Substitute the expression for x into the first equation, x squared plus y squared is equal to 625. The result is the quantity 300 divided by y end quantity squared plus y squared is equal to 625. Simplify. The result is the quantity 90,000 divided by y squared end quantity plus y squared is equal to 625. Multiply each side of the equation by squared to clear the fractions. The result is 90,00 plus y to the fourth power is equal to 625 y squared. Put in standard form. The result is y to the fourth power minus 625 y squared plus 90,000 is equal to 0. Solve by factoring. The factored equation is the quantity y squared minus 225 times the quantity y squared minus 400 is equal to 0. The result is y squared minus 225 is equal to 0 or y squared minus 400 is equal to 0. They simplify to y squared is equal to 225 or y squared is equal to 400. The results are y is equal to plus or minus 15 or y is equal to plus or minus 20. Since y is a side of a rectangle, we discard the negative values and keep y is equal to 15 and y is equal to 20. Substitute back into the second equation, x y is equal to 300. For y is equal to 15, x times 15 is equal to 300, which simplifies to x is equal to 20. For y is equal to 20, x times 20 is equal to 300, which simplifies to x is equal to 15. If the length is 15 inches, the width is 20 inches. If the length s 20 inches, the width is 15 inches.\" data-label=\"\"><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Identify what we are looking for.<\/td><td data-valign=\"top\" data-align=\"left\">The length and width of the rectangle<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Define the variables.<\/td><td data-valign=\"top\" data-align=\"left\">Let \\(x=\\) width of the rectangle<span data-type=\"newline\"><br \/><\/span>\\(\\phantom{\\rule{1.5em}{0ex}}y=\\) length of the rectangle<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Draw a diagram to help visualize the situation.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163873644886\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_008a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"center\">Area is 300 square inches.<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Translate the information into a system of<span data-type=\"newline\"><br \/><\/span>equations.<\/td><td data-valign=\"top\" data-align=\"left\">The diagonal of the right triangle is 25 inches.<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163873595568\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_008b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"center\">The area of the rectangle is 300 square inches.<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163873657015\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_008c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Solve the system using substitution.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163873817843\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_008d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Solve the second equation for <em data-effect=\"italics\">x<\/em>.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163873674419\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_008e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Substitute <em data-effect=\"italics\">x<\/em> into the first equation.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163873634060\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_008f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163873639131\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_008g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Simplify.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163873583212\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_008h_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Multiply by \\({y}^{2}\\) to clear the fractions.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163873861928\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_008i_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Put in standard form.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163869114900\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_008j_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Solve by factoring.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163873881836\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_008k_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163870549939\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_008l_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163873860650\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_008m_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Since <em data-effect=\"italics\">y<\/em> is a side of the rectangle, we discard<span data-type=\"newline\"><br \/><\/span>the negative values.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163873582629\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_008n_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Substitute back into the second equation.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163873866291\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_008o_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163873674785\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_008p_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\">If the length is 15 inches, the width is 20 inches.<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\">If the length is 20 inches, the width is 15 inches.<\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1163873662011\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1163873662015\"><div data-type=\"problem\" id=\"fs-id1163873686856\"><p id=\"fs-id1163873686858\">Edgar purchased a small 20\u201d TV for his garage. The size of a TV is measured on the diagonal of the screen. The screen also has an area of 192 square inches. What are the length and width of the TV screen?<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163873558684\"><p id=\"fs-id1163873899045\">If the length is 12 inches, the width is 16 inches. If the length is 16 inches, the width is 12 inches.<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1163873631303\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1163874018118\"><div data-type=\"problem\" id=\"fs-id1163874018120\"><p id=\"fs-id1163874018122\">The Harper family purchased a small microwave for their family room. The diagonal of the door measures 15 inches. The door also has an area of 108 square inches. What are the length and width of the microwave door?<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163873814313\"><p id=\"fs-id1163873850275\">If the length is 12 inches, the width is 9 inches. If the length is 9 inches, the width is 12 inches.<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1163873558688\" class=\"media-2\"><p id=\"fs-id1163873558692\">Access these online resources for additional instructions and practice with solving nonlinear equations.<\/p><ul id=\"fs-id1163873793432\" data-display=\"block\"><li><a href=\"https:\/\/openstax.org\/l\/37nonsyseq\">Nonlinear Systems of Equations<\/a><\/li><li><a href=\"https:\/\/openstax.org\/l\/37nonsyseq2\">Solve a System of Nonlinear Equations<\/a><\/li><li><a href=\"https:\/\/openstax.org\/l\/37nonsyselim\">Solve a System of Nonlinear Equations by Elimination<\/a><\/li><li><a href=\"https:\/\/openstax.org\/l\/37nonsysapps\">System of Nonlinear Equations \u2013 Area and Perimeter Application<\/a><\/li><\/ul><\/div><\/div><div class=\"textbox\" data-depth=\"1\" id=\"fs-id1163873869190\"><h3 data-type=\"title\">Key Concepts<\/h3><ul id=\"fs-id1163870152077\" data-bullet-style=\"bullet\"><li><strong data-effect=\"bold\">How to solve a system of nonlinear equations by graphing.<\/strong><ol id=\"fs-id1163873781999\" type=\"1\" class=\"stepwise\"><li>Identify the graph of each equation. Sketch the possible options for intersection.<\/li><li>Graph the first equation.<\/li><li>Graph the second equation on the same rectangular coordinate system.<\/li><li>Determine whether the graphs intersect.<\/li><li>Identify the points of intersection.<\/li><li>Check that each ordered pair is a solution to both original equations.<\/li><\/ol><\/li><li><strong data-effect=\"bold\">How to solve a system of nonlinear equations by substitution.<\/strong><ol id=\"fs-id1163873520227\" type=\"1\" class=\"stepwise\"><li>Identify the graph of each equation. Sketch the possible options for intersection.<span data-type=\"newline\"><br \/><\/span><span data-type=\"newline\"><br \/><\/span><span data-type=\"newline\"><br \/><\/span><\/li><li>Solve one of the equations for either variable.<\/li><li>Substitute the expression from Step 2 into the other equation.<\/li><li>Solve the resulting equation.<\/li><li>Substitute each solution in Step 4 into one of the original equations to find the other variable.<\/li><li>Write each solution as an ordered pair.<\/li><li>Check that each ordered pair is a solution to <strong data-effect=\"bold\">both<\/strong> original equations.<\/li><\/ol><\/li><li><strong data-effect=\"bold\">How to solve a system of equations by elimination.<\/strong><ol id=\"fs-id1163873539463\" type=\"1\" class=\"stepwise\"><li>Identify the graph of each equation. Sketch the possible options for intersection.<\/li><li>Write both equations in standard form.<\/li><li>Make the coefficients of one variable opposites.<span data-type=\"newline\"><br \/><\/span>Decide which variable you will eliminate.<span data-type=\"newline\"><br \/><\/span>Multiply one or both equations so that the coefficients of that variable are opposites.<\/li><li>Add the equations resulting from Step 3 to eliminate one variable.<\/li><li>Solve for the remaining variable.<\/li><li>Substitute each solution from Step 5 into one of the original equations. Then solve for the other variable.<\/li><li>Write each solution as an ordered pair.<\/li><li>Check that each ordered pair is a solution to <strong data-effect=\"bold\">both<\/strong> original equations.<\/li><\/ol><\/li><\/ul><\/div><div class=\"textbox\" data-depth=\"1\" id=\"fs-id1163873870804\"><h3 data-type=\"title\">Section Exercises<\/h3><div class=\"practice-perfect\" data-depth=\"2\" id=\"fs-id1163873724382\"><h4 data-type=\"title\">Practice Makes Perfect<\/h4><p id=\"fs-id1163873665575\"><strong data-effect=\"bold\">Solve a System of Nonlinear Equations Using Graphing<\/strong><\/p><p id=\"fs-id1163870242552\">In the following exercises, solve the system of equations by using graphing.<\/p><div data-type=\"exercise\" id=\"fs-id1163870551733\"><div data-type=\"problem\" id=\"fs-id1163870551736\"><p id=\"fs-id1163870551738\">\\(\\left\\{\\begin{array}{c}y=2x+2\\hfill \\\\ y=\\text{\u2212}{x}^{2}+2\\hfill \\end{array}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163873792143\"><div data-type=\"problem\" id=\"fs-id1163873792145\"><p id=\"fs-id1163873795593\">\\(\\left\\{\\begin{array}{c}y=6x-4\\hfill \\\\ y=2{x}^{2}\\hfill \\end{array}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163873666879\"><span data-type=\"media\" id=\"fs-id1163870335164\" data-alt=\"This graph shows the equations of a system, y is equal to 6 x minus 4 which is a line and y is equal to 2 x squared which is a parabola, on the x y-coordinate plane. The vertex of the parabola is (0, 0) and the parabola opens upward. The line has a slope of 6. The line and parabola intersect at the points (1, 2) and (2, 8), which are labeled. The solutions are (1, 2) and (2, 8).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_306_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows the equations of a system, y is equal to 6 x minus 4 which is a line and y is equal to 2 x squared which is a parabola, on the x y-coordinate plane. The vertex of the parabola is (0, 0) and the parabola opens upward. The line has a slope of 6. The line and parabola intersect at the points (1, 2) and (2, 8), which are labeled. The solutions are (1, 2) and (2, 8).\" \/><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163869399259\"><div data-type=\"problem\" id=\"fs-id1163869399261\"><p id=\"fs-id1163869399263\">\\(\\left\\{\\begin{array}{c}x+y=2\\hfill \\\\ x={y}^{2}\\hfill \\end{array}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163873798566\"><div data-type=\"problem\" id=\"fs-id1163873798568\"><p id=\"fs-id1163873798570\">\\(\\left\\{\\begin{array}{c}x-y=-2\\hfill \\\\ x={y}^{2}\\hfill \\end{array}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163873655465\"><span data-type=\"media\" id=\"fs-id1163873872288\" data-alt=\"This graph shows the equations of a system, x minus y is equal to negative 2 which is a line and x is equal to y squared which is a rightward-opening parabola, on the x y-coordinate plane. The vertex of the parabola is (0, 0) and it passes through the points (1, 1) and (1, negative 1). The line has a slope of 1 and a y-intercept at 2. The line and parabola do not intersect, so the system has no solution.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_308_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows the equations of a system, x minus y is equal to negative 2 which is a line and x is equal to y squared which is a rightward-opening parabola, on the x y-coordinate plane. The vertex of the parabola is (0, 0) and it passes through the points (1, 1) and (1, negative 1). The line has a slope of 1 and a y-intercept at 2. The line and parabola do not intersect, so the system has no solution.\" \/><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163874018820\"><div data-type=\"problem\" id=\"fs-id1163873806975\"><p id=\"fs-id1163873806977\">\\(\\left\\{\\begin{array}{c}y=\\frac{3}{2}x+3\\hfill \\\\ y=\\text{\u2212}{x}^{2}+2\\hfill \\end{array}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163870407302\"><div data-type=\"problem\" id=\"fs-id1163873507028\"><p id=\"fs-id1163873507030\">\\(\\left\\{\\begin{array}{c}y=x-1\\hfill \\\\ y={x}^{2}+1\\hfill \\end{array}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163870335080\"><span data-type=\"media\" id=\"fs-id1163870335083\" data-alt=\"This graph shows the equations of a system, y is x minus 1 which is a line and y is equal to x squared plus 1 which is an upward-opening parabola, on the x y-coordinate plane. The vertex of the parabola is (0, 1) and it passes through the points (negative 1, 2) and (1, 2). The line has a slope of 1 and a y-intercept at negative 1. The line and parabola do not intersect, so the system has no solution.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_310_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows the equations of a system, y is x minus 1 which is a line and y is equal to x squared plus 1 which is an upward-opening parabola, on the x y-coordinate plane. The vertex of the parabola is (0, 1) and it passes through the points (negative 1, 2) and (1, 2). The line has a slope of 1 and a y-intercept at negative 1. The line and parabola do not intersect, so the system has no solution.\" \/><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163870516038\"><div data-type=\"problem\" id=\"fs-id1163870516040\"><p id=\"fs-id1163870516043\">\\(\\left\\{\\begin{array}{c}x=-2\\hfill \\\\ {x}^{2}+{y}^{2}=4\\hfill \\end{array}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163873660218\"><div data-type=\"problem\" id=\"fs-id1163873660220\"><p id=\"fs-id1163870619439\">\\(\\left\\{\\begin{array}{c}y=-4\\hfill \\\\ {x}^{2}+{y}^{2}=16\\hfill \\end{array}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163874018254\"><span data-type=\"media\" id=\"fs-id1163873518533\" data-alt=\"This graph shows the equations of a system, x is equal to negative 2 which is a line and x squared plus y squared is equal to 16 which is a circle, on the x y-coordinate plane. The line is horizontal. The center of the circle is (0, 0) and the radius of the circle is 4. The line and circle intersect at (negative 2, 0), so the solution of the system is (negative 2, 0).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_312_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows the equations of a system, x is equal to negative 2 which is a line and x squared plus y squared is equal to 16 which is a circle, on the x y-coordinate plane. The line is horizontal. The center of the circle is (0, 0) and the radius of the circle is 4. The line and circle intersect at (negative 2, 0), so the solution of the system is (negative 2, 0).\" \/><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163873912376\"><div data-type=\"problem\" id=\"fs-id1163870357562\"><p id=\"fs-id1163870357565\">\\(\\left\\{\\begin{array}{c}x=2\\hfill \\\\ {\\left(x+2\\right)}^{2}+{\\left(y+3\\right)}^{2}=16\\hfill \\end{array}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163873791403\"><div data-type=\"problem\" id=\"fs-id1163873659526\"><p id=\"fs-id1163873659528\">\\(\\left\\{\\begin{array}{c}y=-1\\hfill \\\\ {\\left(x-2\\right)}^{2}+{\\left(y-4\\right)}^{2}=25\\hfill \\end{array}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163870324576\"><span data-type=\"media\" id=\"fs-id1163870324580\" data-alt=\"This graph shows the equations of a system, x is equal to 2 which is a line and the quantity x minus 2 end quantity squared plus the quantity y minus 4 end quantity squared is equal to 25 which is a circle, on the x y-coordinate plane. The line is horizontal. The center of the circle is (2, 4) and the radius of the circle is 5. The line and circle intersect at (2, negative 1), so the solution of the system is (2, negative 1).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_314_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows the equations of a system, x is equal to 2 which is a line and the quantity x minus 2 end quantity squared plus the quantity y minus 4 end quantity squared is equal to 25 which is a circle, on the x y-coordinate plane. The line is horizontal. The center of the circle is (2, 4) and the radius of the circle is 5. The line and circle intersect at (2, negative 1), so the solution of the system is (2, negative 1).\" \/><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163873638220\"><div data-type=\"problem\" id=\"fs-id1163873638222\"><p id=\"fs-id1163873962919\">\\(\\left\\{\\begin{array}{c}y=-2x+4\\hfill \\\\ y=\\sqrt[]{x}+1\\hfill \\end{array}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163873863087\"><div data-type=\"problem\" id=\"fs-id1163873863090\"><p id=\"fs-id1163873863092\">\\(\\left\\{\\begin{array}{c}y=-\\frac{1}{2}x+2\\hfill \\\\ y=\\sqrt[]{x}-2\\hfill \\end{array}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163873673781\"><span data-type=\"media\" id=\"fs-id1163873796259\" data-alt=\"This graph shows the equations of a system, y is equal to negative one-half x plus 2 which is a line and the y is equal to the square root of x minus 2, on the x y-coordinate plane. The curve for y is equal to the square root of x minus 2 The curve for y is equal to the square root of x plus 1 where x is greater than or equal to 0 and y is greater than or equal to negative 2. The line and square root curve intersect at (4, 0), so the solution is (4, 0).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_316_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows the equations of a system, y is equal to negative one-half x plus 2 which is a line and the y is equal to the square root of x minus 2, on the x y-coordinate plane. The curve for y is equal to the square root of x minus 2 The curve for y is equal to the square root of x plus 1 where x is greater than or equal to 0 and y is greater than or equal to negative 2. The line and square root curve intersect at (4, 0), so the solution is (4, 0).\" \/><\/span><\/div><\/div><p id=\"fs-id1163869163549\"><strong data-effect=\"bold\">Solve a System of Nonlinear Equations Using Substitution<\/strong><\/p><p id=\"fs-id1163873632875\">In the following exercises, solve the system of equations by using substitution.<\/p><div data-type=\"exercise\" id=\"fs-id1163873628568\"><div data-type=\"problem\" id=\"fs-id1163873628570\"><p id=\"fs-id1163870381243\">\\(\\left\\{\\begin{array}{c}{x}^{2}+4{y}^{2}=4\\hfill \\\\ y=\\frac{1}{2}x-1\\hfill \\end{array}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163873817958\"><div data-type=\"problem\" id=\"fs-id1163873817960\"><p id=\"fs-id1163873817962\">\\(\\left\\{\\begin{array}{c}9{x}^{2}+{y}^{2}=9\\hfill \\\\ y=3x+3\\hfill \\end{array}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163873644302\"><p id=\"fs-id1163873644304\">\\(\\left(-1,0\\right),\\left(0,3\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163873730327\"><div data-type=\"problem\" id=\"fs-id1163873730330\"><p id=\"fs-id1163873730332\">\\(\\left\\{\\begin{array}{c}9{x}^{2}+{y}^{2}=9\\hfill \\\\ y=x+3\\hfill \\end{array}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163873796662\"><div data-type=\"problem\" id=\"fs-id1163873796664\"><p id=\"fs-id1163873657340\">\\(\\left\\{\\begin{array}{c}9{x}^{2}+4{y}^{2}=36\\hfill \\\\ x=2\\hfill \\end{array}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163870406996\"><p id=\"fs-id1163870406998\">\\(\\left(2,0\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163873897281\"><div data-type=\"problem\" id=\"fs-id1163869549732\"><p id=\"fs-id1163869549734\">\\(\\left\\{\\begin{array}{c}4{x}^{2}+{y}^{2}=4\\hfill \\\\ y=4\\hfill \\end{array}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163873627432\"><div data-type=\"problem\" id=\"fs-id1163873627435\"><p id=\"fs-id1163873678509\">\\(\\left\\{\\begin{array}{c}{x}^{2}+{y}^{2}=169\\hfill \\\\ x=12\\hfill \\end{array}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163873533384\"><p id=\"fs-id1163873533386\">\\(\\left(12,-5\\right),\\left(12,5\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163873521592\"><div data-type=\"problem\" id=\"fs-id1163873521594\"><p id=\"fs-id1163873521597\">\\(\\left\\{\\begin{array}{c}3{x}^{2}-y=0\\hfill \\\\ y=2x-1\\hfill \\end{array}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163873874004\"><div data-type=\"problem\" id=\"fs-id1163873874006\"><p id=\"fs-id1163873715863\">\\(\\left\\{\\begin{array}{c}2{y}^{2}-x=0\\hfill \\\\ y=x+1\\hfill \\end{array}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163873782950\"><p id=\"fs-id1163873782953\">No solution<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163873525451\"><div data-type=\"problem\" id=\"fs-id1163873525453\"><p id=\"fs-id1163873525455\">\\(\\left\\{\\begin{array}{c}y={x}^{2}+3\\hfill \\\\ y=x+3\\hfill \\end{array}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163870459435\"><div data-type=\"problem\" id=\"fs-id1163870459437\"><p id=\"fs-id1163870459439\">\\(\\left\\{\\begin{array}{c}y={x}^{2}-4\\hfill \\\\ y=x-4\\hfill \\end{array}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163873821472\"><p id=\"fs-id1163873821475\">\\(\\left(0,-4\\right),\\left(1,-3\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163873539125\"><div data-type=\"problem\" id=\"fs-id1163873539127\"><p id=\"fs-id1163873539129\">\\(\\left\\{\\begin{array}{c}{x}^{2}+{y}^{2}=25\\hfill \\\\ x-y=1\\hfill \\end{array}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163873636025\"><div data-type=\"problem\" id=\"fs-id1163873514452\"><p id=\"fs-id1163873514454\">\\(\\left\\{\\begin{array}{c}{x}^{2}+{y}^{2}=25\\hfill \\\\ 2x+y=10\\hfill \\end{array}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163873785724\"><p id=\"fs-id1163870376594\">\\(\\left(3,4\\right),\\left(5,0\\right)\\)<\/p><\/div><\/div><p id=\"fs-id1163873677678\"><strong data-effect=\"bold\">Solve a System of Nonlinear Equations Using Elimination<\/strong><\/p><p id=\"fs-id1163873766386\">In the following exercises, solve the system of equations by using elimination.<\/p><div data-type=\"exercise\" id=\"fs-id1163873652213\"><div data-type=\"problem\" id=\"fs-id1163873652215\"><p id=\"fs-id1163873652217\">\\(\\left\\{\\begin{array}{c}{x}^{2}+{y}^{2}=16\\hfill \\\\ {x}^{2}-2y=8\\hfill \\end{array}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163873821209\"><div data-type=\"problem\" id=\"fs-id1163869201418\"><p id=\"fs-id1163869201420\">\\(\\left\\{\\begin{array}{c}{x}^{2}+{y}^{2}=16\\hfill \\\\ {x}^{2}-y=4\\hfill \\end{array}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163873628434\"><p id=\"fs-id1163873628436\">\\(\\left(0,-4\\right),\\left(\\text{\u2212}\\sqrt{7},3\\right),\\left(\\sqrt{7},3\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163873871033\"><div data-type=\"problem\" id=\"fs-id1163869406155\"><p id=\"fs-id1163869406157\">\\(\\left\\{\\begin{array}{c}{x}^{2}+{y}^{2}=4\\hfill \\\\ {x}^{2}+2y=1\\hfill \\end{array}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163873814916\"><div data-type=\"problem\" id=\"fs-id1163873866077\"><p id=\"fs-id1163873866079\">\\(\\left\\{\\begin{array}{c}{x}^{2}+{y}^{2}=4\\hfill \\\\ {x}^{2}-y=2\\hfill \\end{array}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163873606493\"><p id=\"fs-id1163873606495\">\\(\\left(0,-2\\right),\\left(\\text{\u2212}\\sqrt{3},1\\right),\\left(\\sqrt{3},1\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163873800365\"><div data-type=\"problem\" id=\"fs-id1163873766052\"><p id=\"fs-id1163873766054\">\\(\\left\\{\\begin{array}{c}{x}^{2}+{y}^{2}=9\\hfill \\\\ {x}^{2}-y=3\\hfill \\end{array}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163873757961\"><div data-type=\"problem\" id=\"fs-id1163873627473\"><p id=\"fs-id1163873627475\">\\(\\left\\{\\begin{array}{c}{x}^{2}+{y}^{2}=4\\hfill \\\\ {y}^{2}-x=2\\hfill \\end{array}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163873935242\"><p id=\"fs-id1163873935244\">\\(\\left(-2,0\\right),\\left(1,\\text{\u2212}\\sqrt{3}\\right),\\left(1,\\sqrt{3}\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163873558582\"><div data-type=\"problem\" id=\"fs-id1163873766374\"><p id=\"fs-id1163873766376\">\\(\\left\\{\\begin{array}{c}{x}^{2}+{y}^{2}=25\\hfill \\\\ 2{x}^{2}-3{y}^{2}=5\\hfill \\end{array}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163873914313\"><div data-type=\"problem\" id=\"fs-id1163873914315\"><p id=\"fs-id1163870621083\">\\(\\left\\{\\begin{array}{c}{x}^{2}+{y}^{2}=20\\hfill \\\\ {x}^{2}-{y}^{2}=-12\\hfill \\end{array}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163869407389\"><p id=\"fs-id1163869407391\">\\(\\left(-2,-4\\right),\\left(-2,4\\right),\\left(2,-4\\right),\\left(2,4\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163873806539\"><div data-type=\"problem\" id=\"fs-id1163873766314\"><p id=\"fs-id1163873766316\">\\(\\left\\{\\begin{array}{c}{x}^{2}+{y}^{2}=13\\hfill \\\\ {x}^{2}-{y}^{2}=5\\hfill \\end{array}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163870413759\"><div data-type=\"problem\" id=\"fs-id1163873796453\"><p id=\"fs-id1163873796455\">\\(\\left\\{\\begin{array}{c}{x}^{2}+{y}^{2}=16\\hfill \\\\ {x}^{2}-{y}^{2}=16\\hfill \\end{array}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163873853964\"><p id=\"fs-id1163873663891\">\\(\\left(-4,0\\right),\\left(4,0\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163870264792\"><div data-type=\"problem\" id=\"fs-id1163870264795\"><p id=\"fs-id1163873788892\">\\(\\left\\{\\begin{array}{c}4{x}^{2}+9{y}^{2}=36\\hfill \\\\ 2{x}^{2}-9{y}^{2}=18\\hfill \\end{array}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163873853215\"><div data-type=\"problem\" id=\"fs-id1163873853217\"><p id=\"fs-id1163873853219\">\\(\\left\\{\\begin{array}{c}{x}^{2}-{y}^{2}=3\\hfill \\\\ 2{x}^{2}+{y}^{2}=6\\hfill \\end{array}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163873596207\"><p id=\"fs-id1163873596209\">\\(\\left(\\text{\u2212}\\sqrt{3},0\\right),\\left(\\sqrt{3},0\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163873805546\"><div data-type=\"problem\" id=\"fs-id1163873823299\"><p id=\"fs-id1163873823301\">\\(\\left\\{\\begin{array}{c}4{x}^{2}-{y}^{2}=4\\hfill \\\\ 4{x}^{2}+{y}^{2}=4\\hfill \\end{array}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163873686685\"><div data-type=\"problem\" id=\"fs-id1163873871039\"><p id=\"fs-id1163873871041\">\\(\\left\\{\\begin{array}{c}{x}^{2}-{y}^{2}=-5\\hfill \\\\ 3{x}^{2}+2{y}^{2}=30\\hfill \\end{array}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163873765849\"><p id=\"fs-id1163873765851\">\\(\\left(-2,-3\\right),\\left(-2,3\\right),\\left(2,-3\\right),\\left(2,3\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163869408844\"><div data-type=\"problem\" id=\"fs-id1163869407406\"><p id=\"fs-id1163869407408\">\\(\\left\\{\\begin{array}{c}{x}^{2}-{y}^{2}=1\\hfill \\\\ {x}^{2}-2y=4\\hfill \\end{array}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163873643106\"><div data-type=\"problem\" id=\"fs-id1163873643108\"><p id=\"fs-id1163873679494\">\\(\\left\\{\\begin{array}{c}2{x}^{2}+{y}^{2}=11\\hfill \\\\ {x}^{2}+3{y}^{2}=28\\hfill \\end{array}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163873624493\"><p id=\"fs-id1163869163401\">\\(\\left(-1,-3\\right),\\left(-1,3\\right),\\left(1,-3\\right),\\left(1,3\\right)\\)<\/p><\/div><\/div><p id=\"fs-id1163873649089\"><strong data-effect=\"bold\">Use a System of Nonlinear Equations to Solve Applications<\/strong><\/p><p id=\"fs-id1163873853528\">In the following exercises, solve the problem using a system of equations.<\/p><div data-type=\"exercise\" id=\"fs-id1163873853531\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1163873898786\"><p id=\"fs-id1163873898789\">The sum of two numbers is \\(-6\\) and the product is 8. Find the numbers.<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163873801053\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1163873801056\"><p id=\"fs-id1163873606050\">The sum of two numbers is 11 and the product is \\(-42.\\) Find the numbers.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163869163582\"><p id=\"fs-id1163873662389\">\\(-3\\) and 14<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163873520191\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1163873657308\"><p id=\"fs-id1163873657310\">The sum of the squares of two numbers is 65. The difference of the number is 3. Find the numbers.<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163869164255\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1163873660716\"><p id=\"fs-id1163873660718\">The sum of the squares of two numbers is 113. The difference of the number is 1. Find the numbers.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163873804464\"><p id=\"fs-id1163873804466\">\\(-7\\) and \\(-8\\) or 8 and 7<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163873632453\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1163873632455\"><p id=\"fs-id1163873632457\">The difference of the squares of two numbers is 15. The difference of twice the square of the first number and the square of the second number is 30. Find the numbers.<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163874017665\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1163874017668\"><p id=\"fs-id1163874017670\">The difference of the squares of two numbers is 20. The difference of the square of the first number and twice the square of the second number is 4. Find the numbers.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163870551949\"><p id=\"fs-id1163873998028\">\\(-6\\) and \\(-4\\) or \\(-6\\) and 4 or 6 and \\(-4\\) or 6 and 4<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163873858060\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1163873858063\"><p id=\"fs-id1163873858065\">The perimeter of a rectangle is 32 inches and its area is 63 square inches. Find the length and width of the rectangle.<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163873863908\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1163873636531\"><p id=\"fs-id1163873636534\">The perimeter of a rectangle is 52 cm and its area is 165 \\({\\text{cm}}^{2}.\\) Find the length and width of the rectangle.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163873821577\"><p id=\"fs-id1163873951878\">If the length is 11 cm, the width is 15 cm. If the length is 15 cm, the width is 11 cm.<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163873951883\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1163873862658\"><p id=\"fs-id1163873862661\">Dion purchased a new microwave. The diagonal of the door measures 17 inches. The door also has an area of 120 square inches. What are the length and width of the microwave door?<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163873787464\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1163869435881\"><p id=\"fs-id1163869435883\">Jules purchased a microwave for his kitchen. The diagonal of the front of the microwave measures 26 inches. The front also has an area of 240 square inches. What are the length and width of the microwave?<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163870547544\"><p id=\"fs-id1163870547546\">If the length is 10 inches, the width is 24 inches. If the length is 24 inches, the width is 10 inches.<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163870335196\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1163873669566\"><p id=\"fs-id1163873669568\">Roman found a widescreen TV on sale, but isn\u2019t sure if it will fit his entertainment center. The TV is 60\u201d. The size of a TV is measured on the diagonal of the screen and a widescreen has a length that is larger than the width. The screen also has an area of 1728 square inches. His entertainment center has an insert for the TV with a length of 50 inches and width of 40 inches. What are the length and width of the TV screen and will it fit Roman\u2019s entertainment center?<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163873557299\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1163873557301\"><p id=\"fs-id1163873795201\">Donnette found a widescreen TV at a garage sale, but isn\u2019t sure if it will fit her entertainment center. The TV is 50\u201d. The size of a TV is measured on the diagonal of the screen and a widescreen has a length that is larger than the width. The screen also has an area of 1200 square inches. Her entertainment center has an insert for the TV with a length of 38 inches and width of 27 inches. What are the length and width of the TV screen and will it fit Donnette\u2019s entertainment center?<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163873795205\"><p id=\"fs-id1163873654397\">The length is 40 inches and the width is 30 inches. The TV will not fit Donnette\u2019s entertainment center.<\/p><\/div><\/div><\/div><div class=\"writing\" data-depth=\"2\" id=\"fs-id1163873863799\"><h4 data-type=\"title\">Writing Exercises<\/h4><div data-type=\"exercise\" id=\"fs-id1163873758518\"><div data-type=\"problem\" id=\"fs-id1163873758520\"><p id=\"fs-id1163870644374\">In your own words, explain the advantages and disadvantages of solving a system of equations by graphing.<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163873644838\"><div data-type=\"problem\"><p id=\"fs-id1163873644842\">Explain in your own words how to solve a system of equations using substitution.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163873668743\"><p id=\"fs-id1163873668745\">Answers will vary.<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163873786868\"><div data-type=\"problem\" id=\"fs-id1163873786870\"><p id=\"fs-id1163873801806\">Explain in your own words how to solve a system of equations using elimination.<\/p><\/div><\/div><div data-type=\"exercise\"><div data-type=\"problem\"><p id=\"fs-id1163870292674\">A circle and a parabola can intersect in ways that would result in 0, 1, 2, 3, or 4 solutions. Draw a sketch of each of the possibilities.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163870350290\"><p id=\"fs-id1163870350292\">Answers will vary.<\/p><\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1163873616410\"><h4 data-type=\"title\">Self Check<\/h4><p id=\"fs-id1163873784103\"><span class=\"token\">\u24d0<\/span> After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.<\/p><span data-type=\"media\" id=\"fs-id1163873660154\" data-alt=\"This table has four columns and five rows. The first row is a header and it labels each column, &#x201c;I can&#x2026;&#x201d;, &#x201c;Confidently,&#x201d; &#x201c;With some help,&#x201d; and &#x201c;No-I don&#x2019;t get it!&#x201d; In row 2, the I can was solve a system of nonlinear equations using graphing. In row 3, the I can solve a system of nonlinear equations using substitution. In row 4, the I can was solve a system of a nonlinear equations using the elimination. In row 5, the I can was use a system of nonlinear equations to solve applications.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_201_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This table has four columns and five rows. The first row is a header and it labels each column, &#x201c;I can&#x2026;&#x201d;, &#x201c;Confidently,&#x201d; &#x201c;With some help,&#x201d; and &#x201c;No-I don&#x2019;t get it!&#x201d; In row 2, the I can was solve a system of nonlinear equations using graphing. In row 3, the I can solve a system of nonlinear equations using substitution. In row 4, the I can was solve a system of a nonlinear equations using the elimination. In row 5, the I can was use a system of nonlinear equations to solve applications.\" \/><\/span><p id=\"fs-id1163873714806\"><span class=\"token\">\u24d1<\/span> After looking at the checklist, do you think you are well-prepared for the next section? Why or why not?<\/p><\/div><\/div><div class=\"review-exercises\" data-depth=\"1\" id=\"fs-id1163874006441\"><h3 data-type=\"title\">Chapter Review Exercises<\/h3><div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1163874006445\"><h4 data-type=\"title\"><a href=\"\/contents\/30062189-1923-4bf7-902b-9f2691a64c71\" class=\"target-chapter\">Distance and Midpoint Formulas; Circles<\/a><\/h4><p id=\"fs-id1163873659904\"><strong data-effect=\"bold\">Use the Distance Formula<\/strong><\/p><p id=\"fs-id1163873798182\">In the following exercises, find the distance between the points. Round to the nearest tenth if needed.<\/p><div data-type=\"exercise\" id=\"fs-id1163873798186\"><div data-type=\"problem\" id=\"fs-id1163870357236\"><p id=\"fs-id1163870357238\">\\(\\left(-5,1\\right)\\) and \\(\\left(-1,4\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163873807388\"><div data-type=\"problem\" id=\"fs-id1163873807390\"><p id=\"fs-id1163873807392\">\\(\\left(-2,5\\right)\\) and \\(\\left(1,5\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163873823025\"><p id=\"fs-id1163873823027\">\\(d=3\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163866979320\"><div data-type=\"problem\" id=\"fs-id1163866979322\"><p id=\"fs-id1163870376214\">\\(\\left(8,2\\right)\\) and \\(\\left(-7,-3\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163873866500\"><div data-type=\"problem\" id=\"fs-id1163873857670\"><p id=\"fs-id1163873857672\">\\(\\left(1,-4\\right)\\) and \\(\\left(5,-5\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163870548914\"><p id=\"fs-id1163869099050\">\\(d=\\sqrt{17},d\\approx 4.1\\)<\/p><\/div><\/div><p id=\"fs-id1163873604614\"><strong data-effect=\"bold\">Use the Midpoint Formula<\/strong><\/p><p id=\"fs-id1163873702172\">In the following exercises, find the midpoint of the line segments whose endpoints are given.<\/p><div data-type=\"exercise\" id=\"fs-id1163866979854\"><div data-type=\"problem\" id=\"fs-id1163866979856\"><p id=\"fs-id1163866979858\">\\(\\left(-2,-6\\right)\\) and \\(\\left(-4,-2\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163869435324\"><div data-type=\"problem\" id=\"fs-id1163869435326\"><p id=\"fs-id1163873924830\">\\(\\left(3,7\\right)\\) and \\(\\left(5,1\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163873639312\"><p id=\"fs-id1163873639314\">\\(\\left(4,4\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163873608722\"><div data-type=\"problem\" id=\"fs-id1163873882098\"><p id=\"fs-id1163873882100\">\\(\\left(-8,-10\\right)\\) and \\(\\left(9,5\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163873750610\"><div data-type=\"problem\" id=\"fs-id1163873742563\"><p id=\"fs-id1163873742565\">\\(\\left(-3,2\\right)\\) and \\(\\left(6,-9\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163873645302\"><p id=\"fs-id1163873645305\">\\(\\left(\\frac{3}{2},-\\frac{7}{2}\\right)\\)<\/p><\/div><\/div><p id=\"fs-id1163870547955\"><strong data-effect=\"bold\">Write the Equation of a Circle in Standard Form<\/strong><\/p><p id=\"fs-id1163870547960\">In the following exercises, write the standard form of the equation of the circle with the given information.<\/p><div data-type=\"exercise\" id=\"fs-id1163873812775\"><div data-type=\"problem\" id=\"fs-id1163873812777\"><p id=\"fs-id1163873812779\">radius is 15 and center is \\(\\left(0,0\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163873673838\"><div data-type=\"problem\" id=\"fs-id1163873814747\"><p id=\"fs-id1163873814750\">radius is \\(\\sqrt{7}\\) and center is \\(\\left(0,0\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163870510888\"><p id=\"fs-id1163873607179\">\\({x}^{2}+{y}^{2}=7\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163874046327\"><div data-type=\"problem\" id=\"fs-id1163874046330\"><p id=\"fs-id1163874046332\">radius is 9 and center is \\(\\left(-3,5\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163873606476\"><div data-type=\"problem\" id=\"fs-id1163873606478\"><p id=\"fs-id1163873606481\">radius is 7 and center is \\(\\left(-2,-5\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163873655226\"><p id=\"fs-id1163873655228\">\\({\\left(x+2\\right)}^{2}+{\\left(y+5\\right)}^{2}=49\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163873624748\"><div data-type=\"problem\" id=\"fs-id1163873882075\"><p id=\"fs-id1163873882078\">center is \\(\\left(3,6\\right)\\) and a point on the circle is \\(\\left(3,-2\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163873616884\"><div data-type=\"problem\" id=\"fs-id1163873616886\"><p id=\"fs-id1163873645451\">center is \\(\\left(2,2\\right)\\) and a point on the circle is \\(\\left(4,4\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163873661694\"><p id=\"fs-id1163873660619\">\\({\\left(x-2\\right)}^{2}+{\\left(y-2\\right)}^{2}=8\\)<\/p><\/div><\/div><p id=\"fs-id1163870550111\"><strong data-effect=\"bold\">Graph a Circle<\/strong><\/p><p id=\"fs-id1163870516094\">In the following exercises, <span class=\"token\">\u24d0<\/span> find the center and radius, then <span class=\"token\">\u24d1<\/span> graph each circle.