{"id":2373,"date":"2018-12-11T13:41:12","date_gmt":"2018-12-11T18:41:12","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/chapter\/solve-systems-of-linear-equations-with-two-variables\/"},"modified":"2018-12-11T13:41:12","modified_gmt":"2018-12-11T18:41:12","slug":"solve-systems-of-linear-equations-with-two-variables","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/chapter\/solve-systems-of-linear-equations-with-two-variables\/","title":{"raw":"Solve Systems of Linear Equations with Two Variables","rendered":"Solve Systems of Linear Equations with Two Variables"},"content":{"raw":"\n[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>Determine whether an ordered pair is a solution of a system of equations<\/li><li>Solve a system of linear equations by graphing<\/li><li>Solve a system of equations by substitution<\/li><li>Solve a system of equations by elimination<\/li><li>Choose the most convenient method to solve a system of linear equations<\/li><\/ul><\/div><div data-type=\"note\" id=\"fs-id1167830925402\" class=\"be-prepared\"><p id=\"fs-id1167835305070\">Before you get started, take this readiness quiz.<\/p><ol id=\"fs-id1167831116154\" type=\"1\"><li>For the equation \\(y=\\frac{2}{3}x-4,\\)<div data-type=\"newline\"><br><\/div> <span class=\"token\">\u24d0<\/span> Is \\(\\left(6,0\\right)\\) a solution? <span class=\"token\">\u24d1<\/span> Is \\(\\left(-3,-2\\right)\\) a solution?<div data-type=\"newline\"><br><\/div> If you missed this problem, review <a href=\"\/contents\/26e8f94c-1f76-46ec-8e6c-344f06971cf5#fs-id1167835400321\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li><li>Find the slope and <em data-effect=\"italics\">y<\/em>-intercept of the line \\(3x-y=12.\\)<div data-type=\"newline\"><br><\/div> If you missed this problem, review <a href=\"\/contents\/c7953cb6-51e3-48e7-9969-821f34daec42#fs-id1167835342973\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li><li>Find the <em data-effect=\"italics\">x-<\/em> and <em data-effect=\"italics\">y<\/em>-intercepts of the line \\(2x-3y=12.\\)<div data-type=\"newline\"><br><\/div> If you missed this problem, review <a href=\"\/contents\/26e8f94c-1f76-46ec-8e6c-344f06971cf5#fs-id1167827987818\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li><\/ol><\/div><div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167835596566\"><h3 data-type=\"title\">Determine Whether an Ordered Pair is a Solution of a System of Equations<\/h3><p>In <a href=\"\/contents\/85b55407-c981-455f-9c47-d72340bd1dbb\" class=\"target-chapter\">Solving Linear Equations<\/a>, we learned how to solve linear equations with one variable. Now we will work with two or more linear equations grouped together, which is known as a <span data-type=\"term\">system of linear equations<\/span>.<\/p><div data-type=\"note\"><div data-type=\"title\">System of Linear Equations<\/div><p id=\"fs-id1167834506071\">When two or more linear equations are grouped together, they form a <strong data-effect=\"bold\">system of linear equations<\/strong>.<\/p><\/div><p>In this section, we will focus our work on systems of two linear equations in two unknowns. We will solve larger systems of equations later in this chapter.<\/p><p id=\"fs-id1167831040311\">An example of a system of two linear equations is shown below. We use a brace to show the two equations are grouped together to form a system of equations.<\/p><div data-type=\"equation\" class=\"unnumbered\" data-label=\"\">\\(\\left\\{\\begin{array}{c}2x+y=7\\hfill \\\\ x-2y=6\\hfill \\end{array}\\)<\/div><p id=\"fs-id1167835513953\">A linear equation in two variables, such as \\(2x+y=7,\\) has an infinite number of solutions. Its graph is a line. Remember, every point on the line is a solution to the equation and every solution to the equation is a point on the line.<\/p><p id=\"fs-id1167835301937\">To solve a system of two linear equations, we want to find the values of the variables that are solutions to <em data-effect=\"italics\">both<\/em> equations. In other words, we are looking for the ordered pairs \\(\\left(x,y\\right)\\) that make both equations true. These are called the <span data-type=\"term\">solutions of a system of equations<\/span>.<\/p><div data-type=\"note\" id=\"fs-id1167834063240\"><div data-type=\"title\">Solutions of a System of Equations<\/div><p id=\"fs-id1167831884388\">The <strong data-effect=\"bold\">solutions of a system of equations<\/strong> are the values of the variables that make <em data-effect=\"italics\">all<\/em> the equations true. A solution of a system of two linear equations is represented by an ordered pair \\(\\left(x,y\\right).\\)<\/p><\/div><p id=\"fs-id1167835167507\">To determine if an ordered pair is a solution to a system of two equations, we substitute the values of the variables into each equation. If the ordered pair makes both equations true, it is a solution to the system.<\/p><div data-type=\"example\" id=\"fs-id1167835326515\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167835360973\"><div data-type=\"problem\" id=\"fs-id1167832056984\"><p id=\"fs-id1167834190204\">Determine whether the ordered pair is a solution to the system \\(\\left\\{\\begin{array}{c}x-y=-1\\hfill \\\\ 2x-y=-5\\hfill \\end{array}.\\)<\/p><p id=\"fs-id1167835483822\"><span class=\"token\">\u24d0<\/span>\\(\\left(-2,-1\\right)\\)<span class=\"token\">\u24d1<\/span>\\(\\left(-4,-3\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834397737\"><p id=\"fs-id1167835330326\"><span class=\"token\">\u24d0<\/span><\/p><div data-type=\"newline\"><br><\/div><span data-type=\"media\" id=\"fs-id1167834135019\" data-alt=\"The equations are x minus y equals minus 1 and 2 x minus y equals minus 5. We substitute x equal to minus 2 and y equal to minus 1 into both equations. So, x minus y equals minus 1 becomes minus 2 minus open parentheses minus 1 close parentheses equal to or not equal to minus 1. Simplifying, we get minus 1 equals minus 1 which is correct. The equation 2 x minus y equals minus 5 becomes 2 times minus 2 minus open parentheses minus 1 close parentheses equal to or not equal to minus 5. Simplifying, we get 5 not equal to minus 5. Hence, the ordered pair minus 2, minus 1 does not make both equations true. So, it is not a solution.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_01_001_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The equations are x minus y equals minus 1 and 2 x minus y equals minus 5. We substitute x equal to minus 2 and y equal to minus 1 into both equations. So, x minus y equals minus 1 becomes minus 2 minus open parentheses minus 1 close parentheses equal to or not equal to minus 1. Simplifying, we get minus 1 equals minus 1 which is correct. The equation 2 x minus y equals minus 5 becomes 2 times minus 2 minus open parentheses minus 1 close parentheses equal to or not equal to minus 5. Simplifying, we get 5 not equal to minus 5. Hence, the ordered pair minus 2, minus 1 does not make both equations true. So, it is not a solution.\"><\/span><p id=\"fs-id1167831824467\"><span class=\"token\">\u24d1<\/span><\/p><div data-type=\"newline\"><br><\/div><span data-type=\"media\" data-alt=\"We substitute x equal to minus 4 and y equal to minus 3 into both equations. So, x minus y equals minus 1 becomes minus 4 minus open parentheses minus 3 close parentheses equal to or not equal to minus 1. Simplifying, we get minus 1 equals minus 1, which is correct. The equation 2 x minus y equals minus 5 becomes 2 times minus 4 minus open parentheses minus 3 close parentheses equal to or not equal to minus 5. Simplifying, we get minus 5 equals minus 5, which is correct. The ordered pair minus 4, minus 3 does make both equations true. Hence, it is a solution.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_01_002_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"We substitute x equal to minus 4 and y equal to minus 3 into both equations. So, x minus y equals minus 1 becomes minus 4 minus open parentheses minus 3 close parentheses equal to or not equal to minus 1. Simplifying, we get minus 1 equals minus 1, which is correct. The equation 2 x minus y equals minus 5 becomes 2 times minus 4 minus open parentheses minus 3 close parentheses equal to or not equal to minus 5. Simplifying, we get minus 5 equals minus 5, which is correct. The ordered pair minus 4, minus 3 does make both equations true. Hence, it is a solution.\"><\/span><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167832066187\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167834346678\"><div data-type=\"problem\" id=\"fs-id1167832056851\"><p>Determine whether the ordered pair is a solution to the system \\(\\begin{array}{c}\\left\\{\\begin{array}{c}3x+y=0\\hfill \\\\ x+2y=-5\\hfill \\end{array}.\\hfill \\\\ \\hfill \\end{array}\\)<\/p><p><span class=\"token\">\u24d0<\/span>\\(\\left(1,-3\\right)\\)<span class=\"token\">\u24d1<\/span>\\(\\left(0,0\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835253906\"><p id=\"fs-id1167835287970\"><span class=\"token\">\u24d0<\/span> yes <span class=\"token\">\u24d1<\/span> no<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167834132168\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167835331532\"><div data-type=\"problem\"><p id=\"fs-id1167832052650\">Determine whether the ordered pair is a solution to the system \\(\\begin{array}{c}\\left\\{\\begin{array}{c}x-3y=-8\\hfill \\\\ -3x-y=4\\hfill \\end{array}.\\hfill \\\\ \\hfill \\end{array}\\)<\/p><p id=\"fs-id1167831921117\"><span class=\"token\">\u24d0<\/span>\\(\\left(2,-2\\right)\\)<span class=\"token\">\u24d1<\/span>\\(\\left(-2,2\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835545439\"><p id=\"fs-id1167826967139\"><span class=\"token\">\u24d0<\/span> no <span class=\"token\">\u24d1<\/span> yes<\/p><\/div><\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167832086919\"><h3 data-type=\"title\">Solve a System of Linear Equations by Graphing<\/h3><p id=\"fs-id1167831893670\">In this section, we will use three methods to solve a system of linear equations. The first method we\u2019ll use is graphing.<\/p><p id=\"fs-id1167831239781\">The graph of a linear equation is a line. Each point on the line is a solution to the equation. For a system of two equations, we will graph two lines. Then we can see all the points that are solutions to each equation. And, by finding what the lines have in common, we\u2019ll find the solution to the system.<\/p><p id=\"fs-id1167835267322\">Most linear equations in one variable have one solution, but we saw that some equations, called contradictions, have no solutions and for other equations, called identities, all numbers are solutions.<\/p><p id=\"fs-id1167832082005\">Similarly, when we solve a system of two linear equations represented by a graph of two lines in the same plane, there are three possible cases, as shown.<\/p><div class=\"bc-figure figure\" id=\"CNX_IntAlg_Figure_04_01_003\"><span data-type=\"media\" id=\"fs-id1167834188679\" data-alt=\"Figure shows three graphs. In the first, the lines intersect at point 3, minus 1. The intersecting lines have one point in common. There is one solution to the system. In the second graph, the lines are parallel. Parallel lines have no points in common. There is no solution to the system. The third graph has only one line. Here, both equations give the same line. Because we have only one line, there are infinite many solutions.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_01_003_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Figure shows three graphs. In the first, the lines intersect at point 3, minus 1. The intersecting lines have one point in common. There is one solution to the system. In the second graph, the lines are parallel. Parallel lines have no points in common. There is no solution to the system. The third graph has only one line. Here, both equations give the same line. Because we have only one line, there are infinite many solutions.\"><\/span><\/div><p id=\"fs-id1167834053646\">Each time we demonstrate a new method, we will use it on the same system of linear equations. At the end of the section you\u2019ll decide which method was the most convenient way to solve this system.<\/p><div data-type=\"example\" id=\"fs-id1167834279490\" class=\"textbox textbox--examples\"><div data-type=\"title\">How to Solve a System of Equations by Graphing<\/div><div data-type=\"exercise\"><div data-type=\"problem\" id=\"fs-id1167832057850\"><p id=\"fs-id1167835280797\">Solve the system by graphing \\(\\left\\{\\begin{array}{c}2x+y=7\\hfill \\\\ x-2y=6\\hfill \\end{array}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167832041475\"><span data-type=\"media\" data-alt=\"Step 1 is to graph the first equation. To graph the first line, write the equation in slope intercept form. So, 2 x plus y equals 7 becomes y equal to minus 2 x plus 7. Here, m is minus 2 and b is 7. So the graph will be a line with slope equal to minus 2 and y intercept equal to 7.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_01_004a_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 1 is to graph the first equation. To graph the first line, write the equation in slope intercept form. So, 2 x plus y equals 7 becomes y equal to minus 2 x plus 7. Here, m is minus 2 and b is 7. So the graph will be a line with slope equal to minus 2 and y intercept equal to 7.\"><\/span><span data-type=\"media\" id=\"fs-id1167835174193\" data-alt=\"Step 2 is to graph the second equation on the same rectangular coordinate system. To graph the second line, use intercepts. For x minus 2y equals 6, the intercepts are 0, minus 3 and 6, 0.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_01_004b_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 2 is to graph the second equation on the same rectangular coordinate system. To graph the second line, use intercepts. For x minus 2y equals 6, the intercepts are 0, minus 3 and 6, 0.\"><\/span><span data-type=\"media\" id=\"fs-id1167835321614\" data-alt=\"Step 3 is to determine whether the lines intersect, are parallel, or are the same line. Here, they intersect.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_01_004c_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 3 is to determine whether the lines intersect, are parallel, or are the same line. Here, they intersect.\"><\/span><span data-type=\"media\" id=\"fs-id1167832138637\" data-alt=\"Step 4 is to identify the solution to the system. If the lines intersect, identify the point of intersection. The lines intersect at 4, minus 1. Now, check to make sure it is a solution to both equations. When x and y are substituted with 4 and minus 1 respectively, both equations hold true. This is the solution to the system. In step 4, if the lines are parallel, the system has no solution and if the lines are the same, the system has an infinite number of solutions.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_01_004d_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 4 is to identify the solution to the system. If the lines intersect, identify the point of intersection. The lines intersect at 4, minus 1. Now, check to make sure it is a solution to both equations. When x and y are substituted with 4 and minus 1 respectively, both equations hold true. This is the solution to the system. In step 4, if the lines are parallel, the system has no solution and if the lines are the same, the system has an infinite number of solutions.\"><\/span><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167832195749\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167835367659\"><div data-type=\"problem\" id=\"fs-id1167826864846\"><p id=\"fs-id1167834233816\">Solve the system by graphing: \\(\\left\\{\\begin{array}{c}x-3y=-3\\hfill \\\\ x+y=5\\hfill \\end{array}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167826799434\"><p id=\"fs-id1167828401865\">\\(\\left(3,2\\right)\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167832065745\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167834289447\"><div data-type=\"problem\" id=\"fs-id1167835368946\"><p id=\"fs-id1167831823922\">Solve the system by graphing: \\(\\left\\{\\begin{array}{c}-x+y=1\\hfill \\\\ 3x+2y=12\\hfill \\end{array}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835350532\"><p id=\"fs-id1167835213780\">\\(\\left(2,3\\right)\\)<\/p><\/div><\/div><\/div><p id=\"fs-id1167826804684\">The steps to use to solve a system of linear equations by graphing are shown here.<\/p><div data-type=\"note\" id=\"fs-id1167835310198\" class=\"howto\"><div data-type=\"title\">Solve a system of linear equations by graphing.<\/div><ol type=\"1\" class=\"stepwise\"><li>Graph the first equation.<\/li><li>Graph the second equation on the same rectangular coordinate system.<\/li><li>Determine whether the lines intersect, are parallel, or are the same line.<\/li><li>Identify the solution to the system. <ul class=\"open-circle\"><li>If the lines intersect, identify the point of intersection. This is the solution to the system.<\/li><li>If the lines are parallel, the system has no solution.<\/li><li>If the lines are the same, the system has an infinite number of solutions.<\/li><\/ul><\/li><li>Check the solution in both equations.<\/li><\/ol><\/div><p id=\"fs-id1167832055393\">In the next example, we\u2019ll first re-write the equations into slope\u2013intercept form as this will make it easy for us to quickly graph the lines.<\/p><div data-type=\"example\" id=\"fs-id1167835410272\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167834185475\"><div data-type=\"problem\" id=\"fs-id1167830706036\"><p id=\"fs-id1167831879885\">Solve the system by graphing: \\(\\left\\{\\begin{array}{c}3x+y=-1\\hfill \\\\ 2x+y=0\\hfill \\end{array}.\\) <\/p><\/div><div data-type=\"solution\" id=\"fs-id1167832042531\"><p id=\"fs-id1167834065071\">We\u2019ll solve both of these equations for \\(y\\) so that we can easily graph them using their slopes and <em data-effect=\"italics\">y<\/em>-intercepts.<\/p><table id=\"fs-id1167826993913\" class=\"unnumbered unstyled can-break\" summary=\"The equations are 3 x plus y equals minus 1 and 2 x plus y equals 0. Solving the first equation for y, we get y equal to minus 3 x minus 1. So, slope is minus 3 and y intercept is minus 1. Similarly, for the second equation, we get y equal to minus 2 x. So, slope is minus 2 and y intercept is 0. Graphing the lines, we get the point of intersection minus 1, 2. To check the solution in both equations, we substitute minus 1 and 2 for x and y respectively. Both equations hold true. The solution is minus 1, 2.\" data-label=\"\"><tbody><tr><td><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834161795\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_01_005b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Solve the first equation for <em data-effect=\"italics\">y<\/em>.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167826804609\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_01_005c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Find the slope and <em data-effect=\"italics\">y<\/em>-intercept.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835327264\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_01_005d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Solve the second equation for <em data-effect=\"italics\">y<\/em>.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834224888\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_01_005e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Find the slope and <em data-effect=\"italics\">y<\/em>-intercept.<\/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\/2018\/12\/CNX_IntAlg_Figure_04_01_005f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Graph the lines.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835422879\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_01_005g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Determine the point of intersection.<\/td><td data-valign=\"top\" data-align=\"left\">The lines intersect at \\(\\left(-1,2\\right).\\)<\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Check the solution in both equations.<div data-type=\"newline\"><br><\/div><div data-type=\"newline\"><br><\/div><span data-type=\"media\" id=\"fs-id1167835369250\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_01_005a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><td><\/td><\/tr><tr><td><\/td><td data-valign=\"top\" data-align=\"left\">The solution is \\(\\left(-1,2\\right).\\)<\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167832086965\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167835639953\"><div data-type=\"problem\" id=\"fs-id1167835332201\"><p id=\"fs-id1167835329488\">Solve the system by graphing: \\(\\left\\{\\begin{array}{c}-x+y=1\\hfill \\\\ 2x+y=10\\hfill \\end{array}.\\)<\/p><\/div><div data-type=\"solution\"><p id=\"fs-id1167834459142\">\\(\\left(3,4\\right)\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167835366362\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167832056565\"><div data-type=\"problem\" id=\"fs-id1167834301215\"><p id=\"fs-id1167834185253\">Solve the system by graphing: \\(\\left\\{\\begin{array}{c}2x+y=6\\hfill \\\\ x+y=1\\hfill \\end{array}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167832058926\"><p id=\"fs-id1167835262504\">\\(\\left(5,-4\\right)\\)<\/p><\/div><\/div><\/div><p id=\"fs-id1167834462906\">In all the systems of linear equations so far, the lines intersected and the solution was one point. In the next two examples, we\u2019ll look at a system of equations that has no solution and at a system of equations that has an infinite number of solutions.<\/p><div data-type=\"example\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167835609626\"><div data-type=\"problem\" id=\"fs-id1167834222410\"><p id=\"fs-id1167835171687\">Solve the system by graphing: \\(\\left\\{\\begin{array}{c}y=\\frac{1}{2}x-3\\hfill \\\\ x-2y=4\\hfill \\end{array}.\\) <\/p><\/div><div data-type=\"solution\" id=\"fs-id1167827943084\"><table id=\"fs-id1167834537193\" class=\"unnumbered unstyled can-break\" summary=\"The equations are y equal to half x minus 3 and x minus 2y equal to 4. The slope and y intercept of the first equation are half and minus 3 respectively. To graph the second equation, we will use the intercepts x equal to 0 when y equal to minus 2 and x equal to 4 when y equal to 0. We graph the lines to determine the point of intersection. The lines are parallel. Since no point is on both lines, there is no ordered pair that makes both equations true. There is no solution to this system.\" data-label=\"\"><tbody><tr><td><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835340829\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_01_006a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">To graph the first equation, we will use its<div data-type=\"newline\"><br><\/div>slope and <em data-effect=\"italics\">y<\/em>-intercept.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834195984\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_01_006b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td>To graph the second equation, we will use<div data-type=\"newline\"><br><\/div>the intercepts.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835365745\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_01_006c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834309752\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_01_006d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Graph the lines.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835595311\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_01_006e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Determine the points of intersection.<\/td><td data-valign=\"top\" data-align=\"left\">The lines are parallel.<div data-type=\"newline\"><br><\/div> Since no point is on both lines, there is no<div data-type=\"newline\"><br><\/div>ordered pair that makes both equations<div data-type=\"newline\"><br><\/div>true. There is no solution to this system.<\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167835329190\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167835354645\"><div data-type=\"problem\" id=\"fs-id1167830702759\"><p id=\"fs-id1167835341562\">Solve the system by graphing: \\(\\left\\{\\begin{array}{c}y=-\\frac{1}{4}x+2\\hfill \\\\ x+4y=-8\\hfill \\end{array}.\\) <\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835262981\"><p>no solution<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167835364524\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167834448636\"><div data-type=\"problem\" id=\"fs-id1167835350341\"><p id=\"fs-id1167834279269\">Solve the system by graphing: \\(\\left\\{\\begin{array}{c}y=3x-1\\hfill \\\\ 6x-2y=6\\hfill \\end{array}.\\) <\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835423011\"><p id=\"fs-id1167834396875\">no solution<\/p><\/div><\/div><\/div><p id=\"fs-id1168754384001\">Sometimes the equations in a system represent the same line. Since every point on the line makes both equations true, there are infinitely many ordered pairs that make both equations true. There are infinitely many solutions to the system.<\/p><div data-type=\"example\" id=\"fs-id1167835418145\" class=\"textbox textbox--examples\"><div data-type=\"exercise\"><div data-type=\"problem\" id=\"fs-id1167834431373\"><p id=\"fs-id1167835479000\">Solve the system by graphing: \\(\\left\\{\\begin{array}{c}y=2x-3\\hfill \\\\ -6x+3y=-9\\hfill \\end{array}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835307320\"><table id=\"fs-id1167835327859\" class=\"unnumbered unstyled can-break\" summary=\"The equations are y equal to 2 x minus 3 and minus 6x plus 3y equal to minus 9. The slope and y intercept of the first equation are 2 and minus 3 respectively. In the second equation, when x is 0, y is minus 3 and when y is 0, x is 3 by 2. We use these points for graphing the lines to determine the point of intersection. The lines are the same. Since every point on the line makes both equations true, there are infinite many ordered pairs that make both equations true. There are infinite many solutions to this system.\" data-label=\"\"><tbody><tr><td><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834191115\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_01_007a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Find the slope and <em data-effect=\"italics\">y<\/em>-intercept of the first equation.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835334409\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_01_007b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Find the intercepts of the second equation.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167831891526\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_01_007c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835240354\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_01_007d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Graph the lines.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835258752\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_01_007e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td><\/td><td data-valign=\"top\" data-align=\"left\">The lines are the same!<div data-type=\"newline\"><br><\/div>Since every point on the line makes both<div data-type=\"newline\"><br><\/div>equations true, there are infinitely many<div data-type=\"newline\"><br><\/div>ordered pairs that make both equations true.<div data-type=\"newline\"><br><\/div>There are infinitely many solutions to this system.<\/td><\/tr><\/tbody><\/table><p id=\"fs-id1167835238066\">If you write the second equation in slope-intercept form, you may recognize that the equations have the same slope and same <em data-effect=\"italics\">y<\/em>-intercept.<\/p><\/div><\/div><\/div><div data-type=\"note\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167835420544\"><div data-type=\"problem\" id=\"fs-id1167835288349\"><p id=\"fs-id1167826978106\">Solve the system by graphing: \\(\\left\\{\\begin{array}{c}y=-3x-6\\hfill \\\\ 6x+2y=-12\\hfill \\end{array}.\\) <\/p><\/div><div data-type=\"solution\" id=\"fs-id1167832043521\"><p id=\"fs-id1167834459122\">infinitely many solutions<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167835186730\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167834516134\"><div data-type=\"problem\" id=\"fs-id1167831887685\"><p id=\"fs-id1167830866127\">Solve the system by graphing: \\(\\left\\{\\begin{array}{c}y=\\frac{1}{2}x-4\\hfill \\\\ 2x-4y=16\\hfill \\end{array}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167826782418\"><p id=\"fs-id1167830838187\">infinitely many solutions<\/p><\/div><\/div><\/div><p id=\"fs-id1167831954237\">When we graphed the second line in the last example, we drew it right over the first line. We say the two lines are <span data-type=\"term\">coincident<\/span>. Coincident lines have the same slope and same <em data-effect=\"italics\">y-<\/em>intercept.<\/p><div data-type=\"note\" id=\"fs-id1167835530462\"><div data-type=\"title\">Coincident Lines<\/div><p id=\"fs-id1167835351616\"><strong data-effect=\"bold\">Coincident lines<\/strong> have the same slope and same <em data-effect=\"italics\">y-<\/em>intercept.<\/p><\/div><p id=\"fs-id1167831191430\">The systems of equations in <a href=\"#fs-id1167834279490\" class=\"autogenerated-content\">(Figure)<\/a> and <a href=\"#fs-id1167835410272\" class=\"autogenerated-content\">(Figure)<\/a> each had two intersecting lines. Each system had one solution.<\/p><p id=\"fs-id1167827987930\">In <a href=\"#fs-id1167835418145\" class=\"autogenerated-content\">(Figure)<\/a>, the equations gave coincident lines, and so the system had infinitely many solutions.<\/p><p id=\"fs-id1167834063977\">The systems in those three examples had at least one solution. A system of equations that has at least one solution is called a <em data-effect=\"italics\">consistent<\/em> system.<\/p><p id=\"fs-id1167831880100\">A system with parallel lines, like <a href=\"#fs-id1167835307740\" class=\"autogenerated-content\">(Figure)<\/a>, has no solution. We call a system of equations like this <em data-effect=\"italics\">inconsistent.<\/em> It has no solution.<\/p><div data-type=\"note\" id=\"fs-id1167835370836\"><div data-type=\"title\">Consistent and Inconsistent Systems<\/div><p id=\"fs-id1167835379459\">A <span data-type=\"term\">consistent system of equations<\/span> is a system of equations with at least one solution.<\/p><p id=\"fs-id1167835239043\">An <span data-type=\"term\">inconsistent system of equations<\/span> is a system of equations with no solution.<\/p><\/div><p id=\"fs-id1167835343554\">We also categorize the equations in a system of equations by calling the equations <em data-effect=\"italics\">independent<\/em> or <em data-effect=\"italics\">dependent<\/em>. If two equations are independent, they each have their own set of solutions. Intersecting lines and parallel lines are independent.<\/p><p>If two equations are dependent, all the solutions of one equation are also solutions of the other equation. When we graph two dependent equations, we get coincident lines.<\/p><p id=\"fs-id1167832057993\">Let\u2019s sum this up by looking at the graphs of the three types of systems. See below and <a href=\"#fs-id1167832096973\" class=\"autogenerated-content\">(Figure)<\/a>.<\/p><span data-type=\"media\" id=\"fs-id1167835170706\" data-alt=\"The figure shows three graphs. The first one has two intersecting line. The second one has two parallel lines. The third one has only one line. This is labeled coincident.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_01_008_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows three graphs. The first one has two intersecting line. The second one has two parallel lines. The third one has only one line. This is labeled coincident.\"><\/span><table id=\"fs-id1167832096973\" summary=\"The table shows that intersecting lines have 1 solution that is consistent and independent. Parallel lines have no solution and are inconsistent and independent. Coincident lines have infinitely many solutions and are consistent and dependent.\"><thead><tr><th data-valign=\"middle\" data-align=\"left\">Lines<\/th><th data-valign=\"middle\" data-align=\"left\">Intersecting<\/th><th data-valign=\"middle\" data-align=\"left\">Parallel<\/th><th data-valign=\"middle\" data-align=\"left\">Coincident<\/th><\/tr><\/thead><tbody><tr><td data-valign=\"middle\" data-align=\"left\"><strong data-effect=\"bold\">Number of solutions<\/strong><\/td><td data-valign=\"middle\" data-align=\"left\">1 point<\/td><td data-valign=\"middle\" data-align=\"left\">No solution<\/td><td data-valign=\"middle\" data-align=\"left\">Infinitely many<\/td><\/tr><tr><td data-valign=\"middle\" data-align=\"left\"><strong data-effect=\"bold\">Consistent\/inconsistent<\/strong><\/td><td data-valign=\"middle\" data-align=\"left\">Consistent<\/td><td data-valign=\"middle\" data-align=\"left\">Inconsistent<\/td><td data-valign=\"middle\" data-align=\"left\">Consistent<\/td><\/tr><tr><td data-valign=\"middle\" data-align=\"left\"><strong data-effect=\"bold\">Dependent\/ independent<\/strong><\/td><td data-valign=\"middle\" data-align=\"left\">Independent<\/td><td data-valign=\"middle\" data-align=\"left\">Independent<\/td><td data-valign=\"middle\" data-align=\"left\">Dependent<\/td><\/tr><\/tbody><\/table><div data-type=\"example\" id=\"fs-id1167834191318\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167834156729\"><div data-type=\"problem\" id=\"fs-id1167834184039\"><p id=\"fs-id1167834189749\">Without graphing, determine the number of solutions and then classify the system of equations.