<\/p><div data-type=\"exercise\" id=\"fs-id1163873526037\"><div data-type=\"problem\" id=\"fs-id1163873837511\"><p id=\"fs-id1163873837513\">\\(2{x}^{2}+2{y}^{2}=450\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163873801972\"><div data-type=\"problem\" id=\"fs-id1163873801974\"><p id=\"fs-id1163873606015\">\\(3{x}^{2}+3{y}^{2}=432\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163873731680\"><p id=\"fs-id1163873731682\"><span class=\"token\">\u24d0<\/span> radius: 12, center: \\(\\left(0,0\\right)\\)<span data-type=\"newline\"><br \/><\/span> <span class=\"token\">\u24d1<\/span><span data-type=\"newline\"><br \/><\/span><\/p><span data-type=\"media\" id=\"fs-id1163873678491\" data-alt=\"The figure shows a circle graphed on the x y coordinate plane. The x-axis of the plane runs from negative 20 to 20. The y-axis of the plane runs from negative 15 to 15. The center of the circle is (0, 0) and the radius of the circle is 12.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_318_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows a circle graphed on the x y coordinate plane. The x-axis of the plane runs from negative 20 to 20. The y-axis of the plane runs from negative 15 to 15. The center of the circle is (0, 0) and the radius of the circle is 12.\" \/><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163873823003\"><div data-type=\"problem\" id=\"fs-id1163873823006\"><p id=\"fs-id1163873823008\">\\({\\left(x+3\\right)}^{2}+{\\left(y-5\\right)}^{2}=81\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163870266037\"><div data-type=\"problem\" id=\"fs-id1163870266039\"><p id=\"fs-id1163873716963\">\\({\\left(x+2\\right)}^{2}+{\\left(y+5\\right)}^{2}=49\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163870398885\"><p id=\"fs-id1163870398888\"><span class=\"token\">\u24d0<\/span> radius: 7, center: \\(\\left(-2,-5\\right)\\)<span data-type=\"newline\"><br \/><\/span> <span class=\"token\">\u24d1<\/span><span data-type=\"newline\"><br \/><\/span><\/p><span data-type=\"media\" id=\"fs-id1163873783987\" data-alt=\"The figure shows a circle graphed on the x y coordinate plane. The x-axis of the plane runs from negative 20 to 20. The y-axis of the plane runs from negative 15 to 15. The center of the circle is (negative 2, negative 5) and the radius of the circle is 7.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_320_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows a circle graphed on the x y coordinate plane. The x-axis of the plane runs from negative 20 to 20. The y-axis of the plane runs from negative 15 to 15. The center of the circle is (negative 2, negative 5) and the radius of the circle is 7.\" \/><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163873558348\"><div data-type=\"problem\" id=\"fs-id1163873632142\"><p id=\"fs-id1163873632144\">\\({x}^{2}+{y}^{2}-6x-12y-19=0\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163873853290\"><div data-type=\"problem\" id=\"fs-id1163873853292\"><p id=\"fs-id1163873631018\">\\({x}^{2}+{y}^{2}-4y-60=0\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163869138122\"><p id=\"fs-id1163874044723\"><span class=\"token\">\u24d0<\/span> radius: 8, center: \\(\\left(0,2\\right)\\)<span data-type=\"newline\"><br \/><\/span> <span class=\"token\">\u24d1<\/span><span data-type=\"newline\"><br \/><\/span><\/p><span data-type=\"media\" id=\"fs-id1163873657446\" data-alt=\"The figure shows a circle graphed on the x y coordinate plane. The x-axis of the plane runs from negative 20 to 20. The y-axis of the plane runs from negative 15 to 15. The center of the circle is (0, 2) and the radius of the circle is 8.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_322_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows a circle graphed on the x y coordinate plane. The x-axis of the plane runs from negative 20 to 20. The y-axis of the plane runs from negative 15 to 15. The center of the circle is (0, 2) and the radius of the circle is 8.\" \/><\/span><\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1163870623961\"><h4 data-type=\"title\"><a href=\"\/contents\/789eaa88-3770-4bb0-9039-ff1146680681\" class=\"target-chapter\">Parabolas<\/a><\/h4><p id=\"fs-id1163873764394\"><strong data-effect=\"bold\">Graph Vertical Parabolas<\/strong><\/p><p id=\"fs-id1163870484482\">In the following exercises, graph each equation by using its properties.<\/p><div data-type=\"exercise\" id=\"fs-id1163870484485\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1163869163587\"><p id=\"fs-id1163869163590\">\\(y={x}^{2}+4x-3\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163870510844\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1163870510846\"><p id=\"fs-id1163870510848\">\\(y=2{x}^{2}+10x+7\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163873639811\"><span data-type=\"media\" id=\"fs-id1163873639814\" data-alt=\"The figure shows an upward-opening parabola graphed on the x y coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 7 to 7. The vertex is (negative five-halves, negative eleven-halves) and the parabola passes through the points (negative 4, negative 1) and (negative 1, negative 1).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_324_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows an upward-opening parabola graphed on the x y coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 7 to 7. The vertex is (negative five-halves, negative eleven-halves) and the parabola passes through the points (negative 4, negative 1) and (negative 1, negative 1).\" \/><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163873817175\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1163873817177\"><p id=\"fs-id1163873817180\">\\(y=-6{x}^{2}+12x-1\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163873853748\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1163873853750\"><p id=\"fs-id1163873853752\">\\(y=\\text{\u2212}{x}^{2}+10x\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163873872896\"><span data-type=\"media\" id=\"fs-id1163873872899\" data-alt=\"The figure shows a downward-opening parabola graphed on the x y coordinate plane. The x-axis of the plane runs from negative 36 to 36. The y-axis of the plane runs from negative 26 to 26. The vertex is (5, 25) and the parabola passes through the points (2, 16) and (8, 16).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_326_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows a downward-opening parabola graphed on the x y coordinate plane. The x-axis of the plane runs from negative 36 to 36. The y-axis of the plane runs from negative 26 to 26. The vertex is (5, 25) and the parabola passes through the points (2, 16) and (8, 16).\" \/><\/span><\/div><\/div><p id=\"fs-id1163873663657\">In the following exercises, <span class=\"token\">\u24d0<\/span> write the equation in standard form, then <span class=\"token\">\u24d1<\/span> use properties of the standard form to graph the equation.<\/p><div data-type=\"exercise\" id=\"fs-id1163873882772\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1163873882775\"><p id=\"fs-id1163873882777\">\\(y={x}^{2}+4x+7\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163873812536\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1163873812538\"><p id=\"fs-id1163873662802\">\\(y=2{x}^{2}-4x-2\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163873822689\"><p id=\"fs-id1163873822691\"><span class=\"token\">\u24d0<\/span>\\(y=2{\\left(x-1\\right)}^{2}-4\\)<span data-type=\"newline\"><br \/><\/span><span class=\"token\">\u24d1<\/span><span data-type=\"newline\"><br \/><\/span><\/p><span data-type=\"media\" id=\"fs-id1163870644905\" data-alt=\"The figure shows an upward-opening parabola graphed on the x y coordinate plane. The x-axis of the plane runs from negative 22 to 22. The y-axis of the plane runs from negative 16 to 16. The vertex is (1, negative 4) and the parabola passes through the points (0, negative 2) and (2, negative 2).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_328_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows an upward-opening parabola graphed on the x y coordinate plane. The x-axis of the plane runs from negative 22 to 22. The y-axis of the plane runs from negative 16 to 16. The vertex is (1, negative 4) and the parabola passes through the points (0, negative 2) and (2, negative 2).\" \/><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163873782264\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1163873782266\"><p id=\"fs-id1163873752504\">\\(y=-3{x}^{2}-18x-29\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163870547472\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1163870547474\"><p id=\"fs-id1163873797140\">\\(y=\\text{\u2212}{x}^{2}+12x-35\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163873866506\"><p id=\"fs-id1163873866509\"><span class=\"token\">\u24d0<\/span>\\(y=\\text{\u2212}{\\left(x-6\\right)}^{2}+1\\)<span data-type=\"newline\"><br \/><\/span><span class=\"token\">\u24d1<\/span><span data-type=\"newline\"><br \/><\/span><\/p><span data-type=\"media\" id=\"fs-id1163873651660\" data-alt=\"The figure shows a downward-opening parabola graphed on the x y coordinate plane. The x-axis of the plane runs from negative 60 to 60. The y-axis of the plane runs from negative 46 to 46. The vertex is (6, 1) and the parabola passes through the points (5, 0) and (7, 0).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_330_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows a downward-opening parabola graphed on the x y coordinate plane. The x-axis of the plane runs from negative 60 to 60. The y-axis of the plane runs from negative 46 to 46. The vertex is (6, 1) and the parabola passes through the points (5, 0) and (7, 0).\" \/><\/span><\/div><\/div><p id=\"fs-id1163873657481\"><strong data-effect=\"bold\">Graph Horizontal Parabolas<\/strong><\/p><p id=\"fs-id1163873608732\">In the following exercises, graph each equation by using its properties.<\/p><div data-type=\"exercise\" id=\"fs-id1163873608736\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1163873768714\"><p id=\"fs-id1163873768716\">\\(x=2{y}^{2}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163873663351\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1163873663353\"><p id=\"fs-id1163873616997\">\\(x=2{y}^{2}+4y+6\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163873855596\"><span data-type=\"media\" id=\"fs-id1163873855599\" data-alt=\"The figure shows a rightward-opening parabola graphed on the x y coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 8 to 8. The vertex is (4, negative 1) and the parabola passes through the points (6, 0) and (6, negative 2).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_332_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows a rightward-opening parabola graphed on the x y coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 8 to 8. The vertex is (4, negative 1) and the parabola passes through the points (6, 0) and (6, negative 2).\" \/><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163869435866\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1163869435868\"><p id=\"fs-id1163869346255\">\\(x=\\text{\u2212}{y}^{2}+2y-4\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163873784033\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1163873784035\"><p id=\"fs-id1163870386273\">\\(x=-3{y}^{2}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163873703287\"><span data-type=\"media\" id=\"fs-id1163873703290\" data-alt=\"The figure shows a leftward-opening parabola graphed on the x y coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 8 to 8. The vertex is (0, 0) and the parabola passes through the points (negative 3, 1) and (negative 3, negative 1).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_334_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows a leftward-opening parabola graphed on the x y coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 8 to 8. The vertex is (0, 0) and the parabola passes through the points (negative 3, 1) and (negative 3, negative 1).\" \/><\/span><\/div><\/div><p id=\"fs-id1163866980050\">In the following exercises, <span class=\"token\">\u24d0<\/span> write the equation in standard form, then <span class=\"token\">\u24d1<\/span> use properties of the standard form to graph the equation.<\/p><div data-type=\"exercise\" id=\"fs-id1163873635535\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1163869409805\"><p id=\"fs-id1163869409807\">\\(x=4{y}^{2}+8y\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163873809661\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1163873809663\"><p id=\"fs-id1163873809665\">\\(x={y}^{2}+4y+5\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163870551528\"><p id=\"fs-id1163869091463\"><span class=\"token\">\u24d0<\/span>\\(x={\\left(y+2\\right)}^{2}+1\\)<span data-type=\"newline\"><br \/><\/span><span class=\"token\">\u24d1<\/span><span data-type=\"newline\"><br \/><\/span><\/p><span data-type=\"media\" id=\"fs-id1163873769107\" data-alt=\"The figure shows a rightward-opening parabola graphed on the x y coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 8 to 8. The vertex is (1, negative 2) and the parabola passes through the points (5, 0) and (5, negative 4).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_336_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows a rightward-opening parabola graphed on the x y coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 8 to 8. The vertex is (1, negative 2) and the parabola passes through the points (5, 0) and (5, negative 4).\" \/><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163873861448\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1163870359109\"><p id=\"fs-id1163870359112\">\\(x=\\text{\u2212}{y}^{2}-6y-7\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163873785782\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1163873757150\"><p id=\"fs-id1163873757152\">\\(x=-2{y}^{2}+4y\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163870489447\"><p id=\"fs-id1163870489449\"><span class=\"token\">\u24d0<\/span>\\(x=-2{\\left(y-1\\right)}^{2}+2\\)<span data-type=\"newline\"><br \/><\/span><span class=\"token\">\u24d1<\/span><span data-type=\"newline\"><br \/><\/span><\/p><span data-type=\"media\" id=\"fs-id1163870291308\" data-alt=\"The figure shows a leftward-opening parabola graphed on the x y coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 8 to 8. The vertex is (2, negative 3) and the parabola passes through the points (0, 2) and (0, 0).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_338_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows a leftward-opening parabola graphed on the x y coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 8 to 8. The vertex is (2, negative 3) and the parabola passes through the points (0, 2) and (0, 0).\" \/><\/span><\/div><\/div><p id=\"fs-id1163873673095\"><strong data-effect=\"bold\">Solve Applications with Parabolas<\/strong><\/p><p id=\"fs-id1163873520546\">In the following exercises, create the equation of the parabolic arch formed in the foundation of the bridge shown. Give the answer in standard form.<\/p><div data-type=\"exercise\" id=\"fs-id1163873520550\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1163873520552\"><span data-type=\"media\" id=\"fs-id1163873657328\" data-alt=\"The figure shows a parabolic arch formed in the foundation of the bridge. The arch is 5 feet high and 20 feet wide.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_202_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows a parabolic arch formed in the foundation of the bridge. The arch is 5 feet high and 20 feet wide.\" \/><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163873761318\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1163873761320\"><span data-type=\"media\" id=\"fs-id1163873761322\" data-alt=\"The figure shows a parabolic arch formed in the foundation of the bridge. The arch is 25 feet high and 30 feet wide.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_203_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows a parabolic arch formed in the foundation of the bridge. The arch is 25 feet high and 30 feet wide.\" \/><\/span><\/div><div data-type=\"solution\" id=\"fs-id1163873764370\"><p id=\"fs-id1163873764372\">\\(y=-\\frac{1}{9}{x}^{2}+\\frac{10}{3}x\\)<\/p><\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1163870644426\"><h4 data-type=\"title\"><a href=\"\/contents\/4c350ea6-1dd0-4a11-a80a-9a5f02497a87\" class=\"target-chapter\">Ellipses<\/a><\/h4><p id=\"fs-id1163870357359\"><strong data-effect=\"bold\">Graph an Ellipse with Center at the Origin<\/strong><\/p><p id=\"fs-id1163870357366\">In the following exercises, graph each ellipse.<\/p><div data-type=\"exercise\" id=\"fs-id1163873629400\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1163873629402\"><p id=\"fs-id1163873629404\">\\(\\frac{{x}^{2}}{36}+\\frac{{y}^{2}}{25}=1\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163873814467\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1163869162846\"><p id=\"fs-id1163869162848\">\\(\\frac{{x}^{2}}{4}+\\frac{{y}^{2}}{81}=1\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163873862770\"><span data-type=\"media\" id=\"fs-id1163873862773\" data-alt=\"The figure shows an ellipse graphed on the x y coordinate plane. The x-axis of the plane runs from negative 14 to 14. The y-axis of the plane runs from negative 10 to 10. The ellipse has a center at (0, 0), a vertical major axis, vertices at (0, plus or minus 9), and co-vertices at (plus or minus 2, 0).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_340_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows an ellipse graphed on the x y coordinate plane. The x-axis of the plane runs from negative 14 to 14. The y-axis of the plane runs from negative 10 to 10. The ellipse has a center at (0, 0), a vertical major axis, vertices at (0, plus or minus 9), and co-vertices at (plus or minus 2, 0).\" \/><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163873896958\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1163873668245\"><p id=\"fs-id1163873668247\">\\(49{x}^{2}+64{y}^{2}=3136\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163873790620\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1163873790622\"><p id=\"fs-id1163873790624\">\\(9{x}^{2}+{y}^{2}=9\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163873863215\"><span data-type=\"media\" id=\"fs-id1163873863218\" data-alt=\"The figure shows an ellipse graphed on the x y coordinate plane. The x-axis of the plane runs from negative 9 to 9. The y-axis of the plane runs from negative 7 to 7. The ellipse has a center at (0, 0), a vertical major axis, vertices at (0, plus or minus 3), and co-vertices at (plus or minus 1, 0).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_342_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows an ellipse graphed on the x y coordinate plane. The x-axis of the plane runs from negative 9 to 9. The y-axis of the plane runs from negative 7 to 7. The ellipse has a center at (0, 0), a vertical major axis, vertices at (0, plus or minus 3), and co-vertices at (plus or minus 1, 0).\" \/><\/span><\/div><\/div><p id=\"fs-id1163870694967\"><strong data-effect=\"bold\">Find the Equation of an Ellipse with Center at the Origin<\/strong><\/p><p id=\"fs-id1163870694974\">In the following exercises, find the equation of the ellipse shown in the graph.<\/p><div data-type=\"exercise\" id=\"fs-id1163873880357\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1163873880359\"><span data-type=\"media\" id=\"fs-id1163873880361\" data-alt=\"The figure shows an ellipse graphed on the x y coordinate plane. The ellipse has a center at (0, 0), a horizontal major axis, vertices at (plus or minus 10, 0), and co-vertices at (0, plus or minus 4).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_204_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows an ellipse graphed on the x y coordinate plane. The ellipse has a center at (0, 0), a horizontal major axis, vertices at (plus or minus 10, 0), and co-vertices at (0, plus or minus 4).\" \/><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163873856127\" class=\"material-set-2\"><div data-type=\"problem\"><span data-type=\"media\" id=\"fs-id1163873800254\" data-alt=\"The figure shows an ellipse graphed on the x y coordinate plane. The ellipse has a center at (0, 0), a vertical major axis, vertices at (0, plus or minus 8), and co-vertices at (plus or minus 6, 0).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_205_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows an ellipse graphed on the x y coordinate plane. The ellipse has a center at (0, 0), a vertical major axis, vertices at (0, plus or minus 8), and co-vertices at (plus or minus 6, 0).\" \/><\/span><\/div><div data-type=\"solution\" id=\"fs-id1163870695358\"><p id=\"fs-id1163870695360\">\\(\\frac{{x}^{2}}{36}+\\frac{{y}^{2}}{64}=1\\)<\/p><\/div><\/div><p id=\"fs-id1163873769016\"><strong data-effect=\"bold\">Graph an Ellipse with Center Not at the Origin<\/strong><\/p><p>In the following exercises, graph each ellipse.<\/p><div data-type=\"exercise\" id=\"fs-id1163873607106\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1163873607108\"><p id=\"fs-id1163873607110\">\\(\\frac{{\\left(x-1\\right)}^{2}}{25}+\\frac{{\\left(y-6\\right)}^{2}}{4}=1\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163873715946\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1163873715949\"><p id=\"fs-id1163873715951\">\\(\\frac{{\\left(x+4\\right)}^{2}}{16}+\\frac{{\\left(y+1\\right)}^{2}}{9}=1\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163873637692\"><span data-type=\"media\" id=\"fs-id1163873637695\" data-alt=\"The figure shows an ellipse graphed on the x y coordinate plane. The x-axis of the plane runs from negative 14 to 14. The y-axis of the plane runs from negative 10 to 10. The ellipse has a center at (negative 4, negative 1), a horizontal major axis, vertices at (negative 8, negative 1) and (0, negative 1) and co-vertices at (negative 4, 2) and (negative 4, negative 4).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_344_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows an ellipse graphed on the x y coordinate plane. The x-axis of the plane runs from negative 14 to 14. The y-axis of the plane runs from negative 10 to 10. The ellipse has a center at (negative 4, negative 1), a horizontal major axis, vertices at (negative 8, negative 1) and (0, negative 1) and co-vertices at (negative 4, 2) and (negative 4, negative 4).\" \/><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163873855121\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1163873855123\"><p id=\"fs-id1163873627563\">\\(\\frac{{\\left(x-5\\right)}^{2}}{16}+\\frac{{\\left(y+3\\right)}^{2}}{36}=1\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163873792951\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1163873792953\"><p id=\"fs-id1163873792955\">\\(\\frac{{\\left(x+3\\right)}^{2}}{9}+\\frac{{\\left(y-2\\right)}^{2}}{25}=1\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163873822877\"><span data-type=\"media\" id=\"fs-id1163873822880\" data-alt=\"The figure shows an ellipse graphed on the x y coordinate plane. The x-axis of the plane runs from negative 14 to 14. The y-axis of the plane runs from negative 10 to 10. The ellipse has a center at (negative 3, 2), a vertical major axis, vertices at (negative 3, 7) and (negative 3, negative 3) and co-vertices at (negative 6, 2) and (0, 2).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_346_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows an ellipse graphed on the x y coordinate plane. The x-axis of the plane runs from negative 14 to 14. The y-axis of the plane runs from negative 10 to 10. The ellipse has a center at (negative 3, 2), a vertical major axis, vertices at (negative 3, 7) and (negative 3, negative 3) and co-vertices at (negative 6, 2) and (0, 2).\" \/><\/span><\/div><\/div><p id=\"fs-id1163873745244\">In the following exercises, <span class=\"token\">\u24d0<\/span> write the equation in standard form and <span class=\"token\">\u24d1<\/span> graph.<\/p><div data-type=\"exercise\" id=\"fs-id1163873604412\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1163873604414\"><p id=\"fs-id1163873604416\">\\({x}^{2}+{y}^{2}+12x+40y+120=0\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163873654528\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1163873654530\"><p id=\"fs-id1163873654532\">\\(25{x}^{2}+4{y}^{2}-150x-56y+321=0\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163873999255\"><p id=\"fs-id1163873799839\"><span class=\"token\">\u24d0<\/span>\\(\\frac{{\\left(x-3\\right)}^{2}}{4}+\\frac{{\\left(y-7\\right)}^{2}}{25}=1\\)<span data-type=\"newline\"><br \/><\/span><span class=\"token\">\u24d1<\/span><span data-type=\"newline\"><br \/><\/span><\/p><span data-type=\"media\" id=\"fs-id1163873786610\" data-alt=\"The figure shows an ellipse graphed on the x y coordinate plane. The x-axis of the plane runs from negative 18 to 18. The y-axis of the plane runs from negative 14 to 14. The ellipse has a center at (3, 7), a vertical major axis, vertices at (3, 2) and (3, 12) and co-vertices at (negative 1, 7) and (5, 7).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_348_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows an ellipse graphed on the x y coordinate plane. The x-axis of the plane runs from negative 18 to 18. The y-axis of the plane runs from negative 14 to 14. The ellipse has a center at (3, 7), a vertical major axis, vertices at (3, 2) and (3, 12) and co-vertices at (negative 1, 7) and (5, 7).\" \/><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163873559682\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1163873559684\"><p id=\"fs-id1163873559687\">\\(25{x}^{2}+4{y}^{2}+150x+125=0\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163870221437\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1163870221439\"><p id=\"fs-id1163870221441\">\\(4{x}^{2}+9{y}^{2}-126x+405=0\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163869409421\"><p id=\"fs-id1163869409424\"><span class=\"token\">\u24d0<\/span>\\(\\frac{{x}^{2}}{9}+\\frac{{\\left(y-7\\right)}^{2}}{4}=1\\)<span data-type=\"newline\"><br \/><\/span><span class=\"token\">\u24d1<\/span><span data-type=\"newline\"><br \/><\/span><\/p><span data-type=\"media\" id=\"fs-id1163873854352\" data-alt=\"The figure shows an ellipse graphed on the x y coordinate plane. The x-axis of the plane runs from negative 15 to 15. The y-axis of the plane runs from negative 11 to 11. The ellipse has a center at (0, 7), a horizontal major axis, vertices at (3, 7) and (negative 3, 7) and co-vertices at (0, 5) and (0, 9).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_350_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows an ellipse graphed on the x y coordinate plane. The x-axis of the plane runs from negative 15 to 15. The y-axis of the plane runs from negative 11 to 11. The ellipse has a center at (0, 7), a horizontal major axis, vertices at (3, 7) and (negative 3, 7) and co-vertices at (0, 5) and (0, 9).\" \/><\/span><\/div><\/div><p id=\"fs-id1163870547451\"><strong data-effect=\"bold\">Solve Applications with Ellipses<\/strong><\/p><p id=\"fs-id1163870547457\">In the following exercises, write the equation of the ellipse described.<\/p><div data-type=\"exercise\" id=\"fs-id1163873642289\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1163873642291\"><p id=\"fs-id1163873642293\">A comet moves in an elliptical orbit around a sun. The closest the comet gets to the sun is approximately 10 AU and the furthest is approximately 90 AU. The sun is one of the foci of the elliptical orbit. Letting the ellipse center at the origin and labeling the axes in AU, the orbit will look like the figure below. Use the graph to write an equation for the elliptical orbit of the comet.<\/p><span data-type=\"media\" id=\"fs-id1163873642300\" data-alt=\"The figure shows a model of an elliptical orbit around the sun on the x y coordinate plane. The ellipse has a center at (0, 0), a horizontal major axis, vertices marked at (plus or minus 50, 0), the sun marked as a foci and labeled (50, 0), the closest distance the comet is from the sun marked as 10 A U, and the farthest a comet is from the sun marked as 90 A U.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_206_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows a model of an elliptical orbit around the sun on the x y coordinate plane. The ellipse has a center at (0, 0), a horizontal major axis, vertices marked at (plus or minus 50, 0), the sun marked as a foci and labeled (50, 0), the closest distance the comet is from the sun marked as 10 A U, and the farthest a comet is from the sun marked as 90 A U.\" \/><\/span><\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1163873694035\"><h4 data-type=\"title\"><a href=\"\/contents\/5784f55d-62cb-474f-aa4f-747760be4966\" class=\"target-chapter\">Hyperbolas<\/a><\/h4><p id=\"fs-id1163873694046\"><strong data-effect=\"bold\">Graph a Hyperbola with Center at \\(\\left(0,0\\right)\\)<\/strong><\/p><p id=\"fs-id1163873749063\">In the following exercises, graph.<\/p><div data-type=\"exercise\" id=\"fs-id1163873749066\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1163873750108\"><p id=\"fs-id1163873750110\">\\(\\frac{{x}^{2}}{25}-\\frac{{y}^{2}}{9}=1\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163873632366\"><span data-type=\"media\" id=\"fs-id1163873632369\" data-alt=\"The figure shows a hyperbola graphed on the x y coordinate plane. The x-axis of the plane runs from negative 12 to 12. The y-axis of the plane runs from negative 9 to 9. The hyperbola has a center at (0, 0) and branches that pass through the vertices (plus or minus 5, 0), and that open left and right.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_351_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows a hyperbola graphed on the x y coordinate plane. The x-axis of the plane runs from negative 12 to 12. The y-axis of the plane runs from negative 9 to 9. The hyperbola has a center at (0, 0) and branches that pass through the vertices (plus or minus 5, 0), and that open left and right.\" \/><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163873732163\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1163873732165\"><p id=\"fs-id1163873732167\">\\(\\frac{{y}^{2}}{49}-\\frac{{x}^{2}}{16}=1\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163873748757\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1163873748760\"><p id=\"fs-id1163873748762\">\\(9{y}^{2}-16{x}^{2}=144\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163870410768\"><span data-type=\"media\" id=\"fs-id1163870410771\" data-alt=\"The figure shows a hyperbola graphed on the x y coordinate plane. The x-axis of the plane runs from negative 19 to 19. The y-axis of the plane runs from negative 15 to 15. The hyperbola has a center at (0, 0) and branches that pass through the vertices (0, plus or minus 4), and that open up and down.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_353_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows a hyperbola graphed on the x y coordinate plane. The x-axis of the plane runs from negative 19 to 19. The y-axis of the plane runs from negative 15 to 15. The hyperbola has a center at (0, 0) and branches that pass through the vertices (0, plus or minus 4), and that open up and down.\" \/><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163870694898\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1163870694900\"><p id=\"fs-id1163870694902\">\\(16{x}^{2}-4{y}^{2}=64\\)<\/p><\/div><\/div><p id=\"fs-id1163873870185\"><strong data-effect=\"bold\">Graph a Hyperbola with Center at \\(\\left(h,k\\right)\\)<\/strong><\/p><p id=\"fs-id1163873912716\">In the following exercises, graph.<\/p><div data-type=\"exercise\" id=\"fs-id1163873912719\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1163873912721\"><p id=\"fs-id1163873912723\">\\(\\frac{{\\left(x+1\\right)}^{2}}{4}-\\frac{{\\left(y+1\\right)}^{2}}{9}=1\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163873625380\"><span data-type=\"media\" id=\"fs-id1163873811040\" data-alt=\"The figure shows a hyperbola graphed on the x y coordinate plane. The x-axis of the plane runs from negative 14 to 14. The y-axis of the plane runs from negative 10 to 10. The hyperbola has a center at (negative 1, negative 1) and branches that pass through the vertices (negative 3, negative 1) and (1, negative 1), and that open left and right.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_355_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows a hyperbola graphed on the x y coordinate plane. The x-axis of the plane runs from negative 14 to 14. The y-axis of the plane runs from negative 10 to 10. The hyperbola has a center at (negative 1, negative 1) and branches that pass through the vertices (negative 3, negative 1) and (1, negative 1), and that open left and right.\" \/><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163870463544\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1163870463546\"><p id=\"fs-id1163870463548\">\\(\\frac{{\\left(x-2\\right)}^{2}}{4}-\\frac{{\\left(y-3\\right)}^{2}}{16}=1\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163870411387\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1163870411389\"><p id=\"fs-id1163870551219\">\\(\\frac{{\\left(y+2\\right)}^{2}}{9}-\\frac{{\\left(x+1\\right)}^{2}}{9}=1\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163870413000\"><span data-type=\"media\" id=\"fs-id1163870413003\" data-alt=\"The figure shows a hyperbola graphed on the x y coordinate plane. The x-axis of the plane runs from negative 14 to 14. The y-axis of the plane runs from negative 10 to 10. The hyperbola has a center at (negative 1, negative 2) and branches that pass through the vertices (negative 1, 1) and (negative 1, negative 5), and that open up and down.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_357_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows a hyperbola graphed on the x y coordinate plane. The x-axis of the plane runs from negative 14 to 14. The y-axis of the plane runs from negative 10 to 10. The hyperbola has a center at (negative 1, negative 2) and branches that pass through the vertices (negative 1, 1) and (negative 1, negative 5), and that open up and down.\" \/><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163873837838\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1163873761350\"><p id=\"fs-id1163873761352\">\\(\\frac{{\\left(y-1\\right)}^{2}}{25}-\\frac{{\\left(x-2\\right)}^{2}}{9}=1\\)<\/p><\/div><\/div><p id=\"fs-id1163873784040\">In the following exercises, <span class=\"token\">\u24d0<\/span> write the equation in standard form and <span class=\"token\">\u24d1<\/span> graph.<\/p><div data-type=\"exercise\" id=\"fs-id1163870463202\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1163870463204\"><p id=\"fs-id1163870463206\">\\(4{x}^{2}-16{y}^{2}+8x+96y-204=0\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163873606412\"><p id=\"fs-id1163870644973\"><span class=\"token\">\u24d0<\/span>\\(\\frac{{\\left(x+1\\right)}^{2}}{16}-\\frac{{\\left(y-3\\right)}^{2}}{4}=1\\)<span data-type=\"newline\"><br \/><\/span><span class=\"token\">\u24d1<\/span><span data-type=\"newline\"><br \/><\/span><\/p><span data-type=\"media\" id=\"fs-id1163873784848\" data-alt=\"The figure shows a hyperbola graphed on the x y coordinate plane. The x-axis of the plane runs from negative 14 to 14. The y-axis of the plane runs from negative 10 to 10. The hyperbola has a center at (negative 1, 3) and branches that pass through the vertices (negative 5, 3) and (3, 3), and that open left and right.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_359_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows a hyperbola graphed on the x y coordinate plane. The x-axis of the plane runs from negative 14 to 14. The y-axis of the plane runs from negative 10 to 10. The hyperbola has a center at (negative 1, 3) and branches that pass through the vertices (negative 5, 3) and (3, 3), and that open left and right.\" \/><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163873641825\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1163873641827\"><p id=\"fs-id1163873641829\">\\(16{x}^{2}-4{y}^{2}-64x-24y-36=0\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163873789523\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1163873744004\"><p id=\"fs-id1163873744006\">\\(4{y}^{2}-16{x}^{2}+32x-8y-76=0\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163873785159\"><p id=\"fs-id1163873785161\"><span class=\"token\">\u24d0<\/span>\\(\\frac{{\\left(y-1\\right)}^{2}}{16}-\\frac{{\\left(x-1\\right)}^{2}}{4}=1\\)<span data-type=\"newline\"><br \/><\/span><span class=\"token\">\u24d1<\/span><span data-type=\"newline\"><br \/><\/span><\/p><span data-type=\"media\" id=\"fs-id1163870504857\" data-alt=\"The figure shows a hyperbola graphed on the x y coordinate plane. The x-axis of the plane runs from negative 14 to 14. The y-axis of the plane runs from negative 10 to 10. The hyperbola has a center at (1, 1) and branches that pass through the vertices (1, negative 3) and (1, 5), and that open up and down.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_361_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows a hyperbola graphed on the x y coordinate plane. The x-axis of the plane runs from negative 14 to 14. The y-axis of the plane runs from negative 10 to 10. The hyperbola has a center at (1, 1) and branches that pass through the vertices (1, negative 3) and (1, 5), and that open up and down.\" \/><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163873866658\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1163873866660\"><p id=\"fs-id1163873866662\">\\(36{y}^{2}-16{x}^{2}-96x+216y-396=0\\)<\/p><\/div><\/div><p id=\"fs-id1163874000994\"><strong data-effect=\"bold\">Identify the Graph of each Equation as a Circle, Parabola, Ellipse, or Hyperbola<\/strong><\/p><p id=\"fs-id1163870303496\">In the following exercises, identify the type of graph.<\/p><div data-type=\"exercise\" id=\"fs-id1163870303499\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1163870303501\"><p id=\"fs-id1163870303503\"><span data-type=\"newline\"><br \/><\/span><span class=\"token\">\u24d0<\/span>\\(16{y}^{2}-9{x}^{2}-36x-96y-36=0\\)<span data-type=\"newline\"><br \/><\/span><span class=\"token\">\u24d1<\/span>\\({x}^{2}+{y}^{2}-4x+10y-7=0\\)<span data-type=\"newline\"><br \/><\/span><span class=\"token\">\u24d2<\/span>\\(y={x}^{2}-2x+3\\)<span data-type=\"newline\"><br \/><\/span><span class=\"token\">\u24d3<\/span>\\(25{x}^{2}+9{y}^{2}=225\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163873863167\"><p id=\"fs-id1163873863169\"><span class=\"token\">\u24d0<\/span> hyperbola <span class=\"token\">\u24d1<\/span> circle <span class=\"token\">\u24d2<\/span> parabola <span class=\"token\">\u24d3<\/span> ellipse<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163873594904\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1163873594906\"><p id=\"fs-id1163870376191\"><span data-type=\"newline\"><br \/><\/span><span class=\"token\">\u24d0<\/span>\\({x}^{2}+{y}^{2}+4x-10y+25=0\\)<span data-type=\"newline\"><br \/><\/span><span class=\"token\">\u24d1<\/span>\\({y}^{2}-{x}^{2}-4y+2x-6=0\\)<span data-type=\"newline\"><br \/><\/span><span class=\"token\">\u24d2<\/span>\\(x=-{y}^{2}-2y+3\\)<span data-type=\"newline\"><br \/><\/span><span class=\"token\">\u24d3<\/span>\\(16{x}^{2}+9{y}^{2}=144\\)<\/p><\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1163869201327\"><h4 data-type=\"title\"><a href=\"\/contents\/c2649a40-6525-4e7f-a3b4-31f5d8a52d29\" class=\"target-chapter\">Solve Systems of Nonlinear Equations<\/a><\/h4><p id=\"fs-id1163873596258\"><strong data-effect=\"bold\">Solve a System of Nonlinear Equations Using Graphing<\/strong><\/p><p id=\"fs-id1163869200663\">In the following exercises, solve the system of equations by using graphing.<\/p><div data-type=\"exercise\" id=\"fs-id1163869200666\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1163869200668\"><p id=\"fs-id1163869200670\">\\(\\left\\{\\begin{array}{c}3{x}^{2}-y=0\\hfill \\\\ y=2x-1\\hfill \\end{array}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163873715314\"><span data-type=\"media\" id=\"fs-id1163873715317\" data-alt=\"The figure shows a parabola and line graphed on the x y coordinate plane. The x-axis of the plane runs from negative 5 to 5. The y-axis of the plane runs from negative 4 to 4. The parabola has a vertex at (0, 0) and opens upward. The line has a slope of 2 with a y-intercept at negative 1. The parabola and line do not intersect, so the system has no solution.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_363_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows a parabola and line graphed on the x y coordinate plane. The x-axis of the plane runs from negative 5 to 5. The y-axis of the plane runs from negative 4 to 4. The parabola has a vertex at (0, 0) and opens upward. The line has a slope of 2 with a y-intercept at negative 1. The parabola and line do not intersect, so the system has no solution.\" \/><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163873957473\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1163873957475\"><p id=\"fs-id1163873957478\">\\(\\left\\{\\begin{array}{c}y={x}^{2}-4\\hfill \\\\ y=x-4\\hfill \\end{array}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163873674120\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1163873674122\"><p id=\"fs-id1163873674124\">\\(\\left\\{\\begin{array}{c}{x}^{2}+{y}^{2}=169\\hfill \\\\ x=12\\hfill \\end{array}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163867240312\"><span data-type=\"media\" id=\"fs-id1163873559641\" data-alt=\"The figure shows a circle and line graphed on the x y coordinate plane. The x-axis of the plane runs from negative 20 to 20. The y-axis of the plane runs from negative 15 to 15. The circle has a center at (0, 0) and a radius of 13. The line is vertical. The circle and line intersect at the points (12, 5) and (12, negative 5), which are labeled. The solution of the system is (12, 5) and (12, negative 5)\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_365_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows a circle and line graphed on the x y coordinate plane. The x-axis of the plane runs from negative 20 to 20. The y-axis of the plane runs from negative 15 to 15. The circle has a center at (0, 0) and a radius of 13. The line is vertical. The circle and line intersect at the points (12, 5) and (12, negative 5), which are labeled. The solution of the system is (12, 5) and (12, negative 5)\" \/><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163873702909\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1163873702911\"><p id=\"fs-id1163873702913\">\\(\\left\\{\\begin{array}{c}{x}^{2}+{y}^{2}=25\\hfill \\\\ y=-5\\hfill \\end{array}\\)<\/p><\/div><\/div><p id=\"fs-id1163869197899\"><strong data-effect=\"bold\">Solve a System of Nonlinear Equations Using Substitution<\/strong><\/p><p id=\"fs-id1163873666962\">In the following exercises, solve the system of equations by using substitution.<\/p><div data-type=\"exercise\" id=\"fs-id1163873666965\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1163873666967\"><p id=\"fs-id1163873666969\">\\(\\left\\{\\begin{array}{c}y={x}^{2}+3\\hfill \\\\ y=-2x+2\\hfill \\end{array}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163873640694\"><p id=\"fs-id1163873640696\">\\(\\left(-1,4\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163873914601\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1163870629110\"><p id=\"fs-id1163870629112\">\\(\\left\\{\\begin{array}{c}{x}^{2}+{y}^{2}=4\\hfill \\\\ x-y=4\\hfill \\end{array}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163873745211\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1163873745213\"><p id=\"fs-id1163873745216\">\\(\\left\\{\\begin{array}{c}9{x}^{2}+4{y}^{2}=36\\hfill \\\\ y-x=5\\hfill \\end{array}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163874017889\"><p id=\"fs-id1163874017891\">No solution<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163873715143\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1163873715145\"><p id=\"fs-id1163873715148\">\\(\\left\\{\\begin{array}{c}{x}^{2}+4{y}^{2}=4\\hfill \\\\ 2x-y=1\\hfill \\end{array}\\)<\/p><\/div><\/div><p id=\"fs-id1163873912526\"><strong data-effect=\"bold\">Solve a System of Nonlinear Equations Using Elimination<\/strong><\/p><p id=\"fs-id1163873912532\">In the following exercises, solve the system of equations by using elimination.<\/p><div data-type=\"exercise\" id=\"fs-id1163873912535\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1163873715345\"><p id=\"fs-id1163873715347\">\\(\\left\\{\\begin{array}{c}{x}^{2}+{y}^{2}=16\\hfill \\\\ {x}^{2}-2y-1=0\\hfill \\end{array}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163873776424\"><p id=\"fs-id1163873776426\">\\(\\left(\\text{\u2212}\\sqrt{7},3\\right),\\left(\\sqrt{7},3\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163873644532\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1163873644534\"><p id=\"fs-id1163873714956\">\\(\\left\\{\\begin{array}{c}{x}^{2}-{y}^{2}=5\\hfill \\\\ -2{x}^{2}-3{y}^{2}=-30\\hfill \\end{array}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163873807235\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1163873807238\"><p id=\"fs-id1163873647430\">\\(\\left\\{\\begin{array}{c}4{x}^{2}+9{y}^{2}=36\\hfill \\\\ 3{y}^{2}-4x=12\\hfill \\end{array}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163873822657\"><p id=\"fs-id1163873822660\">\\(\\left(-3,0\\right),\\left(0,-2\\right),\\left(0,2\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163873686892\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1163873686894\"><p id=\"fs-id1163873686896\">\\(\\left\\{\\begin{array}{c}{x}^{2}+{y}^{2}=14\\hfill \\\\ {x}^{2}-{y}^{2}=16\\hfill \\end{array}\\)<\/p><\/div><\/div><p id=\"fs-id1163873634509\"><strong data-effect=\"bold\">Use a System of Nonlinear Equations to Solve Applications<\/strong><\/p><p id=\"fs-id1163870491200\">In the following exercises, solve the problem using a system of equations.<\/p><div data-type=\"exercise\" id=\"fs-id1163870491203\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1163870491205\"><p id=\"fs-id1163870491207\">The sum of the squares of two numbers is 25. The difference of the numbers is 1. Find the numbers.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163870550952\"><p id=\"fs-id1163870550955\">\\(-3\\) and \\(-4\\) or 4 and 3<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163870557522\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1163870557525\"><p id=\"fs-id1163870557527\">The difference of the squares of two numbers is 45. The difference of the square of the first number and twice the square of the second number is 9. Find the numbers.<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163873620872\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1163873620874\"><p id=\"fs-id1163873620876\">The perimeter of a rectangle is 58 meters and its area is 210 square meters. Find the length and width of the rectangle.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163873793305\"><p id=\"fs-id1163873793307\">If the length is 14 inches, the width is 15 inches. If the length is 15 inches, the width is 14 inches.<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163873793313\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1163873793315\"><p id=\"fs-id1163873758176\">Colton purchased a larger microwave for his kitchen. The diagonal of the front of the microwave measures 34 inches. The front also has an area of 480 square inches. What are the length and width of the microwave?<\/p><\/div><\/div><\/div><\/div><div class=\"practice-test\" data-depth=\"1\" id=\"fs-id1163873803334\"><h3 data-type=\"title\">Practice Test<\/h3><p id=\"fs-id1163873759011\">In the following exercises, find the distance between the points and the midpoint of the line segment with the given endpoints. Round to the nearest tenth as needed.<\/p><div data-type=\"exercise\" id=\"fs-id1163873759015\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1163873759018\"><p id=\"fs-id1163873759020\">\\(\\left(-4,-3\\right)\\) and \\(\\left(-10,-11\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163873748881\"><p id=\"fs-id1163873748883\">distance: 10, midpoint: \\(\\left(-7,-7\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163873764012\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1163873764014\"><p id=\"fs-id1163873764016\">\\(\\left(6,8\\right)\\) and \\(\\left(-5,-3\\right)\\)<\/p><\/div><\/div><p id=\"fs-id1163873760568\">In the following exercises, write the standard form of the equation of the circle with the given information.<\/p><div data-type=\"exercise\" id=\"fs-id1163873760572\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1163873760574\"><p id=\"fs-id1163873760577\">radius is 11 and center is \\(\\left(0,0\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163873795836\"><p id=\"fs-id1163873795838\">\\({x}^{2}+{y}^{2}=121\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163870454177\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1163870454179\"><p id=\"fs-id1163870454181\">radius is 12 and center is \\(\\left(10,-2\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163873862647\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1163873804357\"><p id=\"fs-id1163873804359\">center is \\(\\left(-2,3\\right)\\) and a point on the circle is \\(\\left(2,-3\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163873677840\"><p id=\"fs-id1163873609057\">\\({\\left(x+2\\right)}^{2}+{\\left(y-3\\right)}^{2}=52\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163873807699\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1163873807701\"><p id=\"fs-id1163870461912\">Find the equation of the ellipse shown in the graph.<\/p><span data-type=\"media\" id=\"fs-id1163870461916\" data-alt=\"The figure shows an ellipse graphed on the x y coordinate plane. The ellipse has a center at (0, 0), a vertical major axis, vertices at (0, plus or minus 10), and co-vertices at (plus or minus 6, 0).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_207_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows an ellipse graphed on the x y coordinate plane. The ellipse has a center at (0, 0), a vertical major axis, vertices at (0, plus or minus 10), and co-vertices at (plus or minus 6, 0).\" \/><\/span><\/div><\/div><p id=\"fs-id1163873749967\">In the following exercises, <span class=\"token\">\u24d0<\/span> identify the type of graph of each equation as a circle, parabola, ellipse, or hyperbola, and <span class=\"token\">\u24d1<\/span> graph the equation.<\/p><div data-type=\"exercise\" id=\"fs-id1163873703072\"><div data-type=\"problem\" id=\"fs-id1163873703075\"><p id=\"fs-id1163873703077\">\\(4{x}^{2}+49{y}^{2}=196\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163873608706\"><p id=\"fs-id1163869098961\"><span class=\"token\">\u24d0<\/span> ellipse<span data-type=\"newline\"><br \/><\/span> <span class=\"token\">\u24d1<\/span><span data-type=\"newline\"><br \/><\/span><\/p><span data-type=\"media\" id=\"fs-id1163873794720\" data-alt=\"The figure shows an ellipse graphed on the x y coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 8 to 8. The ellipse has a center at (0, 0), a horizontal major axis, vertices at (plus or minus 7, 0) and co-vertices at (0, plus or minus 2).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_367_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows an ellipse graphed on the x y coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 8 to 8. The ellipse has a center at (0, 0), a horizontal major axis, vertices at (plus or minus 7, 0) and co-vertices at (0, plus or minus 2).\" \/><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163870488379\"><div data-type=\"problem\" id=\"fs-id1163870488381\"><p id=\"fs-id1163870488383\">\\(y=3{\\left(x-2\\right)}^{2}-2\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163873946136\"><div data-type=\"problem\" id=\"fs-id1163873946138\"><p id=\"fs-id1163873782530\">\\(3{x}^{2}+3{y}^{2}=27\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163873869362\"><p id=\"fs-id1163873869365\"><span class=\"token\">\u24d0<\/span> circle<span data-type=\"newline\"><br \/><\/span> <span class=\"token\">\u24d1<\/span><span data-type=\"newline\"><br \/><\/span><\/p><span data-type=\"media\" id=\"fs-id1163873799871\" data-alt=\"The figure shows a circle graphed on the x y coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 8 to 8. The parabola circle has a center at (0, 0) and a radius of 3.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_369_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows a circle graphed on the x y coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 8 to 8. The parabola circle has a center at (0, 0) and a radius of 3.\" \/><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163873822380\"><div data-type=\"problem\" id=\"fs-id1163873822382\"><p id=\"fs-id1163873822384\">\\(\\frac{{y}^{2}}{100}-\\frac{{x}^{2}}{36}=1\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163873763888\"><div data-type=\"problem\" id=\"fs-id1163873763890\"><p id=\"fs-id1163873763892\">\\(\\frac{{x}^{2}}{16}+\\frac{{y}^{2}}{81}=1\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163870462864\"><p id=\"fs-id1163870462866\"><span class=\"token\">\u24d0<\/span> ellipse<span data-type=\"newline\"><br \/><\/span> <span class=\"token\">\u24d1<\/span><span data-type=\"newline\"><br \/><\/span><\/p><span data-type=\"media\" id=\"fs-id1163873820914\" data-alt=\"The figure shows an ellipse graphed on the x y coordinate plane. The x-axis of the plane runs from negative 14 to 14. The y-axis of the plane runs from negative 10 to 10. The ellipse has a center at (0, 0), a vertical major axis, vertices at (0, plus or minus 9) and co-vertices at (plus or minus 4, 0).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_371_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows an ellipse graphed on the x y coordinate plane. The x-axis of the plane runs from negative 14 to 14. The y-axis of the plane runs from negative 10 to 10. The ellipse has a center at (0, 0), a vertical major axis, vertices at (0, plus or minus 9) and co-vertices at (plus or minus 4, 0).\" \/><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163869585713\"><div data-type=\"problem\" id=\"fs-id1163869585715\"><p id=\"fs-id1163869585718\">\\(x=2{y}^{2}+10y+7\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163873786999\"><div data-type=\"problem\" id=\"fs-id1163873606436\"><p id=\"fs-id1163873606438\">\\(64{x}^{2}-9{y}^{2}=576\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163873655737\"><p id=\"fs-id1163873655739\"><span class=\"token\">\u24d0<\/span> hyperbola<span data-type=\"newline\"><br \/><\/span> <span class=\"token\">\u24d1<\/span><span data-type=\"newline\"><br \/><\/span><\/p><span data-type=\"media\" id=\"fs-id1163873764181\" data-alt=\"The figure shows a hyperbola graphed on the x y coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 8 to 8. The hyperbola has a center at (0, 0) and branches that pass through the vertices (plus or minus 3, 0) and that open left and right.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_373_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows a hyperbola graphed on the x y coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 8 to 8. The hyperbola has a center at (0, 0) and branches that pass through the vertices (plus or minus 3, 0) and that open left and right.\" \/><\/span><\/div><\/div><p id=\"fs-id1163873807731\">In the following exercises, <span class=\"token\">\u24d0<\/span> identify the type of graph of each equation as a circle, parabola, ellipse, or hyperbola, <span class=\"token\">\u24d1<\/span> write the equation in standard form, and <span class=\"token\">\u24d2<\/span> graph the equation.<\/p><div data-type=\"exercise\" id=\"fs-id1163870462179\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1163873675733\"><p id=\"fs-id1163873675735\">\\(25{x}^{2}+64{y}^{2}+200x-256y-944=0\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163867247197\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1163873715039\"><p id=\"fs-id1163873715042\">\\({x}^{2}+{y}^{2}+10x+6y+30=0\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163873896962\"><p id=\"fs-id1163873896964\"><span class=\"token\">\u24d0<\/span> circle<span data-type=\"newline\"><br \/><\/span> <span class=\"token\">\u24d1<\/span> \\({\\left(x+5\\right)}^{2}+{\\left(y+3\\right)}^{2}=4\\)<span data-type=\"newline\"><br \/><\/span> <span class=\"token\">\u24d2<\/span><span data-type=\"newline\"><br \/><\/span><\/p><span data-type=\"media\" id=\"fs-id1163874043629\" data-alt=\"The figure shows a circle graphed on the x y coordinate plane. The x-axis of the plane runs from negative 14 to 14. The y-axis of the plane runs from negative 10 to 10. The circle has a center at (negative 5, negative 3) and a radius 2.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_375_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows a circle graphed on the x y coordinate plane. The x-axis of the plane runs from negative 14 to 14. The y-axis of the plane runs from negative 10 to 10. The circle has a center at (negative 5, negative 3) and a radius 2.\" \/><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163870550229\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1163870550231\"><p id=\"fs-id1163870547415\">\\(x=\\text{\u2212}{y}^{2}+2y-4\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163873864767\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1163873864768\"><p id=\"fs-id1163873657428\">\\(9{x}^{2}-25{y}^{2}-36x-50y-214=0\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163873791817\"><p id=\"fs-id1163873791819\"><span class=\"token\">\u24d0<\/span> hyperbola<span data-type=\"newline\"><br \/><\/span> <span class=\"token\">\u24d1<\/span> \\(\\frac{{\\left(x-2\\right)}^{2}}{25}-\\frac{{\\left(y+1\\right)}^{2}}{9}=1\\)<span data-type=\"newline\"><br \/><\/span> <span class=\"token\">\u24d2<\/span><span data-type=\"newline\"><br \/><\/span><\/p><span data-type=\"media\" id=\"fs-id1163873850471\" data-alt=\"The figure shows a hyperbola graphed on the x y coordinate plane. The x-axis of the plane runs from negative 14 to 14. The y-axis of the plane runs from negative 10 to 10. The hyperbola has a center at (2, negative 1) and branches that pass through the vertices (negative 3, negative 1) and (7, negative 1) that open left and right.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_377_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows a hyperbola graphed on the x y coordinate plane. The x-axis of the plane runs from negative 14 to 14. The y-axis of the plane runs from negative 10 to 10. The hyperbola has a center at (2, negative 1) and branches that pass through the vertices (negative 3, negative 1) and (7, negative 1) that open left and right.\" \/><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163870219064\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1163870219067\"><p id=\"fs-id1163870219069\">\\(y={x}^{2}+6x+8\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163873799945\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1163873799948\"><p id=\"fs-id1163870497826\">Solve the nonlinear system of equations by graphing:<span data-type=\"newline\"><br \/><\/span>\\(\\left\\{\\begin{array}{c}3{y}^{2}-x=0\\hfill \\\\ y=-2x-1\\hfill \\end{array}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163873796407\"><p id=\"fs-id1163873796409\">No solution<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163873796414\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1163873750851\"><p id=\"fs-id1163873750853\">Solve the nonlinear system of equations using substitution:<span data-type=\"newline\"><br \/><\/span>\\(\\left\\{\\begin{array}{c}{x}^{2}+{y}^{2}=8\\hfill \\\\ y=\\text{\u2212}x-4\\hfill \\end{array}.\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163873782385\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1163873782388\"><p id=\"fs-id1163873782390\">Solve the nonlinear system of equations using elimination:<span data-type=\"newline\"><br \/><\/span>\\(\\left\\{\\begin{array}{c}{x}^{2}+9{y}^{2}=9\\hfill \\\\ 2{x}^{2}-9{y}^{2}=18\\hfill \\end{array}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163873798322\"><p id=\"fs-id1163873798324\">\\(\\left(0,-3\\right),\\left(0,3\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163873788823\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1163873626844\"><p id=\"fs-id1163873626846\">Create the equation of the parabolic arch formed in the foundation of the bridge shown. Give the answer in \\(y=a{x}^{2}+bx+c\\) form.<\/p><span data-type=\"media\" id=\"fs-id1163873675193\" data-alt=\"The figure shows a parabolic arch formed in the foundation of the bridge. The arch is 10 feet high and 30 feet wide.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_208_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows a parabolic arch formed in the foundation of the bridge. The arch is 10 feet high and 30 feet wide.\" \/><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163870620577\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1163870620579\"><p id=\"fs-id1163870620581\">A comet moves in an elliptical orbit around a sun. The closest the comet gets to the sun is approximately 20 AU and the furthest is approximately 70 AU. The sun is one of the foci of the elliptical orbit. Letting the ellipse center at the origin and labeling the axes in AU, the orbit will look like the figure below. Use the graph to write an equation for the elliptical orbit of the comet.<\/p><span data-type=\"media\" id=\"fs-id1163873914128\" data-alt=\"The figure shows a model of an elliptical orbit around the sun on the x y coordinate plane. The ellipse has a center at (0, 0), a horizontal major axis, vertices marked at (plus or minus 45, 0), the sun marked as a foci and labeled (25, 0), the closest distance the comet is from the sun marked as 20 A U, and the farthest a comet is from the sun marked as 70 A U.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_209_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows a model of an elliptical orbit around the sun on the x y coordinate plane. The ellipse has a center at (0, 0), a horizontal major axis, vertices marked at (plus or minus 45, 0), the sun marked as a foci and labeled (25, 0), the closest distance the comet is from the sun marked as 20 A U, and the farthest a comet is from the sun marked as 70 A U.\" \/><\/span><\/div><div data-type=\"solution\" id=\"fs-id1163870228951\"><p id=\"fs-id1163870228953\">\\(\\frac{{x}^{2}}{2025}+\\frac{{y}^{2}}{1400}=1\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163873821558\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1163873821561\"><p id=\"fs-id1163870170382\">The sum of two numbers is 22 and the product is \\(-240.\\) Find the numbers.<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163873587977\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1163873783582\"><p id=\"fs-id1163873783584\">For her birthday, Olive\u2019s grandparents bought her a new widescreen TV. Before opening it she wants to make sure it will fit her entertainment center. The TV is 55\u201d. The size of a TV is measured on the diagonal of the screen and a widescreen has a length that is larger than the width. The screen also has an area of 1452 square inches. Her entertainment center has an insert for the TV with a length of 50 inches and width of 40 inches. What are the length and width of the TV screen and will it fit Olive\u2019s entertainment center?<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163873783586\"><p id=\"fs-id1163873783588\">The length is 44 inches and the width is 33 inches. The TV will fit Olive\u2019s entertainment center.<\/p><\/div><\/div><\/div><div data-type=\"glossary\" class=\"textbox shaded\"><h3 data-type=\"glossary-title\">Glossary<\/h3><dl id=\"fs-id1163870346283\"><dt>system of nonlinear equations<\/dt><dd id=\"fs-id1163870346288\">A system of nonlinear equations is a system where at least one of the equations is not linear.<\/dd><\/dl><\/div>","rendered":"<div class=\"textbox textbox--learning-objectives\">\n<h3 itemprop=\"educationalUse\">Learning Objectives<\/h3>\n<p>By the end of this section, you will be able to: <\/p>\n<ul>\n<li>Solve a system of nonlinear equations using graphing<\/li>\n<li>Solve a system of nonlinear equations using substitution<\/li>\n<li>Solve a system of nonlinear equations using elimination<\/li>\n<li>Use a system of nonlinear equations to solve applications<\/li>\n<\/ul>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1163873512681\" class=\"be-prepared\">\n<ol id=\"fs-id1163873600190\" type=\"1\">\n<li>Solve the system by graphing: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-94872336f7803e39f2f581a4e139df1d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#120;&#45;&#51;&#121;&#61;&#45;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#120;&#43;&#121;&#61;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"128\" style=\"vertical-align: -17px;\" \/><span data-type=\"newline\"><br \/><\/span> If you missed this problem, review <a href=\"\/contents\/a236420d-b79a-43e3-8f7c-1b8e18c5045e#fs-id1167834279490\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li>\n<li>Solve the system by substitution: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0fe9f402aea22a044ca78c2cd4ceee1d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#120;&#45;&#52;&#121;&#61;&#45;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#45;&#51;&#120;&#43;&#52;&#121;&#61;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"137\" style=\"vertical-align: -17px;\" \/><span data-type=\"newline\"><br \/><\/span> If you missed this problem, review <a href=\"\/contents\/a236420d-b79a-43e3-8f7c-1b8e18c5045e#fs-id1167835328722\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li>\n<li>Solve the system by elimination: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fa22941b62dc37bffd4d1da148442863_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#51;&#120;&#45;&#52;&#121;&#61;&#45;&#57;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#53;&#120;&#43;&#51;&#121;&#61;&#49;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"137\" style=\"vertical-align: -17px;\" \/><span data-type=\"newline\"><br \/><\/span> If you missed this problem, review <a href=\"\/contents\/a236420d-b79a-43e3-8f7c-1b8e18c5045e#fs-id1167834121139\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li>\n<\/ol>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1163873610411\">\n<h3 data-type=\"title\">Solve a System of Nonlinear Equations Using Graphing<\/h3>\n<p id=\"fs-id1163873783473\">We learned how to solve systems of linear equations with two variables by graphing, substitution and elimination. We will be using these same methods as we look at nonlinear systems of equations with two equations and two variables. A <strong data-effect=\"bold\">system of nonlinear equations<\/strong> is a system where at least one of the equations is not linear.<\/p>\n<p id=\"fs-id1163873731492\">For example each of the following systems is a <span data-type=\"term\">system of nonlinear equations<\/span>.<\/p>\n<div data-type=\"equation\" id=\"fs-id1163873644550\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-35ab3e699f07db9803ab286bebffb438_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#61;&#57;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#121;&#61;&#57;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#57;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#61;&#57;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#121;&#61;&#51;&#120;&#45;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#120;&#43;&#121;&#61;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#121;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"630\" style=\"vertical-align: -17px;\" \/><\/div>\n<div data-type=\"note\" id=\"fs-id1163873820382\">\n<div data-type=\"title\">System of Nonlinear Equations<\/div>\n<p id=\"fs-id1163873946484\">A <strong data-effect=\"bold\">system of nonlinear equations<\/strong> is a system where at least one of the equations is not linear.<\/p>\n<\/div>\n<p id=\"fs-id1163873850941\">Just as with systems of linear equations, a solution of a nonlinear system is an ordered pair that makes both equations true. In a nonlinear system, there may be more than one solution. We will see this as we solve a system of nonlinear equations by graphing.<\/p>\n<p id=\"fs-id1163873940693\">When we solved systems of linear equations, the solution of the system was the point of intersection of the two lines. With systems of nonlinear equations, the graphs may be circles, parabolas or hyperbolas and there may be several points of intersection, and so several solutions. Once you identify the graphs, visualize the different ways the graphs could intersect and so how many solutions there might be.<\/p>\n<p id=\"fs-id1163873892611\">To solve systems of nonlinear equations by graphing, we use basically the same steps as with systems of linear equations modified slightly for nonlinear equations. The steps are listed below for reference.<\/p>\n<div data-type=\"note\" id=\"fs-id1163873798518\" class=\"howto\">\n<div data-type=\"title\">Solve a system of nonlinear equations by graphing.<\/div>\n<ol id=\"fs-id1163870411020\" type=\"1\" class=\"stepwise\">\n<li>Identify the graph of each equation. Sketch the possible options for intersection.<\/li>\n<li>Graph the first equation.<\/li>\n<li>Graph the second equation on the same rectangular coordinate system.<\/li>\n<li>Determine whether the graphs intersect.<\/li>\n<li>Identify the points of intersection.<\/li>\n<li>Check that each ordered pair is a solution to both original equations.<\/li>\n<\/ol>\n<\/div>\n<div data-type=\"example\" id=\"fs-id1163870369820\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1163873620675\">\n<div data-type=\"problem\" id=\"fs-id1163873632057\">\n<p id=\"fs-id1163873866469\">Solve the system by graphing: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8635e1aa39a01876683757083f458837_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#120;&#45;&#121;&#61;&#45;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#121;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"119\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163870381445\">\n<table id=\"fs-id1163873912472\" class=\"unnumbered unstyled can-break\" summary=\"Identify each graph of the system x minus y is equal to negative 2, which is a line, and y is equal to x squared, which is a parabola. Sketch the possible options for intersection of a parabola and a line. When a parabola and a line do not intersect, there the system has 0 solutions. When a parabola and line intersect at a single point, the system has one solution. When a parabola and line intersect at two points, the system has two solutions. Graph the line x minus y is equal to negative two. The slope intercept form of the line is y is equal to x plus 2. Graph the parabola, x squared. On a coordinate plane, the line has a slope of 1 and a y-intercept of 2 and the parabola has a vertex at (0, 0) and opens upward. They appear to be (2, 4) and (negative 1, 1). Check to make sure each solution makes both equations true. For (2, 4), is 2 minus 4 equal to negative 2? Negative is equal to negative 2. For (2, 4), is 4 equal to 2 squared? 4 is equal to 4. For (negative 1, 1), is negative 1 minus 1 equal to negative 2? Negative 2 is equal to negative 2. For (negative 1, 1), is 1 equal to the square of negative 1. 1 is equal to 1. The solutions are (2, 4) and (negative 1, 1).\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Identify each graph.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-efdc580787be7dd2a596cecb825db59d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#125;&#120;&#45;&#121;&#61;&#45;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#105;&#110;&#101;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#121;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#112;&#97;&#114;&#97;&#98;&#111;&#108;&#97;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"207\" style=\"vertical-align: -17px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Sketch the possible options for<span data-type=\"newline\"><br \/><\/span>intersection of a parabola and a line.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163873819964\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_001a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Graph the line, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-84a726312ebb6c0046b1fe20928ad42d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#45;&#121;&#61;&#45;&#50;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"92\" style=\"vertical-align: -4px;\" \/><span data-type=\"newline\"><br \/><\/span>Slope-intercept form <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c627f7b2155631f7f49af33dcf088ea6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#120;&#43;&#50;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"78\" style=\"vertical-align: -4px;\" \/><span data-type=\"newline\"><br \/><\/span>Graph the parabola, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5369a264cf669bc443af157f81deefe6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"55\" style=\"vertical-align: -4px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163874012484\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_001b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Identify the points of intersection.<\/td>\n<td data-valign=\"top\" data-align=\"left\">The points of intersection appear to be <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-65c84a7e1d884e51e9ff8e8338318a74_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#44;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a6b0134658a3dfa9a545776158ba6096_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#44;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"59\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Check to make sure each solution makes<span data-type=\"newline\"><br \/><\/span>both equations true.<span data-type=\"newline\"><br \/><\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-65c84a7e1d884e51e9ff8e8338318a74_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#44;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/><span data-type=\"newline\"><br \/><\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b6d92e6feeefe68efe3502908b42850f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#120;&#45;&#121;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#45;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#121;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#50;&#45;&#52;&#38;&#32;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#63;&#125;&#123;&#61;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#45;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#52;&#38;&#32;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#63;&#125;&#123;&#61;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#123;&#50;&#125;&#94;&#123;&#50;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#50;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#45;&#50;&#10003;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#52;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#52;&#10003;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"66\" width=\"231\" style=\"vertical-align: -26px;\" \/><span data-type=\"newline\"><br \/><\/span><span data-type=\"newline\"><br \/><\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c1d1c22dc2eabfdd66670fe21fd738b8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#44;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"51\" style=\"vertical-align: -4px;\" \/><span data-type=\"newline\"><br \/><\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-de52a0f19b7653abfae72264a4c9b610_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#51;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#120;&#45;&#121;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#45;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#121;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#49;&#45;&#49;&#38;&#32;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#63;&#125;&#123;&#61;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#45;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#49;&#38;&#32;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#63;&#125;&#123;&#61;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#50;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#45;&#50;&#10003;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#49;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#49;&#10003;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"66\" width=\"269\" style=\"vertical-align: -26px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\">The solutions are <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-65c84a7e1d884e51e9ff8e8338318a74_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#44;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a6b0134658a3dfa9a545776158ba6096_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#44;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"59\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1163873625836\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1163873812945\">\n<div data-type=\"problem\" id=\"fs-id1163870547413\">\n<p id=\"fs-id1163873506131\">Solve the system by graphing: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a0aacecac0c651ff305a57597dfaa9ad_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#120;&#43;&#121;&#61;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#121;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"113\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163870335137\"><span data-type=\"media\" id=\"fs-id1163873507545\" data-alt=\"This graph shows the equations of a system, x plus y is equal to 4 and y is equal x squared plus 2, and the x y-coordinate plane. The line has a slope of negative 1 and a y-intercept at 4. The vertex of the parabola is (0, 2) and opens upward. The line and parabola intersect at the points (negative 2, 6) and (1, 3), which are labeled.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_301_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows the equations of a system, x plus y is equal to 4 and y is equal x squared plus 2, and the x y-coordinate plane. The line has a slope of negative 1 and a y-intercept at 4. The vertex of the parabola is (0, 2) and opens upward. The line and parabola intersect at the points (negative 2, 6) and (1, 3), which are labeled.\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1163873507073\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1163873860966\">\n<div data-type=\"problem\" id=\"fs-id1163870619657\">\n<p id=\"fs-id1163870644518\">Solve the system by graphing: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0cce1cff26b7aaaa7f70502cf9daf89f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#120;&#45;&#121;&#61;&#45;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#121;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"119\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163873534978\"><span data-type=\"media\" id=\"fs-id1163873864859\" data-alt=\"This graph shows the equations of a system, x minus y is equal to negative 1 and y is equal to negative x squared plus three, and the x y-coordinate plane. The line has a slope of 1 and a y-intercept at 1. The vertex of the parabola is (0, negative 3) and opens upward. The line and parabola intersect at the points (negative 2, negative 1) and (1, 2), which are labeled.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_302_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows the equations of a system, x minus y is equal to negative 1 and y is equal to negative x squared plus three, and the x y-coordinate plane. The line has a slope of 1 and a y-intercept at 1. The vertex of the parabola is (0, negative 3) and opens upward. The line and parabola intersect at the points (negative 2, negative 1) and (1, 2), which are labeled.\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1165926714193\">To identify the graph of each equation, keep in mind the characteristics of the <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b40448f90dbf1bf9cce1035e2f3b1120_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"17\" style=\"vertical-align: 0px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f72e617ab66ab04529ce474aaeeba224_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#121;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"16\" style=\"vertical-align: -4px;\" \/> terms of each conic.<\/p>\n<div data-type=\"example\" id=\"fs-id1163873789993\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1163873807796\">\n<div data-type=\"problem\" id=\"fs-id1163873635154\">\n<p id=\"fs-id1163873530732\">Solve the system by graphing: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f3b01ceb917bca5e23fac213f66ff43f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#121;&#61;&#45;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#61;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"209\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163873679329\">\n<table id=\"fs-id1163873668699\" class=\"unnumbered unstyled can-break\" summary=\"Identify each graph of the system y is equal to negative 1, which is a line, and the quantity x minus 2 squared plus the quantity y plus 3 squared is equal to 4, which is a circle. Sketch the possible options for the intersection of a circle and a line. When a circle and a line do not intersect, there the system has 0 solutions. When a circle and line intersect at a single point, the system has one solution. When a circle and line intersect at two points, the system has two solutions. Graph the circle, the quantity x minus 2 squared plus the quantity y plus 3 squared is equal to 4. Its center is (2, negative 3) and it has a radius 2 units. Graph the line, y is equal to negative 1. It is a horizontal line. Identify the points of intersection. The point of intersection appears to be (2, negative 1). Check to make sure the solution makes both equations true. Substitute the coordinates from (2, negative 1) into the quantity x minus 2 squared plus the quantity y plus 3 squared is equal to 4. Is the quantity 2 minus 2 squared plus the quantity negative 1 plus 3 squared equal to 4? Is 0 squared plus 2 squared equal to 4? 4 is equal to 4. Substitute the coordinates from (2, negative 1) into y is equal to negative 1. Negative 1 is equal to negative 1. The solution is (2, negative 1).\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Identify each graph.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-922811ede19e310d0d0aa74633186b3a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#125;&#121;&#61;&#45;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#105;&#110;&#101;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#61;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#99;&#105;&#114;&#99;&#108;&#101;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"270\" style=\"vertical-align: -17px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Sketch the possible options for the<span data-type=\"newline\"><br \/><\/span>intersection of a circle and a line.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163873865333\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_002a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Graph the circle, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-36c521f8fe0092ce475a3654f9940383_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#61;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"177\" style=\"vertical-align: -4px;\" \/><span data-type=\"newline\"><br \/><\/span>Center: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1564d1d2328bb6bd9e7b30e6d573d2fb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#44;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"52\" style=\"vertical-align: -4px;\" \/> radius: 2<span data-type=\"newline\"><br \/><\/span>Graph the line, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1144e182f26c1fd5166b5411a3ed3cd4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#45;&#49;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"60\" style=\"vertical-align: -4px;\" \/><span data-type=\"newline\"><br \/><\/span>It is a horizontal line.