<\/p><p id=\"fs-id1167834429136\"><span class=\"token\">\u24d0<\/span>\\(\\left\\{\\begin{array}{c}y=3x-1\\hfill \\\\ 6x-2y=12\\hfill \\end{array}\\)<span class=\"token\">\u24d1<\/span>\\(\\left\\{\\begin{array}{c}2x+y=-3\\hfill \\\\ x-5y=5\\hfill \\end{array}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167831882192\"><p id=\"fs-id1167835363341\"><span class=\"token\">\u24d0<\/span> We will compare the slopes and intercepts of the two lines.<\/p><p id=\"fs-id1167835342275\">\\(\\begin{array}{cccc}\\begin{array}{}\\\\ \\\\ \\\\ \\\\ \\hfill \\text{The first equation is already in slope-intercept form.}\\end{array}\\hfill &amp; &amp; &amp; \\phantom{\\rule{7.7em}{0ex}}\\begin{array}{c}\\left\\{\\begin{array}{ccc}\\hfill y&amp; =\\hfill &amp; 3x-1\\hfill \\\\ \\hfill 6x-2y&amp; =\\hfill &amp; 12\\hfill \\end{array}\\hfill \\\\ \\phantom{\\rule{3.1em}{0ex}}y\\phantom{\\rule{0.5em}{0ex}}=\\phantom{\\rule{0.5em}{0ex}}3x-1\\hfill \\end{array}\\hfill \\end{array}\\)<\/p><p>\\(\\begin{array}{cccc}\\begin{array}{c}\\text{Write the second equation in slope-intercept form.}\\hfill \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\text{Find the slope and intercept of each line.}\\hfill \\\\ \\\\ \\\\ \\\\ \\end{array}\\hfill &amp; &amp; &amp; \\phantom{\\rule{2em}{0ex}}\\begin{array}{cccccccc}&amp; &amp; &amp; &amp; &amp; \\hfill 6x-2y&amp; =\\hfill &amp; 12\\hfill \\\\ &amp; &amp; &amp; &amp; &amp; \\hfill -2y&amp; =\\hfill &amp; -6x+12\\hfill \\\\ &amp; &amp; &amp; &amp; &amp; \\hfill \\frac{-2y}{-2}&amp; =\\hfill &amp; \\frac{-6x+12}{-2}\\hfill \\\\ &amp; &amp; &amp; &amp; &amp; \\hfill y&amp; =\\hfill &amp; 3x-6\\hfill \\\\ \\hfill y&amp; =\\hfill &amp; 3x-1\\hfill &amp; &amp; &amp; \\hfill y&amp; =\\hfill &amp; 3x-6\\hfill \\\\ \\hfill m&amp; =\\hfill &amp; 3\\hfill &amp; &amp; &amp; \\hfill m&amp; =\\hfill &amp; 3\\hfill \\\\ \\hfill b&amp; =\\hfill &amp; -1\\hfill &amp; &amp; &amp; \\hfill b&amp; =\\hfill &amp; -6\\hfill \\end{array}\\hfill \\\\ &amp; &amp; &amp; \\begin{array}{c}\\text{Since the slopes are the same and}\\phantom{\\rule{0.2em}{0ex}}y\\text{-intercepts are}\\hfill \\\\ \\text{different, the lines are parallel.}\\hfill \\end{array}\\hfill \\end{array}\\)<\/p><p id=\"fs-id1167834367141\">A system of equations whose graphs are parallel lines has no solution and is inconsistent and independent.<\/p><p id=\"fs-id1167826995612\"><span class=\"token\">\u24d1<\/span> We will compare the slope and intercepts of the two lines.<\/p><p id=\"fs-id1167834084804\">\\(\\begin{array}{cccc}&amp; &amp; &amp; \\left\\{\\begin{array}{ccc}\\hfill 2x+y&amp; =\\hfill &amp; -3\\hfill \\\\ \\hfill x-5y&amp; =\\hfill &amp; 5\\hfill \\end{array}\\hfill \\\\ \\begin{array}{c}\\text{Write both equations in slope\u2013intercept form.}\\hfill \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\end{array}\\hfill &amp; &amp; &amp; \\phantom{\\rule{0.5em}{0ex}}\\begin{array}{cccccccc}\\hfill 2x+y&amp; =\\hfill &amp; -3\\hfill &amp; &amp; &amp; \\hfill x-5y&amp; =\\hfill &amp; 5\\hfill \\\\ \\hfill y&amp; =\\hfill &amp; -2x-3\\hfill &amp; &amp; &amp; \\hfill -5y&amp; =\\hfill &amp; \\text{\u2212}x+5\\hfill \\\\ &amp; &amp; &amp; &amp; &amp; \\hfill \\frac{-5y}{-5}&amp; =\\hfill &amp; \\frac{-x+5}{-5}\\hfill \\\\ &amp; &amp; &amp; &amp; &amp; \\hfill y&amp; =\\hfill &amp; \\frac{1}{5}x-1\\hfill \\end{array}\\hfill \\\\ \\begin{array}{c}\\text{Find the slope and intercept of each line.}\\hfill \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\end{array}\\hfill &amp; &amp; &amp; \\phantom{\\rule{2em}{0ex}}\\begin{array}{cccccccccccc}\\hfill y&amp; =\\hfill &amp; -2x-3\\hfill &amp; &amp; &amp; &amp; &amp; &amp; &amp; \\phantom{\\rule{2em}{0ex}}\\hfill y&amp; =\\hfill &amp; \\frac{1}{5}x-1\\hfill \\\\ \\hfill m&amp; =\\hfill &amp; -2\\hfill &amp; &amp; &amp; &amp; &amp; &amp; &amp; \\hfill m&amp; =\\hfill &amp; \\frac{1}{5}\\hfill \\\\ \\hfill b&amp; =\\hfill &amp; -3\\hfill &amp; &amp; &amp; &amp; &amp; &amp; &amp; \\hfill b&amp; =\\hfill &amp; -1\\hfill \\end{array}\\hfill \\\\ &amp; &amp; &amp; \\text{Since the slopes are different, the lines intersect.}\\hfill \\end{array}\\)<\/p><p id=\"fs-id1167835419049\">A system of equations whose graphs are intersect has 1 solution and is consistent and independent.<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167832151520\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167832151097\"><div data-type=\"problem\" id=\"fs-id1167832151099\"><p id=\"fs-id1167835347863\">Without graphing, determine the number of solutions and then classify the system of equations.<\/p><p id=\"fs-id1167835347866\"><span class=\"token\">\u24d0<\/span>\\(\\left\\{\\begin{array}{c}y=-2x-4\\hfill \\\\ 4x+2y=9\\hfill \\end{array}\\)<span class=\"token\">\u24d1<\/span>\\(\\left\\{\\begin{array}{c}3x+2y=2\\hfill \\\\ 2x+y=1\\hfill \\end{array}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167832076368\"><p id=\"fs-id1167832076370\"><span class=\"token\">\u24d0<\/span> no solution, inconsistent, independent <span class=\"token\">\u24d1<\/span> one solution, consistent, independent<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167831910867\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167835356197\"><div data-type=\"problem\" id=\"fs-id1167835356199\"><p id=\"fs-id1167835376695\">Without graphing, determine the number of solutions and then classify the system of equations.<\/p><p id=\"fs-id1167835376698\"><span class=\"token\">\u24d0<\/span>\\(\\left\\{\\begin{array}{c}y=\\frac{1}{3}x-5\\hfill \\\\ x-3y=6\\hfill \\end{array}\\)<span class=\"token\">\u24d1<\/span>\\(\\left\\{\\begin{array}{c}x+4y=12\\hfill \\\\ -x+y=3\\hfill \\end{array}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834059399\"><p id=\"fs-id1167834059401\"><span class=\"token\">\u24d0<\/span> no solution, inconsistent, independent <span class=\"token\">\u24d1<\/span> one solution, consistent, independent<\/p><\/div><\/div><\/div><p id=\"fs-id1167832060470\">Solving systems of linear equations by graphing is a good way to visualize the types of solutions that may result. However, there are many cases where solving a system by graphing is inconvenient or imprecise. If the graphs extend beyond the small grid with <em data-effect=\"italics\">x<\/em> and <em data-effect=\"italics\">y<\/em> both between \\(-10\\) and 10, graphing the lines may be cumbersome. And if the solutions to the system are not integers, it can be hard to read their values precisely from a graph.<\/p><\/div><div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167834233994\"><h3 data-type=\"title\">Solve a System of Equations by Substitution<\/h3><p id=\"fs-id1167834234000\">We will now solve systems of linear equations by the substitution method.<\/p><p id=\"fs-id1167835334266\">We will use the same system we used first for graphing.<\/p><div data-type=\"equation\" class=\"unnumbered\" data-label=\"\">\\(\\left\\{\\begin{array}{c}2x+y=7\\hfill \\\\ x-2y=6\\hfill \\end{array}\\)<\/div><p id=\"fs-id1167827943012\">We will first solve one of the equations for either <em data-effect=\"italics\">x<\/em> or <em data-effect=\"italics\">y<\/em>. We can choose either equation and solve for either variable\u2014but we\u2019ll try to make a choice that will keep the work easy.<\/p><p id=\"fs-id1167834063721\">Then we substitute that expression into the other equation. The result is an equation with just one variable\u2014and we know how to solve those!<\/p><p id=\"fs-id1167834301188\">After we find the value of one variable, we will substitute that value into one of the original equations and solve for the other variable. Finally, we check our solution and make sure it makes both equations true.<\/p><div data-type=\"example\" id=\"fs-id1167835328722\" class=\"textbox textbox--examples\"><div data-type=\"title\">How to Solve a System of Equations by Substitution<\/div><div data-type=\"exercise\"><div data-type=\"problem\" id=\"fs-id1167835328726\"><p id=\"fs-id1167826802362\">Solve the system by substitution: \\(\\left\\{\\begin{array}{c}2x+y=7\\hfill \\\\ x-2y=6\\hfill \\end{array}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835332322\"><span data-type=\"media\" id=\"fs-id1167835332325\" data-alt=\"The equations are 2 x plus y equals 7 and x minus 2y equals 6. Step 1 is to solve one of the equations for either variable. We\u2019ll solve the first equation for y. We get y equals 7 minus 2 x.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_01_009a_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The equations are 2 x plus y equals 7 and x minus 2y equals 6. Step 1 is to solve one of the equations for either variable. We\u2019ll solve the first equation for y. We get y equals 7 minus 2 x.\"><\/span><span data-type=\"media\" data-alt=\"In step 2, substitute the expression from step 1 into the other equation. We replace y in the second equation with the expression 7 minus 2 x. So, we get x minus 2 open parentheses 7 minus 2 x close parentheses equals 6.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_01_009b_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"In step 2, substitute the expression from step 1 into the other equation. We replace y in the second equation with the expression 7 minus 2 x. So, we get x minus 2 open parentheses 7 minus 2 x close parentheses equals 6.\"><\/span><span data-type=\"media\" id=\"fs-id1167835198382\" data-alt=\"Step 3 is to solve the resulting equation. Now we have an equation with just 1 variable. We solve it to get x equal to 4.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_01_009c_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 3 is to solve the resulting equation. Now we have an equation with just 1 variable. We solve it to get x equal to 4.\"><\/span><span data-type=\"media\" id=\"fs-id1167835229691\" data-alt=\"Step 4 is to substitute the solution in step 3 into one of the original equations to find the other variable. We\u2019ll use the first equation and replace x with 4. We get, 2 times 4 plus y equals 7. Simplifying, we get y equal to minus 1.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_01_009d_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 4 is to substitute the solution in step 3 into one of the original equations to find the other variable. We\u2019ll use the first equation and replace x with 4. We get, 2 times 4 plus y equals 7. Simplifying, we get y equal to minus 1.\"><\/span><span data-type=\"media\" id=\"fs-id1167832066001\" data-alt=\"Step 5 is to write the solution as an ordered pair. The ordered pair is 4, minus 1.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_01_009e_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 5 is to write the solution as an ordered pair. The ordered pair is 4, minus 1.\"><\/span><span data-type=\"media\" id=\"fs-id1167831086582\" data-alt=\"Step 6 is to check that the ordered pair is a solution to both original equations. To do that we Substitute x equal to 4 and y equal to minus 1 into both equations and make sure they are both true.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_01_009f_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 6 is to check that the ordered pair is a solution to both original equations. To do that we Substitute x equal to 4 and y equal to minus 1 into both equations and make sure they are both true.\"><\/span><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167834099143\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167835594824\"><div data-type=\"problem\" id=\"fs-id1167835594827\"><p id=\"fs-id1167830964185\">Solve the system by substitution: \\(\\left\\{\\begin{array}{c}-2x+y=-11\\hfill \\\\ x+3y=9\\hfill \\end{array}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834533520\"><p id=\"fs-id1167834533523\">\\(\\left(6,1\\right)\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167835363503\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167835363506\"><div data-type=\"problem\" id=\"fs-id1167835363508\"><p id=\"fs-id1167835363510\">Solve the system by substitution: \\(\\left\\{\\begin{array}{c}2x+y=-1\\hfill \\\\ 4x+3y=3\\hfill \\end{array}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167832016014\"><p id=\"fs-id1167832016016\">\\(\\left(-3,5\\right)\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167830925047\" class=\"howto\"><div data-type=\"title\">Solve a system of equations by substitution.<\/div><ol id=\"fs-id1167830925054\" type=\"1\" class=\"stepwise\"><li>Solve one of the equations for either variable.<\/li><li>Substitute the expression from Step 1 into the other equation.<\/li><li>Solve the resulting equation.<\/li><li>Substitute the solution in Step 3 into either of the original equations to find the other variable.<\/li><li>Write the solution as an ordered pair.<\/li><li>Check that the ordered pair is a solution to <strong data-effect=\"bold\">both<\/strong> original equations.<\/li><\/ol><\/div><p id=\"fs-id1167835415479\">Be very careful with the signs in the next example.<\/p><div data-type=\"example\" id=\"fs-id1167835415482\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167832057897\"><div data-type=\"problem\" id=\"fs-id1167832057899\"><p id=\"fs-id1167832057901\">Solve the system by substitution: \\(\\left\\{\\begin{array}{c}4x+2y=4\\hfill \\\\ 6x-y=8\\hfill \\end{array}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835595520\"><p id=\"fs-id1167835595522\">We need to solve one equation for one variable. We will solve the first equation for <em data-effect=\"italics\">y<\/em>.<\/p><table id=\"fs-id1167826998205\" class=\"unnumbered unstyled can-break\" summary=\"The equations are 4x plus 2y equals 4 and 6x minus y equals 8. Solving the first equation for y, we get y equal to minus 2 x plus 2. Substituting this in the second equation, we get 6 x minus open parentheses minus 2 x plus 2 close parentheses equals 8. Solving for x, we get x equal to 5 by 4. Substituting this in the first equation, and solving for y, we get y equal to minus 1 by 2. The ordered pair is 5 by 4, minus 1 by 2. Check the ordered pair in both equations. Both hold true. Hence, that is the solution.\" data-label=\"\"><tbody><tr><td><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835254444\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_01_010b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Solve the first equation for <em data-effect=\"italics\">y<\/em>.<div data-type=\"newline\"><br><\/div>Substitute \\(-2x+2\\) for <em data-effect=\"italics\">y<\/em> in the second equation.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835379481\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_01_010c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Replace the <em data-effect=\"italics\">y<\/em> with \\(-2x+2.\\)<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167831239705\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_01_010d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><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-id1167834432973\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_01_010e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Substitute \\(x=\\frac{5}{4}\\) into \\(4x+2y=4\\) to find <em data-effect=\"italics\">y<\/em>.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835609313\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_01_010f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td><\/td><td data-valign=\"top\" data-align=\"left\">The ordered pair is \\(\\left(\\frac{5}{4},-\\frac{1}{2}\\right).\\)<\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Check the ordered pair in both equations.<div data-type=\"newline\"><br><\/div><div data-type=\"newline\"><br><\/div><span data-type=\"media\" id=\"fs-id1167835175410\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_01_010a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><td><\/td><\/tr><tr><td><\/td><td data-valign=\"top\" data-align=\"left\">The solution is \\(\\left(\\frac{5}{4},-\\frac{1}{2}\\right).\\)<\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167826782993\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167826782996\"><div data-type=\"problem\" id=\"fs-id1167826782998\"><p id=\"fs-id1167826783000\">Solve the system by substitution: \\(\\left\\{\\begin{array}{c}x-4y=-4\\hfill \\\\ -3x+4y=0\\hfill \\end{array}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834523798\"><p id=\"fs-id1167834523800\">\\(\\left(2,\\frac{3}{2}\\right)\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167834536415\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167835360488\"><div data-type=\"problem\" id=\"fs-id1167835360490\"><p id=\"fs-id1167835360492\">Solve the system by substitution: \\(\\left\\{\\begin{array}{c}4x-y=0\\hfill \\\\ 2x-3y=5\\hfill \\end{array}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835351996\"><p id=\"fs-id1167835351998\">\\(\\left(-\\frac{1}{2},-2\\right)\\)<\/p><\/div><\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167834222378\"><h3 data-type=\"title\">Solve a System of Equations by Elimination<\/h3><p id=\"fs-id1167834132598\">We have solved systems of linear equations by graphing and by substitution. Graphing works well when the variable coefficients are small and the solution has integer values. Substitution works well when we can easily solve one equation for one of the variables and not have too many fractions in the resulting expression.<\/p><p id=\"fs-id1167834132604\">The third method of solving systems of linear equations is called the Elimination Method. When we solved a system by substitution, we started with two equations and two variables and reduced it to one equation with one variable. This is what we\u2019ll do with the elimination method, too, but we\u2019ll have a different way to get there.<\/p><p id=\"fs-id1167835510985\">The Elimination Method is based on the Addition Property of Equality. The Addition Property of Equality says that when you add the same quantity to both sides of an equation, you still have equality. We will extend the Addition Property of Equality to say that when you add equal quantities to both sides of an equation, the results are equal.<\/p><p id=\"fs-id1167832055312\">For any expressions <em data-effect=\"italics\">a, b, c,<\/em> and <em data-effect=\"italics\">d<\/em>.<\/p><div data-type=\"equation\" id=\"fs-id1167831911684\" class=\"unnumbered\" data-label=\"\">\\(\\begin{array}{cccccc}\\text{if}\\hfill &amp; &amp; &amp; \\hfill a&amp; =\\hfill &amp; b\\hfill \\\\ \\text{and}\\hfill &amp; &amp; &amp; \\hfill c&amp; =\\hfill &amp; d\\hfill \\\\ \\text{then}\\hfill &amp; &amp; &amp; \\hfill a+c&amp; =\\hfill &amp; b+d.\\hfill \\end{array}\\)<\/div><p id=\"fs-id1167835376848\">To solve a system of equations by elimination, we start with both equations in standard form. Then we decide which variable will be easiest to eliminate. How do we decide? We want to have the coefficients of one variable be opposites, so that we can add the equations together and eliminate that variable.<\/p><p id=\"fs-id1167835340855\">Notice how that works when we add these two equations together:<\/p><div data-type=\"equation\" id=\"fs-id1167831847197\" class=\"unnumbered\" data-label=\"\">\\(\\begin{array}{c}\\left\\{\\underset{\\text{\u2014\u2014\u2014\u2014\u2014}}{\\begin{array}{c}3x+y=5\\hfill \\\\ 2x-y=0\\hfill \\end{array}}\\hfill \\\\ \\\\ \\phantom{\\rule{0.8em}{0ex}}5x\\phantom{\\rule{1.8em}{0ex}}=5\\hfill \\end{array}\\)<\/div><p id=\"fs-id1167835338949\">The <em data-effect=\"italics\">y<\/em>\u2019s add to zero and we have one equation with one variable.<\/p><p id=\"fs-id1167834376400\">Let\u2019s try another one:<\/p><div data-type=\"equation\" id=\"fs-id1167834063624\" class=\"unnumbered\" data-label=\"\">\\(\\left\\{\\begin{array}{c}\\phantom{\\rule{0.5em}{0ex}}x+4y=2\\hfill \\\\ 2x+5y=-2\\hfill \\end{array}\\)<\/div><p id=\"fs-id1167835336856\">This time we don\u2019t see a variable that can be immediately eliminated if we add the equations.<\/p><p id=\"fs-id1167834195035\">But if we multiply the first equation by \\(-2,\\) we will make the coefficients of <em data-effect=\"italics\">x<\/em> opposites. We must multiply every term on both sides of the equation by \\(-2.\\)<\/p><span data-type=\"media\" id=\"fs-id1167834193440\" data-alt=\"Minus 2 open parentheses x plus 4y close parentheses is minus 2 times 2. And, 2 x plus 5y is minus 2.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_01_011_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Minus 2 open parentheses x plus 4y close parentheses is minus 2 times 2. And, 2 x plus 5y is minus 2.\"><\/span><p id=\"fs-id1167828420195\">Then rewrite the system of equations.<\/p><span data-type=\"media\" id=\"fs-id1167828420197\" data-alt=\"Minus 2 x minus 8y is minus 4 and 2 x plus 5y is minus 2.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_01_012_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Minus 2 x minus 8y is minus 4 and 2 x plus 5y is minus 2.\"><\/span><p id=\"fs-id1167826997236\">Now we see that the coefficients of the <em data-effect=\"italics\">x<\/em> terms are opposites, so <em data-effect=\"italics\">x<\/em> will be eliminated when we add these two equations.<\/p><span data-type=\"media\" id=\"fs-id1167831031275\" data-alt=\"Minus 2 x minus 8y is minus 4 and 2 x plus 5y is minus 2. Adding these, we get minus 3y equals minus 6.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_01_013_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Minus 2 x minus 8y is minus 4 and 2 x plus 5y is minus 2. Adding these, we get minus 3y equals minus 6.\"><\/span><p id=\"fs-id1167834324678\">Once we get an equation with just one variable, we solve it. Then we substitute that value into one of the original equations to solve for the remaining variable. And, as always, we check our answer to make sure it is a solution to both of the original equations.<\/p><p id=\"fs-id1167834121134\">Now we\u2019ll see how to use elimination to solve the same system of equations we solved by graphing and by substitution.<\/p><div data-type=\"example\" id=\"fs-id1167834121139\" class=\"textbox textbox--examples\"><div data-type=\"title\">How to Solve a System of Equations by Elimination<\/div><div data-type=\"exercise\" id=\"fs-id1167834053780\"><div data-type=\"problem\" id=\"fs-id1167834053782\"><p id=\"fs-id1167834053784\">Solve the system by elimination: \\(\\left\\{\\begin{array}{c}2x+y=7\\hfill \\\\ x-2y=6\\hfill \\end{array}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167832055991\"><span data-type=\"media\" id=\"fs-id1167832055992\" data-alt=\"The equations are 2 x plus y equals 7 and x minus 2y equals 6. Step 1 is to write both equations in standard form. Both equations are in standard form, Ax plus By equals C. If any coefficients are fractions, clear them. There are no fractions.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_01_014a_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The equations are 2 x plus y equals 7 and x minus 2y equals 6. Step 1 is to write both equations in standard form. Both equations are in standard form, Ax plus By equals C. If any coefficients are fractions, clear them. There are no fractions.\"><\/span><span data-type=\"media\" id=\"fs-id1167835489057\" data-alt=\"Step 2 is to make the coefficients of one variable opposites. First decide which variable you will eliminate. Multiply one or both equations so that the coefficients of that variable are opposites. We can eliminate the y\u2019s by multiplying the first equation by 2. We get 4x plus 2y equals 14.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_01_014b_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 2 is to make the coefficients of one variable opposites. First decide which variable you will eliminate. Multiply one or both equations so that the coefficients of that variable are opposites. We can eliminate the y\u2019s by multiplying the first equation by 2. We get 4x plus 2y equals 14.\"><\/span><span data-type=\"media\" id=\"fs-id1167835375465\" data-alt=\"Step 3 is to add the equations resulting from step 2 to eliminate one variable. Adding, we get 5x equals 20.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_01_014c_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 3 is to add the equations resulting from step 2 to eliminate one variable. Adding, we get 5x equals 20.\"><\/span><span data-type=\"media\" id=\"fs-id1167828436322\" data-alt=\"Step 4 is to solve for the remaining variable. Solving for x, we get x equals 4.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_01_014d_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 4 is to solve for the remaining variable. Solving for x, we get x equals 4.\"><\/span><span data-type=\"media\" id=\"fs-id1167835218199\" data-alt=\"Step 5 is to substitute the solution from step 4 into one of the original equations. Then solve for the other variable. Substituting x equal to 4 into the second equation, we get 4 minus 2y equals 6. Solving for y, we get y equal to minus 1.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_01_014e_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 5 is to substitute the solution from step 4 into one of the original equations. Then solve for the other variable. Substituting x equal to 4 into the second equation, we get 4 minus 2y equals 6. Solving for y, we get y equal to minus 1.\"><\/span><span data-type=\"media\" id=\"fs-id1167826782824\" data-alt=\"Step 6 is to write the solution as an ordered pair. Here, the ordered pair is 4, minus 1.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_01_014f_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 6 is to write the solution as an ordered pair. Here, the ordered pair is 4, minus 1.\"><\/span><span data-type=\"media\" id=\"fs-id1167835312274\" data-alt=\"Step 7 is to check that the ordered pair is a solution to both original equations. The ordered pair makes both original equations true.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_01_014g_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 7 is to check that the ordered pair is a solution to both original equations. The ordered pair makes both original equations true.\"><\/span><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167834413553\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167835433665\"><div data-type=\"problem\" id=\"fs-id1167835433667\"><p>Solve the system by elimination: \\(\\left\\{\\begin{array}{c}3x+y=5\\hfill \\\\ 2x-3y=7\\hfill \\end{array}.\\) <\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834161733\"><p id=\"fs-id1167834161734\">\\(\\left(2,-1\\right)\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167831031022\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167832043684\"><div data-type=\"problem\" id=\"fs-id1167832043686\"><p id=\"fs-id1167832043689\">Solve the system by elimination: \\(\\left\\{\\begin{array}{c}4x+y=-5\\hfill \\\\ -2x-2y=-2\\hfill \\end{array}.\\) <\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834134812\"><p id=\"fs-id1167834134813\">\\(\\left(-2,3\\right)\\)<\/p><\/div><\/div><\/div><p id=\"fs-id1167831921387\">The steps are listed here for easy reference.<\/p><div data-type=\"note\" id=\"fs-id1167835347820\" class=\"howto\"><div data-type=\"title\">Solve a system of equations by elimination.<\/div><ol id=\"fs-id1167835268056\" type=\"1\" class=\"stepwise\"><li>Write both equations in standard form. If any coefficients are fractions, clear them.<\/li><li>Make the coefficients of one variable opposites. <ul id=\"fs-id1167835355835\" data-bullet-style=\"bullet\"><li>Decide which variable you will eliminate.<\/li><li>Multiply one or both equations so that the coefficients of that variable are opposites.<\/li><\/ul><\/li><li>Add the equations resulting from Step 2 to eliminate one variable.<\/li><li>Solve for the remaining variable.<\/li><li>Substitute the solution from Step 4 into one of the original equations. Then solve for the other variable.<\/li><li>Write the solution as an ordered pair.<\/li><li>Check that the ordered pair is a solution to <strong data-effect=\"bold\">both<\/strong> original equations.<\/li><\/ol><\/div><p id=\"fs-id1167831835904\">Now we\u2019ll do an example where we need to multiply both equations by constants in order to make the coefficients of one variable opposites.<\/p><div data-type=\"example\" id=\"fs-id1167831835910\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167835183559\"><div data-type=\"problem\" id=\"fs-id1167835183561\"><p id=\"fs-id1167835183563\">Solve the system by elimination: \\(\\left\\{\\begin{array}{c}4x-3y=9\\hfill \\\\ 7x+2y=-6\\hfill \\end{array}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834133101\"><p id=\"fs-id1167834133104\">In this example, we cannot multiply just one equation by any constant to get opposite coefficients. So we will strategically multiply both equations by different constants to get the opposites.<\/p><table id=\"fs-id1167834536402\" class=\"unnumbered unstyled can-break\" summary=\"The equations are 4x minus 3y equals 9 and 7x plus 2y equals minus 6. Both equations are in standard form. To get opposite coefficients of y, we will multiply the first equation by 2 and the second equation by 3. We get 2 open parentheses 4x minus 3y close parentheses equals 2 times 9 and 3 open parentheses 7x plus 2y close parentheses equals 3 times minus 6. Simplifying both, we get 8x minus 6y equals 18 and 21x plus 6y equals minus 18. Adding the two equations to eliminate y, and solving for x, we get x equal to 0. Substituting this into one of the original equations and solving for y, we get y equal to minus 3. The ordered pair of the solution is 0, minus 3. Check that the ordered pair is a solution to both original equations.\" data-label=\"\"><tbody><tr><td><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167832226710\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_01_015b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Both equations are in standard form.<div data-type=\"newline\"><br><\/div>To get opposite coefficients of <em data-effect=\"italics\">y<\/em>, we will<div data-type=\"newline\"><br><\/div>multiply the first equation by 2 and the<div data-type=\"newline\"><br><\/div>second equation by 3.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835509822\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_01_015c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Simplify.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167832058367\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_01_015d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Add the two equations to eliminate <em data-effect=\"italics\">y<\/em>.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835287480\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_01_015e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Solve for <em data-effect=\"italics\">x<\/em>.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167830961361\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_01_015f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Substitute \\(x=0\\) into one of the original equations.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835200374\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_01_015g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><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-id1167835333944\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_01_015h_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Write the solution as an ordered pair.<\/td><td data-valign=\"top\" data-align=\"left\">The ordered pair is \\(\\left(0,-3\\right).\\)<\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Check that the ordered pair is a solution to<div data-type=\"newline\"><br><\/div><strong data-effect=\"bold\">both<\/strong> original equations.<div data-type=\"newline\"><br><\/div><div data-type=\"newline\"><br><\/div><span data-type=\"media\" id=\"fs-id1167835351833\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_01_015a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><td><\/td><\/tr><tr><td><\/td><td data-valign=\"top\" data-align=\"left\">The solution is \\(\\left(0,-3\\right).\\)<\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167826996182\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167835244250\"><div data-type=\"problem\" id=\"fs-id1167835244252\"><p id=\"fs-id1167835244254\">Solve the system by elimination: \\(\\left\\{\\begin{array}{c}3x-4y=-9\\hfill \\\\ 5x+3y=14\\hfill \\end{array}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834340042\"><p id=\"fs-id1167834340045\">\\(\\left(1,3\\right)\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167835163286\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167835163290\"><div data-type=\"problem\" id=\"fs-id1167835163292\"><p id=\"fs-id1167834583891\">Solve each system by elimination: \\(\\left\\{\\begin{array}{c}7x+8y=4\\hfill \\\\ 3x-5y=27\\hfill \\end{array}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167831888060\"><p id=\"fs-id1167831888061\">\\(\\left(4,-3\\right)\\)<\/p><\/div><\/div><\/div><p id=\"fs-id1167830757705\">When the system of equations contains fractions, we will first clear the fractions by multiplying each equation by the LCD of all the fractions in the equation.