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163873810952\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_002b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Identify the points of intersection.<\/td>\n<td data-valign=\"top\" data-align=\"left\">The point of intersection appears to be <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9ed3f84e4d9fd37e7c85bf5177357fe0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#44;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"59\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Check to make sure the solution makes<span data-type=\"newline\"><br \/><\/span>both equations true.<span data-type=\"newline\"><br \/><\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-25c4864d7eaa7ae5b2fe81ae29cf46af_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#44;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"52\" style=\"vertical-align: -4px;\" \/><span data-type=\"newline\"><br \/><\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2231827ac2da27fc3157ee0aeab1fe18_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#121;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#45;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#38;&#32;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#63;&#125;&#123;&#61;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#49;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#45;&#49;&#10003;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#38;&#32;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#63;&#125;&#123;&#61;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#52;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#52;&#10003;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"95\" width=\"351\" style=\"vertical-align: -40px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\">The solution is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9ed3f84e4d9fd37e7c85bf5177357fe0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#44;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"59\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1163873514495\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1163873549503\">\n<div data-type=\"problem\" id=\"fs-id1163873782736\">\n<p id=\"fs-id1163873539054\">Solve the system by graphing: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6d01de577ae96acf9755882383e7ab0d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#120;&#61;&#45;&#54;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#61;&#57;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"209\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163873798472\"><span data-type=\"media\" id=\"fs-id1163873857093\" data-alt=\"This graph shows the equations of a system, x is equal to negative 6 and the quantity x plus 3 squared plus the quantity y minus 1 squared is equal to 9, which is a circle, on the x y-coordinate plane. The line is a vertical line. The center of the circle is (negative 3, 1) and it has a radius of 3 units. The point of intersection between the line and circle is (negative 6, 1).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_303_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows the equations of a system, x is equal to negative 6 and the quantity x plus 3 squared plus the quantity y minus 1 squared is equal to 9, which is a circle, on the x y-coordinate plane. The line is a vertical line. The center of the circle is (negative 3, 1) and it has a radius of 3 units. The point of intersection between the line and circle is (negative 6, 1).\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1163870590841\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1163873770551\">\n<div data-type=\"problem\" id=\"fs-id1163870661129\">\n<p id=\"fs-id1163873672805\">Solve the system by graphing: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d37b1a662d727c4ad29c7ceeb6dd3a0a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#121;&#61;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#61;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"209\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163870548959\"><span data-type=\"media\" id=\"fs-id1163873760122\" data-alt=\"This graph shows the equations of a system, y is equal to negative 4 and the quantity x minus 2 squared plus the quantity y plus 3 squared is equal to 4, which is a circle, on the x y-coordinate plane. The line is a horizontal line. The center of the circle is (2, negative 3) and it has a radius of 2 units. There is no point of intersection between the line and circle, so the system has no solution.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_304_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows the equations of a system, y is equal to negative 4 and the quantity x minus 2 squared plus the quantity y plus 3 squared is equal to 4, which is a circle, on the x y-coordinate plane. The line is a horizontal line. The center of the circle is (2, negative 3) and it has a radius of 2 units. There is no point of intersection between the line and circle, so the system has no solution.\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1163873506935\">\n<h3 data-type=\"title\">Solve a System of Nonlinear Equations Using Substitution<\/h3>\n<p id=\"fs-id1163873621055\">The graphing method works well when the points of intersection are integers and so easy to read off the graph. But more often it is difficult to read the coordinates of the points of intersection. The substitution method is an algebraic method that will work well in many situations. It works especially well when it is easy to solve one of the equations for one of the variables.<\/p>\n<p id=\"fs-id1163870411088\">The substitution method is very similar to the substitution method that we used for systems of linear equations. The steps are listed below for reference.<\/p>\n<div data-type=\"note\" id=\"fs-id1163873858056\" class=\"howto\">\n<div data-type=\"title\">Solve a system of nonlinear equations by substitution.<\/div>\n<ol id=\"fs-id1163873621537\" type=\"1\" class=\"stepwise\">\n<li>Identify the graph of each equation. Sketch the possible options for intersection.<\/li>\n<li>Solve one of the equations for either variable.<\/li>\n<li>Substitute the expression from Step 2 into the other equation.<\/li>\n<li>Solve the resulting equation.<\/li>\n<li>Substitute each solution in Step 4 into one of the original equations to find the other variable.<\/li>\n<li>Write each solution as an ordered pair.<\/li>\n<li>Check that each ordered pair is a solution to <strong data-effect=\"bold\">both<\/strong> original equations.<\/li>\n<\/ol>\n<\/div>\n<div data-type=\"example\" id=\"fs-id1163873865634\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1163873506444\">\n<div data-type=\"problem\" id=\"fs-id1163869585875\">\n<p id=\"fs-id1163873660517\">Solve the system by using substitution: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a8d0cfd5e5eebd221379d7f676b37bd1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#57;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#61;&#57;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#121;&#61;&#51;&#120;&#45;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"129\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163873587705\">\n<table id=\"fs-id1163873798427\" class=\"unnumbered unstyled can-break\" summary=\"Identify each graph of the system, 9 x squared plus y squared is equal to 9, which is an ellipse, and y is equal to 3 x minus 3, which is a line. Sketch the possible options for intersection of an ellipse and a line. When an ellipse and a line do not intersect, the system has 0 solutions. When an ellipse and line intersect at a single point, the system has one solution. When an ellipse and line intersect at two points, the system has two solutions. The equation y is equal to 3 x minus 3 is solved for y already. Substitute 3 x minus 3 for y in the equation, 9 x squared plus y squared is equal to 9. Solve the equation for x. 9 x squared plus the quantity 3 x minus 3 end quantity squared is equal to 9. 9 x squared plus 9 x squared minus 18 x plus 9 is equal to 9. 18 x squared minus 18 x is equal to 0. 18 x times the quantity x minus 1) is equal to 0. So, x is equal to 0 or x is equal to 1. Substitute x is equal to 0 and x is equal to 1 into y is equal to 3 x minus 3 to find y. For x is equal to 0, the result is y is equal to 3 times 0 minus 3, which simplifies to y is equal to 3. For x is equal to 1, the result is y is equal to 3 times 1 minus 3, which simplifies to is equal to 0. The ordered pairs are (0, negative 3) and (1, 0). Check both ordered pairs in both equations. Substitute the coordinates in (0, negative 3) in 9 x squared plus y squared is equal to 9. Is 9 times 0 squared plus negative 3 squared equal to 9? Is 0 plus 9 equal to 9? 9 is equal to 9. Substitute the coordinates in (0, negative 3) in y is equal to 3 x minus 3. Is negative 3 equal to 3 times 0 minus 3? Is negative 3 equal to 0 minus 3? Negative 3 is equal to negative 3. Substitute the coordinates in (1, 0) in 9 x squared plus y squared is equal to 9. Is 9 times 1 squared plus 0 squared equal to 9? Is 9 plus 0 equal to 9? 9 is equal to 9. Substitute the coordinates in (1, 0) in y is equal to 3 x minus 3. Is 0 equal to 3 times 1 minus 3? Is 0 equal to 3 minus 3? 0 is equal to 3. The solutions are (0, negative 3) and (1, 0).\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Identify each graph.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-34f5a42aab6b6feaef58edd2f28af4da_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#125;&#57;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#61;&#57;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#101;&#108;&#108;&#105;&#112;&#115;&#101;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#121;&#61;&#51;&#120;&#45;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#105;&#110;&#101;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"198\" style=\"vertical-align: -17px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Sketch the possible options for intersection of an<span data-type=\"newline\"><br \/><\/span>ellipse and a line.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163869070525\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_003a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">The equation <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-136231896a68e1172a30037a43831cbe_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#51;&#120;&#45;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"83\" style=\"vertical-align: -4px;\" \/> is solved for <em data-effect=\"italics\">y<\/em>.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_003b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163873889838\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_003c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Substitute <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ffc22ef281e4992b6497f3af089f1cca_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#120;&#45;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"50\" style=\"vertical-align: 0px;\" \/> for <em data-effect=\"italics\">y<\/em> in the first equation.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163873635449\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_003d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Solve the equation for <em data-effect=\"italics\">x<\/em>.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163873676950\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_003e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163870638800\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_003f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Substitute <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8203ced39e0cdafefa708857c7ec2264_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"43\" style=\"vertical-align: 0px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3330a01aa4d7d81947b71297d8623d3b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"42\" style=\"vertical-align: -1px;\" \/> into <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-136231896a68e1172a30037a43831cbe_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#51;&#120;&#45;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"83\" style=\"vertical-align: -4px;\" \/> to find <em data-effect=\"italics\">y<\/em>.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163870660800\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_003g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163870359253\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_003h_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\">The ordered pairs are <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3228e365cc5c34a8d1ac63acea4d0408_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"59\" style=\"vertical-align: -4px;\" \/> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-579e1f7c2d61eab1dffcc2d5d02de664_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"45\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Check <strong data-effect=\"bold\">both<\/strong> ordered pairs in <strong data-effect=\"bold\">both<\/strong> equations.<span data-type=\"newline\"><br \/><\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-017f20ea0c6fda3470cedb20ea0b5537_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"52\" style=\"vertical-align: -4px;\" \/><span data-type=\"newline\"><br \/><\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-eb5694a7e66c29defd25493dc78a08cf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#125;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#57;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#57;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#121;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#51;&#120;&#45;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#57;&middot;&#123;&#48;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#38;&#32;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#63;&#125;&#123;&#61;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#57;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#51;&#38;&#32;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#63;&#125;&#123;&#61;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#51;&middot;&#48;&#45;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#48;&#43;&#57;&#38;&#32;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#63;&#125;&#123;&#61;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#57;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#51;&#38;&#32;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#63;&#125;&#123;&#61;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#48;&#45;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#57;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#57;&#10003;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#51;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#45;&#51;&#10003;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"897\" width=\"92\" style=\"vertical-align: -452px;\" \/><span data-type=\"newline\"><br \/><\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d6c757b148db3a6707e7e1f199d80f9f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/><span data-type=\"newline\"><br \/><\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-aa58e18bf463c006429dc937045477fd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#51;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#57;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#57;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#56;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#121;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#51;&#120;&#45;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#57;&middot;&#123;&#49;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#48;&#125;&#94;&#123;&#50;&#125;&#38;&#32;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#63;&#125;&#123;&#61;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#57;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#56;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#48;&#38;&#32;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#63;&#125;&#123;&#61;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#51;&middot;&#49;&#45;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#57;&#43;&#48;&#38;&#32;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#63;&#125;&#123;&#61;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#57;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#56;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#48;&#38;&#32;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#63;&#125;&#123;&#61;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#51;&#45;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#57;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#57;&#10003;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#56;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#48;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#48;&#10003;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"92\" width=\"321\" style=\"vertical-align: -38px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\">The solutions are <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7a56ef6c20718032eaea8a14999feb28_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"109\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1163873808977\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1163873599263\">\n<div data-type=\"problem\" id=\"fs-id1163870152141\">\n<p id=\"fs-id1163873860353\">Solve the system by using substitution: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f4cf003306c5eb4fba9f548626a3dc2a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#57;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#61;&#57;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#121;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#120;&#45;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"129\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163873535063\">\n<p id=\"fs-id1163869190295\">No solution<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1163873808862\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1163873526545\">\n<div data-type=\"problem\" id=\"fs-id1163873791536\">\n<p id=\"fs-id1163873679187\">Solve the system by using substitution: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f7c013b591f5160d9ddcf2b46d855c2d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#52;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#61;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#121;&#61;&#120;&#43;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"129\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163870359066\">\n<p id=\"fs-id1163873999144\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b479db28fd829def74da807429f229d9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#53;&#125;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#54;&#125;&#123;&#53;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"107\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1165926668831\">So far, each system of nonlinear equations has had at least one solution. The next example will show another option.<\/p>\n<div data-type=\"example\" id=\"fs-id1163873668205\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1163873821502\">\n<div data-type=\"problem\" id=\"fs-id1163873940603\">\n<p id=\"fs-id1163870550662\">Solve the system by using substitution: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f3dc99eba677cde62e211884cd67429f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#121;&#61;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#121;&#61;&#120;&#45;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"113\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163873679567\">\n<table id=\"fs-id1163870358764\" class=\"unnumbered unstyled can-break\" summary=\"Identify each graph of the system, x squared minus y is equal to 0, which is a parabola, and y is equal to x minus 2, which is a line. Sketch the possible options for intersection of a parabola and a line. When a parabola and a line do not intersect, the system has 0 solutions. When a parabola and line intersect at a single point, the system has one solution. When a parabola and line intersect at two points, the system has two solutions. The equation y is equal to x minus 2 is solved for y already. Substitute the expression, x minus 2, for y into the equation x squared minus y is equal to 0. The result is x squared minus the quantity x minus 2 is equal to 0, which simplifies to x squared minus x plus 2 is equal to 0. Solve the equation for x. It doesn&#x2019;t factor easily, so we can check the discriminant, which is given by b squared minus 4 a c. Negative 1 squared minus 4 times 1 times 2 simplifies to negative 7. The discriminant is negative, so there is no real solution. The system has no solution.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Identify each graph.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6c2642a66f9f904d586790f21dc22ee7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#125;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#121;&#61;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#112;&#97;&#114;&#97;&#98;&#111;&#108;&#97;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#121;&#61;&#120;&#45;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#105;&#110;&#101;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"201\" style=\"vertical-align: -17px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Sketch the possible options for<span data-type=\"newline\"><br \/><\/span>intersection of a parabola and a line<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163873602664\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_004a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">The equation <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f348093a4067ac94fdac2f12af87e5d4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#120;&#45;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"73\" style=\"vertical-align: -4px;\" \/> is solved for <em data-effect=\"italics\">y<\/em>.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163873783295\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_004b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163873606631\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_004c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Substitute <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-79c51dce88ed8deaaab9733f311dd7ba_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#45;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"40\" style=\"vertical-align: 0px;\" \/> for <em data-effect=\"italics\">y<\/em> in the first equation.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163873862652\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_004d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Solve the equation for <em data-effect=\"italics\">x<\/em>.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163873753879\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_004e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">This doesn\u2019t factor easily, so we can<span data-type=\"newline\"><br \/><\/span>check the discriminant.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-526e3a6211da60ada5a35ae0f4747c50_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#97;&#99;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&middot;&#49;&middot;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#48;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#45;&#55;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"83\" width=\"91\" style=\"vertical-align: -34px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\">The discriminant is negative, so there is no real solution.<span data-type=\"newline\"><br \/><\/span>The system has no solution.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1163873514214\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1163873676484\">\n<div data-type=\"problem\" id=\"fs-id1163866890428\">\n<p id=\"fs-id1163873791504\">Solve the system by using substitution: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-321857cbbf26de061c5e8f7d080944b8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#121;&#61;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#121;&#61;&#50;&#120;&#45;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"114\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163873796772\">\n<p id=\"fs-id1163873525731\">No solution<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1163870170606\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1163870491247\">\n<div data-type=\"problem\" id=\"fs-id1163873769859\">\n<p id=\"fs-id1163870386337\">Solve the system by using substitution: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f12ba76a83ad47a67dab41e101b2dc83_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#120;&#61;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#121;&#61;&#51;&#120;&#45;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"114\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163873781771\">\n<p id=\"fs-id1163873521469\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a775ea16c94320150285549ab8e52d22_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#57;&#125;&#44;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#44;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"107\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1163873642794\">\n<h3 data-type=\"title\">Solve a System of Nonlinear Equations Using Elimination<\/h3>\n<p id=\"fs-id1163873782962\">When we studied systems of linear equations, we used the method of elimination to solve the system. We can also use elimination to solve systems of nonlinear equations. It works well when the equations have both variables squared. When using elimination, we try to make the coefficients of one variable to be opposites, so when we add the equations together, that variable is eliminated.<\/p>\n<p id=\"fs-id1163870547923\">The elimination method is very similar to the elimination method that we used for systems of linear equations. The steps are listed for reference.<\/p>\n<div data-type=\"note\" id=\"fs-id1163873659109\" class=\"howto\">\n<div data-type=\"title\">Solve a system of equations by elimination.<\/div>\n<ol id=\"fs-id1163873782036\" type=\"1\" class=\"stepwise\">\n<li>Identify the graph of each equation. Sketch the possible options for intersection.<\/li>\n<li>Write both equations in standard form.<\/li>\n<li>Make the coefficients of one variable opposites.<span data-type=\"newline\"><br \/><\/span>Decide which variable you will eliminate.<span data-type=\"newline\"><br \/><\/span>Multiply one or both equations so that the coefficients of that variable are opposites.<\/li>\n<li>Add the equations resulting from Step 3 to eliminate one variable.<\/li>\n<li>Solve for the remaining variable.<\/li>\n<li>Substitute each solution from Step 5 into one of the original equations. Then solve for the other variable.<\/li>\n<li>Write each solution as an ordered pair.<\/li>\n<li>Check that each ordered pair is a solution to <strong data-effect=\"bold\">both<\/strong> original equations.<\/li>\n<\/ol>\n<\/div>\n<div data-type=\"example\" id=\"fs-id1163873603178\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1163873534849\">\n<div data-type=\"problem\" id=\"fs-id1163873544878\">\n<p id=\"fs-id1163873757998\">Solve the system by elimination: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2bf7be9c13df91a70b0aa243dbf10083_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#61;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#121;&#61;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"120\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163873788806\">\n<table id=\"fs-id1163873892760\" class=\"unnumbered unstyled can-break\" summary=\"Identify each graph of the system, x squared plus y squared is equal to 4, which is a circle, and x squared minus y is equal to 4, which is a parabola. Sketch the possible options for intersection of a circle and a parabola. When a circle and a parabola do not intersect, the system has 0 solutions. When circle and a parabola intersect at a single point, the system has one solution. When a circle and a parabola intersect at two points, the system has two solutions. When a circle and a parabola intersect at three points, the system has three solutions. When a circle and a parabola intersect at four points, the system has four solutions. Both equations are already in standard form. To get opposite coefficients of x squared, we will multiply the second equation by negative 1. The system is now x squared plus y squared is equal to 4 and negative 1 times the quantity x squared minus y is equal to negative 1 times 4. Simplify. The system is now x squared plus y squared is equal to 4 and negative x squared plus y is equal to negative 4. Add the two equations to eliminate x squared. The result is y squared plus y is equal to 0. Solve for y. Write the equation as y times the quantity y plus 1 is equal to 0. The result is y is equal to 0 and y plus 1 is equal to 0, which simplifies to y is equal to negative 1. Substitute y is equal to 0 and y is equal to negative 1 into one of the original equations. For y is equal to 0, x squared minus y is equal to 4 becomes x squared minus 0 is equal to 4. It simplifies to x squared equals 4, and then x is equal to plus or minus 2. For y is equal to negative 1, x squared minus y is equal to 4 becomes x squared minus negative 1 is equal to 4, which simplifies to x squared is equal to 3. The result is x is equal to plus or minus square root of 3. Write each solution as an ordered pair. The ordered pairs are (negative 2, 0), (2, 0), (square root of 3, negative 1), and (negative square root of 3, negative 1). Check that each ordered pair is a solution to both original equations. We will leave the checks for each of the four solutions to you. The solutions are (negative 2, 0), (2, 0), (square root of 3, negative 1), and (negative square root of 3, negative 1).\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Identify each graph.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163873812129\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_005a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Sketch the possible options for<span data-type=\"newline\"><br \/><\/span>intersection of a circle and a parabola.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163873925043\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_005b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Both equations are in standard form.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163873632086\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_005c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">To get opposite coefficients of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2f82905c002b530c14921e8d459fe64b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"22\" style=\"vertical-align: -4px;\" \/><span data-type=\"newline\"><br \/><\/span>we will multiply the second equation by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fc343c70c6ffe28a6cd57dfb53f250b3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"26\" style=\"vertical-align: -1px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163873606298\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_005d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163873661413\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_005e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Add the two equations to eliminate <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4ecbdeaa97c968725a27882437601678_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"22\" style=\"vertical-align: 0px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163873667924\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_005f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Solve for <em data-effect=\"italics\">y<\/em>.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163873664895\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_005g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163873625115\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_005h_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Substitute <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5e8ef70615fdaee8588017ac1fdd2da0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"42\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9f371b4e77462e28d9f6119571c92982_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"55\" style=\"vertical-align: -4px;\" \/> into one of<span data-type=\"newline\"><br \/><\/span>the original equations. Then solve for <em data-effect=\"italics\">x<\/em>.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163873866321\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_005i_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163873854943\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_005j_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Write each solution as an ordered pair.<\/td>\n<td data-valign=\"top\" data-align=\"center\">The ordered pairs are<span data-type=\"newline\"><br \/><\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-dd6f8f312ab67ad0422a4959540654f3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"51\" style=\"vertical-align: -4px;\" \/> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4d2abda359ee91996db7fe7bb27fa18d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"45\" style=\"vertical-align: -4px;\" \/><span data-type=\"newline\"><br \/><\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-109600de3da5e0a372cfb50e9c30ec23_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#115;&#113;&#114;&#116;&#123;&#51;&#125;&#44;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#92;&#115;&#113;&#114;&#116;&#123;&#51;&#125;&#44;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"140\" style=\"vertical-align: -7px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Check that each ordered pair is a<span data-type=\"newline\"><br \/><\/span>solution to <strong data-effect=\"bold\">both<\/strong> original equations.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">We will leave the checks for each of<span data-type=\"newline\"><br \/><\/span>the four solutions to you.<\/td>\n<td data-valign=\"top\" data-align=\"center\">The solutions are <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4050f6611d38892e64c174797e0a0e29_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"59\" style=\"vertical-align: -4px;\" \/> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b9bfe5f0e85bf2ee8d0ce478041861f9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"45\" style=\"vertical-align: -4px;\" \/> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7fcd9a39816a54b822e2ce0d23f88283_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#115;&#113;&#114;&#116;&#123;&#51;&#125;&#44;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"75\" style=\"vertical-align: -7px;\" \/> and<span data-type=\"newline\"><br \/><\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-904359b83bd1fdb0d4efcdca7329b199_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#92;&#115;&#113;&#114;&#116;&#123;&#51;&#125;&#44;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"75\" style=\"vertical-align: -7px;\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1163870291935\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1163873809154\">\n<div data-type=\"problem\" id=\"fs-id1163873809156\">\n<p id=\"fs-id1163873533770\">Solve the system by elimination: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ed4b78da64b4801901e98c270f56263d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#61;&#57;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#121;&#61;&#57;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"120\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163873741414\">\n<p id=\"fs-id1163873741416\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c34a5ef345dfba3201a759017250f7d3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#125;&#44;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#125;&#44;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"295\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1163873666234\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1163873998114\">\n<div data-type=\"problem\" id=\"fs-id1163873998116\">\n<p id=\"fs-id1163870346533\">Solve the system by elimination: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0961b3fb9bc0c82fa4dd46ba237e66ab_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#61;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#45;&#120;&#43;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#61;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"127\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163873751619\">\n<p id=\"fs-id1163873658452\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4f147ca5e969ac7db6d2dbc0364953ef_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"166\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1165926616304\">There are also four options when we consider a circle and a hyperbola.<\/p>\n<div data-type=\"example\" id=\"fs-id1163873606122\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1163873635082\">\n<div data-type=\"problem\" id=\"fs-id1163873635084\">\n<p id=\"fs-id1163873791386\">Solve the system by elimination: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-118d529ee9afef2f2ddcd49908057b31_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#61;&#55;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#61;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"120\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163869407987\">\n<table id=\"fs-id1163873660150\" class=\"unnumbered unstyled can-break\" summary=\"Identify each graph of the system, x squared plus y squared is equal to 7, which is a circle, and x squared minus y squared is equal to 1, which is a hyperbola. Sketch the possible options for intersection of a circle and a hyperbola. When a circle and a hyperbola do not intersect, the system has 0 solutions. When circle and a hyperbola intersect at a single point, the system has one solution. When a circle and a hyperbola intersect at two points, the system has two solutions. When a circle and a hyperbola intersect at three points, the system has three solutions. When a circle and a hyperbola intersect at four points, the system has four solutions. Both equations are already in standard form. The coefficients of y squared are opposite, so we will add the equations. The result is 2 x squared is equal to 8. Simplify. The result is x squared is equal to 4, which further simplifies to x is equal to plus or minus 2. Substitute x is equal to 2 and x is equal to negative 2 into one of the original equations. Then solve for y. For x is equal 2, x squared plus y squared is equal to 7 becomes 2 squared plus y squared is equal to 7. 4 plus y squared is equal to 7, which simplifies to y squared is equal to 3. So, the result is y is equal to plus or minus square root of 3. For x is equal to negative 2, x squared plus y squared is equal to 7 becomes negative 2 squared plus y squared is equal to 7. 4 plus y squared is equal to 7, which simplifies to y squared is equal to 3. The result is y is equal to plus or minus square root of 3. Write each solution as an ordered pair. The ordered pairs are (negative 2, square root of 3), (negative 2, negative square root of 3), (2, square root of 3), and (2, negative square root of 3). Check that the ordered pair is a solution to both original equations. We will leave the checks for each of the four solutions to you. The solutions are (negative 2, square root of 3), (negative 2, negative square root of 3), (2, square root of 3), and (2, negative square root of 3).\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Identify each graph.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5b027e9dbade67b401a0bce5af3bf50a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#125;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#61;&#55;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#99;&#105;&#114;&#99;&#108;&#101;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#61;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#104;&#121;&#112;&#101;&#114;&#98;&#111;&#108;&#97;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"218\" style=\"vertical-align: -17px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Sketch the possible options for intersection<span data-type=\"newline\"><br \/><\/span>of a circle and hyperbola.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1163873807546\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_006_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Both equations are in standard form.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-40c8a82539561a9243a05dfab37f1c0d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#61;&#55;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#61;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"109\" style=\"vertical-align: -17px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">The coefficients of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f72e617ab66ab04529ce474aaeeba224_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#121;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"16\" style=\"vertical-align: -4px;\" \/> are opposite, so we<span data-type=\"newline\"><br \/><\/span>will add the equations.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fc961399991fa63e277e4a6c5691ee25_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#117;&#110;&#100;&#101;&#114;&#115;&#101;&#116;&#123;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#125;&#123;&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#61;&#55;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#61;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#50;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#61;&#56;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"114\" width=\"113\" style=\"vertical-align: -49px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9511addee50feedeab9bc926aeb942e9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#120;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&plusmn;&#125;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"37\" width=\"73\" style=\"vertical-align: -11px;\" \/><span data-type=\"newline\"><br \/><\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-454b3c02ba6a0dafe6bd3ddfc0b30155_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#50;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#120;&#61;&#45;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"125\" style=\"vertical-align: 0px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Substitute <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c657687cbbf5ea9a7545edb42190e592_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"42\" style=\"vertical-align: 0px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-01f282abd343bbe6b83c45e54b86c6ed_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#45;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"56\" style=\"vertical-align: 0px;\" \/> into one of the<span data-type=\"newline\"><br \/><\/span>original equations. Then solve for <em data-effect=\"italics\">y<\/em>.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a0f549536fe0a54e5cfc919c51fde8b1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#55;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#55;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#50;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#55;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#55;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#52;&#43;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#55;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#52;&#43;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#55;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#121;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&plusmn;&#125;&#92;&#115;&#113;&#114;&#116;&#123;&#51;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#121;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&plusmn;&#125;&#92;&#115;&#113;&#114;&#116;&#123;&#51;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"109\" width=\"344\" style=\"vertical-align: -49px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Write each solution as an ordered pair.<\/td>\n<td data-valign=\"top\" data-align=\"center\">The ordered pairs are <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fc7f1ebac9f7b596ed3ffcfcb7b346b4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#44;&#92;&#115;&#113;&#114;&#116;&#123;&#51;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"75\" style=\"vertical-align: -7px;\" \/> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fd011bbc1779568c0b6dbd7cab7fd889_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#44;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#92;&#115;&#113;&#114;&#116;&#123;&#51;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"75\" style=\"vertical-align: -7px;\" \/><span data-type=\"newline\"><br \/><\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8a29d191aa5dea8784a51c252919a63d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#44;&#92;&#115;&#113;&#114;&#116;&#123;&#51;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"62\" style=\"vertical-align: -7px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-45407d3d84740ec4620320e6a109b884_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#44;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#92;&#115;&#113;&#114;&#116;&#123;&#51;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"62\" style=\"vertical-align: -7px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Check that the ordered pair is a solution to<span data-type=\"newline\"><br \/><\/span><strong data-effect=\"bold\">both<\/strong> original equations.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">We will leave the checks for each of the four<span data-type=\"newline\"><br \/><\/span>solutions to you.<\/td>\n<td data-valign=\"top\" data-align=\"center\">The solutions are <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fc7f1ebac9f7b596ed3ffcfcb7b346b4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#44;&#92;&#115;&#113;&#114;&#116;&#123;&#51;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"75\" style=\"vertical-align: -7px;\" \/> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fd011bbc1779568c0b6dbd7cab7fd889_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#44;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#92;&#115;&#113;&#114;&#116;&#123;&#51;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"75\" style=\"vertical-align: -7px;\" \/> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8a29d191aa5dea8784a51c252919a63d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#44;&#92;&#115;&#113;&#114;&#116;&#123;&#51;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"62\" style=\"vertical-align: -7px;\" \/><span data-type=\"newline\"><br \/><\/span>and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-45407d3d84740ec4620320e6a109b884_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#44;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#92;&#115;&#113;&#114;&#116;&#123;&#51;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"62\" style=\"vertical-align: -7px;\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1163864794326\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1163873670070\">\n<div data-type=\"problem\" id=\"fs-id1163869152940\">\n<p id=\"fs-id1163869152942\">Solve the system by elimination: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f1f7f18d2389c4243e41cc6e764281ea_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#61;&#50;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#61;&#55;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"129\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163873621488\">\n<p id=\"fs-id1163870228725\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bfc51cbdef7488ec453e196dea33900f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#44;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#44;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#44;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#44;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"243\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1163873633652\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1163873633655\">\n<div data-type=\"problem\" id=\"fs-id1163873633657\">\n<p id=\"fs-id1163873665020\">Solve the system by elimination: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-626a0597a0ad3394e35dc9a51471da49_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#61;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#61;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"120\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163873632586\">\n<p id=\"fs-id1163873632588\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-92cd9f3095952526d2cd4df85f12f729_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"102\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1163873764789\">\n<h3 data-type=\"title\">Use a System of Nonlinear Equations to Solve Applications<\/h3>\n<p id=\"fs-id1163870170091\">Systems of nonlinear equations can be used to model and solve many applications. We will look at an everyday geometric situation as our example.<\/p>\n<div data-type=\"example\" id=\"fs-id1163873625652\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1163873625654\">\n<div data-type=\"problem\" id=\"fs-id1163873625656\">\n<p id=\"fs-id1163873919072\">The difference of the squares of two numbers is 15. The sum of the numbers is 5. Find the numbers.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163873919076\">\n<table id=\"fs-id1163873651790\" class=\"unnumbered unstyled can-break\" summary=\"Identify what we are looking for. We are looking for two different numbers. Define the variables. Let x be equal to the first number. Let y be equal to the second number. Translate the information into a system of equations. The first sentence is &#x2018;The difference of the squares of two numbers is 15.&#x2019; Represent it with x squared minus y squared is equal to 15. The second sentence is &#x2018;The sum of the numbers is 5.&#x2019; Represent it with x plus y is equal to 5. The equations x squared minus y squared is equal to 15 and x plus y is equal to 5 form the system. Solve the system by substitution. Solve the second equation for x. The result is x is equal to 5 minus y. Substitute x into the first equation, x squared minus y squared is equal to 15. It becomes the quantity 5 minus y squared minus y squared is equal to 15. Expand and simplify. The result is the quantity 25 minus 10 y plus y squared end quantity minus y squared is equal to 15. 25 minus 10 y plus y squared minus y squared is equal to 15. 25 minus 10 y is equal to 15. Solve for y. Negative y is equal to negative 10. The result is y is equal to 1. Substitute back into the second equation, x plus y is equal to 5. It becomes x plus 1 is equal to 5, which simplifies to x is equal to 4. The numbers are 1 and 4.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Identify what we are looking for.<\/td>\n<td data-valign=\"top\" data-align=\"left\">Two different numbers.<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Define the variables.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e7bbcde7229c9d7d6f7f2b6793961e97_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"28\" style=\"vertical-align: 0px;\" \/> first number<span data-type=\"newline\"><br \/><\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a1f949b5a11b007e850aa6730272a9a1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"27\" style=\"vertical-align: -4px;\" \/> second number<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Translate the information into a system of<span data-type=\"newline\"><br \/><\/span>equations.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">First sentence.<\/td>\n<td data-valign=\"top\" data-align=\"justify\">The difference of the squares of two numbers is 15.<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163873595897\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_007a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Second sentence.<\/td>\n<td data-valign=\"top\" data-align=\"left\">The sum of the numbers is 5.<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163873632234\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_007b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Solve the system by substitution<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163873864424\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_007c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Solve the second equation for <em data-effect=\"italics\">x<\/em>.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163873652600\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_007d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Substitute <em data-effect=\"italics\">x<\/em> into the first equation.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163870645034\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_007e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163873752067\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_007f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Expand and simplify.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163873639497\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_007g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163873679776\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_007h_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Solve for <em data-effect=\"italics\">y<\/em>.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163873668569\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_007i_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163873660235\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_007j_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Substitute back into the second equation.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163869405960\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_007k_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163873869881\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_007l_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"center\">The numbers are 1 and 4.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1163873850715\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1163873605130\">\n<div data-type=\"problem\" id=\"fs-id1163873605132\">\n<p id=\"fs-id1163873605134\">The difference of the squares of two numbers is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-26aff4c04eae778ebc9635b3b38db373_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#50;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"35\" style=\"vertical-align: 0px;\" \/> The sum of the numbers is 10. Find the numbers.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163873674089\">\n<p id=\"fs-id1163873674091\">4 and 6<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1163873645610\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1163873817518\">\n<div data-type=\"problem\" id=\"fs-id1163873817521\">\n<p id=\"fs-id1163873892044\">The difference of the squares of two numbers is 35. The sum of the numbers is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fc343c70c6ffe28a6cd57dfb53f250b3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"26\" style=\"vertical-align: -1px;\" \/> Find the numbers.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163870406471\">\n<p id=\"fs-id1163870406473\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bb9e52ddecc045b16dbf509b1c89f11b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"31\" style=\"vertical-align: -1px;\" \/> and 17<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"example\" id=\"fs-id1163870487923\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1163870487925\">\n<div data-type=\"problem\" id=\"fs-id1163870487927\">\n<p id=\"fs-id1163870271495\">Myra purchased a small 25\u201d TV for her kitchen. The size of a TV is measured on the diagonal of the screen. The screen also has an area of 300 square inches. What are the length and width of the TV screen?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163870593254\">\n<table id=\"fs-id1163870593257\" class=\"unnumbered unstyled can-break\" summary=\"Identify what we are looking for. We are looking for the length and width of the rectangle. Define the variables. Let x be equal to the width of the rectangle. Let y be equal to the length of the rectangle. Draw a diagram to help visualize the situation. The diagram is a rectangle with the width labeled x, the length labeled y, and its diagonal labeled 25 inches. Translate the information into a system of equations. &#x2018;The diagonal of the right triangle is 25 is represented by the equation x squared plus y squared is equal to 25 squared, which is simplified to x squared plus y squared is equal to 625. &#x2018;The area of the rectangle is 300&#x2019; is represented by the equation x y is equal to 300. The equations form the system x squared plus y squared is equal to 625 and x y is equal to 300. Solve the system using substitution. Solve x y is equal to 300 for x. The result is x is equal to 300 divided by y. Substitute the expression for x into the first equation, x squared plus y squared is equal to 625. The result is the quantity 300 divided by y end quantity squared plus y squared is equal to 625. Simplify. The result is the quantity 90,000 divided by y squared end quantity plus y squared is equal to 625. Multiply each side of the equation by squared to clear the fractions. The result is 90,00 plus y to the fourth power is equal to 625 y squared. Put in standard form. The result is y to the fourth power minus 625 y squared plus 90,000 is equal to 0. Solve by factoring. The factored equation is the quantity y squared minus 225 times the quantity y squared minus 400 is equal to 0. The result is y squared minus 225 is equal to 0 or y squared minus 400 is equal to 0. They simplify to y squared is equal to 225 or y squared is equal to 400. The results are y is equal to plus or minus 15 or y is equal to plus or minus 20. Since y is a side of a rectangle, we discard the negative values and keep y is equal to 15 and y is equal to 20. Substitute back into the second equation, x y is equal to 300. For y is equal to 15, x times 15 is equal to 300, which simplifies to x is equal to 20. For y is equal to 20, x times 20 is equal to 300, which simplifies to x is equal to 15. If the length is 15 inches, the width is 20 inches. If the length s 20 inches, the width is 15 inches.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Identify what we are looking for.<\/td>\n<td data-valign=\"top\" data-align=\"left\">The length and width of the rectangle<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Define the variables.<\/td>\n<td data-valign=\"top\" data-align=\"left\">Let <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e7bbcde7229c9d7d6f7f2b6793961e97_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"28\" style=\"vertical-align: 0px;\" \/> width of the rectangle<span data-type=\"newline\"><br \/><\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-300a486b79d88c3b7696ef4b8978050b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#121;&#61;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"27\" style=\"vertical-align: -4px;\" \/> length of the rectangle<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Draw a diagram to help visualize the situation.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163873644886\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_008a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"center\">Area is 300 square inches.<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Translate the information into a system of<span data-type=\"newline\"><br \/><\/span>equations.<\/td>\n<td data-valign=\"top\" data-align=\"left\">The diagonal of the right triangle is 25 inches.<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163873595568\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_008b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"center\">The area of the rectangle is 300 square inches.<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163873657015\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_008c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Solve the system using substitution.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163873817843\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_008d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Solve the second equation for <em data-effect=\"italics\">x<\/em>.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163873674419\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_008e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Substitute <em data-effect=\"italics\">x<\/em> into the first equation.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163873634060\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_008f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163873639131\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_008g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163873583212\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_008h_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Multiply by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f72e617ab66ab04529ce474aaeeba224_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#121;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"16\" style=\"vertical-align: -4px;\" \/> to clear the fractions.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163873861928\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_008i_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Put in standard form.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163869114900\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_008j_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Solve by factoring.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163873881836\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_008k_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163870549939\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_008l_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163873860650\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_008m_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Since <em data-effect=\"italics\">y<\/em> is a side of the rectangle, we discard<span data-type=\"newline\"><br \/><\/span>the negative values.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163873582629\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_008n_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Substitute back into the second equation.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163873866291\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_008o_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163873674785\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_008p_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\">If the length is 15 inches, the width is 20 inches.<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\">If the length is 20 inches, the width is 15 inches.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1163873662011\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1163873662015\">\n<div data-type=\"problem\" id=\"fs-id1163873686856\">\n<p id=\"fs-id1163873686858\">Edgar purchased a small 20\u201d TV for his garage. The size of a TV is measured on the diagonal of the screen. The screen also has an area of 192 square inches. What are the length and width of the TV screen?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163873558684\">\n<p id=\"fs-id1163873899045\">If the length is 12 inches, the width is 16 inches. If the length is 16 inches, the width is 12 inches.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1163873631303\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1163874018118\">\n<div data-type=\"problem\" id=\"fs-id1163874018120\">\n<p id=\"fs-id1163874018122\">The Harper family purchased a small microwave for their family room. The diagonal of the door measures 15 inches. The door also has an area of 108 square inches. What are the length and width of the microwave door?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163873814313\">\n<p id=\"fs-id1163873850275\">If the length is 12 inches, the width is 9 inches. If the length is 9 inches, the width is 12 inches.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1163873558688\" class=\"media-2\">\n<p id=\"fs-id1163873558692\">Access these online resources for additional instructions and practice with solving nonlinear equations.<\/p>\n<ul id=\"fs-id1163873793432\" data-display=\"block\">\n<li><a href=\"https:\/\/openstax.org\/l\/37nonsyseq\">Nonlinear Systems of Equations<\/a><\/li>\n<li><a href=\"https:\/\/openstax.org\/l\/37nonsyseq2\">Solve a System of Nonlinear Equations<\/a><\/li>\n<li><a href=\"https:\/\/openstax.org\/l\/37nonsyselim\">Solve a System of Nonlinear Equations by Elimination<\/a><\/li>\n<li><a href=\"https:\/\/openstax.org\/l\/37nonsysapps\">System of Nonlinear Equations \u2013 Area and Perimeter Application<\/a><\/li>\n<\/ul>\n<\/div>\n<\/div>\n<div class=\"textbox\" data-depth=\"1\" id=\"fs-id1163873869190\">\n<h3 data-type=\"title\">Key Concepts<\/h3>\n<ul id=\"fs-id1163870152077\" data-bullet-style=\"bullet\">\n<li><strong data-effect=\"bold\">How to solve a system of nonlinear equations by graphing.<\/strong>\n<ol id=\"fs-id1163873781999\" type=\"1\" class=\"stepwise\">\n<li>Identify the graph of each equation. Sketch the possible options for intersection.<\/li>\n<li>Graph the first equation.<\/li>\n<li>Graph the second equation on the same rectangular coordinate system.<\/li>\n<li>Determine whether the graphs intersect.<\/li>\n<li>Identify the points of intersection.<\/li>\n<li>Check that each ordered pair is a solution to both original equations.<\/li>\n<\/ol>\n<\/li>\n<li><strong data-effect=\"bold\">How to solve a system of nonlinear equations by substitution.<\/strong>\n<ol id=\"fs-id1163873520227\" type=\"1\" class=\"stepwise\">\n<li>Identify the graph of each equation. Sketch the possible options for intersection.<span data-type=\"newline\"><br \/><\/span><span data-type=\"newline\"><br \/><\/span><span data-type=\"newline\"><br \/><\/span><\/li>\n<li>Solve one of the equations for either variable.<\/li>\n<li>Substitute the expression from Step 2 into the other equation.<\/li>\n<li>Solve the resulting equation.<\/li>\n<li>Substitute each solution in Step 4 into one of the original equations to find the other variable.<\/li>\n<li>Write each solution as an ordered pair.<\/li>\n<li>Check that each ordered pair is a solution to <strong data-effect=\"bold\">both<\/strong> original equations.<\/li>\n<\/ol>\n<\/li>\n<li><strong data-effect=\"bold\">How to solve a system of equations by elimination.<\/strong>\n<ol id=\"fs-id1163873539463\" type=\"1\" class=\"stepwise\">\n<li>Identify the graph of each equation. Sketch the possible options for intersection.<\/li>\n<li>Write both equations in standard form.<\/li>\n<li>Make the coefficients of one variable opposites.<span data-type=\"newline\"><br \/><\/span>Decide which variable you will eliminate.<span data-type=\"newline\"><br \/><\/span>Multiply one or both equations so that the coefficients of that variable are opposites.<\/li>\n<li>Add the equations resulting from Step 3 to eliminate one variable.<\/li>\n<li>Solve for the remaining variable.<\/li>\n<li>Substitute each solution from Step 5 into one of the original equations. Then solve for the other variable.<\/li>\n<li>Write each solution as an ordered pair.<\/li>\n<li>Check that each ordered pair is a solution to <strong data-effect=\"bold\">both<\/strong> original equations.<\/li>\n<\/ol>\n<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox\" data-depth=\"1\" id=\"fs-id1163873870804\">\n<h3 data-type=\"title\">Section Exercises<\/h3>\n<div class=\"practice-perfect\" data-depth=\"2\" id=\"fs-id1163873724382\">\n<h4 data-type=\"title\">Practice Makes Perfect<\/h4>\n<p id=\"fs-id1163873665575\"><strong data-effect=\"bold\">Solve a System of Nonlinear Equations Using Graphing<\/strong><\/p>\n<p id=\"fs-id1163870242552\">In the following exercises, solve the system of equations by using graphing.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1163870551733\">\n<div data-type=\"problem\" id=\"fs-id1163870551736\">\n<p id=\"fs-id1163870551738\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-73e36b8f5b7c549bd59e44fc2bb6518d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#121;&#61;&#50;&#120;&#43;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#121;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"101\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163873792143\">\n<div data-type=\"problem\" id=\"fs-id1163873792145\">\n<p id=\"fs-id1163873795593\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d620d9a30c1f0a21bb9e4445df36d071_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#121;&#61;&#54;&#120;&#45;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#121;&#61;&#50;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"102\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163873666879\"><span data-type=\"media\" id=\"fs-id1163870335164\" data-alt=\"This graph shows the equations of a system, y is equal to 6 x minus 4 which is a line and y is equal to 2 x squared which is a parabola, on the x y-coordinate plane. The vertex of the parabola is (0, 0) and the parabola opens upward. The line has a slope of 6. The line and parabola intersect at the points (1, 2) and (2, 8), which are labeled. The solutions are (1, 2) and (2, 8).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_306_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows the equations of a system, y is equal to 6 x minus 4 which is a line and y is equal to 2 x squared which is a parabola, on the x y-coordinate plane. The vertex of the parabola is (0, 0) and the parabola opens upward. The line has a slope of 6. The line and parabola intersect at the points (1, 2) and (2, 8), which are labeled. The solutions are (1, 2) and (2, 8).\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163869399259\">\n<div data-type=\"problem\" id=\"fs-id1163869399261\">\n<p id=\"fs-id1163869399263\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-290e4f2bb006d9eaca6866aab6ca033f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#120;&#43;&#121;&#61;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#120;&#61;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"92\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163873798566\">\n<div data-type=\"problem\" id=\"fs-id1163873798568\">\n<p id=\"fs-id1163873798570\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4c5545ed54690ec05f29b0209d460e01_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#120;&#45;&#121;&#61;&#45;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#120;&#61;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"106\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163873655465\"><span data-type=\"media\" id=\"fs-id1163873872288\" data-alt=\"This graph shows the equations of a system, x minus y is equal to negative 2 which is a line and x is equal to y squared which is a rightward-opening parabola, on the x y-coordinate plane. The vertex of the parabola is (0, 0) and it passes through the points (1, 1) and (1, negative 1). The line has a slope of 1 and a y-intercept at 2. The line and parabola do not intersect, so the system has no solution.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_308_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows the equations of a system, x minus y is equal to negative 2 which is a line and x is equal to y squared which is a rightward-opening parabola, on the x y-coordinate plane. The vertex of the parabola is (0, 0) and it passes through the points (1, 1) and (1, negative 1). The line has a slope of 1 and a y-intercept at 2. The line and parabola do not intersect, so the system has no solution.\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163874018820\">\n<div data-type=\"problem\" id=\"fs-id1163873806975\">\n<p id=\"fs-id1163873806977\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d4868b35e56960b09975ee00b3f01f35_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#121;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#50;&#125;&#120;&#43;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#121;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"104\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163870407302\">\n<div data-type=\"problem\" id=\"fs-id1163873507028\">\n<p id=\"fs-id1163873507030\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-dc52864b945807b67da2514c309456ee_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#121;&#61;&#120;&#45;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#121;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"100\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163870335080\"><span data-type=\"media\" id=\"fs-id1163870335083\" data-alt=\"This graph shows the equations of a system, y is x minus 1 which is a line and y is equal to x squared plus 1 which is an upward-opening parabola, on the x y-coordinate plane. The vertex of the parabola is (0, 1) and it passes through the points (negative 1, 2) and (1, 2). The line has a slope of 1 and a y-intercept at negative 1. The line and parabola do not intersect, so the system has no solution.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_310_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows the equations of a system, y is x minus 1 which is a line and y is equal to x squared plus 1 which is an upward-opening parabola, on the x y-coordinate plane. The vertex of the parabola is (0, 1) and it passes through the points (negative 1, 2) and (1, 2). The line has a slope of 1 and a y-intercept at negative 1. The line and parabola do not intersect, so the system has no solution.\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163870516038\">\n<div data-type=\"problem\" id=\"fs-id1163870516040\">\n<p id=\"fs-id1163870516043\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b75a31b20a3399bad880068652600f46_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#120;&#61;&#45;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#61;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"109\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163873660218\">\n<div data-type=\"problem\" id=\"fs-id1163873660220\">\n<p id=\"fs-id1163870619439\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2a4a9924766903fffc72be26bb8588ad_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#121;&#61;&#45;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#61;&#49;&#54;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"118\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163874018254\"><span data-type=\"media\" id=\"fs-id1163873518533\" data-alt=\"This graph shows the equations of a system, x is equal to negative 2 which is a line and x squared plus y squared is equal to 16 which is a circle, on the x y-coordinate plane. The line is horizontal. The center of the circle is (0, 0) and the radius of the circle is 4. The line and circle intersect at (negative 2, 0), so the solution of the system is (negative 2, 0).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_312_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows the equations of a system, x is equal to negative 2 which is a line and x squared plus y squared is equal to 16 which is a circle, on the x y-coordinate plane. The line is horizontal. The center of the circle is (0, 0) and the radius of the circle is 4. The line and circle intersect at (negative 2, 0), so the solution of the system is (negative 2, 0).\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163873912376\">\n<div data-type=\"problem\" id=\"fs-id1163870357562\">\n<p id=\"fs-id1163870357565\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cfbafb8d64fbad19bf2cef81b350dbb7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#120;&#61;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#61;&#49;&#54;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"206\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163873791403\">\n<div data-type=\"problem\" id=\"fs-id1163873659526\">\n<p id=\"fs-id1163873659528\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6c2862728106f316b1547d0c0c8e83f5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#121;&#61;&#45;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#61;&#50;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"205\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163870324576\"><span data-type=\"media\" id=\"fs-id1163870324580\" data-alt=\"This graph shows the equations of a system, x is equal to 2 which is a line and the quantity x minus 2 end quantity squared plus the quantity y minus 4 end quantity squared is equal to 25 which is a circle, on the x y-coordinate plane. The line is horizontal. The center of the circle is (2, 4) and the radius of the circle is 5. The line and circle intersect at (2, negative 1), so the solution of the system is (2, negative 1).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_314_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows the equations of a system, x is equal to 2 which is a line and the quantity x minus 2 end quantity squared plus the quantity y minus 4 end quantity squared is equal to 25 which is a circle, on the x y-coordinate plane. The line is horizontal. The center of the circle is (2, 4) and the radius of the circle is 5. The line and circle intersect at (2, negative 1), so the solution of the system is (2, negative 1).\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163873638220\">\n<div data-type=\"problem\" id=\"fs-id1163873638222\">\n<p id=\"fs-id1163873962919\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e0370aea31c092612e0d24a1d2b343ad_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#121;&#61;&#45;&#50;&#120;&#43;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#121;&#61;&#92;&#115;&#113;&#114;&#116;&#91;&#93;&#123;&#120;&#125;&#43;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"116\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163873863087\">\n<div data-type=\"problem\" id=\"fs-id1163873863090\">\n<p id=\"fs-id1163873863092\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-37888da2bbd1eaf95255e6fbe4540de1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#121;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#120;&#43;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#121;&#61;&#92;&#115;&#113;&#114;&#116;&#91;&#93;&#123;&#120;&#125;&#45;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"117\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163873673781\"><span data-type=\"media\" id=\"fs-id1163873796259\" data-alt=\"This graph shows the equations of a system, y is equal to negative one-half x plus 2 which is a line and the y is equal to the square root of x minus 2, on the x y-coordinate plane. The curve for y is equal to the square root of x minus 2 The curve for y is equal to the square root of x plus 1 where x is greater than or equal to 0 and y is greater than or equal to negative 2. The line and square root curve intersect at (4, 0), so the solution is (4, 0).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_316_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows the equations of a system, y is equal to negative one-half x plus 2 which is a line and the y is equal to the square root of x minus 2, on the x y-coordinate plane. The curve for y is equal to the square root of x minus 2 The curve for y is equal to the square root of x plus 1 where x is greater than or equal to 0 and y is greater than or equal to negative 2. The line and square root curve intersect at (4, 0), so the solution is (4, 0).\" \/><\/span><\/div>\n<\/div>\n<p id=\"fs-id1163869163549\"><strong data-effect=\"bold\">Solve a System of Nonlinear Equations Using Substitution<\/strong><\/p>\n<p id=\"fs-id1163873632875\">In the following exercises, solve the system of equations by using substitution.