<\/p><div data-type=\"example\" id=\"fs-id1167834535350\" class=\"textbox textbox--examples\"><div data-type=\"exercise\"><div data-type=\"problem\" id=\"fs-id1167834535354\"><p id=\"fs-id1167831117475\">Solve the system by elimination: \\(\\left\\{\\begin{array}{c}x+\\frac{1}{2}y=6\\hfill \\\\ \\frac{3}{2}x+\\frac{2}{3}y=\\frac{17}{2}\\hfill \\end{array}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167832041346\"><p id=\"fs-id1167832041348\">In this example, both equations have fractions. Our first step will be to multiply each equation by the LCD of all the fractions in the equation to clear the fractions.<\/p><table id=\"fs-id1167834464514\" class=\"unnumbered unstyled can-break\" summary=\"The equations are x plus half y equals 6 and 3 by 2 x plus2 by 3 y equals 17 by 2. To clear the fractions, multiply each equation by its LCD. Multiplying the first equation by 2 the second one by 6, and simplifying both, we get 2 x plus y equals 12 and 9x plus 4y equals 51. Now that both equations are in standard form, we can eliminate y by multiplying the top equation by minus 4. This becomes minus 8x minus 4y equals minus 48. Adding this to 9x plus 4y equals 51, we get x equal to 3. Substituting this into either of the original equations, we get y equal to 6. The ordered pair is 3, 6. It is a solution to both original equations.\" data-label=\"\"><tbody><tr><td><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835575772\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_01_016b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">To clear the fractions, multiply each<div data-type=\"newline\"><br><\/div>equation by its LCD.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167832151434\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_01_016c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Simplify.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834064043\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_01_016d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Now we are ready to eliminate one<div data-type=\"newline\"><br><\/div>of the variables. Notice that both equations are in<div data-type=\"newline\"><br><\/div>standard form.<\/td><td><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">We can eliminate \\(y\\) by multiplying the top equation by \\(-4.\\)<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835353312\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_01_016e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Simplify and add.<div data-type=\"newline\"><br><\/div><div data-type=\"newline\"><br><\/div>Substitute \\(x=3\\) into one of the original equations.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167828421222\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_01_016f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Solve for \\(y\\).<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835589634\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_01_016g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><div data-type=\"newline\"><br><\/div><span data-type=\"media\" id=\"fs-id1167834184112\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_01_016h_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><div data-type=\"newline\"><br><\/div><span data-type=\"media\" id=\"fs-id1167835327424\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_01_016i_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Write the solution as an ordered pair.<\/td><td data-valign=\"top\" data-align=\"left\">The ordered pair is \\(\\left(3,6\\right).\\)<\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Check that the ordered pair is a solution to<div data-type=\"newline\"><br><\/div>both original equations.<div data-type=\"newline\"><br><\/div><div data-type=\"newline\"><br><\/div><span data-type=\"media\" id=\"fs-id1167835190458\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_01_016a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><td><\/td><\/tr><tr><td><\/td><td data-valign=\"top\" data-align=\"left\">The solution is \\(\\left(3,6\\right).\\)<\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167835309662\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167835342662\"><div data-type=\"problem\" id=\"fs-id1167835342664\"><p id=\"fs-id1167835342666\">Solve each system by elimination: \\(\\left\\{\\begin{array}{c}\\frac{1}{3}x-\\frac{1}{2}y=1\\hfill \\\\ \\frac{3}{4}x-y=\\frac{5}{2}\\hfill \\end{array}.\\) <\/p><\/div><div data-type=\"solution\" id=\"fs-id1167831921076\"><p id=\"fs-id1167831921079\">\\(\\left(6,2\\right)\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167831890568\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167835341021\"><div data-type=\"problem\"><p id=\"fs-id1167835341025\">Solve each system by elimination: \\(\\left\\{\\begin{array}{c}x+\\frac{3}{5}y=-\\frac{1}{5}\\hfill \\\\ -\\frac{1}{2}x-\\frac{2}{3}y=\\frac{5}{6}\\hfill \\end{array}.\\) <\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834190571\"><p id=\"fs-id1167834190573\">\\(\\left(1,-2\\right)\\)<\/p><\/div><\/div><\/div><p id=\"fs-id1167831163799\">When we solved the system by graphing, we saw that not all systems of linear equations have a single ordered pair as a solution. When the two equations were really the same line, there were infinitely many solutions. We called that a consistent system. When the two equations described parallel lines, there was no solution. We called that an inconsistent system.<\/p><p id=\"fs-id1167831163806\">The same is true using substitution or elimination. If the equation at the end of substitution or elimination is a true statement, we have a consistent but dependent system and the system of equations has infinitely many solutions. If the equation at the end of substitution or elimination is a false statement, we have an inconsistent system and the system of equations has no solution.<\/p><div data-type=\"example\" id=\"fs-id1167832138940\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167832138943\"><div data-type=\"problem\" id=\"fs-id1167835380388\"><p id=\"fs-id1167835380390\">Solve the system by elimination: \\(\\left\\{\\begin{array}{c}3x+4y=12\\hfill \\\\ y=3-\\frac{3}{4}x\\hfill \\end{array}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167831117510\"><p id=\"fs-id1167831117512\">\\(\\begin{array}{cccc}&amp; &amp; &amp; \\left\\{\\begin{array}{c}3x+4y=12\\hfill \\\\ y=3-\\frac{3}{4}x\\hfill \\end{array}\\hfill \\\\ \\\\ \\text{Write the second equation in standard form.}\\hfill &amp; &amp; &amp; \\left\\{\\begin{array}{c}3x+4y=12\\hfill \\\\ \\frac{3}{4}x+y=3\\hfill \\end{array}\\hfill \\\\ \\\\ \\begin{array}{c}\\text{Clear the fractions by multiplying the}\\hfill \\\\ \\text{second equation by 4.}\\hfill \\end{array}\\hfill &amp; &amp; &amp; \\left\\{\\begin{array}{c}3x+4y=12\\hfill \\\\ 4\\left(\\frac{3}{4}x+y\\right)=4\\left(3\\right)\\hfill \\end{array}\\hfill \\\\ \\\\ \\text{Simplify.}\\hfill &amp; &amp; &amp; \\left\\{\\begin{array}{c}3x+4y=12\\hfill \\\\ 3x+4y=12\\hfill \\end{array}\\hfill \\\\ \\\\ \\begin{array}{c}\\text{To eliminate a variable, we multiply the}\\hfill \\\\ \\text{second equation by}\\phantom{\\rule{0.2em}{0ex}}-1.\\phantom{\\rule{0.2em}{0ex}}\\text{Simplify and add.}\\hfill \\end{array}\\hfill &amp; &amp; &amp; \\begin{array}{c}\\underset{______________}{\\left\\{\\begin{array}{c}\\phantom{\\rule{0.6em}{0ex}}3x+4y=12\\hfill \\\\ -3x-4y=-12\\hfill \\end{array}}\\hfill \\\\ \\hfill \\phantom{\\rule{1em}{0ex}}0=0\\hfill \\end{array}\\hfill \\end{array}\\)<\/p><p id=\"fs-id1167831086799\">This is a true statement. The equations are consistent but dependent. Their graphs would be the same line. The system has infinitely many solutions.<\/p><p id=\"fs-id1167831833311\">After we cleared the fractions in the second equation, did you notice that the two equations were the same? That means we have coincident lines.<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167835357545\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167835357549\"><div data-type=\"problem\" id=\"fs-id1167835357552\"><p id=\"fs-id1167834066022\">Solve the system by elimination: \\(\\left\\{\\begin{array}{c}5x-3y=15\\hfill \\\\ y=-5+\\frac{5}{3}x\\hfill \\end{array}.\\) <\/p><\/div><div data-type=\"solution\" id=\"fs-id1167831823837\"><p id=\"fs-id1167835379275\">infinitely many solutions<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167835379282\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167835357672\"><div data-type=\"problem\" id=\"fs-id1167835357674\"><p id=\"fs-id1167835357677\">Solve the system by elimination: \\(\\left\\{\\begin{array}{c}x+2y=6\\hfill \\\\ y=-\\frac{1}{2}x+3\\hfill \\end{array}.\\) <\/p><\/div><div data-type=\"solution\" id=\"fs-id1167831920066\"><p id=\"fs-id1167831920068\">infinitely many solutions<\/p><\/div><\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167826783829\"><h3 data-type=\"title\">Choose the Most Convenient Method to Solve a System of Linear Equations<\/h3><p id=\"fs-id1167826783834\">When you solve a system of linear equations in in an application, you will not be told which method to use. You will need to make that decision yourself. So you\u2019ll want to choose the method that is easiest to do and minimizes your chance of making mistakes.<\/p><div data-type=\"equation\" id=\"fs-id1171792628785\" class=\"unnumbered\" data-label=\"\">\\(\\begin{array}{c}\\mathbf{\\text{Choose the Most Convenient Method to Solve a System of Linear Equations}}\\hfill \\\\ \\begin{array}{ccccccc}\\underset{\\text{\u2014\u2014\u2014\u2014}}{\\mathbf{\\text{Graphing}}}\\hfill &amp; &amp; &amp; \\underset{\\text{\u2014\u2014\u2014\u2014\u2014}}{\\mathbf{\\text{Substitution}}}\\hfill &amp; &amp; &amp; \\underset{\\text{\u2014\u2014\u2014\u2014\u2014}}{\\mathbf{\\text{Elimination}}}\\hfill \\\\ \\text{Use when you need a}\\hfill &amp; &amp; &amp; \\text{Use when one equation is}\\hfill &amp; &amp; &amp; \\text{Use when the equations are}\\hfill \\\\ \\text{picture of the situation.}\\hfill &amp; &amp; &amp; \\text{already solved or can be}\\hfill &amp; &amp; &amp; \\text{in standard form.}\\hfill \\\\ &amp; &amp; &amp; \\text{easily solved for one}\\hfill &amp; &amp; &amp; \\\\ &amp; &amp; &amp; \\text{variable.}\\hfill &amp; &amp; &amp; \\end{array}\\hfill \\end{array}\\)<\/div><div data-type=\"example\" id=\"fs-id1167834525591\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167834525593\"><div data-type=\"problem\" id=\"fs-id1167834525595\"><p id=\"fs-id1167834525597\">For each system of linear equations, decide whether it would be more convenient to solve it by substitution or elimination. Explain your answer.<\/p><p id=\"fs-id1167831112489\"><span class=\"token\">\u24d0<\/span>\\(\\left\\{\\begin{array}{c}3x+8y=40\\hfill \\\\ 7x-4y=-32\\hfill \\end{array}\\)<span class=\"token\">\u24d1<\/span>\\(\\left\\{\\begin{array}{c}5x+6y=12\\hfill \\\\ y=\\frac{2}{3}x-1\\hfill \\end{array}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834273719\"><p id=\"fs-id1167834273721\"><span class=\"token\">\u24d0<\/span><\/p><div data-type=\"equation\" class=\"unnumbered\" data-label=\"\">\\(\\left\\{\\begin{array}{c}3x+8y=40\\hfill \\\\ 7x-4y=-32\\hfill \\end{array}\\)<\/div><p id=\"fs-id1167835355915\">Since both equations are in standard form, using elimination will be most convenient.<\/p><p id=\"fs-id1167835355918\"><span class=\"token\">\u24d1<\/span><\/p><div data-type=\"equation\" id=\"fs-id1167827940624\" class=\"unnumbered\" data-label=\"\">\\(\\left\\{\\begin{array}{c}5x+6y=12\\hfill \\\\ y=\\frac{2}{3}x-1\\hfill \\end{array}\\)<\/div><p id=\"fs-id1167835419071\">Since one equation is already solved for <em data-effect=\"italics\">y<\/em>, using substitution will be most convenient.<\/p><\/div><\/div><\/div><div data-type=\"note\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167835360936\"><div data-type=\"problem\" id=\"fs-id1167835360938\"><p id=\"fs-id1167835331536\">For each system of linear equations decide whether it would be more convenient to solve it by substitution or elimination. Explain your answer.<\/p><p id=\"fs-id1167835331541\"><span class=\"token\">\u24d0<\/span>\\(\\left\\{\\begin{array}{c}4x-5y=-32\\hfill \\\\ 3x+2y=-1\\hfill \\end{array}\\)<span class=\"token\">\u24d1<\/span>\\(\\left\\{\\begin{array}{c}x=2y-1\\hfill \\\\ 3x-5y=-7\\hfill \\end{array}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835235457\"><p id=\"fs-id1167835235460\"><span class=\"token\">\u24d0<\/span> Since both equations are in standard form, using elimination will be most convenient. <span class=\"token\">\u24d1<\/span> Since one equation is already solved for <em data-effect=\"italics\">x<\/em>, using substitution will be most convenient.<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167826798760\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167826798764\"><div data-type=\"problem\" id=\"fs-id1167826857360\"><p id=\"fs-id1167826857362\">For each system of linear equations decide whether it would be more convenient to solve it by substitution or elimination. Explain your answer.<\/p><p id=\"fs-id1167826857367\"><span class=\"token\">\u24d0<\/span>\\(\\left\\{\\begin{array}{c}y=2x-1\\hfill \\\\ 3x-4y=-6\\hfill \\end{array}\\)<span class=\"token\">\u24d1<\/span>\\(\\left\\{\\begin{array}{c}6x-2y=12\\hfill \\\\ 3x+7y=-13\\hfill \\end{array}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834049030\"><p id=\"fs-id1167834049032\"><span class=\"token\">\u24d0<\/span> Since one equation is already solved for <em data-effect=\"italics\">y<\/em>, using substitution will be most convenient. <span class=\"token\">\u24d1<\/span> Since both equations are in standard form, using elimination will be most convenient.<\/p><\/div><\/div><\/div><\/div><div class=\"textbox\" data-depth=\"1\" id=\"fs-id1167835378580\"><h3 data-type=\"title\">Key Concepts<\/h3><ul id=\"fs-id1167835234062\" data-bullet-style=\"bullet\"><li><strong data-effect=\"bold\">How to solve a system of linear equations by graphing.<\/strong><ol id=\"fs-id1167831921610\" type=\"1\" class=\"stepwise\"><li>Graph the first equation.<\/li><li>Graph the second equation on the same rectangular coordinate system.<\/li><li>Determine whether the lines intersect, are parallel, or are the same line.<\/li><li>Identify the solution to the system.<div data-type=\"newline\"><br><\/div> If the lines intersect, identify the point of intersection. This is the solution to the system.<div data-type=\"newline\"><br><\/div> If the lines are parallel, the system has no solution.<div data-type=\"newline\"><br><\/div> If the lines are the same, the system has an infinite number of solutions.<\/li><li>Check the solution in both equations.<\/li><\/ol><\/li><li><strong data-effect=\"bold\">How to solve a system of equations by substitution.<\/strong><ol id=\"fs-id1167835330573\" type=\"1\" class=\"stepwise\"><li>Solve one of the equations for either variable.<\/li><li>Substitute the expression from Step 1 into the other equation.<\/li><li>Solve the resulting equation.<\/li><li>Substitute the solution in Step 3 into either of the original equations to find the other variable.<\/li><li>Write the solution as an ordered pair.<\/li><li>Check that the 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-id1167835305193\" type=\"1\" class=\"stepwise\"><li>Write both equations in standard form. If any coefficients are fractions, clear them.<\/li><li>Make the coefficients of one variable opposites.<div data-type=\"newline\"><br><\/div>Decide which variable you will eliminate.<div data-type=\"newline\"><br><\/div>Multiply one or both equations so that the coefficients of that variable are opposites.<\/li><li>Add the equations resulting from Step 2 to eliminate one variable.<\/li><li>Solve for the remaining variable.<\/li><li>Substitute the solution from Step 4 into one of the original equations. Then solve for the other variable.<\/li><li>Write the solution as an ordered pair.<\/li><li>Check that the ordered pair is a solution to <strong data-effect=\"bold\">both<\/strong> original equations.<div data-type=\"newline\"><br><\/div> \\(\\begin{array}{}\\\\ \\\\ \\mathbf{\\text{Choose the Most Convenient Method to Solve a System of Linear Equations}}\\hfill \\\\ \\begin{array}{ccccccc}\\underset{\\text{\u2014\u2014\u2014\u2014\u2014}}{\\mathbf{\\text{Graphing}}}\\hfill &amp; &amp; &amp; \\underset{\\text{\u2014\u2014\u2014\u2014\u2014\u2014\u2014}}{\\mathbf{\\text{Substitution}}}\\hfill &amp; &amp; &amp; \\underset{\\text{\u2014\u2014\u2014\u2014\u2014\u2014\u2014}}{\\mathbf{\\text{Elimination}}}\\hfill \\\\ \\begin{array}{c}\\text{Use when you need a}\\hfill \\\\ \\text{picture of the situation.}\\hfill \\end{array}\\hfill &amp; &amp; &amp; \\begin{array}{c}\\text{Use when one equation is}\\hfill \\\\ \\text{already solved or can be}\\hfill \\\\ \\text{easily solved for one}\\hfill \\\\ \\text{variable.}\\hfill \\end{array}\\hfill &amp; &amp; &amp; \\begin{array}{c}\\text{Use when the equations are}\\hfill \\\\ \\text{in standard form.}\\hfill \\end{array}\\hfill \\end{array}\\hfill \\end{array}\\)<\/li><\/ol><\/li><\/ul><\/div><div class=\"textbox\" data-depth=\"1\" id=\"fs-id1167835634244\"><div class=\"practice-perfect\" data-depth=\"2\" id=\"fs-id1167834279700\"><h4 data-type=\"title\">Practice Makes Perfect<\/h4><p id=\"fs-id1167835595230\"><strong data-effect=\"bold\">Determine Whether an Ordered Pair is a Solution of a System of Equations<\/strong><\/p><p id=\"fs-id1167834124355\">In the following exercises, determine if the following points are solutions to the given system of equations.<\/p><div data-type=\"exercise\" id=\"fs-id1167834124359\"><div data-type=\"problem\" id=\"fs-id1167834124361\"><p id=\"fs-id1167834063037\">\\(\\left\\{\\begin{array}{c}2x-6y=0\\hfill \\\\ 3x-4y=5\\hfill \\end{array}\\)<\/p><p id=\"fs-id1167831911303\"><span class=\"token\">\u24d0<\/span>\\(\\left(3,1\\right)\\)<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\(\\left(-3,4\\right)\\)<\/div><div data-type=\"solution\" id=\"fs-id1167834423297\"><p id=\"fs-id1167834423299\"><span class=\"token\">\u24d0<\/span> yes <span class=\"token\">\u24d1<\/span> no<\/p><\/div><\/div><div data-type=\"exercise\"><div data-type=\"problem\" id=\"fs-id1167830960525\"><p>\\(\\left\\{\\begin{array}{c}-3x+y=8\\hfill \\\\ -x+2y=-9\\hfill \\end{array}\\)<\/p><p id=\"fs-id1167826874537\"><span class=\"token\">\u24d0<\/span>\\(\\left(-5,-7\\right)\\)<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\(\\left(-5,7\\right)\\)<\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831116465\"><div data-type=\"problem\" id=\"fs-id1167831116468\"><p id=\"fs-id1167831116470\">\\(\\left\\{\\begin{array}{c}x+y=2\\hfill \\\\ y=\\frac{3}{4}x\\hfill \\end{array}\\)<\/p><p id=\"fs-id1167834557190\"><span class=\"token\">\u24d0<\/span>\\(\\left(\\frac{8}{7},\\frac{6}{7}\\right)\\)<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\(\\left(1,\\frac{3}{4}\\right)\\)<\/div><div data-type=\"solution\" id=\"fs-id1167835400298\"><p id=\"fs-id1167835400300\"><span class=\"token\">\u24d0<\/span> yes <span class=\"token\">\u24d1<\/span> no<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167826926965\"><div data-type=\"problem\" id=\"fs-id1167826926967\"><p id=\"fs-id1167834525517\">\\(\\left\\{\\begin{array}{c}2x+3y=6\\hfill \\\\ y=\\frac{2}{3}x+2\\hfill \\end{array}\\)<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d0<\/span>\\(\\left(-6,2\\right)\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\(\\left(-3,4\\right)\\)<\/div><\/div><p id=\"fs-id1167830705743\"><strong data-effect=\"bold\">Solve a System of Linear Equations by Graphing<\/strong><\/p><p id=\"fs-id1167834130183\">In the following exercises, solve the following systems of equations by graphing.<\/p><div data-type=\"exercise\" id=\"fs-id1167830924068\"><div data-type=\"problem\" id=\"fs-id1167830924070\"><p id=\"fs-id1167830924073\">\\(\\left\\{\\begin{array}{c}3x+y=-3\\hfill \\\\ 2x+3y=5\\hfill \\end{array}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167832052855\"><p id=\"fs-id1167831880957\">\\(\\left(-3,2\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167830865575\"><div data-type=\"problem\" id=\"fs-id1167830865578\"><p id=\"fs-id1167830865580\">\\(\\left\\{\\begin{array}{c}-x+y=2\\hfill \\\\ 2x+y=-4\\hfill \\end{array}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831969744\"><div data-type=\"problem\" id=\"fs-id1167831969746\"><p id=\"fs-id1167832086997\">\\(\\left\\{\\begin{array}{c}y=x+2\\hfill \\\\ y=-2x+2\\hfill \\end{array}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167826997095\"><p id=\"fs-id1167835206118\">\\(\\left(0,2\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834184570\"><div data-type=\"problem\" id=\"fs-id1167834184573\"><p id=\"fs-id1167834184575\">\\(\\left\\{\\begin{array}{c}y=x-2\\hfill \\\\ y=-3x+2\\hfill \\end{array}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835236187\"><div data-type=\"problem\" id=\"fs-id1167835236189\"><p id=\"fs-id1167835236191\">\\(\\left\\{\\begin{array}{c}y=\\phantom{\\rule{0.2em}{0ex}}\\text{}\\phantom{\\rule{0.2em}{0ex}}\\frac{3}{2}x+1\\hfill \\\\ y=-\\frac{1}{2}x+5\\hfill \\end{array}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835510445\"><p id=\"fs-id1167835510447\">\\(\\left(2,4\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835215816\"><div data-type=\"problem\" id=\"fs-id1167835215818\"><p>\\(\\left\\{\\begin{array}{c}y=\\frac{2}{3}x-2\\hfill \\\\ y=-\\frac{1}{3}x-5\\hfill \\end{array}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831891501\"><div data-type=\"problem\" id=\"fs-id1167835350488\"><p id=\"fs-id1167835350490\">\\(\\left\\{\\begin{array}{c}x+y=-4\\hfill \\\\ -x+2y=-2\\hfill \\end{array}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167831115549\"><p id=\"fs-id1167831115551\">\\(\\left(-2,2\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835344480\"><div data-type=\"problem\" id=\"fs-id1167835344482\"><p id=\"fs-id1167831891601\">\\(\\left\\{\\begin{array}{c}-x+3y=3\\hfill \\\\ x+3y=3\\hfill \\end{array}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167832043496\"><div data-type=\"problem\"><p id=\"fs-id1167826801731\">\\(\\left\\{\\begin{array}{c}-2x+3y=3\\hfill \\\\ x+3y=12\\hfill \\end{array}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834431232\"><p id=\"fs-id1167834431234\">\\(\\left(3,3\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834300235\"><div data-type=\"problem\" id=\"fs-id1167834300237\"><p id=\"fs-id1167832055418\">\\(\\left\\{\\begin{array}{c}2x-y=4\\hfill \\\\ 2x+3y=12\\hfill \\end{array}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831837693\"><div data-type=\"problem\" id=\"fs-id1167831837695\"><p id=\"fs-id1167831837697\">\\(\\left\\{\\begin{array}{c}x+3y=-6\\hfill \\\\ y=-\\frac{4}{3}x+4\\hfill \\end{array}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835376871\"><p id=\"fs-id1167831025421\">\\(\\left(6,-4\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167826977938\"><div data-type=\"problem\" id=\"fs-id1167826977940\"><p id=\"fs-id1167826977943\">\\(\\left\\{\\begin{array}{c}-x+2y=-6\\hfill \\\\ y=-\\frac{1}{2}x-1\\hfill \\end{array}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167828240455\"><div data-type=\"problem\" id=\"fs-id1167828240457\"><p id=\"fs-id1167834185690\">\\(\\left\\{\\begin{array}{c}-2x+4y=4\\hfill \\\\ y=\\frac{1}{2}x\\hfill \\end{array}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167831826448\"><p id=\"fs-id1167835237081\">no solution<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835237086\"><div data-type=\"problem\" id=\"fs-id1167835237088\"><p id=\"fs-id1167832055711\">\\(\\left\\{\\begin{array}{c}3x+5y=10\\hfill \\\\ y=-\\frac{3}{5}x+1\\hfill \\end{array}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835359930\"><div data-type=\"problem\" id=\"fs-id1167835359932\"><p>\\(\\left\\{\\begin{array}{c}\\phantom{\\rule{0.2em}{0ex}}4x-3y=8\\hfill \\\\ \\phantom{\\rule{0.2em}{0ex}}8x-6y=14\\hfill \\end{array}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167832036092\"><p id=\"fs-id1167832036094\">no solution<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167830703070\"><div data-type=\"problem\" id=\"fs-id1167830703072\"><p id=\"fs-id1167830703074\">\\(\\left\\{\\begin{array}{c}x+3y=4\\hfill \\\\ -2x-6y=3\\hfill \\end{array}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835380427\"><div data-type=\"problem\" id=\"fs-id1167835380429\"><p id=\"fs-id1167835380431\">\\(\\left\\{\\begin{array}{c}x=-3y+4\\hfill \\\\ 2x+6y=8\\hfill \\end{array}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835479147\"><p id=\"fs-id1167835479149\">infinite solutions<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834294493\"><div data-type=\"problem\"><p>\\(\\left\\{\\begin{array}{c}4x=3y+7\\hfill \\\\ 8x-6y=14\\hfill \\end{array}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167821102414\"><div data-type=\"problem\" id=\"fs-id1167821102417\"><p id=\"fs-id1167821102419\">\\(\\left\\{\\begin{array}{c}2x+\\text{}\\phantom{\\rule{0.2em}{0ex}}y=6\\hfill \\\\ -8x-4y=-24\\hfill \\end{array}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834463108\"><p id=\"fs-id1167834463110\">infinite solutions<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834423546\"><div data-type=\"problem\" id=\"fs-id1167834423548\"><p id=\"fs-id1167834423550\">\\(\\left\\{\\begin{array}{c}5x+\\phantom{\\rule{0.2em}{0ex}}2y=7\\hfill \\\\ -10x-4y=-14\\hfill \\end{array}\\)<\/p><\/div><\/div><p id=\"fs-id1167835353488\">Without graphing, determine the number of solutions and then classify the system of equations.<\/p><div data-type=\"exercise\" id=\"fs-id1167835353491\"><div data-type=\"problem\" id=\"fs-id1167834133342\"><p id=\"fs-id1167834133345\">\\(\\left\\{\\begin{array}{c}y=\\frac{2}{3}x+1\\hfill \\\\ -2x+3y=5\\hfill \\end{array}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835422585\"><p id=\"fs-id1167835422587\">1 point, consistent and independent<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834129862\"><div data-type=\"problem\" id=\"fs-id1167834129864\"><p id=\"fs-id1167834129866\">\\(\\left\\{\\begin{array}{c}y=\\frac{3}{2}x+1\\hfill \\\\ 2x-3y=7\\hfill \\end{array}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167832067470\"><div data-type=\"problem\" id=\"fs-id1167832067472\"><p id=\"fs-id1167832067474\">\\(\\left\\{\\begin{array}{c}5x+3y=4\\hfill \\\\ 2x-3y=5\\hfill \\end{array}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835310613\"><p id=\"fs-id1167831882494\">1 point, consistent and independent<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831882499\"><div data-type=\"problem\" id=\"fs-id1167831882501\"><p id=\"fs-id1167831882503\">\\(\\left\\{\\begin{array}{c}y=-\\frac{1}{2}x+5\\hfill \\\\ x+2y=10\\hfill \\end{array}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831066235\"><div data-type=\"problem\" id=\"fs-id1167831066237\"><p id=\"fs-id1167831066239\">\\(\\left\\{\\begin{array}{c}5x-2y=10\\hfill \\\\ y=\\frac{5}{2}x-5\\hfill \\end{array}\\)<\/p><\/div><div data-type=\"solution\"><p id=\"fs-id1167832046536\">infinite solutions, consistent, dependent<\/p><\/div><\/div><p id=\"fs-id1167832046542\"><strong data-effect=\"bold\">Solve a System of Equations by Substitution<\/strong><\/p><p id=\"fs-id1167834556545\">In the following exercises, solve the systems of equations by substitution.<\/p><div data-type=\"exercise\" id=\"fs-id1167834556548\"><div data-type=\"problem\" id=\"fs-id1167834556550\"><p id=\"fs-id1167834556552\">\\(\\left\\{\\begin{array}{c}2x+y=-4\\hfill \\\\ 3x-2y=-6\\hfill \\end{array}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834094784\"><div data-type=\"problem\" id=\"fs-id1167834533392\"><p id=\"fs-id1167834533394\">\\(\\left\\{\\begin{array}{c}2x+y=-2\\hfill \\\\ 3x-y=7\\hfill \\end{array}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835351247\"><p id=\"fs-id1167835351249\">\\(\\left(1,-4\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835340122\"><div data-type=\"problem\" id=\"fs-id1167835340124\"><p id=\"fs-id1167835340126\">\\(\\left\\{\\begin{array}{c}x-2y=-5\\hfill \\\\ 2x-3y=-4\\hfill \\end{array}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831921781\"><div data-type=\"problem\" id=\"fs-id1167831921783\"><p id=\"fs-id1167831921785\">\\(\\left\\{\\begin{array}{c}x-3y=-9\\hfill \\\\ 2x+5y=4\\hfill \\end{array}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834431035\"><p id=\"fs-id1167834299862\">\\(\\left(-3,2\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834314750\"><div data-type=\"problem\" id=\"fs-id1167835348333\"><p id=\"fs-id1167835348335\">\\(\\left\\{\\begin{array}{c}5x-2y=-6\\hfill \\\\ y=3x+3\\hfill \\end{array}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835534326\"><div data-type=\"problem\" id=\"fs-id1167835534328\"><p id=\"fs-id1167835420362\">\\(\\left\\{\\begin{array}{c}-2x+2y=6\\hfill \\\\ y=-3x+1\\hfill \\end{array}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167831943922\"><p id=\"fs-id1167831943924\">\\(\\left(-1\\text{\/}2,5\\text{\/}2\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834422473\"><div data-type=\"problem\" id=\"fs-id1167834422475\"><p id=\"fs-id1167834422478\">\\(\\left\\{\\begin{array}{c}2x+5y=1\\hfill \\\\ y=\\frac{1}{3}x-2\\hfill \\end{array}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835180558\"><div data-type=\"problem\" id=\"fs-id1167835180560\"><p id=\"fs-id1167835180562\">\\(\\left\\{\\begin{array}{c}3x+4y=1\\hfill \\\\ y=-\\frac{2}{5}x+2\\hfill \\end{array}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834532308\"><p id=\"fs-id1167834532310\">\\(\\left(-5,4\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835297443\"><div data-type=\"problem\" id=\"fs-id1167835297445\"><p id=\"fs-id1167835297447\">\\(\\left\\{\\begin{array}{c}2x+y=5\\hfill \\\\ x-2y=-15\\hfill \\end{array}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835320772\"><div data-type=\"problem\" id=\"fs-id1167835320775\"><p id=\"fs-id1167835320777\">\\(\\left\\{\\begin{array}{c}4x+y=10\\hfill \\\\ x-2y=-20\\hfill \\end{array}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835304713\"><p id=\"fs-id1167835304715\">\\(\\left(0,10\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167828447139\"><div data-type=\"problem\" id=\"fs-id1167828447141\"><p id=\"fs-id1167828447143\">\\(\\left\\{\\begin{array}{c}y=-2x-1\\hfill \\\\ y=-\\frac{1}{3}x+4\\hfill \\end{array}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834422758\"><div data-type=\"problem\" id=\"fs-id1167834422760\"><p id=\"fs-id1167834422762\">\\(\\left\\{\\begin{array}{c}y=x-6\\hfill \\\\ y=-\\frac{3}{2}x+4\\hfill \\end{array}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834489757\"><p id=\"fs-id1167834489759\">\\(\\left(4,-2\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834190734\"><div data-type=\"problem\" id=\"fs-id1167834190737\"><p id=\"fs-id1167834190739\">\\(\\left\\{\\begin{array}{c}\\phantom{\\rule{0.2em}{0ex}}\\text{}\\phantom{\\rule{0.2em}{0ex}}x=2y\\hfill \\\\ 4x-8y=0\\hfill \\end{array}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835280872\"><div data-type=\"problem\"><p>\\(\\left\\{\\begin{array}{c}\\phantom{\\rule{0.2em}{0ex}}2x-16y=8\\hfill \\\\ -x-8y=-4\\hfill \\end{array}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835417628\"><p>\\(\\left(4,0\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167830699913\"><div data-type=\"problem\" id=\"fs-id1167830699915\"><p id=\"fs-id1167830699917\">\\(\\left\\{\\begin{array}{c}y=\\frac{7}{8}x+4\\hfill \\\\ -7x+8y=6\\hfill \\end{array}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835304080\"><div data-type=\"problem\" id=\"fs-id1167835304082\"><p>\\(\\left\\{\\begin{array}{c}y=-\\frac{2}{3}x+5\\hfill \\\\ 2x+3y=11\\hfill \\end{array}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835357993\"><p id=\"fs-id1167835357996\">none<\/p><\/div><\/div><p id=\"fs-id1167826863952\"><strong data-effect=\"bold\">Solve a System of Equations by Elimination<\/strong><\/p><p id=\"fs-id1167826863958\">In the following exercises, solve the systems of equations by elimination.<\/p><div data-type=\"exercise\" id=\"fs-id1167826863961\"><div data-type=\"problem\"><p>\\(\\left\\{\\begin{array}{c}5x+2y=2\\hfill \\\\ -3x-y=0\\hfill \\end{array}\\)<\/p><\/div><\/div><div data-type=\"exercise\"><div data-type=\"problem\" id=\"fs-id1167831836066\"><p id=\"fs-id1167831836068\">\\(\\left\\{\\begin{array}{c}6x-5y=-1\\hfill \\\\ 2x+y=13\\hfill \\end{array}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167831825741\"><p id=\"fs-id1167831825743\">\\(\\left(4,5\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831907914\"><div data-type=\"problem\" id=\"fs-id1167831907917\"><p id=\"fs-id1167831907919\">\\(\\left\\{\\begin{array}{c}2x-5y=7\\hfill \\\\ 3x-y=17\\hfill \\end{array}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835363585\"><div data-type=\"problem\" id=\"fs-id1167835363587\"><p id=\"fs-id1167835363589\">\\(\\left\\{\\begin{array}{c}5x-3y=-1\\hfill \\\\ 2x-y=2\\hfill \\end{array}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167831112627\"><p id=\"fs-id1167834593004\">\\(\\left(7,12\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167830705259\"><div data-type=\"problem\" id=\"fs-id1167832058610\"><p>\\(\\left\\{\\begin{array}{c}3x-5y=-9\\hfill \\\\ 5x+2y=16\\hfill \\end{array}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834147076\"><div data-type=\"problem\" id=\"fs-id1167835376196\"><p id=\"fs-id1167835376198\">\\(\\left\\{\\begin{array}{c}4x-3y=3\\hfill \\\\ 2x+5y=-31\\hfill \\end{array}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167826978568\"><p id=\"fs-id1167826978570\">\\(\\left(-3,-5\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167826978378\"><div data-type=\"problem\" id=\"fs-id1167826978380\"><p id=\"fs-id1167831954879\">\\(\\left\\{\\begin{array}{c}3x+8y=-3\\hfill \\\\ 2x+5y=-3\\hfill \\end{array}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834430174\"><div data-type=\"problem\" id=\"fs-id1167834430176\"><p id=\"fs-id1167835166917\">\\(\\left\\{\\begin{array}{c}11x+9y=-5\\hfill \\\\ 7x+5y=-1\\hfill \\end{array}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834195693\"><p id=\"fs-id1167834195695\">\\(\\left(2,-3\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835365118\"><div data-type=\"problem\" id=\"fs-id1167835365120\"><p id=\"fs-id1167835365122\">\\(\\left\\{\\begin{array}{c}3x+8y=67\\hfill \\\\ 5x+3y=60\\hfill \\end{array}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835332982\"><div data-type=\"problem\" id=\"fs-id1167835332984\"><p id=\"fs-id1167835332986\">\\(\\left\\{\\begin{array}{c}2x+9y=-4\\hfill \\\\ 3x+13y=-7\\hfill \\end{array}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834403143\"><p id=\"fs-id1167834403145\">\\(\\left(-11,2\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167832052599\"><div data-type=\"problem\" id=\"fs-id1167832052601\"><p id=\"fs-id1167832052603\">\\(\\left\\{\\begin{array}{c}\\frac{1}{3}x-y=-3\\hfill \\\\ x+\\frac{5}{2}y=2\\hfill \\end{array}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834547275\"><div data-type=\"problem\" id=\"fs-id1167834547278\"><p id=\"fs-id1167834547280\">\\(\\left\\{\\begin{array}{c}x+\\frac{1}{2}y=\\frac{3}{2}\\hfill \\\\ \\frac{1}{5}x-\\frac{1}{5}y=3\\hfill \\end{array}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835345974\"><p id=\"fs-id1167835345976\">\\(\\left(6\\text{\/}-9,24\\text{\/}7\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835370986\"><div data-type=\"problem\" id=\"fs-id1167835370988\"><p id=\"fs-id1167835370990\">\\(\\left\\{\\begin{array}{c}x+\\frac{1}{3}y=-1\\hfill \\\\ \\frac{1}{3}x+\\frac{1}{2}y=1\\hfill \\end{array}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167832076583\"><div data-type=\"problem\" id=\"fs-id1167832076585\"><p id=\"fs-id1167834161573\">\\(\\left\\{\\begin{array}{c}\\frac{1}{3}x-y=-3\\hfill \\\\ \\frac{2}{3}x+\\frac{5}{2}y=3\\hfill \\end{array}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834377203\"><p id=\"fs-id1167834377205\">\\(\\left(-3,2\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835374572\"><div data-type=\"problem\" id=\"fs-id1167835374574\"><p id=\"fs-id1167835374576\">\\(\\left\\{\\begin{array}{c}2x+y=3\\hfill \\\\ 6x+3y=9\\hfill \\end{array}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834120246\"><div data-type=\"problem\" id=\"fs-id1167834120248\"><p id=\"fs-id1167834120250\">\\(\\left\\{\\begin{array}{c}x-4y=-1\\hfill \\\\ -3x+12y=3\\hfill \\end{array}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835370201\"><p id=\"fs-id1167835370203\">infinitely many<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835370208\"><div data-type=\"problem\" id=\"fs-id1167832066123\"><p id=\"fs-id1167832066125\">\\(\\left\\{\\begin{array}{c}-3x-y=8\\hfill \\\\ 6x+2y=-16\\hfill \\end{array}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835348718\"><div data-type=\"problem\" id=\"fs-id1167835348720\"><p id=\"fs-id1167835348722\">\\(\\left\\{\\begin{array}{c}4x+3y=2\\hfill \\\\ 20x+15y=10\\hfill \\end{array}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167826781183\"><p id=\"fs-id1167826781185\">infinitely many<\/p><\/div><\/div><p id=\"fs-id1167826781191\"><strong data-effect=\"bold\">Choose the Most Convenient Method to Solve a System of Linear Equations<\/strong><\/p><p id=\"fs-id1167834431079\">In the following exercises, decide whether it would be more convenient to solve the system of equations by substitution or elimination.<\/p><div data-type=\"exercise\" id=\"fs-id1167834431083\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167835282942\"><p id=\"fs-id1167835282945\"><\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d0<\/span>\\(\\left\\{\\begin{array}{c}8x-15y=-32\\hfill \\\\ 6x+3y=-5\\hfill \\end{array}\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\(\\left\\{\\begin{array}{c}x=4y-3\\hfill \\\\ 4x-2y=-6\\hfill \\end{array}\\)<\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831239296\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167831239299\"><p id=\"fs-id1167826807915\"><\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d0<\/span>\\(\\left\\{\\begin{array}{c}y=7x-5\\hfill \\\\ 3x-2y=16\\hfill \\end{array}\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\(\\left\\{\\begin{array}{c}12x-5y=-42\\hfill \\\\ 3x+7y=-15\\hfill \\end{array}\\)<\/div><div data-type=\"solution\" id=\"fs-id1167835370118\"><p id=\"fs-id1167835370121\"><span class=\"token\">\u24d0<\/span> substitution <span class=\"token\">\u24d1<\/span> elimination<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831920498\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167831920500\"><p id=\"fs-id1167831920502\"><\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d0<\/span>\\(\\left\\{\\begin{array}{c}y=4x+9\\hfill \\\\ 5x-2y=-21\\hfill \\end{array}\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\(\\left\\{\\begin{array}{c}9x-4y=24\\hfill \\\\ 3x+5y=-14\\hfill \\end{array}\\)<\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834394773\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167834394775\"><p id=\"fs-id1167830963602\"><\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d0<\/span>\\(\\left\\{\\begin{array}{c}14x-15y=-30\\hfill \\\\ 7x+2y=10\\hfill \\end{array}\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\(\\left\\{\\begin{array}{c}x=9y-11\\hfill \\\\ 2x-7y=-27\\hfill \\end{array}\\)<\/div><div data-type=\"solution\" id=\"fs-id1167835379819\"><p id=\"fs-id1167835379821\"><span class=\"token\">\u24d0<\/span> elimination <span class=\"token\">\u24d1<\/span> substituion<\/p><\/div><\/div><\/div><div class=\"writing\" data-depth=\"2\" id=\"fs-id1167830984904\"><h4 data-type=\"title\">Writing Exercises<\/h4><div data-type=\"exercise\" id=\"fs-id1167830984912\"><div data-type=\"problem\" id=\"fs-id1167830984914\"><p id=\"fs-id1167828426612\">In a system of linear equations, the two equations have the same intercepts. Describe the possible solutions to the system.<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167830964362\"><div data-type=\"problem\" id=\"fs-id1167830964364\"><p>Solve the system of equations by substitution and explain all your steps in words: \\(\\left\\{\\begin{array}{c}3x+y=12\\hfill \\\\ x=y-8\\hfill \\end{array}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834397152\"><p id=\"fs-id1167834397154\">Answers will vary.<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167832153527\"><div data-type=\"problem\"><p>Solve the system of equations by elimination and explain all your steps in words: \\(\\left\\{\\begin{array}{c}5x+4y=10\\hfill \\\\ 2x=3y+27\\hfill \\end{array}.\\)<\/p><\/div><\/div><div data-type=\"exercise\"><div data-type=\"problem\" id=\"fs-id1167835433652\"><p id=\"fs-id1167835433654\">Solve the system of equations \\(\\left\\{\\begin{array}{c}x+y=10\\hfill \\\\ x-y=6\\hfill \\end{array}\\)<\/p><p id=\"fs-id1167834494857\"><span class=\"token\">\u24d0<\/span> by graphing <span class=\"token\">\u24d1<\/span> by substitution<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d2<\/span> Which method do you prefer? Why?<\/div><div data-type=\"solution\" id=\"fs-id1167834219257\"><p id=\"fs-id1167834219259\">Answers will vary.<\/p><\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1167834219266\"><h4 data-type=\"title\">Self Check<\/h4><p id=\"fs-id1167834397580\">After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.<\/p><span data-type=\"media\" id=\"fs-id1167832060286\" data-alt=\"This table has 4 columns 5 rows and a header row. The header row labels each column: I can, confidently, with some help and no, I don\u2019t get it. The first column has the following statements: determine whether an ordered pair is a solution of a system of equations, solve a system of linear equations by graphing, solve a system of equations by substitution, solve a system of equations by elimination, choose the most convenient method to solve a system of linear equations. The remaining columns are blank.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_01_201_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This table has 4 columns 5 rows and a header row. The header row labels each column: I can, confidently, with some help and no, I don\u2019t get it. The first column has the following statements: determine whether an ordered pair is a solution of a system of equations, solve a system of linear equations by graphing, solve a system of equations by substitution, solve a system of equations by elimination, choose the most convenient method to solve a system of linear equations. The remaining columns are blank.\"><\/span><p id=\"fs-id1167832060283\">If most of your checks were:<\/p><p id=\"fs-id1167832150935\"><strong data-effect=\"bold\">\u2026confidently.<\/strong> Congratulations! You have achieved the objectives in this section. Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific.<\/p><p id=\"fs-id1167832150945\"><strong data-effect=\"bold\">\u2026with some help.<\/strong> This must be addressed quickly because topics you do not master become potholes in your road to success. In math every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Who can you ask for help? Your fellow classmates and instructor are good resources. Is there a place on campus where math tutors are available? Can your study skills be improved?<\/p><p id=\"fs-id1167830699785\"><strong data-effect=\"bold\">\u2026no - I don\u2019t get it!<\/strong> This is a warning sign and you must not ignore it. You should get help right away or you will quickly be overwhelmed. See your instructor as soon as you can to discuss your situation. Together you can come up with a plan to get you the help you need.<\/p><\/div><\/div><div data-type=\"glossary\" class=\"textbox shaded\"><h3 data-type=\"glossary-title\">Glossary<\/h3><dl id=\"fs-id1167835321158\"><dt>coincident lines<\/dt><dd id=\"fs-id1167835321161\">Coincident lines have the same slope and same <em data-effect=\"italics\">y<\/em>-intercept.<\/dd><\/dl><dl id=\"fs-id1167831890606\"><dt>consistent and inconsistent systems<\/dt><dd id=\"fs-id1167831890609\">Consistent system of equations is a system of equations with at least one solution; inconsistent system of equations is a system of equations with no solution.<\/dd><\/dl><dl id=\"fs-id1167831890614\"><dt>solutions of a system of equations<\/dt><dd id=\"fs-id1167834191858\">Solutions of a system of equations are the values of the variables that make <em data-effect=\"italics\">all<\/em> the equations true; solution is represented by an ordered pair \\(\\left(x,y\\right).\\)<\/dd><\/dl><dl id=\"fs-id1167828182376\"><dt>system of linear equations<\/dt><dd id=\"fs-id1167828182379\">When two or more linear equations are grouped together, they form a system of linear equations.<\/dd><\/dl><\/div>\n","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>Determine whether an ordered pair is a solution of a system of equations<\/li>\n<li>Solve a system of linear equations by graphing<\/li>\n<li>Solve a system of equations by substitution<\/li>\n<li>Solve a system of equations by elimination<\/li>\n<li>Choose the most convenient method to solve a system of linear equations<\/li>\n<\/ul>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167830925402\" class=\"be-prepared\">\n<p id=\"fs-id1167835305070\">Before you get started, take this readiness quiz.<\/p>\n<ol id=\"fs-id1167831116154\" type=\"1\">\n<li>For the equation <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-61870c0df3b7092f8e34e17103c593f8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;&#120;&#45;&#52;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"89\" style=\"vertical-align: -6px;\" \/>\n<div data-type=\"newline\"><\/div>\n<p> <span class=\"token\">\u24d0<\/span> Is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d7121f8d5dd68d2f78cd8dd11d35a5dd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/> a solution? <span class=\"token\">\u24d1<\/span> Is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a17ffffadfb8456567f4803e943d15a2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#44;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"65\" style=\"vertical-align: -4px;\" \/> a solution?<\/p>\n<div data-type=\"newline\"><\/div>\n<p> If you missed this problem, review <a href=\"\/contents\/26e8f94c-1f76-46ec-8e6c-344f06971cf5#fs-id1167835400321\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li>\n<li>Find the slope and <em data-effect=\"italics\">y<\/em>-intercept of the line <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-578935c1636be9a00f6f3b61ebdacbef_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#120;&#45;&#121;&#61;&#49;&#50;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"96\" style=\"vertical-align: -4px;\" \/>\n<div data-type=\"newline\"><\/div>\n<p> If you missed this problem, review <a href=\"\/contents\/c7953cb6-51e3-48e7-9969-821f34daec42#fs-id1167835342973\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li>\n<li>Find the <em data-effect=\"italics\">x-<\/em> and <em data-effect=\"italics\">y<\/em>-intercepts of the line <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9a41dd20c17e42002894abd984bd0815_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#120;&#45;&#51;&#121;&#61;&#49;&#50;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"104\" style=\"vertical-align: -4px;\" \/>\n<div data-type=\"newline\"><\/div>\n<p> If you missed this problem, review <a href=\"\/contents\/26e8f94c-1f76-46ec-8e6c-344f06971cf5#fs-id1167827987818\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li>\n<\/ol>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167835596566\">\n<h3 data-type=\"title\">Determine Whether an Ordered Pair is a Solution of a System of Equations<\/h3>\n<p>In <a href=\"\/contents\/85b55407-c981-455f-9c47-d72340bd1dbb\" class=\"target-chapter\">Solving Linear Equations<\/a>, we learned how to solve linear equations with one variable. Now we will work with two or more linear equations grouped together, which is known as a <span data-type=\"term\">system of linear equations<\/span>.<\/p>\n<div data-type=\"note\">\n<div data-type=\"title\">System of Linear Equations<\/div>\n<p id=\"fs-id1167834506071\">When two or more linear equations are grouped together, they form a <strong data-effect=\"bold\">system of linear equations<\/strong>.<\/p>\n<\/div>\n<p>In this section, we will focus our work on systems of two linear equations in two unknowns. We will solve larger systems of equations later in this chapter.<\/p>\n<p id=\"fs-id1167831040311\">An example of a system of two linear equations is shown below. We use a brace to show the two equations are grouped together to form a system of equations.<\/p>\n<div data-type=\"equation\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-67d56114ed51dc4537e8df2392156c29_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;&#120;&#43;&#121;&#61;&#55;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#120;&#45;&#50;&#121;&#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=\"102\" style=\"vertical-align: -17px;\" \/><\/div>\n<p id=\"fs-id1167835513953\">A linear equation in two variables, such as <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-eaf8b19c0a9f71eea2565b29acaee03d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#120;&#43;&#121;&#61;&#55;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"87\" style=\"vertical-align: -4px;\" \/> has an infinite number of solutions. Its graph is a line. Remember, every point on the line is a solution to the equation and every solution to the equation is a point on the line.<\/p>\n<p id=\"fs-id1167835301937\">To solve a system of two linear equations, we want to find the values of the variables that are solutions to <em data-effect=\"italics\">both<\/em> equations. In other words, we are looking for the ordered pairs <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-aee61752ae042431152087f74b766103_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#44;&#121;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"39\" style=\"vertical-align: -4px;\" \/> that make both equations true. These are called the <span data-type=\"term\">solutions of a system of equations<\/span>.<\/p>\n<div data-type=\"note\" id=\"fs-id1167834063240\">\n<div data-type=\"title\">Solutions of a System of Equations<\/div>\n<p id=\"fs-id1167831884388\">The <strong data-effect=\"bold\">solutions of a system of equations<\/strong> are the values of the variables that make <em data-effect=\"italics\">all<\/em> the equations true. A solution of a system of two linear equations is represented by an ordered pair <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0ab61ec2033ed5b69d0447eac5d6a4f8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#44;&#121;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"47\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<p id=\"fs-id1167835167507\">To determine if an ordered pair is a solution to a system of two equations, we substitute the values of the variables into each equation. If the ordered pair makes both equations true, it is a solution to the system.<\/p>\n<div data-type=\"example\" id=\"fs-id1167835326515\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167835360973\">\n<div data-type=\"problem\" id=\"fs-id1167832056984\">\n<p id=\"fs-id1167834190204\">Determine whether the ordered pair is a solution to the system <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-90db7513e0516c68a1bfde9717a49331_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;&#50;&#120;&#45;&#121;&#61;&#45;&#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;\" \/><\/p>\n<p id=\"fs-id1167835483822\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ef9b0ead245d0b508218b1cb59d48442_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#44;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"65\" style=\"vertical-align: -4px;\" \/><span class=\"token\">\u24d1<\/span><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;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834397737\">\n<p id=\"fs-id1167835330326\"><span class=\"token\">\u24d0<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167834135019\" data-alt=\"The equations are x minus y equals minus 1 and 2 x minus y equals minus 5. We substitute x equal to minus 2 and y equal to minus 1 into both equations. So, x minus y equals minus 1 becomes minus 2 minus open parentheses minus 1 close parentheses equal to or not equal to minus 1. Simplifying, we get minus 1 equals minus 1 which is correct. The equation 2 x minus y equals minus 5 becomes 2 times minus 2 minus open parentheses minus 1 close parentheses equal to or not equal to minus 5. Simplifying, we get 5 not equal to minus 5. Hence, the ordered pair minus 2, minus 1 does not make both equations true. So, it is not a solution.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_01_001_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The equations are x minus y equals minus 1 and 2 x minus y equals minus 5. We substitute x equal to minus 2 and y equal to minus 1 into both equations. So, x minus y equals minus 1 becomes minus 2 minus open parentheses minus 1 close parentheses equal to or not equal to minus 1. Simplifying, we get minus 1 equals minus 1 which is correct. The equation 2 x minus y equals minus 5 becomes 2 times minus 2 minus open parentheses minus 1 close parentheses equal to or not equal to minus 5. Simplifying, we get 5 not equal to minus 5. Hence, the ordered pair minus 2, minus 1 does not make both equations true. So, it is not a solution.\" \/><\/span><\/p>\n<p id=\"fs-id1167831824467\"><span class=\"token\">\u24d1<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" data-alt=\"We substitute x equal to minus 4 and y equal to minus 3 into both equations. So, x minus y equals minus 1 becomes minus 4 minus open parentheses minus 3 close parentheses equal to or not equal to minus 1. Simplifying, we get minus 1 equals minus 1, which is correct. The equation 2 x minus y equals minus 5 becomes 2 times minus 4 minus open parentheses minus 3 close parentheses equal to or not equal to minus 5. Simplifying, we get minus 5 equals minus 5, which is correct. The ordered pair minus 4, minus 3 does make both equations true. Hence, it is a solution.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_01_002_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"We substitute x equal to minus 4 and y equal to minus 3 into both equations. So, x minus y equals minus 1 becomes minus 4 minus open parentheses minus 3 close parentheses equal to or not equal to minus 1. Simplifying, we get minus 1 equals minus 1, which is correct. The equation 2 x minus y equals minus 5 becomes 2 times minus 4 minus open parentheses minus 3 close parentheses equal to or not equal to minus 5. Simplifying, we get minus 5 equals minus 5, which is correct. The ordered pair minus 4, minus 3 does make both equations true. Hence, it is a solution.\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167832066187\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167834346678\">\n<div data-type=\"problem\" id=\"fs-id1167832056851\">\n<p>Determine whether the ordered pair is a solution to the system <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7e7a02521946325d1cff4cd47cd1e771_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;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#51;&#120;&#43;&#121;&#61;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#120;&#43;&#50;&#121;&#61;&#45;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#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=\"128\" style=\"vertical-align: -6px;\" \/><\/p>\n<p><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-876cb3caf33e984a34d443f2b79f105e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#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=\"52\" style=\"vertical-align: -4px;\" \/><span class=\"token\">\u24d1<\/span><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-id1167835253906\">\n<p id=\"fs-id1167835287970\"><span class=\"token\">\u24d0<\/span> yes <span class=\"token\">\u24d1<\/span> no<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167834132168\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167835331532\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167832052650\">Determine whether the ordered pair is a solution to the system <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-932dcda31337b3bc10ffff8d91b1bd72_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;&#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;&#56;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#45;&#51;&#120;&#45;&#121;&#61;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#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=\"128\" style=\"vertical-align: -6px;\" \/><\/p>\n<p id=\"fs-id1167831921117\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8fbb5d034d8bc7481119846ba5facddb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#44;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"52\" style=\"vertical-align: -4px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-627391f2fad7216057fc57692a374893_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#44;&#50;&#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 data-type=\"solution\" id=\"fs-id1167835545439\">\n<p id=\"fs-id1167826967139\"><span class=\"token\">\u24d0<\/span> no <span class=\"token\">\u24d1<\/span> yes<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167832086919\">\n<h3 data-type=\"title\">Solve a System of Linear Equations by Graphing<\/h3>\n<p id=\"fs-id1167831893670\">In this section, we will use three methods to solve a system of linear equations. The first method we\u2019ll use is graphing.<\/p>\n<p id=\"fs-id1167831239781\">The graph of a linear equation is a line. Each point on the line is a solution to the equation. For a system of two equations, we will graph two lines. Then we can see all the points that are solutions to each equation. And, by finding what the lines have in common, we\u2019ll find the solution to the system.<\/p>\n<p id=\"fs-id1167835267322\">Most linear equations in one variable have one solution, but we saw that some equations, called contradictions, have no solutions and for other equations, called identities, all numbers are solutions.<\/p>\n<p id=\"fs-id1167832082005\">Similarly, when we solve a system of two linear equations represented by a graph of two lines in the same plane, there are three possible cases, as shown.<\/p>\n<div class=\"bc-figure figure\" id=\"CNX_IntAlg_Figure_04_01_003\"><span data-type=\"media\" id=\"fs-id1167834188679\" data-alt=\"Figure shows three graphs. In the first, the lines intersect at point 3, minus 1. The intersecting lines have one point in common. There is one solution to the system. In the second graph, the lines are parallel. Parallel lines have no points in common. There is no solution to the system. The third graph has only one line. Here, both equations give the same line. Because we have only one line, there are infinite many solutions.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_01_003_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Figure shows three graphs. In the first, the lines intersect at point 3, minus 1. The intersecting lines have one point in common. There is one solution to the system. In the second graph, the lines are parallel. Parallel lines have no points in common. There is no solution to the system. The third graph has only one line. Here, both equations give the same line. Because we have only one line, there are infinite many solutions.\" \/><\/span><\/div>\n<p id=\"fs-id1167834053646\">Each time we demonstrate a new method, we will use it on the same system of linear equations. At the end of the section you\u2019ll decide which method was the most convenient way to solve this system.<\/p>\n<div data-type=\"example\" id=\"fs-id1167834279490\" class=\"textbox textbox--examples\">\n<div data-type=\"title\">How to Solve a System of Equations by Graphing<\/div>\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1167832057850\">\n<p id=\"fs-id1167835280797\">Solve the system by graphing <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-93c77aec095fb80605576f00f515b39d_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;&#120;&#43;&#121;&#61;&#55;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#120;&#45;&#50;&#121;&#61;&#54;&#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-id1167832041475\"><span data-type=\"media\" data-alt=\"Step 1 is to graph the first equation. To graph the first line, write the equation in slope intercept form. So, 2 x plus y equals 7 becomes y equal to minus 2 x plus 7. Here, m is minus 2 and b is 7. So the graph will be a line with slope equal to minus 2 and y intercept equal to 7.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_01_004a_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 1 is to graph the first equation. To graph the first line, write the equation in slope intercept form. So, 2 x plus y equals 7 becomes y equal to minus 2 x plus 7. Here, m is minus 2 and b is 7. So the graph will be a line with slope equal to minus 2 and y intercept equal to 7.\" \/><\/span><span data-type=\"media\" id=\"fs-id1167835174193\" data-alt=\"Step 2 is to graph the second equation on the same rectangular coordinate system. To graph the second line, use intercepts. For x minus 2y equals 6, the intercepts are 0, minus 3 and 6, 0.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_01_004b_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 2 is to graph the second equation on the same rectangular coordinate system. To graph the second line, use intercepts. For x minus 2y equals 6, the intercepts are 0, minus 3 and 6, 0.\" \/><\/span><span data-type=\"media\" id=\"fs-id1167835321614\" data-alt=\"Step 3 is to determine whether the lines intersect, are parallel, or are the same line. Here, they intersect.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_01_004c_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 3 is to determine whether the lines intersect, are parallel, or are the same line. Here, they intersect.\" \/><\/span><span data-type=\"media\" id=\"fs-id1167832138637\" data-alt=\"Step 4 is to identify the solution to the system. If the lines intersect, identify the point of intersection. The lines intersect at 4, minus 1. Now, check to make sure it is a solution to both equations. When x and y are substituted with 4 and minus 1 respectively, both equations hold true. This is the solution to the system. In step 4, if the lines are parallel, the system has no solution and if the lines are the same, the system has an infinite number of solutions.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_01_004d_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 4 is to identify the solution to the system. If the lines intersect, identify the point of intersection. The lines intersect at 4, minus 1. Now, check to make sure it is a solution to both equations. When x and y are substituted with 4 and minus 1 respectively, both equations hold true. This is the solution to the system. In step 4, if the lines are parallel, the system has no solution and if the lines are the same, the system has an infinite number of solutions.\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167832195749\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167835367659\">\n<div data-type=\"problem\" id=\"fs-id1167826864846\">\n<p id=\"fs-id1167834233816\">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;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167826799434\">\n<p id=\"fs-id1167828401865\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f3b8ee106f7e5a95f1dd3e0ab5f16435_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#44;&#50;&#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>\n<div data-type=\"note\" id=\"fs-id1167832065745\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167834289447\">\n<div data-type=\"problem\" id=\"fs-id1167835368946\">\n<p id=\"fs-id1167831823922\">Solve the system by graphing: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b3e1238260018027bf15a64ae7c35021_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;&#45;&#120;&#43;&#121;&#61;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#51;&#120;&#43;&#50;&#121;&#61;&#49;&#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=\"132\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835350532\">\n<p id=\"fs-id1167835213780\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3fca1a384cf5042876a719066cbbb127_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#44;&#51;&#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>\n<p id=\"fs-id1167826804684\">The steps to use to solve a system of linear equations by graphing are shown here.<\/p>\n<div data-type=\"note\" id=\"fs-id1167835310198\" class=\"howto\">\n<div data-type=\"title\">Solve a system of linear equations by graphing.<\/div>\n<ol type=\"1\" class=\"stepwise\">\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 lines intersect, are parallel, or are the same line.<\/li>\n<li>Identify the solution to the system.\n<ul class=\"open-circle\">\n<li>If the lines intersect, identify the point of intersection. This is the solution to the system.<\/li>\n<li>If the lines are parallel, the system has no solution.<\/li>\n<li>If the lines are the same, the system has an infinite number of solutions.<\/li>\n<\/ul>\n<\/li>\n<li>Check the solution in both equations.<\/li>\n<\/ol>\n<\/div>\n<p id=\"fs-id1167832055393\">In the next example, we\u2019ll first re-write the equations into slope\u2013intercept form as this will make it easy for us to quickly graph the lines.