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1163873628568\">\n<div data-type=\"problem\" id=\"fs-id1163873628570\">\n<p id=\"fs-id1163870381243\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3d4372c6942996a9b16dc916f1dbe0e2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#52;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#61;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#121;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#120;&#45;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"117\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163873817958\">\n<div data-type=\"problem\" id=\"fs-id1163873817960\">\n<p id=\"fs-id1163873817962\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b977e74286456fbdcecc83d5fe36b21a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#57;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#61;&#57;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#121;&#61;&#51;&#120;&#43;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"117\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163873644302\">\n<p id=\"fs-id1163873644304\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c7a32c4347dd49726de19d795315072e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"102\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163873730327\">\n<div data-type=\"problem\" id=\"fs-id1163873730330\">\n<p id=\"fs-id1163873730332\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-11a5b27ee4799c4ae04be191b94eb6fa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#57;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#61;&#57;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#121;&#61;&#120;&#43;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"117\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163873796662\">\n<div data-type=\"problem\" id=\"fs-id1163873796664\">\n<p id=\"fs-id1163873657340\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3154ea091215ae3c54a2701e728492ae_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#57;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#52;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#61;&#51;&#54;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#120;&#61;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"135\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163870406996\">\n<p id=\"fs-id1163870406998\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-559928bd7c8949c8342dd73437aef05a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163873897281\">\n<div data-type=\"problem\" id=\"fs-id1163869549732\">\n<p id=\"fs-id1163869549734\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5065f56d00528e61b88813325a1364e2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#52;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#61;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#121;&#61;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"117\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163873627432\">\n<div data-type=\"problem\" id=\"fs-id1163873627435\">\n<p id=\"fs-id1163873678509\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e413bf91800120b9944ce9b8dc56a50e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#61;&#49;&#54;&#57;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#120;&#61;&#49;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"127\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163873533384\">\n<p id=\"fs-id1163873533386\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-74e8bcf1e6576c0db379a32453eeccb6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#50;&#44;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#50;&#44;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"119\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163873521592\">\n<div data-type=\"problem\" id=\"fs-id1163873521594\">\n<p id=\"fs-id1163873521597\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-53ab51c5f1831bc1414f7df501108680_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#51;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#121;&#61;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#121;&#61;&#50;&#120;&#45;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"110\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163873874004\">\n<div data-type=\"problem\" id=\"fs-id1163873874006\">\n<p id=\"fs-id1163873715863\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1244137b50ee7d940acef4e9c0476ff5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#50;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#120;&#61;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#121;&#61;&#120;&#43;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"110\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163873782950\">\n<p id=\"fs-id1163873782953\">No solution<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163873525451\">\n<div data-type=\"problem\" id=\"fs-id1163873525453\">\n<p id=\"fs-id1163873525455\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-562535a2ac5fa61d7ab45bf4c77b9802_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#121;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#121;&#61;&#120;&#43;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"101\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163870459435\">\n<div data-type=\"problem\" id=\"fs-id1163870459437\">\n<p id=\"fs-id1163870459439\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-001c14fc4a0b5ac9c5f4c6dca0fcc40d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#121;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#121;&#61;&#120;&#45;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"101\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163873821472\">\n<p id=\"fs-id1163873821475\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-76d5a5665bd55e2c11a05bcaeafc4dc6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#44;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"116\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163873539125\">\n<div data-type=\"problem\" id=\"fs-id1163873539127\">\n<p id=\"fs-id1163873539129\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f3a377b65ebc5be67f860b4bac8f965f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#61;&#50;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#120;&#45;&#121;&#61;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"117\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163873636025\">\n<div data-type=\"problem\" id=\"fs-id1163873514452\">\n<p id=\"fs-id1163873514454\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1030590761abc2bbd770b06588390a15_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#61;&#50;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#50;&#120;&#43;&#121;&#61;&#49;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"117\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163873785724\">\n<p id=\"fs-id1163870376594\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4b36e9bc6bb9ebb785a76d89ab4774b0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#44;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"88\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1163873677678\"><strong data-effect=\"bold\">Solve a System of Nonlinear Equations Using Elimination<\/strong><\/p>\n<p id=\"fs-id1163873766386\">In the following exercises, solve the system of equations by using elimination.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1163873652213\">\n<div data-type=\"problem\" id=\"fs-id1163873652215\">\n<p id=\"fs-id1163873652217\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3759d237b817c69bffe6669a5d85c280_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#61;&#49;&#54;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#121;&#61;&#56;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"118\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163873821209\">\n<div data-type=\"problem\" id=\"fs-id1163869201418\">\n<p id=\"fs-id1163869201420\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cad74ca5f747fe1297839e35b456aabf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#61;&#49;&#54;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#121;&#61;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"118\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163873628434\">\n<p id=\"fs-id1163873628436\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8f5db119cdcad79f004ba08b527fab42_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#92;&#115;&#113;&#114;&#116;&#123;&#55;&#125;&#44;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#115;&#113;&#114;&#116;&#123;&#55;&#125;&#44;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"185\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163873871033\">\n<div data-type=\"problem\" id=\"fs-id1163869406155\">\n<p id=\"fs-id1163869406157\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-af1b7cc92a7fd549b8c115043974edb7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#61;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#121;&#61;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"109\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163873814916\">\n<div data-type=\"problem\" id=\"fs-id1163873866077\">\n<p id=\"fs-id1163873866079\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7c3b23e94cc8a0811f03243264fbd87d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#61;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#121;&#61;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"109\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163873606493\">\n<p id=\"fs-id1163873606495\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-68573856b189e6057b8c9b4e32d4ab6e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#92;&#115;&#113;&#114;&#116;&#123;&#51;&#125;&#44;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#115;&#113;&#114;&#116;&#123;&#51;&#125;&#44;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"185\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163873800365\">\n<div data-type=\"problem\" id=\"fs-id1163873766052\">\n<p id=\"fs-id1163873766054\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f0f17257f22720b944d1a712835b75ab_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#61;&#57;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#121;&#61;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"109\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163873757961\">\n<div data-type=\"problem\" id=\"fs-id1163873627473\">\n<p id=\"fs-id1163873627475\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-568140c052b742d29f1d397e4d05d1ca_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#61;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#120;&#61;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"109\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163873935242\">\n<p id=\"fs-id1163873935244\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1e32fc9723fcacdecf8b1f386e210e4d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#44;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#92;&#115;&#113;&#114;&#116;&#123;&#51;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#44;&#92;&#115;&#113;&#114;&#116;&#123;&#51;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"185\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163873558582\">\n<div data-type=\"problem\" id=\"fs-id1163873766374\">\n<p id=\"fs-id1163873766376\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2a7e684714e969ec7abb398dd47d5570_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#61;&#50;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#50;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#51;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#61;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"125\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163873914313\">\n<div data-type=\"problem\" id=\"fs-id1163873914315\">\n<p id=\"fs-id1163870621083\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e0b4e454656ab2f7aa1222e91f777652_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#61;&#50;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#61;&#45;&#49;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"131\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163869407389\">\n<p id=\"fs-id1163869407391\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-86cb8bd66d2bdd42f2182eae329167f1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#44;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#44;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#44;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#44;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"243\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163873806539\">\n<div data-type=\"problem\" id=\"fs-id1163873766314\">\n<p id=\"fs-id1163873766316\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-59e8da3a4eda1b4738c54276fcb6da31_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#61;&#49;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#61;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"118\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163870413759\">\n<div data-type=\"problem\" id=\"fs-id1163873796453\">\n<p id=\"fs-id1163873796455\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c62af92a7426bf3319d4930999ae1594_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#61;&#49;&#54;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#61;&#49;&#54;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"118\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163873853964\">\n<p id=\"fs-id1163873663891\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7cf15e29f7e8178c92f5a960a41d8ba0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#52;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"102\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163870264792\">\n<div data-type=\"problem\" id=\"fs-id1163870264795\">\n<p id=\"fs-id1163873788892\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4d719a1eb6614cf66afbc92be2226779_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#52;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#57;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#61;&#51;&#54;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#50;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#57;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#61;&#49;&#56;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"135\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163873853215\">\n<div data-type=\"problem\" id=\"fs-id1163873853217\">\n<p id=\"fs-id1163873853219\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0a69203c55da6075875f650137a5892e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#61;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#50;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#61;&#54;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"117\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163873596207\">\n<p id=\"fs-id1163873596209\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-75e043fb89178f0aac88b77e6b62919c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#92;&#115;&#113;&#114;&#116;&#123;&#51;&#125;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#115;&#113;&#114;&#116;&#123;&#51;&#125;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"120\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163873805546\">\n<div data-type=\"problem\" id=\"fs-id1163873823299\">\n<p id=\"fs-id1163873823301\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e1fd5c7ba269157fc01507e0c8d1f9fa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#52;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#61;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#52;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#61;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"117\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163873686685\">\n<div data-type=\"problem\" id=\"fs-id1163873871039\">\n<p id=\"fs-id1163873871041\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6a2cd24bd184348d235660f6bf48f877_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#61;&#45;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#51;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#61;&#51;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"135\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163873765849\">\n<p id=\"fs-id1163873765851\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-dafb6785c0ea62be2a67ccc774ec580a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#44;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#44;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#44;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#44;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"243\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163869408844\">\n<div data-type=\"problem\" id=\"fs-id1163869407406\">\n<p id=\"fs-id1163869407408\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b7d56b5cbb7e0ed36862e21cd5342c12_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#61;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#121;&#61;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"110\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163873643106\">\n<div data-type=\"problem\" id=\"fs-id1163873643108\">\n<p id=\"fs-id1163873679494\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d6e053e6f126735b1e758bcaa3d035ed_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#50;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#61;&#49;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#51;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#61;&#50;&#56;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"126\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163873624493\">\n<p id=\"fs-id1163869163401\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a98637c1b0e15486ea0dec65d33ec7aa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#44;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#44;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#44;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#44;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"243\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1163873649089\"><strong data-effect=\"bold\">Use a System of Nonlinear Equations to Solve Applications<\/strong><\/p>\n<p id=\"fs-id1163873853528\">In the following exercises, solve the problem using a system of equations.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1163873853531\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1163873898786\">\n<p id=\"fs-id1163873898789\">The sum of two numbers is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4797c874a138ca175d7c2cd8b3ed9a98_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"22\" style=\"vertical-align: 0px;\" \/> and the product is 8. Find the numbers.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163873801053\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1163873801056\">\n<p id=\"fs-id1163873606050\">The sum of two numbers is 11 and the product is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4e39e2a2b1ff28fc8255dbe598b27144_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#52;&#50;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"35\" style=\"vertical-align: -1px;\" \/> Find the numbers.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163869163582\">\n<p id=\"fs-id1163873662389\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-470cb162cf92c55d139f4f69216225e5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"22\" style=\"vertical-align: 0px;\" \/> and 14<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163873520191\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1163873657308\">\n<p id=\"fs-id1163873657310\">The sum of the squares of two numbers is 65. The difference of the number is 3. Find the numbers.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163869164255\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1163873660716\">\n<p id=\"fs-id1163873660718\">The sum of the squares of two numbers is 113. The difference of the number is 1. Find the numbers.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163873804464\">\n<p id=\"fs-id1163873804466\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-00c6d30c5f7439a21caf981437f64be1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"22\" style=\"vertical-align: 0px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ca32393b1b5af7c55a95d89cf9d610f2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"22\" style=\"vertical-align: 0px;\" \/> or 8 and 7<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163873632453\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1163873632455\">\n<p id=\"fs-id1163873632457\">The difference of the squares of two numbers is 15. The difference of twice the square of the first number and the square of the second number is 30. Find the numbers.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163874017665\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1163874017668\">\n<p id=\"fs-id1163874017670\">The difference of the squares of two numbers is 20. The difference of the square of the first number and twice the square of the second number is 4. Find the numbers.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163870551949\">\n<p id=\"fs-id1163873998028\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4797c874a138ca175d7c2cd8b3ed9a98_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"22\" style=\"vertical-align: 0px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-00b9cce9021441b203ec0271d72e6ba2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"22\" style=\"vertical-align: -1px;\" \/> or <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4797c874a138ca175d7c2cd8b3ed9a98_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"22\" style=\"vertical-align: 0px;\" \/> and 4 or 6 and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-00b9cce9021441b203ec0271d72e6ba2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"22\" style=\"vertical-align: -1px;\" \/> or 6 and 4<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163873858060\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1163873858063\">\n<p id=\"fs-id1163873858065\">The perimeter of a rectangle is 32 inches and its area is 63 square inches. Find the length and width of the rectangle.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163873863908\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1163873636531\">\n<p id=\"fs-id1163873636534\">The perimeter of a rectangle is 52 cm and its area is 165 <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d4a0bede27c6010953bfa1a0ebbeb68b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#99;&#109;&#125;&#125;&#94;&#123;&#50;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"34\" style=\"vertical-align: 0px;\" \/> Find the length and width of the rectangle.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163873821577\">\n<p id=\"fs-id1163873951878\">If the length is 11 cm, the width is 15 cm. If the length is 15 cm, the width is 11 cm.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163873951883\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1163873862658\">\n<p id=\"fs-id1163873862661\">Dion purchased a new microwave. The diagonal of the door measures 17 inches. The door also has an area of 120 square inches. What are the length and width of the microwave door?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163873787464\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1163869435881\">\n<p id=\"fs-id1163869435883\">Jules purchased a microwave for his kitchen. The diagonal of the front of the microwave measures 26 inches. The front also has an area of 240 square inches. What are the length and width of the microwave?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163870547544\">\n<p id=\"fs-id1163870547546\">If the length is 10 inches, the width is 24 inches. If the length is 24 inches, the width is 10 inches.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163870335196\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1163873669566\">\n<p id=\"fs-id1163873669568\">Roman found a widescreen TV on sale, but isn\u2019t sure if it will fit his entertainment center. The TV is 60\u201d. The size of a TV is measured on the diagonal of the screen and a widescreen has a length that is larger than the width. The screen also has an area of 1728 square inches. His entertainment center has an insert for the TV with a length of 50 inches and width of 40 inches. What are the length and width of the TV screen and will it fit Roman\u2019s entertainment center?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163873557299\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1163873557301\">\n<p id=\"fs-id1163873795201\">Donnette found a widescreen TV at a garage sale, but isn\u2019t sure if it will fit her entertainment center. The TV is 50\u201d. The size of a TV is measured on the diagonal of the screen and a widescreen has a length that is larger than the width. The screen also has an area of 1200 square inches. Her entertainment center has an insert for the TV with a length of 38 inches and width of 27 inches. What are the length and width of the TV screen and will it fit Donnette\u2019s entertainment center?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163873795205\">\n<p id=\"fs-id1163873654397\">The length is 40 inches and the width is 30 inches. The TV will not fit Donnette\u2019s entertainment center.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"writing\" data-depth=\"2\" id=\"fs-id1163873863799\">\n<h4 data-type=\"title\">Writing Exercises<\/h4>\n<div data-type=\"exercise\" id=\"fs-id1163873758518\">\n<div data-type=\"problem\" id=\"fs-id1163873758520\">\n<p id=\"fs-id1163870644374\">In your own words, explain the advantages and disadvantages of solving a system of equations by graphing.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163873644838\">\n<div data-type=\"problem\">\n<p id=\"fs-id1163873644842\">Explain in your own words how to solve a system of equations using substitution.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163873668743\">\n<p id=\"fs-id1163873668745\">Answers will vary.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163873786868\">\n<div data-type=\"problem\" id=\"fs-id1163873786870\">\n<p id=\"fs-id1163873801806\">Explain in your own words how to solve a system of equations using elimination.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\">\n<div data-type=\"problem\">\n<p id=\"fs-id1163870292674\">A circle and a parabola can intersect in ways that would result in 0, 1, 2, 3, or 4 solutions. Draw a sketch of each of the possibilities.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163870350290\">\n<p id=\"fs-id1163870350292\">Answers will vary.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1163873616410\">\n<h4 data-type=\"title\">Self Check<\/h4>\n<p id=\"fs-id1163873784103\"><span class=\"token\">\u24d0<\/span> After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1163873660154\" data-alt=\"This table has four columns and five rows. The first row is a header and it labels each column, &#x201c;I can&#x2026;&#x201d;, &#x201c;Confidently,&#x201d; &#x201c;With some help,&#x201d; and &#x201c;No-I don&#x2019;t get it!&#x201d; In row 2, the I can was solve a system of nonlinear equations using graphing. In row 3, the I can solve a system of nonlinear equations using substitution. In row 4, the I can was solve a system of a nonlinear equations using the elimination. In row 5, the I can was use a system of nonlinear equations to solve applications.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_201_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This table has four columns and five rows. The first row is a header and it labels each column, &#x201c;I can&#x2026;&#x201d;, &#x201c;Confidently,&#x201d; &#x201c;With some help,&#x201d; and &#x201c;No-I don&#x2019;t get it!&#x201d; In row 2, the I can was solve a system of nonlinear equations using graphing. In row 3, the I can solve a system of nonlinear equations using substitution. In row 4, the I can was solve a system of a nonlinear equations using the elimination. In row 5, the I can was use a system of nonlinear equations to solve applications.\" \/><\/span><\/p>\n<p id=\"fs-id1163873714806\"><span class=\"token\">\u24d1<\/span> After looking at the checklist, do you think you are well-prepared for the next section? Why or why not?<\/p>\n<\/div>\n<\/div>\n<div class=\"review-exercises\" data-depth=\"1\" id=\"fs-id1163874006441\">\n<h3 data-type=\"title\">Chapter Review Exercises<\/h3>\n<div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1163874006445\">\n<h4 data-type=\"title\"><a href=\"\/contents\/30062189-1923-4bf7-902b-9f2691a64c71\" class=\"target-chapter\">Distance and Midpoint Formulas; Circles<\/a><\/h4>\n<p id=\"fs-id1163873659904\"><strong data-effect=\"bold\">Use the Distance Formula<\/strong><\/p>\n<p id=\"fs-id1163873798182\">In the following exercises, find the distance between the points. Round to the nearest tenth if needed.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1163873798186\">\n<div data-type=\"problem\" id=\"fs-id1163870357236\">\n<p id=\"fs-id1163870357238\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-eb0e705487e4783110580c14181b9d4f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#53;&#44;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"51\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b098ad691d3b1e66796376e6000e2385_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#44;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"51\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163873807388\">\n<div data-type=\"problem\" id=\"fs-id1163873807390\">\n<p id=\"fs-id1163873807392\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-edd8d8b57c2815309edbd447803c95fa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#44;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"51\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fc88778f80c6fa1eb186cfd3741de73e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#44;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163873823025\">\n<p id=\"fs-id1163873823027\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c670fab3ff7b014c602cc0ebf5e374b1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#100;&#61;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"42\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163866979320\">\n<div data-type=\"problem\" id=\"fs-id1163866979322\">\n<p id=\"fs-id1163870376214\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d9b49b201a3efa56726407933e068d18_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#56;&#44;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5db13f20e02f8d03333a782732fed899_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#55;&#44;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"65\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163873866500\">\n<div data-type=\"problem\" id=\"fs-id1163873857670\">\n<p id=\"fs-id1163873857672\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-95e282826e52860edf4d8703db75fb7f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#44;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"52\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-38deec632295a22f878bf66affc585e5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#44;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"52\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163870548914\">\n<p id=\"fs-id1163869099050\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7fae2aa2aa77d0ef2ceb0ab47696a811_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#100;&#61;&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#55;&#125;&#44;&#100;&#92;&#97;&#112;&#112;&#114;&#111;&#120;&#32;&#52;&#46;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"128\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1163873604614\"><strong data-effect=\"bold\">Use the Midpoint Formula<\/strong><\/p>\n<p id=\"fs-id1163873702172\">In the following exercises, find the midpoint of the line segments whose endpoints are given.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1163866979854\">\n<div data-type=\"problem\" id=\"fs-id1163866979856\">\n<p id=\"fs-id1163866979858\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1ea573d1f83d98725156be4230a519b6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#44;&#45;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"65\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4ce13ff56205290ac2f68e7ba2f28d73_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#52;&#44;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"65\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163869435324\">\n<div data-type=\"problem\" id=\"fs-id1163869435326\">\n<p id=\"fs-id1163873924830\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-af3fe858ad1de1474e4e0f6749723189_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#44;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b932df7404bba7c4812c40fc6d0d4b4a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#44;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163873639312\">\n<p id=\"fs-id1163873639314\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9f071f7020da53c225631b96b8f9875e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#44;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163873608722\">\n<div data-type=\"problem\" id=\"fs-id1163873882098\">\n<p id=\"fs-id1163873882100\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b4db9d0e6fe1cc60ea7fc1745a9316a0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#56;&#44;&#45;&#49;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"74\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-88195871adecf56ab04003457649ee4f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#57;&#44;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163873750610\">\n<div data-type=\"problem\" id=\"fs-id1163873742563\">\n<p id=\"fs-id1163873742565\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1f307aa6b65fb69739e6c2b7458409de_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#44;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"51\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3976ef1ff68d71667ca3289946f8d0c7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#44;&#45;&#57;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"52\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163873645302\">\n<p id=\"fs-id1163873645305\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-531eacedac1162ab05b3ff89db429f6e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#50;&#125;&#44;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#125;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"55\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1163870547955\"><strong data-effect=\"bold\">Write the Equation of a Circle in Standard Form<\/strong><\/p>\n<p id=\"fs-id1163870547960\">In the following exercises, write the standard form of the equation of the circle with the given information.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1163873812775\">\n<div data-type=\"problem\" id=\"fs-id1163873812777\">\n<p id=\"fs-id1163873812779\">radius is 15 and center is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c53627fd7039dcb62c54d86fe468e6e1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163873673838\">\n<div data-type=\"problem\" id=\"fs-id1163873814747\">\n<p id=\"fs-id1163873814750\">radius is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f55c583bdda023ba48a9c930325a0708_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#55;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"23\" style=\"vertical-align: -2px;\" \/> and center is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c53627fd7039dcb62c54d86fe468e6e1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163870510888\">\n<p id=\"fs-id1163873607179\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c348a52db36500d8158f1ee44da68e5a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#61;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"89\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163874046327\">\n<div data-type=\"problem\" id=\"fs-id1163874046330\">\n<p id=\"fs-id1163874046332\">radius is 9 and center is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5aa29da239e6fed61429835a4b4444af_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#44;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"51\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163873606476\">\n<div data-type=\"problem\" id=\"fs-id1163873606478\">\n<p id=\"fs-id1163873606481\">radius is 7 and center is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0fc65316aa33a9c85778a50c7aec6891_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#44;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"65\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163873655226\">\n<p id=\"fs-id1163873655228\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-240bb3850d7bfdbf4c6a6af546b12066_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#43;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#61;&#52;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"186\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163873624748\">\n<div data-type=\"problem\" id=\"fs-id1163873882075\">\n<p id=\"fs-id1163873882078\">center is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d1fa2be7ed8e95fd6a934ca178fab1d7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#44;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/> and a point on the circle is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-dbaec542907415eac32615dfae0ae911_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#44;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"52\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163873616884\">\n<div data-type=\"problem\" id=\"fs-id1163873616886\">\n<p id=\"fs-id1163873645451\">center is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f78f604644c2cdddbea2fc4d8ad49cca_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#44;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/> and a point on the circle is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9f071f7020da53c225631b96b8f9875e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#44;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163873661694\">\n<p id=\"fs-id1163873660619\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6795dcca3f9eb5640528f1ef00c86ee2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#61;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"177\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1163870550111\"><strong data-effect=\"bold\">Graph a Circle<\/strong><\/p>\n<p id=\"fs-id1163870516094\">In the following exercises, <span class=\"token\">\u24d0<\/span> find the center and radius, then <span class=\"token\">\u24d1<\/span> graph each circle.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1163873526037\">\n<div data-type=\"problem\" id=\"fs-id1163873837511\">\n<p id=\"fs-id1163873837513\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1d77e5ab7c470544bd2b91bcbfe4d4ac_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#61;&#52;&#53;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"125\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163873801972\">\n<div data-type=\"problem\" id=\"fs-id1163873801974\">\n<p id=\"fs-id1163873606015\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-695f3f522161e408e7619189d136f817_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#51;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#61;&#52;&#51;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"124\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163873731680\">\n<p id=\"fs-id1163873731682\"><span class=\"token\">\u24d0<\/span> radius: 12, center: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c53627fd7039dcb62c54d86fe468e6e1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/><span data-type=\"newline\"><br \/><\/span> <span class=\"token\">\u24d1<\/span><span data-type=\"newline\"><br \/><\/span><\/p>\n<p><span data-type=\"media\" id=\"fs-id1163873678491\" data-alt=\"The figure shows a circle graphed on the x y coordinate plane. The x-axis of the plane runs from negative 20 to 20. The y-axis of the plane runs from negative 15 to 15. The center of the circle is (0, 0) and the radius of the circle is 12.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_318_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows a circle graphed on the x y coordinate plane. The x-axis of the plane runs from negative 20 to 20. The y-axis of the plane runs from negative 15 to 15. The center of the circle is (0, 0) and the radius of the circle is 12.\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163873823003\">\n<div data-type=\"problem\" id=\"fs-id1163873823006\">\n<p id=\"fs-id1163873823008\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-111c50dbf513119370f9d4f80ffc0315_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#61;&#56;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"185\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163870266037\">\n<div data-type=\"problem\" id=\"fs-id1163870266039\">\n<p id=\"fs-id1163873716963\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-240bb3850d7bfdbf4c6a6af546b12066_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#43;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#61;&#52;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"186\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163870398885\">\n<p id=\"fs-id1163870398888\"><span class=\"token\">\u24d0<\/span> radius: 7, center: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0fc65316aa33a9c85778a50c7aec6891_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#44;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"65\" style=\"vertical-align: -4px;\" \/><span data-type=\"newline\"><br \/><\/span> <span class=\"token\">\u24d1<\/span><span data-type=\"newline\"><br \/><\/span><\/p>\n<p><span data-type=\"media\" id=\"fs-id1163873783987\" data-alt=\"The figure shows a circle graphed on the x y coordinate plane. The x-axis of the plane runs from negative 20 to 20. The y-axis of the plane runs from negative 15 to 15. The center of the circle is (negative 2, negative 5) and the radius of the circle is 7.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_320_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows a circle graphed on the x y coordinate plane. The x-axis of the plane runs from negative 20 to 20. The y-axis of the plane runs from negative 15 to 15. The center of the circle is (negative 2, negative 5) and the radius of the circle is 7.\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163873558348\">\n<div data-type=\"problem\" id=\"fs-id1163873632142\">\n<p id=\"fs-id1163873632144\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7a8cb2a9a8af14d5e0b0bbd4179ce44a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#54;&#120;&#45;&#49;&#50;&#121;&#45;&#49;&#57;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"218\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163873853290\">\n<div data-type=\"problem\" id=\"fs-id1163873853292\">\n<p id=\"fs-id1163873631018\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a616ed2bf13b82031b7d5ab5055fb3b3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#121;&#45;&#54;&#48;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"168\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163869138122\">\n<p id=\"fs-id1163874044723\"><span class=\"token\">\u24d0<\/span> radius: 8, center: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-398828550549fdd4b2191f8f7cde7bd6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/><span data-type=\"newline\"><br \/><\/span> <span class=\"token\">\u24d1<\/span><span data-type=\"newline\"><br \/><\/span><\/p>\n<p><span data-type=\"media\" id=\"fs-id1163873657446\" data-alt=\"The figure shows a circle graphed on the x y coordinate plane. The x-axis of the plane runs from negative 20 to 20. The y-axis of the plane runs from negative 15 to 15. The center of the circle is (0, 2) and the radius of the circle is 8.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_322_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows a circle graphed on the x y coordinate plane. The x-axis of the plane runs from negative 20 to 20. The y-axis of the plane runs from negative 15 to 15. The center of the circle is (0, 2) and the radius of the circle is 8.\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1163870623961\">\n<h4 data-type=\"title\"><a href=\"\/contents\/789eaa88-3770-4bb0-9039-ff1146680681\" class=\"target-chapter\">Parabolas<\/a><\/h4>\n<p id=\"fs-id1163873764394\"><strong data-effect=\"bold\">Graph Vertical Parabolas<\/strong><\/p>\n<p id=\"fs-id1163870484482\">In the following exercises, graph each equation by using its properties.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1163870484485\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1163869163587\">\n<p id=\"fs-id1163869163590\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-47a775621f68425cfbd70d84bb7e465c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#52;&#120;&#45;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"122\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163870510844\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1163870510846\">\n<p id=\"fs-id1163870510848\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-17b9984babf31cb9ac9b546ee116ae69_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#50;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#48;&#120;&#43;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"140\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163873639811\"><span data-type=\"media\" id=\"fs-id1163873639814\" data-alt=\"The figure shows an upward-opening parabola graphed on the x y coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 7 to 7. The vertex is (negative five-halves, negative eleven-halves) and the parabola passes through the points (negative 4, negative 1) and (negative 1, negative 1).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_324_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows an upward-opening parabola graphed on the x y coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 7 to 7. The vertex is (negative five-halves, negative eleven-halves) and the parabola passes through the points (negative 4, negative 1) and (negative 1, negative 1).\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163873817175\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1163873817177\">\n<p id=\"fs-id1163873817180\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-98e99435e94bcd6ff1b559d6b90656cc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#45;&#54;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#50;&#120;&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"153\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163873853748\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1163873853750\">\n<p id=\"fs-id1163873853752\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-507173c2af60d799a3ed3c38cc5a4e63_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#48;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"100\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163873872896\"><span data-type=\"media\" id=\"fs-id1163873872899\" data-alt=\"The figure shows a downward-opening parabola graphed on the x y coordinate plane. The x-axis of the plane runs from negative 36 to 36. The y-axis of the plane runs from negative 26 to 26. The vertex is (5, 25) and the parabola passes through the points (2, 16) and (8, 16).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_326_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows a downward-opening parabola graphed on the x y coordinate plane. The x-axis of the plane runs from negative 36 to 36. The y-axis of the plane runs from negative 26 to 26. The vertex is (5, 25) and the parabola passes through the points (2, 16) and (8, 16).\" \/><\/span><\/div>\n<\/div>\n<p id=\"fs-id1163873663657\">In the following exercises, <span class=\"token\">\u24d0<\/span> write the equation in standard form, then <span class=\"token\">\u24d1<\/span> use properties of the standard form to graph the equation.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1163873882772\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1163873882775\">\n<p id=\"fs-id1163873882777\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2d9359d026f85ba9ed50232f835c51bc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#52;&#120;&#43;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"122\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163873812536\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1163873812538\">\n<p id=\"fs-id1163873662802\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ebdeed5e981cb73b3fbf45885ed8ae05_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#50;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#120;&#45;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"130\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163873822689\">\n<p id=\"fs-id1163873822691\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4d206818e4ee7e25cf359da850ed5f2b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#50;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#45;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"135\" style=\"vertical-align: -4px;\" \/><span data-type=\"newline\"><br \/><\/span><span class=\"token\">\u24d1<\/span><span data-type=\"newline\"><br \/><\/span><\/p>\n<p><span data-type=\"media\" id=\"fs-id1163870644905\" data-alt=\"The figure shows an upward-opening parabola graphed on the x y coordinate plane. The x-axis of the plane runs from negative 22 to 22. The y-axis of the plane runs from negative 16 to 16. The vertex is (1, negative 4) and the parabola passes through the points (0, negative 2) and (2, negative 2).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_328_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows an upward-opening parabola graphed on the x y coordinate plane. The x-axis of the plane runs from negative 22 to 22. The y-axis of the plane runs from negative 16 to 16. The vertex is (1, negative 4) and the parabola passes through the points (0, negative 2) and (2, negative 2).\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163873782264\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1163873782266\">\n<p id=\"fs-id1163873752504\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8e1c2ae80526432758b548c72be4514b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#45;&#51;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#56;&#120;&#45;&#50;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"163\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163870547472\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1163870547474\">\n<p id=\"fs-id1163873797140\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c791dcd645f9b219e3d710bc99ffaaee_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#50;&#120;&#45;&#51;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"139\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163873866506\">\n<p id=\"fs-id1163873866509\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c23ac59a1b55653315c5764edbe2f06c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"125\" style=\"vertical-align: -4px;\" \/><span data-type=\"newline\"><br \/><\/span><span class=\"token\">\u24d1<\/span><span data-type=\"newline\"><br \/><\/span><\/p>\n<p><span data-type=\"media\" id=\"fs-id1163873651660\" data-alt=\"The figure shows a downward-opening parabola graphed on the x y coordinate plane. The x-axis of the plane runs from negative 60 to 60. The y-axis of the plane runs from negative 46 to 46. The vertex is (6, 1) and the parabola passes through the points (5, 0) and (7, 0).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_330_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows a downward-opening parabola graphed on the x y coordinate plane. The x-axis of the plane runs from negative 60 to 60. The y-axis of the plane runs from negative 46 to 46. The vertex is (6, 1) and the parabola passes through the points (5, 0) and (7, 0).\" \/><\/span><\/div>\n<\/div>\n<p id=\"fs-id1163873657481\"><strong data-effect=\"bold\">Graph Horizontal Parabolas<\/strong><\/p>\n<p id=\"fs-id1163873608732\">In the following exercises, graph each equation by using its properties.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1163873608736\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1163873768714\">\n<p id=\"fs-id1163873768716\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-131a874aaa5ab5d0fa123256a235a21a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#50;&#123;&#121;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"59\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163873663351\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1163873663353\">\n<p id=\"fs-id1163873616997\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e7f3a573659392d55e2b59b3075e514c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#50;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#43;&#52;&#121;&#43;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"130\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163873855596\"><span data-type=\"media\" id=\"fs-id1163873855599\" data-alt=\"The figure shows a rightward-opening parabola graphed on the x y coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 8 to 8. The vertex is (4, negative 1) and the parabola passes through the points (6, 0) and (6, negative 2).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_332_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows a rightward-opening parabola graphed on the x y coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 8 to 8. The vertex is (4, negative 1) and the parabola passes through the points (6, 0) and (6, negative 2).\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163869435866\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1163869435868\">\n<p id=\"fs-id1163869346255\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7ab657cb66ae41b34fe64bf22e518e99_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#121;&#45;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"121\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163873784033\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1163873784035\">\n<p id=\"fs-id1163870386273\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ee8de15ff0b4037d4fb42963061db3ad_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#45;&#51;&#123;&#121;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"73\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163873703287\"><span data-type=\"media\" id=\"fs-id1163873703290\" data-alt=\"The figure shows a leftward-opening parabola graphed on the x y coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 8 to 8. The vertex is (0, 0) and the parabola passes through the points (negative 3, 1) and (negative 3, negative 1).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_334_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows a leftward-opening parabola graphed on the x y coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 8 to 8. The vertex is (0, 0) and the parabola passes through the points (negative 3, 1) and (negative 3, negative 1).\" \/><\/span><\/div>\n<\/div>\n<p id=\"fs-id1163866980050\">In the following exercises, <span class=\"token\">\u24d0<\/span> write the equation in standard form, then <span class=\"token\">\u24d1<\/span> use properties of the standard form to graph the equation.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1163873635535\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1163869409805\">\n<p id=\"fs-id1163869409807\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-36e914bd3fad8f407203f878abaaaee5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#52;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#43;&#56;&#121;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"99\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163873809661\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1163873809663\">\n<p id=\"fs-id1163873809665\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d700d8dc7f259ec8b59ac78e20375986_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#43;&#52;&#121;&#43;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"120\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163870551528\">\n<p id=\"fs-id1163869091463\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-345c91eceb9293356aaa29ed19ba3563_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#43;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"125\" style=\"vertical-align: -4px;\" \/><span data-type=\"newline\"><br \/><\/span><span class=\"token\">\u24d1<\/span><span data-type=\"newline\"><br \/><\/span><\/p>\n<p><span data-type=\"media\" id=\"fs-id1163873769107\" data-alt=\"The figure shows a rightward-opening parabola graphed on the x y coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 8 to 8. The vertex is (1, negative 2) and the parabola passes through the points (5, 0) and (5, negative 4).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_336_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows a rightward-opening parabola graphed on the x y coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 8 to 8. The vertex is (1, negative 2) and the parabola passes through the points (5, 0) and (5, negative 4).\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163873861448\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1163870359109\">\n<p id=\"fs-id1163870359112\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e834cccc6a0985099c82a9e151e977f3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#54;&#121;&#45;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"121\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163873785782\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1163873757150\">\n<p id=\"fs-id1163873757152\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-22b5dda252579a656839fae66b12949e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#45;&#50;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#43;&#52;&#121;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"113\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163870489447\">\n<p id=\"fs-id1163870489449\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9f12e068b6e2d9d80f4f7ea02b83c67e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#45;&#50;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"147\" style=\"vertical-align: -4px;\" \/><span data-type=\"newline\"><br \/><\/span><span class=\"token\">\u24d1<\/span><span data-type=\"newline\"><br \/><\/span><\/p>\n<p><span data-type=\"media\" id=\"fs-id1163870291308\" data-alt=\"The figure shows a leftward-opening parabola graphed on the x y coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 8 to 8. The vertex is (2, negative 3) and the parabola passes through the points (0, 2) and (0, 0).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_338_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows a leftward-opening parabola graphed on the x y coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 8 to 8. The vertex is (2, negative 3) and the parabola passes through the points (0, 2) and (0, 0).\" \/><\/span><\/div>\n<\/div>\n<p id=\"fs-id1163873673095\"><strong data-effect=\"bold\">Solve Applications with Parabolas<\/strong><\/p>\n<p id=\"fs-id1163873520546\">In the following exercises, create the equation of the parabolic arch formed in the foundation of the bridge shown. Give the answer in standard form.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1163873520550\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1163873520552\"><span data-type=\"media\" id=\"fs-id1163873657328\" data-alt=\"The figure shows a parabolic arch formed in the foundation of the bridge. The arch is 5 feet high and 20 feet wide.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_202_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows a parabolic arch formed in the foundation of the bridge. The arch is 5 feet high and 20 feet wide.\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163873761318\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1163873761320\"><span data-type=\"media\" id=\"fs-id1163873761322\" data-alt=\"The figure shows a parabolic arch formed in the foundation of the bridge. The arch is 25 feet high and 30 feet wide.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_203_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows a parabolic arch formed in the foundation of the bridge. The arch is 25 feet high and 30 feet wide.\" \/><\/span><\/div>\n<div data-type=\"solution\" id=\"fs-id1163873764370\">\n<p id=\"fs-id1163873764372\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-acfec78052d3fddf12d3f401171f69be_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#57;&#125;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#48;&#125;&#123;&#51;&#125;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"125\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1163870644426\">\n<h4 data-type=\"title\"><a href=\"\/contents\/4c350ea6-1dd0-4a11-a80a-9a5f02497a87\" class=\"target-chapter\">Ellipses<\/a><\/h4>\n<p id=\"fs-id1163870357359\"><strong data-effect=\"bold\">Graph an Ellipse with Center at the Origin<\/strong><\/p>\n<p id=\"fs-id1163870357366\">In the following exercises, graph each ellipse.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1163873629400\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1163873629402\">\n<p id=\"fs-id1163873629404\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8f344eda5f9dfa1e0cbf72843c5f0542_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#51;&#54;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#50;&#53;&#125;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"88\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163873814467\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1163869162846\">\n<p id=\"fs-id1163869162848\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-54b5411038839baecca34fda791cf1a6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#52;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#56;&#49;&#125;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"88\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163873862770\"><span data-type=\"media\" id=\"fs-id1163873862773\" data-alt=\"The figure shows an ellipse graphed on the x y coordinate plane. The x-axis of the plane runs from negative 14 to 14. The y-axis of the plane runs from negative 10 to 10. The ellipse has a center at (0, 0), a vertical major axis, vertices at (0, plus or minus 9), and co-vertices at (plus or minus 2, 0).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_340_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows an ellipse graphed on the x y coordinate plane. The x-axis of the plane runs from negative 14 to 14. The y-axis of the plane runs from negative 10 to 10. The ellipse has a center at (0, 0), a vertical major axis, vertices at (0, plus or minus 9), and co-vertices at (plus or minus 2, 0).\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163873896958\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1163873668245\">\n<p id=\"fs-id1163873668247\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0a3e068243cdf26cf175a4d1a3a68a65_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#57;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#54;&#52;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#61;&#51;&#49;&#51;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"152\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163873790620\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1163873790622\">\n<p id=\"fs-id1163873790624\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c8b151c010e6bcbd9e479cbd01a2b472_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#57;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#61;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"98\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163873863215\"><span data-type=\"media\" id=\"fs-id1163873863218\" data-alt=\"The figure shows an ellipse graphed on the x y coordinate plane. The x-axis of the plane runs from negative 9 to 9. The y-axis of the plane runs from negative 7 to 7. The ellipse has a center at (0, 0), a vertical major axis, vertices at (0, plus or minus 3), and co-vertices at (plus or minus 1, 0).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_342_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows an ellipse graphed on the x y coordinate plane. The x-axis of the plane runs from negative 9 to 9. The y-axis of the plane runs from negative 7 to 7. The ellipse has a center at (0, 0), a vertical major axis, vertices at (0, plus or minus 3), and co-vertices at (plus or minus 1, 0).\" \/><\/span><\/div>\n<\/div>\n<p id=\"fs-id1163870694967\"><strong data-effect=\"bold\">Find the Equation of an Ellipse with Center at the Origin<\/strong><\/p>\n<p id=\"fs-id1163870694974\">In the following exercises, find the equation of the ellipse shown in the graph.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1163873880357\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1163873880359\"><span data-type=\"media\" id=\"fs-id1163873880361\" data-alt=\"The figure shows an ellipse graphed on the x y coordinate plane. The ellipse has a center at (0, 0), a horizontal major axis, vertices at (plus or minus 10, 0), and co-vertices at (0, plus or minus 4).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_204_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows an ellipse graphed on the x y coordinate plane. The ellipse has a center at (0, 0), a horizontal major axis, vertices at (plus or minus 10, 0), and co-vertices at (0, plus or minus 4).\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163873856127\" class=\"material-set-2\">\n<div data-type=\"problem\"><span data-type=\"media\" id=\"fs-id1163873800254\" data-alt=\"The figure shows an ellipse graphed on the x y coordinate plane. The ellipse has a center at (0, 0), a vertical major axis, vertices at (0, plus or minus 8), and co-vertices at (plus or minus 6, 0).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_205_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows an ellipse graphed on the x y coordinate plane. The ellipse has a center at (0, 0), a vertical major axis, vertices at (0, plus or minus 8), and co-vertices at (plus or minus 6, 0).\" \/><\/span><\/div>\n<div data-type=\"solution\" id=\"fs-id1163870695358\">\n<p id=\"fs-id1163870695360\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2551267c3a0b4db8afbcb882507001bd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#51;&#54;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#54;&#52;&#125;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"88\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1163873769016\"><strong data-effect=\"bold\">Graph an Ellipse with Center Not at the Origin<\/strong><\/p>\n<p>In the following exercises, graph each ellipse.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1163873607106\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1163873607108\">\n<p id=\"fs-id1163873607110\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5ac76ab9e6e807ec669cfbf35fc7682a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#50;&#53;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#52;&#125;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"144\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163873715946\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1163873715949\">\n<p id=\"fs-id1163873715951\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0a89f8cfef5da2a1fa3ccc75b9777496_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#49;&#54;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#57;&#125;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"144\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163873637692\"><span data-type=\"media\" id=\"fs-id1163873637695\" data-alt=\"The figure shows an ellipse graphed on the x y coordinate plane. The x-axis of the plane runs from negative 14 to 14. The y-axis of the plane runs from negative 10 to 10. The ellipse has a center at (negative 4, negative 1), a horizontal major axis, vertices at (negative 8, negative 1) and (0, negative 1) and co-vertices at (negative 4, 2) and (negative 4, negative 4).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_344_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows an ellipse graphed on the x y coordinate plane. The x-axis of the plane runs from negative 14 to 14. The y-axis of the plane runs from negative 10 to 10. The ellipse has a center at (negative 4, negative 1), a horizontal major axis, vertices at (negative 8, negative 1) and (0, negative 1) and co-vertices at (negative 4, 2) and (negative 4, negative 4).\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163873855121\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1163873855123\">\n<p id=\"fs-id1163873627563\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-065d49bcf8c64d5df1390153d6170866_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#49;&#54;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#51;&#54;&#125;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"144\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163873792951\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1163873792953\">\n<p id=\"fs-id1163873792955\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7e607d9d7a3061d6223a42c5a08b2b8c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#57;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#50;&#53;&#125;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"144\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163873822877\"><span data-type=\"media\" id=\"fs-id1163873822880\" data-alt=\"The figure shows an ellipse graphed on the x y coordinate plane. The x-axis of the plane runs from negative 14 to 14. The y-axis of the plane runs from negative 10 to 10. The ellipse has a center at (negative 3, 2), a vertical major axis, vertices at (negative 3, 7) and (negative 3, negative 3) and co-vertices at (negative 6, 2) and (0, 2).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_346_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows an ellipse graphed on the x y coordinate plane. The x-axis of the plane runs from negative 14 to 14. The y-axis of the plane runs from negative 10 to 10. The ellipse has a center at (negative 3, 2), a vertical major axis, vertices at (negative 3, 7) and (negative 3, negative 3) and co-vertices at (negative 6, 2) and (0, 2).\" \/><\/span><\/div>\n<\/div>\n<p id=\"fs-id1163873745244\">In the following exercises, <span class=\"token\">\u24d0<\/span> write the equation in standard form and <span class=\"token\">\u24d1<\/span> graph.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1163873604412\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1163873604414\">\n<p id=\"fs-id1163873604416\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-30d3bec006a3231daa70d7c50044407c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#50;&#120;&#43;&#52;&#48;&#121;&#43;&#49;&#50;&#48;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"236\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163873654528\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1163873654530\">\n<p id=\"fs-id1163873654532\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d98de96f69147d56079d724e58bd1972_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#53;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#52;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#53;&#48;&#120;&#45;&#53;&#54;&#121;&#43;&#51;&#50;&#49;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"271\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163873999255\">\n<p id=\"fs-id1163873799839\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1ad607f8f8ce35fbbc6429226912622d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#52;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#50;&#53;&#125;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"144\" style=\"vertical-align: -6px;\" \/><span data-type=\"newline\"><br \/><\/span><span class=\"token\">\u24d1<\/span><span data-type=\"newline\"><br \/><\/span><\/p>\n<p><span data-type=\"media\" id=\"fs-id1163873786610\" data-alt=\"The figure shows an ellipse graphed on the x y coordinate plane. The x-axis of the plane runs from negative 18 to 18. The y-axis of the plane runs from negative 14 to 14. The ellipse has a center at (3, 7), a vertical major axis, vertices at (3, 2) and (3, 12) and co-vertices at (negative 1, 7) and (5, 7).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_348_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows an ellipse graphed on the x y coordinate plane. The x-axis of the plane runs from negative 18 to 18. The y-axis of the plane runs from negative 14 to 14. The ellipse has a center at (3, 7), a vertical major axis, vertices at (3, 2) and (3, 12) and co-vertices at (negative 1, 7) and (5, 7).\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163873559682\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1163873559684\">\n<p id=\"fs-id1163873559687\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-529acd30dbbad00c58eed057dcc5aefe_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#53;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#52;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#53;&#48;&#120;&#43;&#49;&#50;&#53;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"222\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163870221437\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1163870221439\">\n<p id=\"fs-id1163870221441\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d9fb67d37dc2c8211c940a0f27db3faf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#57;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#50;&#54;&#120;&#43;&#52;&#48;&#53;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"214\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163869409421\">\n<p id=\"fs-id1163869409424\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-86d9f7a470b576a6210a3c0ae5bc4d73_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#57;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#52;&#125;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"116\" style=\"vertical-align: -6px;\" \/><span data-type=\"newline\"><br \/><\/span><span class=\"token\">\u24d1<\/span><span data-type=\"newline\"><br \/><\/span><\/p>\n<p><span data-type=\"media\" id=\"fs-id1163873854352\" data-alt=\"The figure shows an ellipse graphed on the x y coordinate plane. The x-axis of the plane runs from negative 15 to 15. The y-axis of the plane runs from negative 11 to 11. The ellipse has a center at (0, 7), a horizontal major axis, vertices at (3, 7) and (negative 3, 7) and co-vertices at (0, 5) and (0, 9).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_350_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows an ellipse graphed on the x y coordinate plane. The x-axis of the plane runs from negative 15 to 15. The y-axis of the plane runs from negative 11 to 11. The ellipse has a center at (0, 7), a horizontal major axis, vertices at (3, 7) and (negative 3, 7) and co-vertices at (0, 5) and (0, 9).\" \/><\/span><\/div>\n<\/div>\n<p id=\"fs-id1163870547451\"><strong data-effect=\"bold\">Solve Applications with Ellipses<\/strong><\/p>\n<p id=\"fs-id1163870547457\">In the following exercises, write the equation of the ellipse described.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1163873642289\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1163873642291\">\n<p id=\"fs-id1163873642293\">A comet moves in an elliptical orbit around a sun. The closest the comet gets to the sun is approximately 10 AU and the furthest is approximately 90 AU. The sun is one of the foci of the elliptical orbit. Letting the ellipse center at the origin and labeling the axes in AU, the orbit will look like the figure below. Use the graph to write an equation for the elliptical orbit of the comet.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1163873642300\" data-alt=\"The figure shows a model of an elliptical orbit around the sun on the x y coordinate plane. The ellipse has a center at (0, 0), a horizontal major axis, vertices marked at (plus or minus 50, 0), the sun marked as a foci and labeled (50, 0), the closest distance the comet is from the sun marked as 10 A U, and the farthest a comet is from the sun marked as 90 A U.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_206_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows a model of an elliptical orbit around the sun on the x y coordinate plane. The ellipse has a center at (0, 0), a horizontal major axis, vertices marked at (plus or minus 50, 0), the sun marked as a foci and labeled (50, 0), the closest distance the comet is from the sun marked as 10 A U, and the farthest a comet is from the sun marked as 90 A U.\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1163873694035\">\n<h4 data-type=\"title\"><a href=\"\/contents\/5784f55d-62cb-474f-aa4f-747760be4966\" class=\"target-chapter\">Hyperbolas<\/a><\/h4>\n<p id=\"fs-id1163873694046\"><strong data-effect=\"bold\">Graph a Hyperbola with Center at <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c53627fd7039dcb62c54d86fe468e6e1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/><\/strong><\/p>\n<p id=\"fs-id1163873749063\">In the following exercises, graph.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1163873749066\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1163873750108\">\n<p id=\"fs-id1163873750110\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-df7b23768af1f23ca1a6e28a1350a219_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#50;&#53;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#57;&#125;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"88\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163873632366\"><span data-type=\"media\" id=\"fs-id1163873632369\" data-alt=\"The figure shows a hyperbola graphed on the x y coordinate plane. The x-axis of the plane runs from negative 12 to 12. The y-axis of the plane runs from negative 9 to 9. The hyperbola has a center at (0, 0) and branches that pass through the vertices (plus or minus 5, 0), and that open left and right.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_351_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows a hyperbola graphed on the x y coordinate plane. The x-axis of the plane runs from negative 12 to 12. The y-axis of the plane runs from negative 9 to 9. The hyperbola has a center at (0, 0) and branches that pass through the vertices (plus or minus 5, 0), and that open left and right.\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163873732163\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1163873732165\">\n<p id=\"fs-id1163873732167\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-357891b85b213b0a0b9d12670f286acc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#52;&#57;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#49;&#54;&#125;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"88\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163873748757\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1163873748760\">\n<p id=\"fs-id1163873748762\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ec5f17c479da0d6748b8edbda6cb1024_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#57;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#54;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#61;&#49;&#52;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"134\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163870410768\"><span data-type=\"media\" id=\"fs-id1163870410771\" data-alt=\"The figure shows a hyperbola graphed on the x y coordinate plane. The x-axis of the plane runs from negative 19 to 19. The y-axis of the plane runs from negative 15 to 15. The hyperbola has a center at (0, 0) and branches that pass through the vertices (0, plus or minus 4), and that open up and down.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_353_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows a hyperbola graphed on the x y coordinate plane. The x-axis of the plane runs from negative 19 to 19. The y-axis of the plane runs from negative 15 to 15. The hyperbola has a center at (0, 0) and branches that pass through the vertices (0, plus or minus 4), and that open up and down.\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163870694898\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1163870694900\">\n<p id=\"fs-id1163870694902\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5fc0bbcf3075e189809581df3a5df90d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#54;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#61;&#54;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"124\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1163873870185\"><strong data-effect=\"bold\">Graph a Hyperbola with Center at <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-772b77341e52468c6e31b5bff8f72528_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#104;&#44;&#107;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"40\" style=\"vertical-align: -4px;\" \/><\/strong><\/p>\n<p id=\"fs-id1163873912716\">In the following exercises, graph.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1163873912719\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1163873912721\">\n<p id=\"fs-id1163873912723\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-77fd854193aa9b33021fc04044411677_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#52;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#57;&#125;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"144\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163873625380\"><span data-type=\"media\" id=\"fs-id1163873811040\" data-alt=\"The figure shows a hyperbola graphed on the x y coordinate plane. The x-axis of the plane runs from negative 14 to 14. The y-axis of the plane runs from negative 10 to 10. The hyperbola has a center at (negative 1, negative 1) and branches that pass through the vertices (negative 3, negative 1) and (1, negative 1), and that open left and right.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_355_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows a hyperbola graphed on the x y coordinate plane. The x-axis of the plane runs from negative 14 to 14. The y-axis of the plane runs from negative 10 to 10. The hyperbola has a center at (negative 1, negative 1) and branches that pass through the vertices (negative 3, negative 1) and (1, negative 1), and that open left and right.\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163870463544\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1163870463546\">\n<p id=\"fs-id1163870463548\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8a8da8781311a271a26a3a789f95a64e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#52;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#49;&#54;&#125;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"144\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163870411387\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1163870411389\">\n<p id=\"fs-id1163870551219\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2251388a462e5ff4a33e2c0638f2cbb0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#43;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#57;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#57;&#125;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"144\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163870413000\"><span data-type=\"media\" id=\"fs-id1163870413003\" data-alt=\"The figure shows a hyperbola graphed on the x y coordinate plane. The x-axis of the plane runs from negative 14 to 14. The y-axis of the plane runs from negative 10 to 10. The hyperbola has a center at (negative 1, negative 2) and branches that pass through the vertices (negative 1, 1) and (negative 1, negative 5), and that open up and down.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_357_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows a hyperbola graphed on the x y coordinate plane. The x-axis of the plane runs from negative 14 to 14. The y-axis of the plane runs from negative 10 to 10. The hyperbola has a center at (negative 1, negative 2) and branches that pass through the vertices (negative 1, 1) and (negative 1, negative 5), and that open up and down.\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163873837838\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1163873761350\">\n<p id=\"fs-id1163873761352\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-750ca9e861d3665d01696b61eebe5f18_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#50;&#53;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#57;&#125;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"144\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1163873784040\">In the following exercises, <span class=\"token\">\u24d0<\/span> write the equation in standard form and <span class=\"token\">\u24d1<\/span> graph.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1163870463202\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1163870463204\">\n<p id=\"fs-id1163870463206\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a8a7c6e3c88a089aeb42081c81018b12_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#54;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#43;&#56;&#120;&#43;&#57;&#54;&#121;&#45;&#50;&#48;&#52;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"253\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163873606412\">\n<p id=\"fs-id1163870644973\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4e513b6414bc3bb4d790a1164fb0fdca_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#49;&#54;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#52;&#125;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"144\" style=\"vertical-align: -7px;\" \/><span data-type=\"newline\"><br \/><\/span><span class=\"token\">\u24d1<\/span><span data-type=\"newline\"><br \/><\/span><\/p>\n<p><span data-type=\"media\" id=\"fs-id1163873784848\" data-alt=\"The figure shows a hyperbola graphed on the x y coordinate plane. The x-axis of the plane runs from negative 14 to 14. The y-axis of the plane runs from negative 10 to 10. The hyperbola has a center at (negative 1, 3) and branches that pass through the vertices (negative 5, 3) and (3, 3), and that open left and right.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_359_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows a hyperbola graphed on the x y coordinate plane. The x-axis of the plane runs from negative 14 to 14. The y-axis of the plane runs from negative 10 to 10. The hyperbola has a center at (negative 1, 3) and branches that pass through the vertices (negative 5, 3) and (3, 3), and that open left and right.