<\/p>\n<div data-type=\"example\" id=\"fs-id1167835410272\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167834185475\">\n<div data-type=\"problem\" id=\"fs-id1167830706036\">\n<p id=\"fs-id1167831879885\">Solve the system by graphing: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f00ab61c2392cd4cd04c9e2e92c6756a_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;&#43;&#121;&#61;&#45;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#50;&#120;&#43;&#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=\"128\" style=\"vertical-align: -17px;\" \/> <\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167832042531\">\n<p id=\"fs-id1167834065071\">We\u2019ll solve both of these equations for <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0af556714940c351c933bba8cf840796_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\" \/> so that we can easily graph them using their slopes and <em data-effect=\"italics\">y<\/em>-intercepts.<\/p>\n<table id=\"fs-id1167826993913\" class=\"unnumbered unstyled can-break\" summary=\"The equations are 3 x plus y equals minus 1 and 2 x plus y equals 0. Solving the first equation for y, we get y equal to minus 3 x minus 1. So, slope is minus 3 and y intercept is minus 1. Similarly, for the second equation, we get y equal to minus 2 x. So, slope is minus 2 and y intercept is 0. Graphing the lines, we get the point of intersection minus 1, 2. To check the solution in both equations, we substitute minus 1 and 2 for x and y respectively. Both equations hold true. The solution is minus 1, 2.\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834161795\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_01_005b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Solve the first equation for <em data-effect=\"italics\">y<\/em>.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167826804609\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_01_005c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Find the slope and <em data-effect=\"italics\">y<\/em>-intercept.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835327264\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_01_005d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Solve the second equation for <em data-effect=\"italics\">y<\/em>.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834224888\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_01_005e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Find the slope and <em data-effect=\"italics\">y<\/em>-intercept.<\/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\/2018\/12\/CNX_IntAlg_Figure_04_01_005f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Graph the lines.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835422879\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_01_005g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Determine the point of intersection.<\/td>\n<td data-valign=\"top\" data-align=\"left\">The lines intersect at <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ef1573c33b4f63176af1871c59d0fba4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#44;&#50;&#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>\n<td data-valign=\"top\" data-align=\"left\">Check the solution in both equations.<\/p>\n<div data-type=\"newline\"><\/div>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167835369250\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_01_005a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/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-ef1573c33b4f63176af1871c59d0fba4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#44;&#50;&#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-id1167832086965\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167835639953\">\n<div data-type=\"problem\" id=\"fs-id1167835332201\">\n<p id=\"fs-id1167835329488\">Solve the system by graphing: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2224b844096f02f3dc2d954ec08eb05d_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;&#45;&#120;&#43;&#121;&#61;&#49;&#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;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"123\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\">\n<p id=\"fs-id1167834459142\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-aecc845ff8fcecb422091e6436adca8d_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;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167835366362\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167832056565\">\n<div data-type=\"problem\" id=\"fs-id1167834301215\">\n<p id=\"fs-id1167834185253\">Solve the system by graphing: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-250a6e43221717bebed7122cb3a5e4b2_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;&#120;&#43;&#121;&#61;&#54;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#120;&#43;&#121;&#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=\"114\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167832058926\">\n<p id=\"fs-id1167835262504\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-12aeb9559a0d3673fa3759c68e52b96b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#44;&#45;&#52;&#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>\n<p id=\"fs-id1167834462906\">In all the systems of linear equations so far, the lines intersected and the solution was one point. In the next two examples, we\u2019ll look at a system of equations that has no solution and at a system of equations that has an infinite number of solutions.<\/p>\n<div data-type=\"example\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167835609626\">\n<div data-type=\"problem\" id=\"fs-id1167834222410\">\n<p id=\"fs-id1167835171687\">Solve the system by graphing: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ea4118511a2c356ccdff5a6a4b720570_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;&#49;&#125;&#123;&#50;&#125;&#120;&#45;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#120;&#45;&#50;&#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=\"44\" width=\"116\" style=\"vertical-align: -17px;\" \/> <\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167827943084\">\n<table id=\"fs-id1167834537193\" class=\"unnumbered unstyled can-break\" summary=\"The equations are y equal to half x minus 3 and x minus 2y equal to 4. The slope and y intercept of the first equation are half and minus 3 respectively. To graph the second equation, we will use the intercepts x equal to 0 when y equal to minus 2 and x equal to 4 when y equal to 0. We graph the lines to determine the point of intersection. The lines are parallel. Since no point is on both lines, there is no ordered pair that makes both equations true. There is no solution to this system.\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835340829\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_01_006a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">To graph the first equation, we will use its<\/p>\n<div data-type=\"newline\"><\/div>\n<p>slope and <em data-effect=\"italics\">y<\/em>-intercept.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834195984\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_01_006b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td>To graph the second equation, we will use<\/p>\n<div data-type=\"newline\"><\/div>\n<p>the intercepts.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835365745\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_01_006c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834309752\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_01_006d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Graph the lines.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835595311\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_01_006e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Determine the points of intersection.<\/td>\n<td data-valign=\"top\" data-align=\"left\">The lines are parallel.<\/p>\n<div data-type=\"newline\"><\/div>\n<p> Since no point is on both lines, there is no<\/p>\n<div data-type=\"newline\"><\/div>\n<p>ordered pair that makes both equations<\/p>\n<div data-type=\"newline\"><\/div>\n<p>true. There is no solution to this system.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167835329190\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167835354645\">\n<div data-type=\"problem\" id=\"fs-id1167830702759\">\n<p id=\"fs-id1167835341562\">Solve the system by graphing: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-93464f18895fb875b7f5621896581607_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;&#52;&#125;&#120;&#43;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#120;&#43;&#52;&#121;&#61;&#45;&#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=\"44\" width=\"130\" style=\"vertical-align: -17px;\" \/> <\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835262981\">\n<p>no solution<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167835364524\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167834448636\">\n<div data-type=\"problem\" id=\"fs-id1167835350341\">\n<p id=\"fs-id1167834279269\">Solve the system by graphing: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3742d0c62589fe07a881cb1860b3a22b_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;&#51;&#120;&#45;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#54;&#120;&#45;&#50;&#121;&#61;&#54;&#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=\"123\" style=\"vertical-align: -17px;\" \/> <\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835423011\">\n<p id=\"fs-id1167834396875\">no solution<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1168754384001\">Sometimes the equations in a system represent the same line. Since every point on the line makes both equations true, there are infinitely many ordered pairs that make both equations true. There are infinitely many solutions to the system.<\/p>\n<div data-type=\"example\" id=\"fs-id1167835418145\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1167834431373\">\n<p id=\"fs-id1167835479000\">Solve the system by graphing: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-007b48e429a6e87df1723b5dc5ca1841_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;&#45;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#45;&#54;&#120;&#43;&#51;&#121;&#61;&#45;&#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=\"150\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835307320\">\n<table id=\"fs-id1167835327859\" class=\"unnumbered unstyled can-break\" summary=\"The equations are y equal to 2 x minus 3 and minus 6x plus 3y equal to minus 9. The slope and y intercept of the first equation are 2 and minus 3 respectively. In the second equation, when x is 0, y is minus 3 and when y is 0, x is 3 by 2. We use these points for graphing the lines to determine the point of intersection. The lines are the same. Since every point on the line makes both equations true, there are infinite many ordered pairs that make both equations true. There are infinite many solutions to this system.\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834191115\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_01_007a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Find the slope and <em data-effect=\"italics\">y<\/em>-intercept of the first equation.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835334409\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_01_007b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Find the intercepts of the second equation.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167831891526\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_01_007c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835240354\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_01_007d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Graph the lines.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835258752\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_01_007e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td data-valign=\"top\" data-align=\"left\">The lines are the same!<\/p>\n<div data-type=\"newline\"><\/div>\n<p>Since every point on the line makes both<\/p>\n<div data-type=\"newline\"><\/div>\n<p>equations true, there are infinitely many<\/p>\n<div data-type=\"newline\"><\/div>\n<p>ordered pairs that make both equations true.<\/p>\n<div data-type=\"newline\"><\/div>\n<p>There are infinitely many solutions to this system.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1167835238066\">If you write the second equation in slope-intercept form, you may recognize that the equations have the same slope and same <em data-effect=\"italics\">y<\/em>-intercept.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167835420544\">\n<div data-type=\"problem\" id=\"fs-id1167835288349\">\n<p id=\"fs-id1167826978106\">Solve the system by graphing: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3d13a2aff02eb9c7d37e43da6e79158c_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;&#51;&#120;&#45;&#54;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#54;&#120;&#43;&#50;&#121;&#61;&#45;&#49;&#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=\"145\" style=\"vertical-align: -17px;\" \/> <\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167832043521\">\n<p id=\"fs-id1167834459122\">infinitely many solutions<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167835186730\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167834516134\">\n<div data-type=\"problem\" id=\"fs-id1167831887685\">\n<p id=\"fs-id1167830866127\">Solve the system by graphing: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f86f12cb8da8da5fd9e0a8cde4db3f9d_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;&#49;&#125;&#123;&#50;&#125;&#120;&#45;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#50;&#120;&#45;&#52;&#121;&#61;&#49;&#54;&#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=\"44\" width=\"132\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167826782418\">\n<p id=\"fs-id1167830838187\">infinitely many solutions<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167831954237\">When we graphed the second line in the last example, we drew it right over the first line. We say the two lines are <span data-type=\"term\">coincident<\/span>. Coincident lines have the same slope and same <em data-effect=\"italics\">y-<\/em>intercept.<\/p>\n<div data-type=\"note\" id=\"fs-id1167835530462\">\n<div data-type=\"title\">Coincident Lines<\/div>\n<p id=\"fs-id1167835351616\"><strong data-effect=\"bold\">Coincident lines<\/strong> have the same slope and same <em data-effect=\"italics\">y-<\/em>intercept.<\/p>\n<\/div>\n<p id=\"fs-id1167831191430\">The systems of equations in <a href=\"#fs-id1167834279490\" class=\"autogenerated-content\">(Figure)<\/a> and <a href=\"#fs-id1167835410272\" class=\"autogenerated-content\">(Figure)<\/a> each had two intersecting lines. Each system had one solution.<\/p>\n<p id=\"fs-id1167827987930\">In <a href=\"#fs-id1167835418145\" class=\"autogenerated-content\">(Figure)<\/a>, the equations gave coincident lines, and so the system had infinitely many solutions.<\/p>\n<p id=\"fs-id1167834063977\">The systems in those three examples had at least one solution. A system of equations that has at least one solution is called a <em data-effect=\"italics\">consistent<\/em> system.<\/p>\n<p id=\"fs-id1167831880100\">A system with parallel lines, like <a href=\"#fs-id1167835307740\" class=\"autogenerated-content\">(Figure)<\/a>, has no solution. We call a system of equations like this <em data-effect=\"italics\">inconsistent.<\/em> It has no solution.<\/p>\n<div data-type=\"note\" id=\"fs-id1167835370836\">\n<div data-type=\"title\">Consistent and Inconsistent Systems<\/div>\n<p id=\"fs-id1167835379459\">A <span data-type=\"term\">consistent system of equations<\/span> is a system of equations with at least one solution.<\/p>\n<p id=\"fs-id1167835239043\">An <span data-type=\"term\">inconsistent system of equations<\/span> is a system of equations with no solution.<\/p>\n<\/div>\n<p id=\"fs-id1167835343554\">We also categorize the equations in a system of equations by calling the equations <em data-effect=\"italics\">independent<\/em> or <em data-effect=\"italics\">dependent<\/em>. If two equations are independent, they each have their own set of solutions. Intersecting lines and parallel lines are independent.<\/p>\n<p>If two equations are dependent, all the solutions of one equation are also solutions of the other equation. When we graph two dependent equations, we get coincident lines.<\/p>\n<p id=\"fs-id1167832057993\">Let\u2019s sum this up by looking at the graphs of the three types of systems. See below and <a href=\"#fs-id1167832096973\" class=\"autogenerated-content\">(Figure)<\/a>.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167835170706\" data-alt=\"The figure shows three graphs. The first one has two intersecting line. The second one has two parallel lines. The third one has only one line. This is labeled coincident.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_01_008_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows three graphs. The first one has two intersecting line. The second one has two parallel lines. The third one has only one line. This is labeled coincident.\" \/><\/span><\/p>\n<table id=\"fs-id1167832096973\" summary=\"The table shows that intersecting lines have 1 solution that is consistent and independent. Parallel lines have no solution and are inconsistent and independent. Coincident lines have infinitely many solutions and are consistent and dependent.\">\n<thead>\n<tr>\n<th data-valign=\"middle\" data-align=\"left\">Lines<\/th>\n<th data-valign=\"middle\" data-align=\"left\">Intersecting<\/th>\n<th data-valign=\"middle\" data-align=\"left\">Parallel<\/th>\n<th data-valign=\"middle\" data-align=\"left\">Coincident<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td data-valign=\"middle\" data-align=\"left\"><strong data-effect=\"bold\">Number of solutions<\/strong><\/td>\n<td data-valign=\"middle\" data-align=\"left\">1 point<\/td>\n<td data-valign=\"middle\" data-align=\"left\">No solution<\/td>\n<td data-valign=\"middle\" data-align=\"left\">Infinitely many<\/td>\n<\/tr>\n<tr>\n<td data-valign=\"middle\" data-align=\"left\"><strong data-effect=\"bold\">Consistent\/inconsistent<\/strong><\/td>\n<td data-valign=\"middle\" data-align=\"left\">Consistent<\/td>\n<td data-valign=\"middle\" data-align=\"left\">Inconsistent<\/td>\n<td data-valign=\"middle\" data-align=\"left\">Consistent<\/td>\n<\/tr>\n<tr>\n<td data-valign=\"middle\" data-align=\"left\"><strong data-effect=\"bold\">Dependent\/ independent<\/strong><\/td>\n<td data-valign=\"middle\" data-align=\"left\">Independent<\/td>\n<td data-valign=\"middle\" data-align=\"left\">Independent<\/td>\n<td data-valign=\"middle\" data-align=\"left\">Dependent<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div data-type=\"example\" id=\"fs-id1167834191318\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167834156729\">\n<div data-type=\"problem\" id=\"fs-id1167834184039\">\n<p id=\"fs-id1167834189749\">Without graphing, determine the number of solutions and then classify the system of equations.<\/p>\n<p id=\"fs-id1167834429136\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9ce555698ccc78e2d269c94c0b277c1a_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;&#51;&#120;&#45;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#54;&#120;&#45;&#50;&#121;&#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=\"119\" style=\"vertical-align: -17px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5a14d11691ed831db79ccb667551d56d_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;&#120;&#43;&#121;&#61;&#45;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#120;&#45;&#53;&#121;&#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=\"116\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167831882192\">\n<p id=\"fs-id1167835363341\"><span class=\"token\">\u24d0<\/span> We will compare the slopes and intercepts of the two lines.<\/p>\n<p id=\"fs-id1167835342275\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f0fcc012e50b82be54a1c9428e5471ac_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;&#125;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#125;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#84;&#104;&#101;&#32;&#102;&#105;&#114;&#115;&#116;&#32;&#101;&#113;&#117;&#97;&#116;&#105;&#111;&#110;&#32;&#105;&#115;&#32;&#97;&#108;&#114;&#101;&#97;&#100;&#121;&#32;&#105;&#110;&#32;&#115;&#108;&#111;&#112;&#101;&#45;&#105;&#110;&#116;&#101;&#114;&#99;&#101;&#112;&#116;&#32;&#102;&#111;&#114;&#109;&#46;&#125;&#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;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#55;&#46;&#55;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#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;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#121;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#51;&#120;&#45;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#54;&#120;&#45;&#50;&#121;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#49;&#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;&#92;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#51;&#46;&#49;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#121;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#61;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#51;&#120;&#45;&#49;&#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=\"84\" width=\"775\" style=\"vertical-align: -47px;\" \/><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-49ca660b26baab8fe982981bd5184a05_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;&#125;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#87;&#114;&#105;&#116;&#101;&#32;&#116;&#104;&#101;&#32;&#115;&#101;&#99;&#111;&#110;&#100;&#32;&#101;&#113;&#117;&#97;&#116;&#105;&#111;&#110;&#32;&#105;&#110;&#32;&#115;&#108;&#111;&#112;&#101;&#45;&#105;&#110;&#116;&#101;&#114;&#99;&#101;&#112;&#116;&#32;&#102;&#111;&#114;&#109;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#70;&#105;&#110;&#100;&#32;&#116;&#104;&#101;&#32;&#115;&#108;&#111;&#112;&#101;&#32;&#97;&#110;&#100;&#32;&#105;&#110;&#116;&#101;&#114;&#99;&#101;&#112;&#116;&#32;&#111;&#102;&#32;&#101;&#97;&#99;&#104;&#32;&#108;&#105;&#110;&#101;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#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;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#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;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#54;&#120;&#45;&#50;&#121;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#49;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#50;&#121;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#45;&#54;&#120;&#43;&#49;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#45;&#50;&#121;&#125;&#123;&#45;&#50;&#125;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#45;&#54;&#120;&#43;&#49;&#50;&#125;&#123;&#45;&#50;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#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;&#54;&#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;&#51;&#120;&#45;&#49;&#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;&#51;&#120;&#45;&#54;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#109;&#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;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#109;&#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;&#98;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#45;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#98;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#45;&#54;&#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;&#92;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#105;&#110;&#99;&#101;&#32;&#116;&#104;&#101;&#32;&#115;&#108;&#111;&#112;&#101;&#115;&#32;&#97;&#114;&#101;&#32;&#116;&#104;&#101;&#32;&#115;&#97;&#109;&#101;&#32;&#97;&#110;&#100;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#121;&#92;&#116;&#101;&#120;&#116;&#123;&#45;&#105;&#110;&#116;&#101;&#114;&#99;&#101;&#112;&#116;&#115;&#32;&#97;&#114;&#101;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#100;&#105;&#102;&#102;&#101;&#114;&#101;&#110;&#116;&#44;&#32;&#116;&#104;&#101;&#32;&#108;&#105;&#110;&#101;&#115;&#32;&#97;&#114;&#101;&#32;&#112;&#97;&#114;&#97;&#108;&#108;&#101;&#108;&#46;&#125;&#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=\"390\" width=\"841\" style=\"vertical-align: -190px;\" \/><\/p>\n<p id=\"fs-id1167834367141\">A system of equations whose graphs are parallel lines has no solution and is inconsistent and independent.<\/p>\n<p id=\"fs-id1167826995612\"><span class=\"token\">\u24d1<\/span> We will compare the slope and intercepts of the two lines.<\/p>\n<p id=\"fs-id1167834084804\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-738b1a82b4ee1c6da0345a16f417b3e5_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;&#125;&#38;&#32;&#38;&#32;&#38;&#32;&#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;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#50;&#120;&#43;&#121;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#45;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#120;&#45;&#53;&#121;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#53;&#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;&#92;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#87;&#114;&#105;&#116;&#101;&#32;&#98;&#111;&#116;&#104;&#32;&#101;&#113;&#117;&#97;&#116;&#105;&#111;&#110;&#115;&#32;&#105;&#110;&#32;&#115;&#108;&#111;&#112;&#101;&#45;&#105;&#110;&#116;&#101;&#114;&#99;&#101;&#112;&#116;&#32;&#102;&#111;&#114;&#109;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#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;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#53;&#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;&#50;&#120;&#43;&#121;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#45;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#120;&#45;&#53;&#121;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#53;&#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;&#45;&#50;&#120;&#45;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#53;&#121;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#120;&#43;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#45;&#53;&#121;&#125;&#123;&#45;&#53;&#125;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#45;&#120;&#43;&#53;&#125;&#123;&#45;&#53;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#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;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#53;&#125;&#120;&#45;&#49;&#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;&#92;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#70;&#105;&#110;&#100;&#32;&#116;&#104;&#101;&#32;&#115;&#108;&#111;&#112;&#101;&#32;&#97;&#110;&#100;&#32;&#105;&#110;&#116;&#101;&#114;&#99;&#101;&#112;&#116;&#32;&#111;&#102;&#32;&#101;&#97;&#99;&#104;&#32;&#108;&#105;&#110;&#101;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#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;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#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;&#99;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#121;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#45;&#50;&#120;&#45;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#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;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#121;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#53;&#125;&#120;&#45;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#109;&#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;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#109;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#53;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#98;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#45;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#98;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#45;&#49;&#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;&#92;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#105;&#110;&#99;&#101;&#32;&#116;&#104;&#101;&#32;&#115;&#108;&#111;&#112;&#101;&#115;&#32;&#97;&#114;&#101;&#32;&#100;&#105;&#102;&#102;&#101;&#114;&#101;&#110;&#116;&#44;&#32;&#116;&#104;&#101;&#32;&#108;&#105;&#110;&#101;&#115;&#32;&#105;&#110;&#116;&#101;&#114;&#115;&#101;&#99;&#116;&#46;&#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=\"458\" width=\"832\" style=\"vertical-align: -223px;\" \/><\/p>\n<p id=\"fs-id1167835419049\">A system of equations whose graphs are intersect has 1 solution and is consistent and independent.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167832151520\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167832151097\">\n<div data-type=\"problem\" id=\"fs-id1167832151099\">\n<p id=\"fs-id1167835347863\">Without graphing, determine the number of solutions and then classify the system of equations.<\/p>\n<p id=\"fs-id1167835347866\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-268ac950028533cb2a6f7e84f4c6d3f3_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;&#45;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#52;&#120;&#43;&#50;&#121;&#61;&#57;&#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;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d78dd6d3e1b54ea94cf3eaff925b2bac_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;&#43;&#50;&#121;&#61;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#50;&#120;&#43;&#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=\"110\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167832076368\">\n<p id=\"fs-id1167832076370\"><span class=\"token\">\u24d0<\/span> no solution, inconsistent, independent <span class=\"token\">\u24d1<\/span> one solution, consistent, independent<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167831910867\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167835356197\">\n<div data-type=\"problem\" id=\"fs-id1167835356199\">\n<p id=\"fs-id1167835376695\">Without graphing, determine the number of solutions and then classify the system of equations.<\/p>\n<p id=\"fs-id1167835376698\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4ef5c3e3e4081958de66696105743f4e_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;&#49;&#125;&#123;&#51;&#125;&#120;&#45;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#120;&#45;&#51;&#121;&#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=\"44\" width=\"103\" style=\"vertical-align: -17px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c42b40080a9bb0c3587b0309d5ab98df_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;&#52;&#121;&#61;&#49;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#45;&#120;&#43;&#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=\"110\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834059399\">\n<p id=\"fs-id1167834059401\"><span class=\"token\">\u24d0<\/span> no solution, inconsistent, independent <span class=\"token\">\u24d1<\/span> one solution, consistent, independent<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167832060470\">Solving systems of linear equations by graphing is a good way to visualize the types of solutions that may result. However, there are many cases where solving a system by graphing is inconvenient or imprecise. If the graphs extend beyond the small grid with <em data-effect=\"italics\">x<\/em> and <em data-effect=\"italics\">y<\/em> both between <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-24329a36a4e10288288979f77a565ca2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"31\" style=\"vertical-align: -1px;\" \/> and 10, graphing the lines may be cumbersome. And if the solutions to the system are not integers, it can be hard to read their values precisely from a graph.<\/p>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167834233994\">\n<h3 data-type=\"title\">Solve a System of Equations by Substitution<\/h3>\n<p id=\"fs-id1167834234000\">We will now solve systems of linear equations by the substitution method.<\/p>\n<p id=\"fs-id1167835334266\">We will use the same system we used first for graphing.<\/p>\n<div data-type=\"equation\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-67d56114ed51dc4537e8df2392156c29_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;&#120;&#43;&#121;&#61;&#55;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#120;&#45;&#50;&#121;&#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=\"102\" style=\"vertical-align: -17px;\" \/><\/div>\n<p id=\"fs-id1167827943012\">We will first solve one of the equations for either <em data-effect=\"italics\">x<\/em> or <em data-effect=\"italics\">y<\/em>. We can choose either equation and solve for either variable\u2014but we\u2019ll try to make a choice that will keep the work easy.<\/p>\n<p id=\"fs-id1167834063721\">Then we substitute that expression into the other equation. The result is an equation with just one variable\u2014and we know how to solve those!<\/p>\n<p id=\"fs-id1167834301188\">After we find the value of one variable, we will substitute that value into one of the original equations and solve for the other variable. Finally, we check our solution and make sure it makes both equations true.