\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163873641825\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1163873641827\">\n<p id=\"fs-id1163873641829\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7138d820dae3d70af19fafcd9f188659_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#54;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#54;&#52;&#120;&#45;&#50;&#52;&#121;&#45;&#51;&#54;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"252\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163873789523\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1163873744004\">\n<p id=\"fs-id1163873744006\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-13b4bf89fcb421091f60bced7c7b3628_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#54;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#51;&#50;&#120;&#45;&#56;&#121;&#45;&#55;&#54;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"245\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163873785159\">\n<p id=\"fs-id1163873785161\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6617264bf3b0ce60a962674f0818da95_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#49;&#54;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#52;&#125;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"144\" style=\"vertical-align: -7px;\" \/><span data-type=\"newline\"><br \/><\/span><span class=\"token\">\u24d1<\/span><span data-type=\"newline\"><br \/><\/span><\/p>\n<p><span data-type=\"media\" id=\"fs-id1163870504857\" data-alt=\"The figure shows a hyperbola graphed on the x y coordinate plane. The x-axis of the plane runs from negative 14 to 14. The y-axis of the plane runs from negative 10 to 10. The hyperbola has a center at (1, 1) and branches that pass through the vertices (1, negative 3) and (1, 5), and that open up and down.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_361_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows a hyperbola graphed on the x y coordinate plane. The x-axis of the plane runs from negative 14 to 14. The y-axis of the plane runs from negative 10 to 10. The hyperbola has a center at (1, 1) and branches that pass through the vertices (1, negative 3) and (1, 5), and that open up and down.\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163873866658\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1163873866660\">\n<p id=\"fs-id1163873866662\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c4eb17277866c655e009bf89c0705bc3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#54;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#54;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#57;&#54;&#120;&#43;&#50;&#49;&#54;&#121;&#45;&#51;&#57;&#54;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"280\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1163874000994\"><strong data-effect=\"bold\">Identify the Graph of each Equation as a Circle, Parabola, Ellipse, or Hyperbola<\/strong><\/p>\n<p id=\"fs-id1163870303496\">In the following exercises, identify the type of graph.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1163870303499\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1163870303501\">\n<p id=\"fs-id1163870303503\"><span data-type=\"newline\"><br \/><\/span><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-11d60db4f69475d56c97b749c334fb18_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#54;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#57;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#51;&#54;&#120;&#45;&#57;&#54;&#121;&#45;&#51;&#54;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"252\" style=\"vertical-align: -4px;\" \/><span data-type=\"newline\"><br \/><\/span><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bc73afbac658af3f7fb1ab28608a4ab9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#120;&#43;&#49;&#48;&#121;&#45;&#55;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"209\" style=\"vertical-align: -4px;\" \/><span data-type=\"newline\"><br \/><\/span><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d19b14a700a5dcb911d0d58348aab1c3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#120;&#43;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"122\" style=\"vertical-align: -4px;\" \/><span data-type=\"newline\"><br \/><\/span><span class=\"token\">\u24d3<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7573fbf621ceac3e135b7e5a7214d030_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#53;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#57;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#61;&#50;&#50;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"133\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163873863167\">\n<p id=\"fs-id1163873863169\"><span class=\"token\">\u24d0<\/span> hyperbola <span class=\"token\">\u24d1<\/span> circle <span class=\"token\">\u24d2<\/span> parabola <span class=\"token\">\u24d3<\/span> ellipse<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163873594904\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1163873594906\">\n<p id=\"fs-id1163870376191\"><span data-type=\"newline\"><br \/><\/span><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3602886ca6a247f8f01a217530da6809_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#43;&#52;&#120;&#45;&#49;&#48;&#121;&#43;&#50;&#53;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"218\" style=\"vertical-align: -4px;\" \/><span data-type=\"newline\"><br \/><\/span><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-16a7b816eaf07bb291c9fa1e46587627_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#121;&#43;&#50;&#120;&#45;&#54;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"200\" style=\"vertical-align: -4px;\" \/><span data-type=\"newline\"><br \/><\/span><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7bea449bf363882d50a5631e2d66ada1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#45;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#121;&#43;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"135\" style=\"vertical-align: -4px;\" \/><span data-type=\"newline\"><br \/><\/span><span class=\"token\">\u24d3<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3d2704801c5f75b76e0a63301dcdc330_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#54;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#57;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#61;&#49;&#52;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"133\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1163869201327\">\n<h4 data-type=\"title\"><a href=\"\/contents\/c2649a40-6525-4e7f-a3b4-31f5d8a52d29\" class=\"target-chapter\">Solve Systems of Nonlinear Equations<\/a><\/h4>\n<p id=\"fs-id1163873596258\"><strong data-effect=\"bold\">Solve a System of Nonlinear Equations Using Graphing<\/strong><\/p>\n<p id=\"fs-id1163869200663\">In the following exercises, solve the system of equations by using graphing.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1163869200666\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1163869200668\">\n<p id=\"fs-id1163869200670\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-53ab51c5f1831bc1414f7df501108680_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#51;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#121;&#61;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#121;&#61;&#50;&#120;&#45;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"110\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163873715314\"><span data-type=\"media\" id=\"fs-id1163873715317\" data-alt=\"The figure shows a parabola and line graphed on the x y coordinate plane. The x-axis of the plane runs from negative 5 to 5. The y-axis of the plane runs from negative 4 to 4. The parabola has a vertex at (0, 0) and opens upward. The line has a slope of 2 with a y-intercept at negative 1. The parabola and line do not intersect, so the system has no solution.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_363_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows a parabola and line graphed on the x y coordinate plane. The x-axis of the plane runs from negative 5 to 5. The y-axis of the plane runs from negative 4 to 4. The parabola has a vertex at (0, 0) and opens upward. The line has a slope of 2 with a y-intercept at negative 1. The parabola and line do not intersect, so the system has no solution.\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163873957473\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1163873957475\">\n<p id=\"fs-id1163873957478\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-001c14fc4a0b5ac9c5f4c6dca0fcc40d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#121;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#121;&#61;&#120;&#45;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"101\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163873674120\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1163873674122\">\n<p id=\"fs-id1163873674124\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e413bf91800120b9944ce9b8dc56a50e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#61;&#49;&#54;&#57;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#120;&#61;&#49;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"127\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163867240312\"><span data-type=\"media\" id=\"fs-id1163873559641\" data-alt=\"The figure shows a circle and line graphed on the x y coordinate plane. The x-axis of the plane runs from negative 20 to 20. The y-axis of the plane runs from negative 15 to 15. The circle has a center at (0, 0) and a radius of 13. The line is vertical. The circle and line intersect at the points (12, 5) and (12, negative 5), which are labeled. The solution of the system is (12, 5) and (12, negative 5)\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_365_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows a circle and line graphed on the x y coordinate plane. The x-axis of the plane runs from negative 20 to 20. The y-axis of the plane runs from negative 15 to 15. The circle has a center at (0, 0) and a radius of 13. The line is vertical. The circle and line intersect at the points (12, 5) and (12, negative 5), which are labeled. The solution of the system is (12, 5) and (12, negative 5)\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163873702909\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1163873702911\">\n<p id=\"fs-id1163873702913\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d5a3937049e62b3423e907f46b94058a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#61;&#50;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#121;&#61;&#45;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"117\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1163869197899\"><strong data-effect=\"bold\">Solve a System of Nonlinear Equations Using Substitution<\/strong><\/p>\n<p id=\"fs-id1163873666962\">In the following exercises, solve the system of equations by using substitution.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1163873666965\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1163873666967\">\n<p id=\"fs-id1163873666969\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fa06f07af16846221ee9710f623e1f56_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#121;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#121;&#61;&#45;&#50;&#120;&#43;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"115\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163873640694\">\n<p id=\"fs-id1163873640696\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b098ad691d3b1e66796376e6000e2385_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#44;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"51\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163873914601\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1163870629110\">\n<p id=\"fs-id1163870629112\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-95660194f20286e50b3e7d9d2996bd43_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#61;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#120;&#45;&#121;&#61;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"109\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163873745211\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1163873745213\">\n<p id=\"fs-id1163873745216\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-39dca0495e842444a5a949575d3df614_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#57;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#52;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#61;&#51;&#54;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#121;&#45;&#120;&#61;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"135\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163874017889\">\n<p id=\"fs-id1163874017891\">No solution<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163873715143\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1163873715145\">\n<p id=\"fs-id1163873715148\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cbbbc7d2dd6ed6a853a78b57a15ffad3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#52;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#61;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#50;&#120;&#45;&#121;&#61;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"117\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1163873912526\"><strong data-effect=\"bold\">Solve a System of Nonlinear Equations Using Elimination<\/strong><\/p>\n<p id=\"fs-id1163873912532\">In the following exercises, solve the system of equations by using elimination.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1163873912535\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1163873715345\">\n<p id=\"fs-id1163873715347\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-67e93e621984b90e5894d2294804b562_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#61;&#49;&#54;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#121;&#45;&#49;&#61;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"140\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163873776424\">\n<p id=\"fs-id1163873776426\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d3b9458541433e3e3ad10a4b320ee665_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#92;&#115;&#113;&#114;&#116;&#123;&#55;&#125;&#44;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#115;&#113;&#114;&#116;&#123;&#55;&#125;&#44;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"120\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163873644532\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1163873644534\">\n<p id=\"fs-id1163873714956\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ad01086e943e39f98de8ec41fab7590c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#61;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#45;&#50;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#51;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#61;&#45;&#51;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"163\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163873807235\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1163873807238\">\n<p id=\"fs-id1163873647430\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2c107504c73b797850ccf63a44797998_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#52;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#57;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#61;&#51;&#54;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#51;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#120;&#61;&#49;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"135\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163873822657\">\n<p id=\"fs-id1163873822660\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-27bf33c855a6a3ad27e22fd205320832_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"166\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163873686892\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1163873686894\">\n<p id=\"fs-id1163873686896\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-931ab439fc2bb6d4d139d7497ed4b11b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#61;&#49;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#61;&#49;&#54;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"118\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1163873634509\"><strong data-effect=\"bold\">Use a System of Nonlinear Equations to Solve Applications<\/strong><\/p>\n<p id=\"fs-id1163870491200\">In the following exercises, solve the problem using a system of equations.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1163870491203\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1163870491205\">\n<p id=\"fs-id1163870491207\">The sum of the squares of two numbers is 25. The difference of the numbers is 1. Find the numbers.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163870550952\">\n<p id=\"fs-id1163870550955\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-470cb162cf92c55d139f4f69216225e5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"22\" style=\"vertical-align: 0px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-00b9cce9021441b203ec0271d72e6ba2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"22\" style=\"vertical-align: -1px;\" \/> or 4 and 3<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163870557522\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1163870557525\">\n<p id=\"fs-id1163870557527\">The difference of the squares of two numbers is 45. The difference of the square of the first number and twice the square of the second number is 9. Find the numbers.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163873620872\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1163873620874\">\n<p id=\"fs-id1163873620876\">The perimeter of a rectangle is 58 meters and its area is 210 square meters. Find the length and width of the rectangle.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163873793305\">\n<p id=\"fs-id1163873793307\">If the length is 14 inches, the width is 15 inches. If the length is 15 inches, the width is 14 inches.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163873793313\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1163873793315\">\n<p id=\"fs-id1163873758176\">Colton purchased a larger microwave for his kitchen. The diagonal of the front of the microwave measures 34 inches. The front also has an area of 480 square inches. What are the length and width of the microwave?<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"practice-test\" data-depth=\"1\" id=\"fs-id1163873803334\">\n<h3 data-type=\"title\">Practice Test<\/h3>\n<p id=\"fs-id1163873759011\">In the following exercises, find the distance between the points and the midpoint of the line segment with the given endpoints. Round to the nearest tenth as needed.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1163873759015\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1163873759018\">\n<p id=\"fs-id1163873759020\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ebe785a1a1f173c2cb47dc17a15cdc63_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#52;&#44;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"65\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b1516f8ea9f287af67a1aad2116169b8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#48;&#44;&#45;&#49;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"83\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163873748881\">\n<p id=\"fs-id1163873748883\">distance: 10, midpoint: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8d61f9f7cc7c9bee7e196b06cac3edea_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#55;&#44;&#45;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"65\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163873764012\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1163873764014\">\n<p id=\"fs-id1163873764016\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6032429e4ed847fa0727a169e71d4eaa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#44;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-eb242707b07d2123762ae0b5253ad3ca_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#53;&#44;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"65\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1163873760568\">In the following exercises, write the standard form of the equation of the circle with the given information.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1163873760572\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1163873760574\">\n<p id=\"fs-id1163873760577\">radius is 11 and center is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c53627fd7039dcb62c54d86fe468e6e1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163873795836\">\n<p id=\"fs-id1163873795838\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7f3ae0f7e2c764004e3be2f110967d1d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#61;&#49;&#50;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"106\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163870454177\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1163870454179\">\n<p id=\"fs-id1163870454181\">radius is 12 and center is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-15407bbefb27e2b45cd1c1dd1bf45584_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#48;&#44;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"61\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163873862647\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1163873804357\">\n<p id=\"fs-id1163873804359\">center is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6fd7f677a681964debbd5fb9bbb3c944_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#44;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"51\" style=\"vertical-align: -4px;\" \/> and a point on the circle is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1564d1d2328bb6bd9e7b30e6d573d2fb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#44;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"52\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163873677840\">\n<p id=\"fs-id1163873609057\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-15770b0b58d824414119bbc2c9d0cfb4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#61;&#53;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"185\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163873807699\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1163873807701\">\n<p id=\"fs-id1163870461912\">Find the equation of the ellipse shown in the graph.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1163870461916\" data-alt=\"The figure shows an ellipse graphed on the x y coordinate plane. The ellipse has a center at (0, 0), a vertical major axis, vertices at (0, plus or minus 10), and co-vertices at (plus or minus 6, 0).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_207_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows an ellipse graphed on the x y coordinate plane. The ellipse has a center at (0, 0), a vertical major axis, vertices at (0, plus or minus 10), and co-vertices at (plus or minus 6, 0).\" \/><\/span><\/div>\n<\/div>\n<p id=\"fs-id1163873749967\">In the following exercises, <span class=\"token\">\u24d0<\/span> identify the type of graph of each equation as a circle, parabola, ellipse, or hyperbola, and <span class=\"token\">\u24d1<\/span> graph the equation.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1163873703072\">\n<div data-type=\"problem\" id=\"fs-id1163873703075\">\n<p id=\"fs-id1163873703077\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e1cde8c200053344d7962f5fe0c445ea_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#52;&#57;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#61;&#49;&#57;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"134\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163873608706\">\n<p id=\"fs-id1163869098961\"><span class=\"token\">\u24d0<\/span> ellipse<span data-type=\"newline\"><br \/><\/span> <span class=\"token\">\u24d1<\/span><span data-type=\"newline\"><br \/><\/span><\/p>\n<p><span data-type=\"media\" id=\"fs-id1163873794720\" data-alt=\"The figure shows an ellipse graphed on the x y coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 8 to 8. The ellipse has a center at (0, 0), a horizontal major axis, vertices at (plus or minus 7, 0) and co-vertices at (0, plus or minus 2).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_367_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows an ellipse graphed on the x y coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 8 to 8. The ellipse has a center at (0, 0), a horizontal major axis, vertices at (plus or minus 7, 0) and co-vertices at (0, plus or minus 2).\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163870488379\">\n<div data-type=\"problem\" id=\"fs-id1163870488381\">\n<p id=\"fs-id1163870488383\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a6cd2ea41c4163e3a59582c0b2b7cedf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#51;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#45;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"134\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163873946136\">\n<div data-type=\"problem\" id=\"fs-id1163873946138\">\n<p id=\"fs-id1163873782530\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d530ba5aa9d5846eedffe1afd9066d11_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#51;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#61;&#50;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"116\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163873869362\">\n<p id=\"fs-id1163873869365\"><span class=\"token\">\u24d0<\/span> circle<span data-type=\"newline\"><br \/><\/span> <span class=\"token\">\u24d1<\/span><span data-type=\"newline\"><br \/><\/span><\/p>\n<p><span data-type=\"media\" id=\"fs-id1163873799871\" data-alt=\"The figure shows a circle graphed on the x y coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 8 to 8. The parabola circle has a center at (0, 0) and a radius of 3.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_369_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows a circle graphed on the x y coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 8 to 8. The parabola circle has a center at (0, 0) and a radius of 3.\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163873822380\">\n<div data-type=\"problem\" id=\"fs-id1163873822382\">\n<p id=\"fs-id1163873822384\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-24d8687fb094e38870ae0731c0b8df06_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#49;&#48;&#48;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#51;&#54;&#125;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"94\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163873763888\">\n<div data-type=\"problem\" id=\"fs-id1163873763890\">\n<p id=\"fs-id1163873763892\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-92d5aac20c5f17dfa09e7ea058cec16a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#49;&#54;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#56;&#49;&#125;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"88\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163870462864\">\n<p id=\"fs-id1163870462866\"><span class=\"token\">\u24d0<\/span> ellipse<span data-type=\"newline\"><br \/><\/span> <span class=\"token\">\u24d1<\/span><span data-type=\"newline\"><br \/><\/span><\/p>\n<p><span data-type=\"media\" id=\"fs-id1163873820914\" data-alt=\"The figure shows an ellipse graphed on the x y coordinate plane. The x-axis of the plane runs from negative 14 to 14. The y-axis of the plane runs from negative 10 to 10. The ellipse has a center at (0, 0), a vertical major axis, vertices at (0, plus or minus 9) and co-vertices at (plus or minus 4, 0).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_371_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows an ellipse graphed on the x y coordinate plane. The x-axis of the plane runs from negative 14 to 14. The y-axis of the plane runs from negative 10 to 10. The ellipse has a center at (0, 0), a vertical major axis, vertices at (0, plus or minus 9) and co-vertices at (plus or minus 4, 0).\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163869585713\">\n<div data-type=\"problem\" id=\"fs-id1163869585715\">\n<p id=\"fs-id1163869585718\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-254a508101587bcffa7568fb09992acf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#50;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#48;&#121;&#43;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"139\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163873786999\">\n<div data-type=\"problem\" id=\"fs-id1163873606436\">\n<p id=\"fs-id1163873606438\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d07b4b0f09697c2851d17f4705e6aebc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#52;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#57;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#61;&#53;&#55;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"134\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163873655737\">\n<p id=\"fs-id1163873655739\"><span class=\"token\">\u24d0<\/span> hyperbola<span data-type=\"newline\"><br \/><\/span> <span class=\"token\">\u24d1<\/span><span data-type=\"newline\"><br \/><\/span><\/p>\n<p><span data-type=\"media\" id=\"fs-id1163873764181\" data-alt=\"The figure shows a hyperbola graphed on the x y coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 8 to 8. The hyperbola has a center at (0, 0) and branches that pass through the vertices (plus or minus 3, 0) and that open left and right.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_373_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows a hyperbola graphed on the x y coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 8 to 8. The hyperbola has a center at (0, 0) and branches that pass through the vertices (plus or minus 3, 0) and that open left and right.\" \/><\/span><\/div>\n<\/div>\n<p id=\"fs-id1163873807731\">In the following exercises, <span class=\"token\">\u24d0<\/span> identify the type of graph of each equation as a circle, parabola, ellipse, or hyperbola, <span class=\"token\">\u24d1<\/span> write the equation in standard form, and <span class=\"token\">\u24d2<\/span> graph the equation.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1163870462179\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1163873675733\">\n<p id=\"fs-id1163873675735\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9273584ed060c013402bb29bbbccc8ba_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#53;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#54;&#52;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#48;&#48;&#120;&#45;&#50;&#53;&#54;&#121;&#45;&#57;&#52;&#52;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"289\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163867247197\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1163873715039\">\n<p id=\"fs-id1163873715042\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-57834271609ed04eade02417cfb457b4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#48;&#120;&#43;&#54;&#121;&#43;&#51;&#48;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"218\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163873896962\">\n<p id=\"fs-id1163873896964\"><span class=\"token\">\u24d0<\/span> circle<span data-type=\"newline\"><br \/><\/span> <span class=\"token\">\u24d1<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-49f68a5e913d741e208beaf683e21d45_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#61;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"177\" style=\"vertical-align: -4px;\" \/><span data-type=\"newline\"><br \/><\/span> <span class=\"token\">\u24d2<\/span><span data-type=\"newline\"><br \/><\/span><\/p>\n<p><span data-type=\"media\" id=\"fs-id1163874043629\" data-alt=\"The figure shows a circle graphed on the x y coordinate plane. The x-axis of the plane runs from negative 14 to 14. The y-axis of the plane runs from negative 10 to 10. The circle has a center at (negative 5, negative 3) and a radius 2.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_375_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows a circle graphed on the x y coordinate plane. The x-axis of the plane runs from negative 14 to 14. The y-axis of the plane runs from negative 10 to 10. The circle has a center at (negative 5, negative 3) and a radius 2.\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163870550229\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1163870550231\">\n<p id=\"fs-id1163870547415\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7ab657cb66ae41b34fe64bf22e518e99_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#121;&#45;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"121\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163873864767\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1163873864768\">\n<p id=\"fs-id1163873657428\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8ae25f49aafc67523fa11c87c5d83374_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#57;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#53;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#51;&#54;&#120;&#45;&#53;&#48;&#121;&#45;&#50;&#49;&#52;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"262\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163873791817\">\n<p id=\"fs-id1163873791819\"><span class=\"token\">\u24d0<\/span> hyperbola<span data-type=\"newline\"><br \/><\/span> <span class=\"token\">\u24d1<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0c960d47d03dbe91667616822e931e16_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#50;&#53;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#57;&#125;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"144\" style=\"vertical-align: -6px;\" \/><span data-type=\"newline\"><br \/><\/span> <span class=\"token\">\u24d2<\/span><span data-type=\"newline\"><br \/><\/span><\/p>\n<p><span data-type=\"media\" id=\"fs-id1163873850471\" data-alt=\"The figure shows a hyperbola graphed on the x y coordinate plane. The x-axis of the plane runs from negative 14 to 14. The y-axis of the plane runs from negative 10 to 10. The hyperbola has a center at (2, negative 1) and branches that pass through the vertices (negative 3, negative 1) and (7, negative 1) that open left and right.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_377_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows a hyperbola graphed on the x y coordinate plane. The x-axis of the plane runs from negative 14 to 14. The y-axis of the plane runs from negative 10 to 10. The hyperbola has a center at (2, negative 1) and branches that pass through the vertices (negative 3, negative 1) and (7, negative 1) that open left and right.\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163870219064\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1163870219067\">\n<p id=\"fs-id1163870219069\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8b68147f1d47d155737aac298dd37be8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#54;&#120;&#43;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"122\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163873799945\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1163873799948\">\n<p id=\"fs-id1163870497826\">Solve the nonlinear system of equations by graphing:<span data-type=\"newline\"><br \/><\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9c2f552069697a11a7eb2357a5b10afa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#51;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#120;&#61;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#121;&#61;&#45;&#50;&#120;&#45;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"128\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163873796407\">\n<p id=\"fs-id1163873796409\">No solution<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163873796414\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1163873750851\">\n<p id=\"fs-id1163873750853\">Solve the nonlinear system of equations using substitution:<span data-type=\"newline\"><br \/><\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-dc5ee69a011569671c66e619df50ffea_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#61;&#56;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#121;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#120;&#45;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"120\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163873782385\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1163873782388\">\n<p id=\"fs-id1163873782390\">Solve the nonlinear system of equations using elimination:<span data-type=\"newline\"><br \/><\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9437a6c10a4d15ca0a7599c7f0cd1fd8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#57;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#61;&#57;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#50;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#57;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#61;&#49;&#56;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"147\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163873798322\">\n<p id=\"fs-id1163873798324\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-256a758deeb6e9eb298e52bdebcbbbb6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"102\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163873788823\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1163873626844\">\n<p id=\"fs-id1163873626846\">Create the equation of the parabolic arch formed in the foundation of the bridge shown. Give the answer in <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-af4106dc6abca5e766ae7ae01481411c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#97;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#98;&#120;&#43;&#99;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"129\" style=\"vertical-align: -4px;\" \/> form.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1163873675193\" data-alt=\"The figure shows a parabolic arch formed in the foundation of the bridge. The arch is 10 feet high and 30 feet wide.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_208_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows a parabolic arch formed in the foundation of the bridge. The arch is 10 feet high and 30 feet wide.\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163870620577\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1163870620579\">\n<p id=\"fs-id1163870620581\">A comet moves in an elliptical orbit around a sun. The closest the comet gets to the sun is approximately 20 AU and the furthest is approximately 70 AU. The sun is one of the foci of the elliptical orbit. Letting the ellipse center at the origin and labeling the axes in AU, the orbit will look like the figure below. Use the graph to write an equation for the elliptical orbit of the comet.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1163873914128\" data-alt=\"The figure shows a model of an elliptical orbit around the sun on the x y coordinate plane. The ellipse has a center at (0, 0), a horizontal major axis, vertices marked at (plus or minus 45, 0), the sun marked as a foci and labeled (25, 0), the closest distance the comet is from the sun marked as 20 A U, and the farthest a comet is from the sun marked as 70 A U.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_05_209_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows a model of an elliptical orbit around the sun on the x y coordinate plane. The ellipse has a center at (0, 0), a horizontal major axis, vertices marked at (plus or minus 45, 0), the sun marked as a foci and labeled (25, 0), the closest distance the comet is from the sun marked as 20 A U, and the farthest a comet is from the sun marked as 70 A U.\" \/><\/span><\/div>\n<div data-type=\"solution\" id=\"fs-id1163870228951\">\n<p id=\"fs-id1163870228953\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-17e7fa2101fda258d4c113b43be1249b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#50;&#48;&#50;&#53;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#49;&#52;&#48;&#48;&#125;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"114\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163873821558\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1163873821561\">\n<p id=\"fs-id1163870170382\">The sum of two numbers is 22 and the product is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-888ff981d3decbd5f8a8cd49d67e1037_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#50;&#52;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"44\" style=\"vertical-align: -1px;\" \/> Find the numbers.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163873587977\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1163873783582\">\n<p id=\"fs-id1163873783584\">For her birthday, Olive\u2019s grandparents bought her a new widescreen TV. Before opening it she wants to make sure it will fit her entertainment center. The TV is 55\u201d. The size of a TV is measured on the diagonal of the screen and a widescreen has a length that is larger than the width. The screen also has an area of 1452 square inches. Her entertainment center has an insert for the TV with a length of 50 inches and width of 40 inches. What are the length and width of the TV screen and will it fit Olive\u2019s entertainment center?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163873783586\">\n<p id=\"fs-id1163873783588\">The length is 44 inches and the width is 33 inches. The TV will fit Olive\u2019s entertainment center.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"glossary\" class=\"textbox shaded\">\n<h3 data-type=\"glossary-title\">Glossary<\/h3>\n<dl id=\"fs-id1163870346283\">\n<dt>system of nonlinear equations<\/dt>\n<dd id=\"fs-id1163870346288\">A system of nonlinear equations is a system where at least one of the equations is not linear.<\/dd>\n<\/dl>\n<\/div>\n","protected":false},"author":103,"menu_order":6,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-15661","chapter","type-chapter","status-publish","hentry"],"part":15253,"_links":{"self":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters\/15661","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/users\/103"}],"version-history":[{"count":0,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters\/15661\/revisions"}],"part":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/parts\/15253"}],"metadata":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters\/15661\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/media?parent=15661"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapter-type?post=15661"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/contributor?post=15661"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/license?post=15661"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}