<\/p>\n<div data-type=\"example\" id=\"fs-id1167835328722\" class=\"textbox textbox--examples\">\n<div data-type=\"title\">How to Solve a System of Equations by Substitution<\/div>\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1167835328726\">\n<p id=\"fs-id1167826802362\">Solve the system by substitution: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-93c77aec095fb80605576f00f515b39d_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;&#120;&#43;&#121;&#61;&#55;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#120;&#45;&#50;&#121;&#61;&#54;&#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-id1167835332322\"><span data-type=\"media\" id=\"fs-id1167835332325\" data-alt=\"The equations are 2 x plus y equals 7 and x minus 2y equals 6. Step 1 is to solve one of the equations for either variable. We\u2019ll solve the first equation for y. We get y equals 7 minus 2 x.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_01_009a_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The equations are 2 x plus y equals 7 and x minus 2y equals 6. Step 1 is to solve one of the equations for either variable. We\u2019ll solve the first equation for y. We get y equals 7 minus 2 x.\" \/><\/span><span data-type=\"media\" data-alt=\"In step 2, substitute the expression from step 1 into the other equation. We replace y in the second equation with the expression 7 minus 2 x. So, we get x minus 2 open parentheses 7 minus 2 x close parentheses equals 6.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_01_009b_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"In step 2, substitute the expression from step 1 into the other equation. We replace y in the second equation with the expression 7 minus 2 x. So, we get x minus 2 open parentheses 7 minus 2 x close parentheses equals 6.\" \/><\/span><span data-type=\"media\" id=\"fs-id1167835198382\" data-alt=\"Step 3 is to solve the resulting equation. Now we have an equation with just 1 variable. We solve it to get x equal to 4.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_01_009c_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 3 is to solve the resulting equation. Now we have an equation with just 1 variable. We solve it to get x equal to 4.\" \/><\/span><span data-type=\"media\" id=\"fs-id1167835229691\" data-alt=\"Step 4 is to substitute the solution in step 3 into one of the original equations to find the other variable. We\u2019ll use the first equation and replace x with 4. We get, 2 times 4 plus y equals 7. Simplifying, we get y equal to minus 1.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_01_009d_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 4 is to substitute the solution in step 3 into one of the original equations to find the other variable. We\u2019ll use the first equation and replace x with 4. We get, 2 times 4 plus y equals 7. Simplifying, we get y equal to minus 1.\" \/><\/span><span data-type=\"media\" id=\"fs-id1167832066001\" data-alt=\"Step 5 is to write the solution as an ordered pair. The ordered pair is 4, minus 1.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_01_009e_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 5 is to write the solution as an ordered pair. The ordered pair is 4, minus 1.\" \/><\/span><span data-type=\"media\" id=\"fs-id1167831086582\" data-alt=\"Step 6 is to check that the ordered pair is a solution to both original equations. To do that we Substitute x equal to 4 and y equal to minus 1 into both equations and make sure they are both true.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_01_009f_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 6 is to check that the ordered pair is a solution to both original equations. To do that we Substitute x equal to 4 and y equal to minus 1 into both equations and make sure they are both true.\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167834099143\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167835594824\">\n<div data-type=\"problem\" id=\"fs-id1167835594827\">\n<p id=\"fs-id1167830964185\">Solve the system by substitution: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-86373ea0b7246f8750b1eb4d0e48edff_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;&#45;&#50;&#120;&#43;&#121;&#61;&#45;&#49;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#120;&#43;&#51;&#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=\"150\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834533520\">\n<p id=\"fs-id1167834533523\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fc9db18ceda8b325515059e9c425b44f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#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>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167835363503\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167835363506\">\n<div data-type=\"problem\" id=\"fs-id1167835363508\">\n<p id=\"fs-id1167835363510\">Solve the system by substitution: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0857ef6adb8d9743ec76ef46e7e7dc11_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;&#120;&#43;&#121;&#61;&#45;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#52;&#120;&#43;&#51;&#121;&#61;&#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=\"128\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167832016014\">\n<p id=\"fs-id1167832016016\"><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>\n<div data-type=\"note\" id=\"fs-id1167830925047\" class=\"howto\">\n<div data-type=\"title\">Solve a system of equations by substitution.<\/div>\n<ol id=\"fs-id1167830925054\" type=\"1\" class=\"stepwise\">\n<li>Solve one of the equations for either variable.<\/li>\n<li>Substitute the expression from Step 1 into the other equation.<\/li>\n<li>Solve the resulting equation.<\/li>\n<li>Substitute the solution in Step 3 into either of the original equations to find the other variable.<\/li>\n<li>Write the solution as an ordered pair.<\/li>\n<li>Check that the ordered pair is a solution to <strong data-effect=\"bold\">both<\/strong> original equations.<\/li>\n<\/ol>\n<\/div>\n<p id=\"fs-id1167835415479\">Be very careful with the signs in the next example.<\/p>\n<div data-type=\"example\" id=\"fs-id1167835415482\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167832057897\">\n<div data-type=\"problem\" id=\"fs-id1167832057899\">\n<p id=\"fs-id1167832057901\">Solve the system by substitution: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bd0172627c8a81b34147e8f0b512e160_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;&#120;&#43;&#50;&#121;&#61;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#54;&#120;&#45;&#121;&#61;&#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=\"123\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835595520\">\n<p id=\"fs-id1167835595522\">We need to solve one equation for one variable. We will solve the first equation for <em data-effect=\"italics\">y<\/em>.<\/p>\n<table id=\"fs-id1167826998205\" class=\"unnumbered unstyled can-break\" summary=\"The equations are 4x plus 2y equals 4 and 6x minus y equals 8. Solving the first equation for y, we get y equal to minus 2 x plus 2. Substituting this in the second equation, we get 6 x minus open parentheses minus 2 x plus 2 close parentheses equals 8. Solving for x, we get x equal to 5 by 4. Substituting this in the first equation, and solving for y, we get y equal to minus 1 by 2. The ordered pair is 5 by 4, minus 1 by 2. Check the ordered pair in both equations. Both hold true. Hence, that is the solution.\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835254444\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_01_010b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Solve the first equation for <em data-effect=\"italics\">y<\/em>.<\/p>\n<div data-type=\"newline\"><\/div>\n<p>Substitute <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cb2dc7153be35a09426026e69b822e19_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#50;&#120;&#43;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"61\" style=\"vertical-align: -2px;\" \/> for <em data-effect=\"italics\">y<\/em> in the second equation.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835379481\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_01_010c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Replace the <em data-effect=\"italics\">y<\/em> with <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-13e32d97f9d5cdd9563b303c52aabd2e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#50;&#120;&#43;&#50;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"66\" style=\"vertical-align: -2px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167831239705\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_01_010d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\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-id1167834432973\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_01_010e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\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-890be6c8bbcdea64df661b746e7455d7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"43\" style=\"vertical-align: -6px;\" \/> into <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8e8598a43925106f10d51781ee855a34_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#120;&#43;&#50;&#121;&#61;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"91\" 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-id1167835609313\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_01_010f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td data-valign=\"top\" data-align=\"left\">The ordered pair is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0d7d76bb674d165b6af23c5553f6a9f3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#52;&#125;&#44;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"64\" style=\"vertical-align: -7px;\" \/><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Check the ordered pair in both equations.<\/p>\n<div data-type=\"newline\"><\/div>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167835175410\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_01_010a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/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-0d7d76bb674d165b6af23c5553f6a9f3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#52;&#125;&#44;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"64\" style=\"vertical-align: -7px;\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167826782993\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167826782996\">\n<div data-type=\"problem\" id=\"fs-id1167826782998\">\n<p id=\"fs-id1167826783000\">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;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834523798\">\n<p id=\"fs-id1167834523800\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2e8fb98e4e344485819d20845a65ffcb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"40\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167834536415\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167835360488\">\n<div data-type=\"problem\" id=\"fs-id1167835360490\">\n<p id=\"fs-id1167835360492\">Solve the system by substitution: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d745ea70b80ecfbf00b3bc2ea7ad7fbb_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;&#120;&#45;&#121;&#61;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#50;&#120;&#45;&#51;&#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=\"123\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835351996\">\n<p id=\"fs-id1167835351998\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ed17114b958601c8882d0e6751166e00_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#44;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"68\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167834222378\">\n<h3 data-type=\"title\">Solve a System of Equations by Elimination<\/h3>\n<p id=\"fs-id1167834132598\">We have solved systems of linear equations by graphing and by substitution. Graphing works well when the variable coefficients are small and the solution has integer values. Substitution works well when we can easily solve one equation for one of the variables and not have too many fractions in the resulting expression.<\/p>\n<p id=\"fs-id1167834132604\">The third method of solving systems of linear equations is called the Elimination Method. When we solved a system by substitution, we started with two equations and two variables and reduced it to one equation with one variable. This is what we\u2019ll do with the elimination method, too, but we\u2019ll have a different way to get there.<\/p>\n<p id=\"fs-id1167835510985\">The Elimination Method is based on the Addition Property of Equality. The Addition Property of Equality says that when you add the same quantity to both sides of an equation, you still have equality. We will extend the Addition Property of Equality to say that when you add equal quantities to both sides of an equation, the results are equal.<\/p>\n<p id=\"fs-id1167832055312\">For any expressions <em data-effect=\"italics\">a, b, c,<\/em> and <em data-effect=\"italics\">d<\/em>.<\/p>\n<div data-type=\"equation\" id=\"fs-id1167831911684\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-18cce786ca36cc10402758347dedeb67_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;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#105;&#102;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#97;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#98;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#110;&#100;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#99;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#100;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#104;&#101;&#110;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#97;&#43;&#99;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#98;&#43;&#100;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"59\" width=\"210\" style=\"vertical-align: -24px;\" \/><\/div>\n<p id=\"fs-id1167835376848\">To solve a system of equations by elimination, we start with both equations in standard form. Then we decide which variable will be easiest to eliminate. How do we decide? We want to have the coefficients of one variable be opposites, so that we can add the equations together and eliminate that variable.<\/p>\n<p id=\"fs-id1167835340855\">Notice how that works when we add these two equations together:<\/p>\n<div data-type=\"equation\" id=\"fs-id1167831847197\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-959d2e7b91cd6276478b7d2f4b653bf3_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;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#117;&#110;&#100;&#101;&#114;&#115;&#101;&#116;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#8212;&#8212;&#8212;&#8212;&#8212;&#125;&#125;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#51;&#120;&#43;&#121;&#61;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#50;&#120;&#45;&#121;&#61;&#48;&#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;&#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;&#53;&#120;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#56;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#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=\"104\" width=\"104\" style=\"vertical-align: -44px;\" \/><\/div>\n<p id=\"fs-id1167835338949\">The <em data-effect=\"italics\">y<\/em>\u2019s add to zero and we have one equation with one variable.<\/p>\n<p id=\"fs-id1167834376400\">Let\u2019s try another one:<\/p>\n<div data-type=\"equation\" id=\"fs-id1167834063624\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-662c6d571d47a1ffeb4532fe8982b447_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;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#120;&#43;&#52;&#121;&#61;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#50;&#120;&#43;&#53;&#121;&#61;&#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=\"43\" width=\"124\" style=\"vertical-align: -17px;\" \/><\/div>\n<p id=\"fs-id1167835336856\">This time we don\u2019t see a variable that can be immediately eliminated if we add the equations.<\/p>\n<p id=\"fs-id1167834195035\">But if we multiply the first equation by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c3bd060f32a71334fb5cdf65d10fc75c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#50;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"26\" style=\"vertical-align: -4px;\" \/> we will make the coefficients of <em data-effect=\"italics\">x<\/em> opposites. We must multiply every term on both sides of the equation by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d36c434a1919e0aaa1a4125fdaa40853_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#50;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"26\" style=\"vertical-align: 0px;\" \/><\/p>\n<p><span data-type=\"media\" id=\"fs-id1167834193440\" data-alt=\"Minus 2 open parentheses x plus 4y close parentheses is minus 2 times 2. And, 2 x plus 5y is minus 2.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_01_011_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Minus 2 open parentheses x plus 4y close parentheses is minus 2 times 2. And, 2 x plus 5y is minus 2.\" \/><\/span><\/p>\n<p id=\"fs-id1167828420195\">Then rewrite the system of equations.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167828420197\" data-alt=\"Minus 2 x minus 8y is minus 4 and 2 x plus 5y is minus 2.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_01_012_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Minus 2 x minus 8y is minus 4 and 2 x plus 5y is minus 2.\" \/><\/span><\/p>\n<p id=\"fs-id1167826997236\">Now we see that the coefficients of the <em data-effect=\"italics\">x<\/em> terms are opposites, so <em data-effect=\"italics\">x<\/em> will be eliminated when we add these two equations.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167831031275\" data-alt=\"Minus 2 x minus 8y is minus 4 and 2 x plus 5y is minus 2. Adding these, we get minus 3y equals minus 6.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_01_013_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Minus 2 x minus 8y is minus 4 and 2 x plus 5y is minus 2. Adding these, we get minus 3y equals minus 6.\" \/><\/span><\/p>\n<p id=\"fs-id1167834324678\">Once we get an equation with just one variable, we solve it. Then we substitute that value into one of the original equations to solve for the remaining variable. And, as always, we check our answer to make sure it is a solution to both of the original equations.<\/p>\n<p id=\"fs-id1167834121134\">Now we\u2019ll see how to use elimination to solve the same system of equations we solved by graphing and by substitution.<\/p>\n<div data-type=\"example\" id=\"fs-id1167834121139\" class=\"textbox textbox--examples\">\n<div data-type=\"title\">How to Solve a System of Equations by Elimination<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834053780\">\n<div data-type=\"problem\" id=\"fs-id1167834053782\">\n<p id=\"fs-id1167834053784\">Solve the system by elimination: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-93c77aec095fb80605576f00f515b39d_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;&#120;&#43;&#121;&#61;&#55;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#120;&#45;&#50;&#121;&#61;&#54;&#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-id1167832055991\"><span data-type=\"media\" id=\"fs-id1167832055992\" data-alt=\"The equations are 2 x plus y equals 7 and x minus 2y equals 6. Step 1 is to write both equations in standard form. Both equations are in standard form, Ax plus By equals C. If any coefficients are fractions, clear them. There are no fractions.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_01_014a_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The equations are 2 x plus y equals 7 and x minus 2y equals 6. Step 1 is to write both equations in standard form. Both equations are in standard form, Ax plus By equals C. If any coefficients are fractions, clear them. There are no fractions.\" \/><\/span><span data-type=\"media\" id=\"fs-id1167835489057\" data-alt=\"Step 2 is to make the coefficients of one variable opposites. First decide which variable you will eliminate. Multiply one or both equations so that the coefficients of that variable are opposites. We can eliminate the y\u2019s by multiplying the first equation by 2. We get 4x plus 2y equals 14.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_01_014b_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 2 is to make the coefficients of one variable opposites. First decide which variable you will eliminate. Multiply one or both equations so that the coefficients of that variable are opposites. We can eliminate the y\u2019s by multiplying the first equation by 2. We get 4x plus 2y equals 14.\" \/><\/span><span data-type=\"media\" id=\"fs-id1167835375465\" data-alt=\"Step 3 is to add the equations resulting from step 2 to eliminate one variable. Adding, we get 5x equals 20.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_01_014c_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 3 is to add the equations resulting from step 2 to eliminate one variable. Adding, we get 5x equals 20.\" \/><\/span><span data-type=\"media\" id=\"fs-id1167828436322\" data-alt=\"Step 4 is to solve for the remaining variable. Solving for x, we get x equals 4.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_01_014d_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 4 is to solve for the remaining variable. Solving for x, we get x equals 4.\" \/><\/span><span data-type=\"media\" id=\"fs-id1167835218199\" data-alt=\"Step 5 is to substitute the solution from step 4 into one of the original equations. Then solve for the other variable. Substituting x equal to 4 into the second equation, we get 4 minus 2y equals 6. Solving for y, we get y equal to minus 1.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_01_014e_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 5 is to substitute the solution from step 4 into one of the original equations. Then solve for the other variable. Substituting x equal to 4 into the second equation, we get 4 minus 2y equals 6. Solving for y, we get y equal to minus 1.\" \/><\/span><span data-type=\"media\" id=\"fs-id1167826782824\" data-alt=\"Step 6 is to write the solution as an ordered pair. Here, the ordered pair is 4, minus 1.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_01_014f_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 6 is to write the solution as an ordered pair. Here, the ordered pair is 4, minus 1.\" \/><\/span><span data-type=\"media\" id=\"fs-id1167835312274\" data-alt=\"Step 7 is to check that the ordered pair is a solution to both original equations. The ordered pair makes both original equations true.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_01_014g_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 7 is to check that the ordered pair is a solution to both original equations. The ordered pair makes both original equations true.\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167834413553\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167835433665\">\n<div data-type=\"problem\" id=\"fs-id1167835433667\">\n<p>Solve the system by elimination: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-89fe536e1fc050ef61f73b66f26d3ed4_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;&#43;&#121;&#61;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#50;&#120;&#45;&#51;&#121;&#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=\"123\" style=\"vertical-align: -17px;\" \/> <\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834161733\">\n<p id=\"fs-id1167834161734\"><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;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167831031022\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167832043684\">\n<div data-type=\"problem\" id=\"fs-id1167832043686\">\n<p id=\"fs-id1167832043689\">Solve the system by elimination: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6bca9f340140c5013983c0e21156c471_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;&#120;&#43;&#121;&#61;&#45;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#45;&#50;&#120;&#45;&#50;&#121;&#61;&#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=\"150\" style=\"vertical-align: -17px;\" \/> <\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834134812\">\n<p id=\"fs-id1167834134813\"><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;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167831921387\">The steps are listed here for easy reference.<\/p>\n<div data-type=\"note\" id=\"fs-id1167835347820\" class=\"howto\">\n<div data-type=\"title\">Solve a system of equations by elimination.<\/div>\n<ol id=\"fs-id1167835268056\" type=\"1\" class=\"stepwise\">\n<li>Write both equations in standard form. If any coefficients are fractions, clear them.<\/li>\n<li>Make the coefficients of one variable opposites.\n<ul id=\"fs-id1167835355835\" data-bullet-style=\"bullet\">\n<li>Decide which variable you will eliminate.<\/li>\n<li>Multiply one or both equations so that the coefficients of that variable are opposites.<\/li>\n<\/ul>\n<\/li>\n<li>Add the equations resulting from Step 2 to eliminate one variable.<\/li>\n<li>Solve for the remaining variable.<\/li>\n<li>Substitute the solution from Step 4 into one of the original equations. Then solve for the other variable.<\/li>\n<li>Write the solution as an ordered pair.<\/li>\n<li>Check that the ordered pair is a solution to <strong data-effect=\"bold\">both<\/strong> original equations.<\/li>\n<\/ol>\n<\/div>\n<p id=\"fs-id1167831835904\">Now we\u2019ll do an example where we need to multiply both equations by constants in order to make the coefficients of one variable opposites.<\/p>\n<div data-type=\"example\" id=\"fs-id1167831835910\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167835183559\">\n<div data-type=\"problem\" id=\"fs-id1167835183561\">\n<p id=\"fs-id1167835183563\">Solve the system by elimination: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6fea373874d6f97446f5b3ddf80f0548_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;&#120;&#45;&#51;&#121;&#61;&#57;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#55;&#120;&#43;&#50;&#121;&#61;&#45;&#54;&#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;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834133101\">\n<p id=\"fs-id1167834133104\">In this example, we cannot multiply just one equation by any constant to get opposite coefficients. So we will strategically multiply both equations by different constants to get the opposites.<\/p>\n<table id=\"fs-id1167834536402\" class=\"unnumbered unstyled can-break\" summary=\"The equations are 4x minus 3y equals 9 and 7x plus 2y equals minus 6. Both equations are in standard form. To get opposite coefficients of y, we will multiply the first equation by 2 and the second equation by 3. We get 2 open parentheses 4x minus 3y close parentheses equals 2 times 9 and 3 open parentheses 7x plus 2y close parentheses equals 3 times minus 6. Simplifying both, we get 8x minus 6y equals 18 and 21x plus 6y equals minus 18. Adding the two equations to eliminate y, and solving for x, we get x equal to 0. Substituting this into one of the original equations and solving for y, we get y equal to minus 3. The ordered pair of the solution is 0, minus 3. Check that the ordered pair is a solution to both original equations.\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167832226710\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_01_015b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Both equations are in standard form.<\/p>\n<div data-type=\"newline\"><\/div>\n<p>To get opposite coefficients of <em data-effect=\"italics\">y<\/em>, we will<\/p>\n<div data-type=\"newline\"><\/div>\n<p>multiply the first equation by 2 and the<\/p>\n<div data-type=\"newline\"><\/div>\n<p>second equation by 3.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835509822\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_01_015c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167832058367\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_01_015d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Add the two equations to eliminate <em data-effect=\"italics\">y<\/em>.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835287480\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_01_015e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Solve for <em data-effect=\"italics\">x<\/em>.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167830961361\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_01_015f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\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;\" \/> into one of the original equations.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835200374\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_01_015g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\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-id1167835333944\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_01_015h_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Write the solution as an ordered pair.<\/td>\n<td data-valign=\"top\" data-align=\"left\">The ordered pair is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2a02f2acb07c1c1b796ba6e9846dec18_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;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"59\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Check that the ordered pair is a solution to<\/p>\n<div data-type=\"newline\"><\/div>\n<p><strong data-effect=\"bold\">both<\/strong> original equations.<\/p>\n<div data-type=\"newline\"><\/div>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167835351833\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_01_015a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/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-2a02f2acb07c1c1b796ba6e9846dec18_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;&#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-id1167826996182\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167835244250\">\n<div data-type=\"problem\" id=\"fs-id1167835244252\">\n<p id=\"fs-id1167835244254\">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;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834340042\">\n<p id=\"fs-id1167834340045\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-450ee1faefebebc715b20a97daae94ad_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#44;&#51;&#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>\n<div data-type=\"note\" id=\"fs-id1167835163286\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167835163290\">\n<div data-type=\"problem\" id=\"fs-id1167835163292\">\n<p id=\"fs-id1167834583891\">Solve each system by elimination: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d26f084254f633bd04d51b68d993b877_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;&#55;&#120;&#43;&#56;&#121;&#61;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#51;&#120;&#45;&#53;&#121;&#61;&#50;&#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=\"132\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167831888060\">\n<p id=\"fs-id1167831888061\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9a73c1812707ce5b0b7341d06a667a00_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#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>\n<\/div>\n<p id=\"fs-id1167830757705\">When the system of equations contains fractions, we will first clear the fractions by multiplying each equation by the LCD of all the fractions in the equation.<\/p>\n<div data-type=\"example\" id=\"fs-id1167834535350\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1167834535354\">\n<p id=\"fs-id1167831117475\">Solve the system by elimination: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-484d3f9546d3eb2628e8b710df9085d7_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;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#121;&#61;&#54;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#50;&#125;&#120;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;&#121;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#55;&#125;&#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=\"44\" width=\"135\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167832041346\">\n<p id=\"fs-id1167832041348\">In this example, both equations have fractions. Our first step will be to multiply each equation by the LCD of all the fractions in the equation to clear the fractions.<\/p>\n<table id=\"fs-id1167834464514\" class=\"unnumbered unstyled can-break\" summary=\"The equations are x plus half y equals 6 and 3 by 2 x plus2 by 3 y equals 17 by 2. To clear the fractions, multiply each equation by its LCD. Multiplying the first equation by 2 the second one by 6, and simplifying both, we get 2 x plus y equals 12 and 9x plus 4y equals 51. Now that both equations are in standard form, we can eliminate y by multiplying the top equation by minus 4. This becomes minus 8x minus 4y equals minus 48. Adding this to 9x plus 4y equals 51, we get x equal to 3. Substituting this into either of the original equations, we get y equal to 6. The ordered pair is 3, 6. It is a solution to both original equations.\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835575772\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_01_016b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">To clear the fractions, multiply each<\/p>\n<div data-type=\"newline\"><\/div>\n<p>equation by its LCD.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167832151434\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_01_016c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834064043\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_01_016d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Now we are ready to eliminate one<\/p>\n<div data-type=\"newline\"><\/div>\n<p>of the variables. Notice that both equations are in<\/p>\n<div data-type=\"newline\"><\/div>\n<p>standard form.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">We can eliminate <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0af556714940c351c933bba8cf840796_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\" \/> by multiplying the top equation by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e344351eb8a6a075b3d1df3943b3d637_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#52;&#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-id1167835353312\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_01_016e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Simplify and add.<\/p>\n<div data-type=\"newline\"><\/div>\n<div data-type=\"newline\"><\/div>\n<p>Substitute <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3573bf1ea4c223bb71878796b2106731_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"43\" style=\"vertical-align: 0px;\" \/> into one of the original equations.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167828421222\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_01_016f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Solve for <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0af556714940c351c933bba8cf840796_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\" \/>.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835589634\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_01_016g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167834184112\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_01_016h_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167835327424\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_01_016i_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Write the solution as an ordered pair.<\/td>\n<td data-valign=\"top\" data-align=\"left\">The ordered pair is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ca2d3282a1ae5ba5304f0dc132237a4c_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;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"45\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Check that the ordered pair is a solution to<\/p>\n<div data-type=\"newline\"><\/div>\n<p>both original equations.<\/p>\n<div data-type=\"newline\"><\/div>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167835190458\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_01_016a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/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-ca2d3282a1ae5ba5304f0dc132237a4c_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;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"45\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167835309662\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167835342662\">\n<div data-type=\"problem\" id=\"fs-id1167835342664\">\n<p id=\"fs-id1167835342666\">Solve each system by elimination: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9315d559fd6fa1b13563f0ea60c52e40_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;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#120;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#121;&#61;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;&#120;&#45;&#121;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#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=\"44\" width=\"127\" style=\"vertical-align: -17px;\" \/> <\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167831921076\">\n<p id=\"fs-id1167831921079\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ccc35b92a6567ced556cb46473589564_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#44;&#50;&#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>\n<div data-type=\"note\" id=\"fs-id1167831890568\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167835341021\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167835341025\">Solve each system by elimination: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-19bee8ebc801732fe262d5a8da036e13_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;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#53;&#125;&#121;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#53;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#120;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;&#121;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#54;&#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=\"44\" width=\"142\" style=\"vertical-align: -17px;\" \/> <\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834190571\">\n<p id=\"fs-id1167834190573\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a685e6a8556e8135bece1243dc70fe1f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#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>\n<p id=\"fs-id1167831163799\">When we solved the system by graphing, we saw that not all systems of linear equations have a single ordered pair as a solution. When the two equations were really the same line, there were infinitely many solutions. We called that a consistent system. When the two equations described parallel lines, there was no solution. We called that an inconsistent system.<\/p>\n<p id=\"fs-id1167831163806\">The same is true using substitution or elimination. If the equation at the end of substitution or elimination is a true statement, we have a consistent but dependent system and the system of equations has infinitely many solutions. If the equation at the end of substitution or elimination is a false statement, we have an inconsistent system and the system of equations has no solution.<\/p>\n<div data-type=\"example\" id=\"fs-id1167832138940\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167832138943\">\n<div data-type=\"problem\" id=\"fs-id1167835380388\">\n<p id=\"fs-id1167835380390\">Solve the system by elimination: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c7971354f2f4bcefb34b8754eb78d741_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;&#43;&#52;&#121;&#61;&#49;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#121;&#61;&#51;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;&#120;&#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=\"132\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167831117510\">\n<p id=\"fs-id1167831117512\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-79f87335133bf01df081fb0fab7d9ce6_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;&#125;&#38;&#32;&#38;&#32;&#38;&#32;&#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;&#43;&#52;&#121;&#61;&#49;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#121;&#61;&#51;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;&#120;&#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;&#92;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#87;&#114;&#105;&#116;&#101;&#32;&#116;&#104;&#101;&#32;&#115;&#101;&#99;&#111;&#110;&#100;&#32;&#101;&#113;&#117;&#97;&#116;&#105;&#111;&#110;&#32;&#105;&#110;&#32;&#115;&#116;&#97;&#110;&#100;&#97;&#114;&#100;&#32;&#102;&#111;&#114;&#109;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#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;&#43;&#52;&#121;&#61;&#49;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;&#120;&#43;&#121;&#61;&#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;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#67;&#108;&#101;&#97;&#114;&#32;&#116;&#104;&#101;&#32;&#102;&#114;&#97;&#99;&#116;&#105;&#111;&#110;&#115;&#32;&#98;&#121;&#32;&#109;&#117;&#108;&#116;&#105;&#112;&#108;&#121;&#105;&#110;&#103;&#32;&#116;&#104;&#101;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#101;&#99;&#111;&#110;&#100;&#32;&#101;&#113;&#117;&#97;&#116;&#105;&#111;&#110;&#32;&#98;&#121;&#32;&#52;&#46;&#125;&#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;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#51;&#120;&#43;&#52;&#121;&#61;&#49;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#52;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;&#120;&#43;&#121;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#52;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#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;&#92;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#105;&#109;&#112;&#108;&#105;&#102;&#121;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#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;&#43;&#52;&#121;&#61;&#49;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#51;&#120;&#43;&#52;&#121;&#61;&#49;&#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;&#92;&#32;&#92;&#92;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#84;&#111;&#32;&#101;&#108;&#105;&#109;&#105;&#110;&#97;&#116;&#101;&#32;&#97;&#32;&#118;&#97;&#114;&#105;&#97;&#98;&#108;&#101;&#44;&#32;&#119;&#101;&#32;&#109;&#117;&#108;&#116;&#105;&#112;&#108;&#121;&#32;&#116;&#104;&#101;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#101;&#99;&#111;&#110;&#100;&#32;&#101;&#113;&#117;&#97;&#116;&#105;&#111;&#110;&#32;&#98;&#121;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#45;&#49;&#46;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#105;&#109;&#112;&#108;&#105;&#102;&#121;&#32;&#97;&#110;&#100;&#32;&#97;&#100;&#100;&#46;&#125;&#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;&#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;&#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;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#54;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#51;&#120;&#43;&#52;&#121;&#61;&#49;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#45;&#51;&#120;&#45;&#52;&#121;&#61;&#45;&#49;&#50;&#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;&#104;&#102;&#105;&#108;&#108;&#32;&#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;&#48;&#61;&#48;&#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=\"367\" width=\"563\" style=\"vertical-align: -176px;\" \/><\/p>\n<p id=\"fs-id1167831086799\">This is a true statement. The equations are consistent but dependent. Their graphs would be the same line. The system has infinitely many solutions.<\/p>\n<p id=\"fs-id1167831833311\">After we cleared the fractions in the second equation, did you notice that the two equations were the same? That means we have coincident lines.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167835357545\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167835357549\">\n<div data-type=\"problem\" id=\"fs-id1167835357552\">\n<p id=\"fs-id1167834066022\">Solve the system by elimination: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c190b34dcdfa8a94d0865fdbd354fa72_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;&#53;&#120;&#45;&#51;&#121;&#61;&#49;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#121;&#61;&#45;&#53;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#51;&#125;&#120;&#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=\"132\" style=\"vertical-align: -17px;\" \/> <\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167831823837\">\n<p id=\"fs-id1167835379275\">infinitely many solutions<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167835379282\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167835357672\">\n<div data-type=\"problem\" id=\"fs-id1167835357674\">\n<p id=\"fs-id1167835357677\">Solve the system by elimination: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4b57e3792ce23516fdd6b10244aea8f2_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;&#50;&#121;&#61;&#54;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#121;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#120;&#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=\"130\" style=\"vertical-align: -17px;\" \/> <\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167831920066\">\n<p id=\"fs-id1167831920068\">infinitely many solutions<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167826783829\">\n<h3 data-type=\"title\">Choose the Most Convenient Method to Solve a System of Linear Equations<\/h3>\n<p id=\"fs-id1167826783834\">When you solve a system of linear equations in in an application, you will not be told which method to use. You will need to make that decision yourself. So you\u2019ll want to choose the method that is easiest to do and minimizes your chance of making mistakes.<\/p>\n<div data-type=\"equation\" id=\"fs-id1171792628785\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-942d56b9cd6376b7feea308958285e76_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;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#67;&#104;&#111;&#111;&#115;&#101;&#32;&#116;&#104;&#101;&#32;&#77;&#111;&#115;&#116;&#32;&#67;&#111;&#110;&#118;&#101;&#110;&#105;&#101;&#110;&#116;&#32;&#77;&#101;&#116;&#104;&#111;&#100;&#32;&#116;&#111;&#32;&#83;&#111;&#108;&#118;&#101;&#32;&#97;&#32;&#83;&#121;&#115;&#116;&#101;&#109;&#32;&#111;&#102;&#32;&#76;&#105;&#110;&#101;&#97;&#114;&#32;&#69;&#113;&#117;&#97;&#116;&#105;&#111;&#110;&#115;&#125;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#92;&#117;&#110;&#100;&#101;&#114;&#115;&#101;&#116;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#8212;&#8212;&#8212;&#8212;&#125;&#125;&#123;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#71;&#114;&#97;&#112;&#104;&#105;&#110;&#103;&#125;&#125;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#117;&#110;&#100;&#101;&#114;&#115;&#101;&#116;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#8212;&#8212;&#8212;&#8212;&#8212;&#125;&#125;&#123;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#117;&#98;&#115;&#116;&#105;&#116;&#117;&#116;&#105;&#111;&#110;&#125;&#125;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#117;&#110;&#100;&#101;&#114;&#115;&#101;&#116;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#8212;&#8212;&#8212;&#8212;&#8212;&#125;&#125;&#123;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#69;&#108;&#105;&#109;&#105;&#110;&#97;&#116;&#105;&#111;&#110;&#125;&#125;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#85;&#115;&#101;&#32;&#119;&#104;&#101;&#110;&#32;&#121;&#111;&#117;&#32;&#110;&#101;&#101;&#100;&#32;&#97;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#85;&#115;&#101;&#32;&#119;&#104;&#101;&#110;&#32;&#111;&#110;&#101;&#32;&#101;&#113;&#117;&#97;&#116;&#105;&#111;&#110;&#32;&#105;&#115;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#85;&#115;&#101;&#32;&#119;&#104;&#101;&#110;&#32;&#116;&#104;&#101;&#32;&#101;&#113;&#117;&#97;&#116;&#105;&#111;&#110;&#115;&#32;&#97;&#114;&#101;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#112;&#105;&#99;&#116;&#117;&#114;&#101;&#32;&#111;&#102;&#32;&#116;&#104;&#101;&#32;&#115;&#105;&#116;&#117;&#97;&#116;&#105;&#111;&#110;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#108;&#114;&#101;&#97;&#100;&#121;&#32;&#115;&#111;&#108;&#118;&#101;&#100;&#32;&#111;&#114;&#32;&#99;&#97;&#110;&#32;&#98;&#101;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#105;&#110;&#32;&#115;&#116;&#97;&#110;&#100;&#97;&#114;&#100;&#32;&#102;&#111;&#114;&#109;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#101;&#97;&#115;&#105;&#108;&#121;&#32;&#115;&#111;&#108;&#118;&#101;&#100;&#32;&#102;&#111;&#114;&#32;&#111;&#110;&#101;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#92;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#118;&#97;&#114;&#105;&#97;&#98;&#108;&#101;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#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=\"134\" width=\"701\" style=\"vertical-align: -61px;\" \/><\/div>\n<div data-type=\"example\" id=\"fs-id1167834525591\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167834525593\">\n<div data-type=\"problem\" id=\"fs-id1167834525595\">\n<p id=\"fs-id1167834525597\">For each system of linear equations, decide whether it would be more convenient to solve it by substitution or elimination. Explain your answer.<\/p>\n<p id=\"fs-id1167831112489\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d2746097fc625dc40233a8104017b121_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;&#43;&#56;&#121;&#61;&#52;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#55;&#120;&#45;&#52;&#121;&#61;&#45;&#51;&#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=\"133\" style=\"vertical-align: -17px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-dedf6d9e8b4daa707f09e1f73d9008f0_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;&#53;&#120;&#43;&#54;&#121;&#61;&#49;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#121;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#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=\"119\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834273719\">\n<p id=\"fs-id1167834273721\"><span class=\"token\">\u24d0<\/span><\/p>\n<div data-type=\"equation\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d2746097fc625dc40233a8104017b121_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;&#43;&#56;&#121;&#61;&#52;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#55;&#120;&#45;&#52;&#121;&#61;&#45;&#51;&#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=\"133\" style=\"vertical-align: -17px;\" \/><\/div>\n<p id=\"fs-id1167835355915\">Since both equations are in standard form, using elimination will be most convenient.<\/p>\n<p id=\"fs-id1167835355918\"><span class=\"token\">\u24d1<\/span><\/p>\n<div data-type=\"equation\" id=\"fs-id1167827940624\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-dedf6d9e8b4daa707f09e1f73d9008f0_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;&#53;&#120;&#43;&#54;&#121;&#61;&#49;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#121;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#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=\"119\" style=\"vertical-align: -17px;\" \/><\/div>\n<p id=\"fs-id1167835419071\">Since one equation is already solved for <em data-effect=\"italics\">y<\/em>, using substitution will be most convenient.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167835360936\">\n<div data-type=\"problem\" id=\"fs-id1167835360938\">\n<p id=\"fs-id1167835331536\">For each system of linear equations decide whether it would be more convenient to solve it by substitution or elimination. Explain your answer.<\/p>\n<p id=\"fs-id1167835331541\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-105f0dd9eea0089e7d13f1596a1b3a61_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;&#120;&#45;&#53;&#121;&#61;&#45;&#51;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#51;&#120;&#43;&#50;&#121;&#61;&#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=\"133\" style=\"vertical-align: -17px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b1313a3247883c741a09911c6033a596_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;&#121;&#45;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#51;&#120;&#45;&#53;&#121;&#61;&#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=\"43\" width=\"125\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835235457\">\n<p id=\"fs-id1167835235460\"><span class=\"token\">\u24d0<\/span> Since both equations are in standard form, using elimination will be most convenient. <span class=\"token\">\u24d1<\/span> Since one equation is already solved for <em data-effect=\"italics\">x<\/em>, using substitution will be most convenient.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167826798760\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167826798764\">\n<div data-type=\"problem\" id=\"fs-id1167826857360\">\n<p id=\"fs-id1167826857362\">For each system of linear equations decide whether it would be more convenient to solve it by substitution or elimination. Explain your answer.<\/p>\n<p id=\"fs-id1167826857367\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c9671a6f423b8567d3562da42657b1e8_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;&#45;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#51;&#120;&#45;&#52;&#121;&#61;&#45;&#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=\"125\" style=\"vertical-align: -17px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6573a379645722deae26918f5319f1a3_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;&#54;&#120;&#45;&#50;&#121;&#61;&#49;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#51;&#120;&#43;&#55;&#121;&#61;&#45;&#49;&#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=\"134\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834049030\">\n<p id=\"fs-id1167834049032\"><span class=\"token\">\u24d0<\/span> Since one equation is already solved for <em data-effect=\"italics\">y<\/em>, using substitution will be most convenient. <span class=\"token\">\u24d1<\/span> Since both equations are in standard form, using elimination will be most convenient.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox\" data-depth=\"1\" id=\"fs-id1167835378580\">\n<h3 data-type=\"title\">Key Concepts<\/h3>\n<ul id=\"fs-id1167835234062\" data-bullet-style=\"bullet\">\n<li><strong data-effect=\"bold\">How to solve a system of linear equations by graphing.<\/strong>\n<ol id=\"fs-id1167831921610\" type=\"1\" class=\"stepwise\">\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 lines intersect, are parallel, or are the same line.<\/li>\n<li>Identify the solution to the system.\n<div data-type=\"newline\"><\/div>\n<p> If the lines intersect, identify the point of intersection. This is the solution to the system.<\/p>\n<div data-type=\"newline\"><\/div>\n<p> If the lines are parallel, the system has no solution.<\/p>\n<div data-type=\"newline\"><\/div>\n<p> If the lines are the same, the system has an infinite number of solutions.<\/li>\n<li>Check the solution in both equations.<\/li>\n<\/ol>\n<\/li>\n<li><strong data-effect=\"bold\">How to solve a system of equations by substitution.<\/strong>\n<ol id=\"fs-id1167835330573\" type=\"1\" class=\"stepwise\">\n<li>Solve one of the equations for either variable.<\/li>\n<li>Substitute the expression from Step 1 into the other equation.<\/li>\n<li>Solve the resulting equation.<\/li>\n<li>Substitute the solution in Step 3 into either of the original equations to find the other variable.<\/li>\n<li>Write the solution as an ordered pair.<\/li>\n<li>Check that the 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-id1167835305193\" type=\"1\" class=\"stepwise\">\n<li>Write both equations in standard form. If any coefficients are fractions, clear them.<\/li>\n<li>Make the coefficients of one variable opposites.\n<div data-type=\"newline\"><\/div>\n<p>Decide which variable you will eliminate.<\/p>\n<div data-type=\"newline\"><\/div>\n<p>Multiply one or both equations so that the coefficients of that variable are opposites.<\/li>\n<li>Add the equations resulting from Step 2 to eliminate one variable.<\/li>\n<li>Solve for the remaining variable.<\/li>\n<li>Substitute the solution from Step 4 into one of the original equations. Then solve for the other variable.<\/li>\n<li>Write the solution as an ordered pair.<\/li>\n<li>Check that the ordered pair is a solution to <strong data-effect=\"bold\">both<\/strong> original equations.\n<div data-type=\"newline\"><\/div>\n<p> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7f2000fce9dae84b262870e9168f9dc1_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;&#92;&#32;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#67;&#104;&#111;&#111;&#115;&#101;&#32;&#116;&#104;&#101;&#32;&#77;&#111;&#115;&#116;&#32;&#67;&#111;&#110;&#118;&#101;&#110;&#105;&#101;&#110;&#116;&#32;&#77;&#101;&#116;&#104;&#111;&#100;&#32;&#116;&#111;&#32;&#83;&#111;&#108;&#118;&#101;&#32;&#97;&#32;&#83;&#121;&#115;&#116;&#101;&#109;&#32;&#111;&#102;&#32;&#76;&#105;&#110;&#101;&#97;&#114;&#32;&#69;&#113;&#117;&#97;&#116;&#105;&#111;&#110;&#115;&#125;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#92;&#117;&#110;&#100;&#101;&#114;&#115;&#101;&#116;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#8212;&#8212;&#8212;&#8212;&#8212;&#125;&#125;&#123;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#71;&#114;&#97;&#112;&#104;&#105;&#110;&#103;&#125;&#125;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#117;&#110;&#100;&#101;&#114;&#115;&#101;&#116;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#125;&#125;&#123;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#117;&#98;&#115;&#116;&#105;&#116;&#117;&#116;&#105;&#111;&#110;&#125;&#125;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#117;&#110;&#100;&#101;&#114;&#115;&#101;&#116;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#125;&#125;&#123;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#69;&#108;&#105;&#109;&#105;&#110;&#97;&#116;&#105;&#111;&#110;&#125;&#125;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#85;&#115;&#101;&#32;&#119;&#104;&#101;&#110;&#32;&#121;&#111;&#117;&#32;&#110;&#101;&#101;&#100;&#32;&#97;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#112;&#105;&#99;&#116;&#117;&#114;&#101;&#32;&#111;&#102;&#32;&#116;&#104;&#101;&#32;&#115;&#105;&#116;&#117;&#97;&#116;&#105;&#111;&#110;&#46;&#125;&#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;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#85;&#115;&#101;&#32;&#119;&#104;&#101;&#110;&#32;&#111;&#110;&#101;&#32;&#101;&#113;&#117;&#97;&#116;&#105;&#111;&#110;&#32;&#105;&#115;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#108;&#114;&#101;&#97;&#100;&#121;&#32;&#115;&#111;&#108;&#118;&#101;&#100;&#32;&#111;&#114;&#32;&#99;&#97;&#110;&#32;&#98;&#101;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#101;&#97;&#115;&#105;&#108;&#121;&#32;&#115;&#111;&#108;&#118;&#101;&#100;&#32;&#102;&#111;&#114;&#32;&#111;&#110;&#101;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#118;&#97;&#114;&#105;&#97;&#98;&#108;&#101;&#46;&#125;&#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;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#85;&#115;&#101;&#32;&#119;&#104;&#101;&#110;&#32;&#116;&#104;&#101;&#32;&#101;&#113;&#117;&#97;&#116;&#105;&#111;&#110;&#115;&#32;&#97;&#114;&#101;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#105;&#110;&#32;&#115;&#116;&#97;&#110;&#100;&#97;&#114;&#100;&#32;&#102;&#111;&#114;&#109;&#46;&#125;&#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;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"134\" width=\"741\" style=\"vertical-align: -83px;\" \/><\/li>\n<\/ol>\n<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox\" data-depth=\"1\" id=\"fs-id1167835634244\">\n<div class=\"practice-perfect\" data-depth=\"2\" id=\"fs-id1167834279700\">\n<h4 data-type=\"title\">Practice Makes Perfect<\/h4>\n<p id=\"fs-id1167835595230\"><strong data-effect=\"bold\">Determine Whether an Ordered Pair is a Solution of a System of Equations<\/strong><\/p>\n<p id=\"fs-id1167834124355\">In the following exercises, determine if the following points are solutions to the given system of equations.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167834124359\">\n<div data-type=\"problem\" id=\"fs-id1167834124361\">\n<p id=\"fs-id1167834063037\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d854d19255e3c6bf6fd214d7e3985a4b_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;&#120;&#45;&#54;&#121;&#61;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#51;&#120;&#45;&#52;&#121;&#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=\"111\" style=\"vertical-align: -17px;\" \/><\/p>\n<p id=\"fs-id1167831911303\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4307e3633578c7e56f8f767895b20497_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#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 data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-862a9525ac8db19009bf877fff4597b4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#44;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"51\" style=\"vertical-align: -4px;\" \/><\/div>\n<div data-type=\"solution\" id=\"fs-id1167834423297\">\n<p id=\"fs-id1167834423299\"><span class=\"token\">\u24d0<\/span> yes <span class=\"token\">\u24d1<\/span> no<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1167830960525\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0aa28865e0789179ebfc94ca371cdce0_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;&#45;&#51;&#120;&#43;&#121;&#61;&#56;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#45;&#120;&#43;&#50;&#121;&#61;&#45;&#57;&#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=\"130\" style=\"vertical-align: -17px;\" \/><\/p>\n<p id=\"fs-id1167826874537\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-974e6af6b3d25b0f2b0be2609051ca66_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#53;&#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 data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3d05d9d115588439e7e8a4631926b118_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#53;&#44;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"51\" style=\"vertical-align: -4px;\" \/><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831116465\">\n<div data-type=\"problem\" id=\"fs-id1167831116468\">\n<p id=\"fs-id1167831116470\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-200c08ae9f5b63c7032ea221a5e44b10_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;&#121;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;&#120;&#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<p id=\"fs-id1167834557190\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-edea40c117a1cc66603e136200800f55_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#56;&#125;&#123;&#55;&#125;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#54;&#125;&#123;&#55;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"42\" style=\"vertical-align: -7px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0b41e214fd7bce792b1d8d237b99971a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"40\" style=\"vertical-align: -7px;\" \/><\/div>\n<div data-type=\"solution\" id=\"fs-id1167835400298\">\n<p id=\"fs-id1167835400300\"><span class=\"token\">\u24d0<\/span> yes <span class=\"token\">\u24d1<\/span> no<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167826926965\">\n<div data-type=\"problem\" id=\"fs-id1167826926967\">\n<p id=\"fs-id1167834525517\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8b24c5b1a159906caede893c131cc922_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;&#120;&#43;&#51;&#121;&#61;&#54;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#121;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;&#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=\"111\" style=\"vertical-align: -17px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-551477e8eec13715fc061945ccc1b984_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#54;&#44;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"51\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-862a9525ac8db19009bf877fff4597b4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#44;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"51\" style=\"vertical-align: -4px;\" \/><\/div>\n<\/div>\n<p id=\"fs-id1167830705743\"><strong data-effect=\"bold\">Solve a System of Linear Equations by Graphing<\/strong><\/p>\n<p id=\"fs-id1167834130183\">In the following exercises, solve the following systems of equations by graphing.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167830924068\">\n<div data-type=\"problem\" id=\"fs-id1167830924070\">\n<p id=\"fs-id1167830924073\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-efc0cfb8676e8a73d602912943d3e935_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;&#43;&#121;&#61;&#45;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#50;&#120;&#43;&#51;&#121;&#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=\"116\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167832052855\">\n<p id=\"fs-id1167831880957\"><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;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167830865575\">\n<div data-type=\"problem\" id=\"fs-id1167830865578\">\n<p id=\"fs-id1167830865580\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3bb19505c791170b269607844a0128b2_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;&#45;&#120;&#43;&#121;&#61;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#50;&#120;&#43;&#121;&#61;&#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=\"116\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831969744\">\n<div data-type=\"problem\" id=\"fs-id1167831969746\">\n<p id=\"fs-id1167832086997\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-083748f580c44ff95b70afa809fc6fa2_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;&#43;&#50;&#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-id1167826997095\">\n<p id=\"fs-id1167835206118\"><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;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834184570\">\n<div data-type=\"problem\" id=\"fs-id1167834184573\">\n<p id=\"fs-id1167834184575\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6fb27a8d2366d597a6936c4a1eb54dde_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;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#121;&#61;&#45;&#51;&#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>\n<div data-type=\"exercise\" id=\"fs-id1167835236187\">\n<div data-type=\"problem\" id=\"fs-id1167835236189\">\n<p id=\"fs-id1167835236191\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d7a4324aba5a0b7bf3b640f279d8722e_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;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#50;&#125;&#120;&#43;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#121;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#120;&#43;&#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=\"44\" width=\"117\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835510445\">\n<p id=\"fs-id1167835510447\"><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;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835215816\">\n<div data-type=\"problem\" id=\"fs-id1167835215818\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b79e9224a38230d50f7943e4bb190eba_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;&#50;&#125;&#123;&#51;&#125;&#120;&#45;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#121;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#120;&#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=\"44\" width=\"117\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831891501\">\n<div data-type=\"problem\" id=\"fs-id1167835350488\">\n<p id=\"fs-id1167835350490\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ef8b059c8d711ea0a24f39236e85586b_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;&#45;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#45;&#120;&#43;&#50;&#121;&#61;&#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=\"43\" width=\"129\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167831115549\">\n<p id=\"fs-id1167831115551\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-627391f2fad7216057fc57692a374893_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#44;&#50;&#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-id1167835344480\">\n<div data-type=\"problem\" id=\"fs-id1167835344482\">\n<p id=\"fs-id1167831891601\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6fddfa34848950190027f39bfeec5091_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;&#45;&#120;&#43;&#51;&#121;&#61;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#120;&#43;&#51;&#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=\"116\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167832043496\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167826801731\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-045e9a4b173ce0a25e47612b3402a1b6_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;&#45;&#50;&#120;&#43;&#51;&#121;&#61;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#120;&#43;&#51;&#121;&#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=\"125\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834431232\">\n<p id=\"fs-id1167834431234\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c4a2258b08828b82f5478b79177f57c2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#44;&#51;&#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-id1167834300235\">\n<div data-type=\"problem\" id=\"fs-id1167834300237\">\n<p id=\"fs-id1167832055418\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-eeef1d76445655d83f14ce5061b5605a_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;&#120;&#45;&#121;&#61;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#50;&#120;&#43;&#51;&#121;&#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=\"119\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831837693\">\n<div data-type=\"problem\" id=\"fs-id1167831837695\">\n<p id=\"fs-id1167831837697\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5e4c61d26bdbedffc33ff75f8e1194f6_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;&#51;&#121;&#61;&#45;&#54;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#121;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#51;&#125;&#120;&#43;&#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-id1167835376871\">\n<p id=\"fs-id1167831025421\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-eb87bfe9ba69a89c4ac12d88180c12e1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#44;&#45;&#52;&#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-id1167826977938\">\n<div data-type=\"problem\" id=\"fs-id1167826977940\">\n<p id=\"fs-id1167826977943\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-52cc8ba9173e0c67e1bb89fd47ff4a0b_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;&#45;&#120;&#43;&#50;&#121;&#61;&#45;&#54;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#121;&#61;&#45;&#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=\"130\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167828240455\">\n<div data-type=\"problem\" id=\"fs-id1167828240457\">\n<p id=\"fs-id1167834185690\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ae045e47076c6ed872912536093bb17f_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;&#45;&#50;&#120;&#43;&#52;&#121;&#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;&#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 data-type=\"solution\" id=\"fs-id1167831826448\">\n<p id=\"fs-id1167835237081\">no solution<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835237086\">\n<div data-type=\"problem\" id=\"fs-id1167835237088\">\n<p id=\"fs-id1167832055711\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-94cbcc702ac4715a9435a9b92bbfd7ca_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;&#43;&#53;&#121;&#61;&#49;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#121;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#53;&#125;&#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=\"120\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835359930\">\n<div data-type=\"problem\" id=\"fs-id1167835359932\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-18a0c60617555af7f7e7a47892541595_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;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#52;&#120;&#45;&#51;&#121;&#61;&#56;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#56;&#120;&#45;&#54;&#121;&#61;&#49;&#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=\"123\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167832036092\">\n<p id=\"fs-id1167832036094\">no solution<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167830703070\">\n<div data-type=\"problem\" id=\"fs-id1167830703072\">\n<p id=\"fs-id1167830703074\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5a3c9084d431ff13056e87ce74069968_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;&#51;&#121;&#61;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#45;&#50;&#120;&#45;&#54;&#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=\"125\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835380427\">\n<div data-type=\"problem\" id=\"fs-id1167835380429\">\n<p id=\"fs-id1167835380431\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a432602acf19e64ccd5de9199215d03a_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;&#51;&#121;&#43;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#50;&#120;&#43;&#54;&#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=\"116\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835479147\">\n<p id=\"fs-id1167835479149\">infinite solutions<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834294493\">\n<div data-type=\"problem\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bf03b9d38d715454310669ffde49e4a0_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;&#120;&#61;&#51;&#121;&#43;&#55;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#56;&#120;&#45;&#54;&#121;&#61;&#49;&#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=\"120\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167821102414\">\n<div data-type=\"problem\" id=\"fs-id1167821102417\">\n<p id=\"fs-id1167821102419\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a69852681ed5b513c4631c1e243abd3a_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;&#120;&#43;&#92;&#116;&#101;&#120;&#116;&#123;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#121;&#61;&#54;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#45;&#56;&#120;&#45;&#52;&#121;&#61;&#45;&#50;&#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=\"148\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834463108\">\n<p id=\"fs-id1167834463110\">infinite solutions<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834423546\">\n<div data-type=\"problem\" id=\"fs-id1167834423548\">\n<p id=\"fs-id1167834423550\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1e83d9274213c2eb5d9d47b04898b616_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;&#53;&#120;&#43;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#50;&#121;&#61;&#55;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#45;&#49;&#48;&#120;&#45;&#52;&#121;&#61;&#45;&#49;&#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=\"157\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167835353488\">Without graphing, determine the number of solutions and then classify the system of equations.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167835353491\">\n<div data-type=\"problem\" id=\"fs-id1167834133342\">\n<p id=\"fs-id1167834133345\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-537cc7431b0791faa5d5e0d218532558_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;&#50;&#125;&#123;&#51;&#125;&#120;&#43;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#45;&#50;&#120;&#43;&#51;&#121;&#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=\"44\" width=\"124\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835422585\">\n<p id=\"fs-id1167835422587\">1 point, consistent and independent<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834129862\">\n<div data-type=\"problem\" id=\"fs-id1167834129864\">\n<p id=\"fs-id1167834129866\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-283d5747048094fcca2b492652c1a7bb_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;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#50;&#120;&#45;&#51;&#121;&#61;&#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=\"44\" width=\"111\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167832067470\">\n<div data-type=\"problem\" id=\"fs-id1167832067472\">\n<p id=\"fs-id1167832067474\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f6ae579b66abad472566994fc930c8df_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;&#53;&#120;&#43;&#51;&#121;&#61;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#50;&#120;&#45;&#51;&#121;&#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=\"111\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835310613\">\n<p id=\"fs-id1167831882494\">1 point, consistent and independent<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831882499\">\n<div data-type=\"problem\" id=\"fs-id1167831882501\">\n<p id=\"fs-id1167831882503\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9b10f580d6d4fb5e32fad3e0cd050d5f_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;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#120;&#43;&#50;&#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=\"44\" width=\"117\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831066235\">\n<div data-type=\"problem\" id=\"fs-id1167831066237\">\n<p id=\"fs-id1167831066239\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-af1350338d4589309e8224f72d2675bb_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;&#53;&#120;&#45;&#50;&#121;&#61;&#49;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#121;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#50;&#125;&#120;&#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=\"120\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\">\n<p id=\"fs-id1167832046536\">infinite solutions, consistent, dependent<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167832046542\"><strong data-effect=\"bold\">Solve a System of Equations by Substitution<\/strong><\/p>\n<p id=\"fs-id1167834556545\">In the following exercises, solve the systems of equations by substitution.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167834556548\">\n<div data-type=\"problem\" id=\"fs-id1167834556550\">\n<p id=\"fs-id1167834556552\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a720fc661ea9c6c17a689b46d7c0c067_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;&#120;&#43;&#121;&#61;&#45;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#51;&#120;&#45;&#50;&#121;&#61;&#45;&#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=\"125\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834094784\">\n<div data-type=\"problem\" id=\"fs-id1167834533392\">\n<p id=\"fs-id1167834533394\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-188fd877d4e5b386fa645a7a2fe81f36_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;&#120;&#43;&#121;&#61;&#45;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#51;&#120;&#45;&#121;&#61;&#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=\"43\" width=\"115\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835351247\">\n<p id=\"fs-id1167835351249\"><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;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835340122\">\n<div data-type=\"problem\" id=\"fs-id1167835340124\">\n<p id=\"fs-id1167835340126\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7ed9df8a733f224c3cdc2d8a5b3e1b10_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;&#50;&#121;&#61;&#45;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#50;&#120;&#45;&#51;&#121;&#61;&#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=\"125\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831921781\">\n<div data-type=\"problem\" id=\"fs-id1167831921783\">\n<p id=\"fs-id1167831921785\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-49ff151ef7419336d89331d2bd5ec391_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;&#57;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#50;&#120;&#43;&#53;&#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=\"116\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834431035\">\n<p id=\"fs-id1167834299862\"><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;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834314750\">\n<div data-type=\"problem\" id=\"fs-id1167835348333\">\n<p id=\"fs-id1167835348335\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-510cae26785f1943a25b251618796a82_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;&#53;&#120;&#45;&#50;&#121;&#61;&#45;&#54;&#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=\"125\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835534326\">\n<div data-type=\"problem\" id=\"fs-id1167835534328\">\n<p id=\"fs-id1167835420362\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f684b7f99dbe9dbde2cd10e9d69a0558_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;&#45;&#50;&#120;&#43;&#50;&#121;&#61;&#54;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#121;&#61;&#45;&#51;&#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=\"125\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167831943922\">\n<p id=\"fs-id1167831943924\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f7efccd3b1dd4bd29145637785ce4ebb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#92;&#116;&#101;&#120;&#116;&#123;&#47;&#125;&#50;&#44;&#53;&#92;&#116;&#101;&#120;&#116;&#123;&#47;&#125;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"87\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834422473\">\n<div data-type=\"problem\" id=\"fs-id1167834422475\">\n<p id=\"fs-id1167834422478\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d21d9d684102e404434819814ffa8eb2_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;&#120;&#43;&#53;&#121;&#61;&#49;&#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;&#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=\"110\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835180558\">\n<div data-type=\"problem\" id=\"fs-id1167835180560\">\n<p id=\"fs-id1167835180562\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3321ebb9af7c155b3a3ed7ad6715894f_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;&#43;&#52;&#121;&#61;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#121;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#53;&#125;&#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=\"117\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834532308\">\n<p id=\"fs-id1167834532310\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cf0ffd7a31e177b4a3caa16a8b3141b5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#53;&#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-id1167835297443\">\n<div data-type=\"problem\" id=\"fs-id1167835297445\">\n<p id=\"fs-id1167835297447\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4048245d8cad85768f817c0a93b3f2b0_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;&#120;&#43;&#121;&#61;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#120;&#45;&#50;&#121;&#61;&#45;&#49;&#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=\"124\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835320772\">\n<div data-type=\"problem\" id=\"fs-id1167835320775\">\n<p id=\"fs-id1167835320777\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-af06bcf41df85e331d0173b8b2377293_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;&#120;&#43;&#121;&#61;&#49;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#120;&#45;&#50;&#121;&#61;&#45;&#50;&#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=\"125\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835304713\">\n<p id=\"fs-id1167835304715\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f16714015fdd77f25981f5ca60f79d0d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#49;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"47\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167828447139\">\n<div data-type=\"problem\" id=\"fs-id1167828447141\">\n<p id=\"fs-id1167828447143\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-dc5071229d9b4477fd38fcdc98244b8c_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;&#45;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#121;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#120;&#43;&#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>\n<div data-type=\"exercise\" id=\"fs-id1167834422758\">\n<div data-type=\"problem\" id=\"fs-id1167834422760\">\n<p id=\"fs-id1167834422762\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e888e2d5c41a84e8f028c640e5500f54_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;&#54;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#121;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#50;&#125;&#120;&#43;&#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-id1167834489757\">\n<p id=\"fs-id1167834489759\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-067f891ede38d609b0614eef084ae132_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#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-id1167834190734\">\n<div data-type=\"problem\" id=\"fs-id1167834190737\">\n<p id=\"fs-id1167834190739\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7948ffccf3ef2c9e111839ebf8955de5_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;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#120;&#61;&#50;&#121;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#52;&#120;&#45;&#56;&#121;&#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=\"111\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835280872\">\n<div data-type=\"problem\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-46a52ab53d1c0950dff66e760207b784_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;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#50;&#120;&#45;&#49;&#54;&#121;&#61;&#56;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#45;&#120;&#45;&#56;&#121;&#61;&#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=\"130\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835417628\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2809647061e2f05aa3080110836f8805_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#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-id1167830699913\">\n<div data-type=\"problem\" id=\"fs-id1167830699915\">\n<p id=\"fs-id1167830699917\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-98c5e14aea7d9f5ac85291dddbde7097_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;&#55;&#125;&#123;&#56;&#125;&#120;&#43;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#45;&#55;&#120;&#43;&#56;&#121;&#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=\"44\" width=\"125\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835304080\">\n<div data-type=\"problem\" id=\"fs-id1167835304082\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ae49d9646d42d8adc80caac054260823_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;&#50;&#125;&#123;&#51;&#125;&#120;&#43;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#50;&#120;&#43;&#51;&#121;&#61;&#49;&#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=\"44\" width=\"119\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835357993\">\n<p id=\"fs-id1167835357996\">none<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167826863952\"><strong data-effect=\"bold\">Solve a System of Equations by Elimination<\/strong><\/p>\n<p id=\"fs-id1167826863958\">In the following exercises, solve the systems of equations by elimination.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167826863961\">\n<div data-type=\"problem\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-328854f38bbcaae2c89cea4fb8eefd4f_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;&#53;&#120;&#43;&#50;&#121;&#61;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#45;&#51;&#120;&#45;&#121;&#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=\"116\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1167831836066\">\n<p id=\"fs-id1167831836068\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-dda8f544c1358f19ec66e7314058b53a_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;&#54;&#120;&#45;&#53;&#121;&#61;&#45;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#50;&#120;&#43;&#121;&#61;&#49;&#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=\"124\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167831825741\">\n<p id=\"fs-id1167831825743\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b385c1ebb27da6a9f60a8f21a49f0483_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#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-id1167831907914\">\n<div data-type=\"problem\" id=\"fs-id1167831907917\">\n<p id=\"fs-id1167831907919\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bc3e2d1d46c0a0ade0b9492ea2c0dd2d_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;&#120;&#45;&#53;&#121;&#61;&#55;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#51;&#120;&#45;&#121;&#61;&#49;&#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=\"43\" width=\"111\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835363585\">\n<div data-type=\"problem\" id=\"fs-id1167835363587\">\n<p id=\"fs-id1167835363589\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-27b3e6ad066bc733ddae7843c538f0f3_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;&#53;&#120;&#45;&#51;&#121;&#61;&#45;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#50;&#120;&#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=\"124\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167831112627\">\n<p id=\"fs-id1167834593004\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c472d4aedfcd5d7e779428d51ee49fae_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#55;&#44;&#49;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"47\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167830705259\">\n<div data-type=\"problem\" id=\"fs-id1167832058610\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-135991749befb3e82b1307c43199bf94_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;&#53;&#121;&#61;&#45;&#57;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#53;&#120;&#43;&#50;&#121;&#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=\"125\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834147076\">\n<div data-type=\"problem\" id=\"fs-id1167835376196\">\n<p id=\"fs-id1167835376198\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c276f096835ad66624bc15dd8cefdccc_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;&#120;&#45;&#51;&#121;&#61;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#50;&#120;&#43;&#53;&#121;&#61;&#45;&#51;&#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=\"133\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167826978568\">\n<p id=\"fs-id1167826978570\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0bd0b31c672d6277d527b16652fb255d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#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>\n<div data-type=\"exercise\" id=\"fs-id1167826978378\">\n<div data-type=\"problem\" id=\"fs-id1167826978380\">\n<p id=\"fs-id1167831954879\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2ccd8d1a3f8aa7e63619e7075daf6d0a_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;&#43;&#56;&#121;&#61;&#45;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#50;&#120;&#43;&#53;&#121;&#61;&#45;&#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=\"125\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834430174\">\n<div data-type=\"problem\" id=\"fs-id1167834430176\">\n<p id=\"fs-id1167835166917\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bac4765568681135dfb678c7b0009428_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;&#49;&#49;&#120;&#43;&#57;&#121;&#61;&#45;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#55;&#120;&#43;&#53;&#121;&#61;&#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=\"133\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834195693\">\n<p id=\"fs-id1167834195695\"><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>\n<div data-type=\"exercise\" id=\"fs-id1167835365118\">\n<div data-type=\"problem\" id=\"fs-id1167835365120\">\n<p id=\"fs-id1167835365122\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7da54d02f8290ddcaa21b4630f851688_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;&#43;&#56;&#121;&#61;&#54;&#55;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#53;&#120;&#43;&#51;&#121;&#61;&#54;&#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=\"120\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835332982\">\n<div data-type=\"problem\" id=\"fs-id1167835332984\">\n<p id=\"fs-id1167835332986\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fb932e549586f6d3fc60793281a89d12_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;&#120;&#43;&#57;&#121;&#61;&#45;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#51;&#120;&#43;&#49;&#51;&#121;&#61;&#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=\"43\" width=\"134\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834403143\">\n<p id=\"fs-id1167834403145\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ad61f0fb262a775c4f3b32185756ebea_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#49;&#44;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"60\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167832052599\">\n<div data-type=\"problem\" id=\"fs-id1167832052601\">\n<p id=\"fs-id1167832052603\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3d2e9ace77e33442a931fb48fe1df16f_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;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#120;&#45;&#121;&#61;&#45;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#120;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#50;&#125;&#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=\"44\" width=\"118\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834547275\">\n<div data-type=\"problem\" id=\"fs-id1167834547278\">\n<p id=\"fs-id1167834547280\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1e348ca4340e4b404df31f424dd13980_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;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#121;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#50;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#53;&#125;&#120;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#53;&#125;&#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=\"44\" width=\"115\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835345974\">\n<p id=\"fs-id1167835345976\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e2962cf78c978777fdc30c7b0f6f073c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#92;&#116;&#101;&#120;&#116;&#123;&#47;&#125;&#45;&#57;&#44;&#50;&#52;&#92;&#116;&#101;&#120;&#116;&#123;&#47;&#125;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"104\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835370986\">\n<div data-type=\"problem\" id=\"fs-id1167835370988\">\n<p id=\"fs-id1167835370990\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ab99aaa67d6d71818391eb90ef54dcf7_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;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#121;&#61;&#45;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#120;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#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=\"44\" width=\"117\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167832076583\">\n<div data-type=\"problem\" id=\"fs-id1167832076585\">\n<p id=\"fs-id1167834161573\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-14165a05b936d72852d5c9fe27d476e5_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;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#120;&#45;&#121;&#61;&#45;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;&#120;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#50;&#125;&#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=\"44\" width=\"118\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834377203\">\n<p id=\"fs-id1167834377205\"><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;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835374572\">\n<div data-type=\"problem\" id=\"fs-id1167835374574\">\n<p id=\"fs-id1167835374576\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-80c4731a316ba331082ef53183022132_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;&#120;&#43;&#121;&#61;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#54;&#120;&#43;&#51;&#121;&#61;&#57;&#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=\"111\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834120246\">\n<div data-type=\"problem\" id=\"fs-id1167834120248\">\n<p id=\"fs-id1167834120250\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-49a15c99c7df16b2ded0cd01d8310e5b_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;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#45;&#51;&#120;&#43;&#49;&#50;&#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=\"134\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835370201\">\n<p id=\"fs-id1167835370203\">infinitely many<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835370208\">\n<div data-type=\"problem\" id=\"fs-id1167832066123\">\n<p id=\"fs-id1167832066125\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bf7e284daf756541b93485cdc1b7c8d2_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;&#45;&#51;&#120;&#45;&#121;&#61;&#56;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#54;&#120;&#43;&#50;&#121;&#61;&#45;&#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=\"134\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835348718\">\n<div data-type=\"problem\" id=\"fs-id1167835348720\">\n<p id=\"fs-id1167835348722\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-dcd3e6daeb7187c48a48cc90a4ed2e24_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;&#120;&#43;&#51;&#121;&#61;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#50;&#48;&#120;&#43;&#49;&#53;&#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=\"138\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167826781183\">\n<p id=\"fs-id1167826781185\">infinitely many<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167826781191\"><strong data-effect=\"bold\">Choose the Most Convenient Method to Solve a System of Linear Equations<\/strong><\/p>\n<p id=\"fs-id1167834431079\">In the following exercises, decide whether it would be more convenient to solve the system of equations by substitution or elimination.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167834431083\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835282942\">\n<p id=\"fs-id1167835282945\">\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0fbcf17fee700e3df2b21aa0979d18f5_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;&#56;&#120;&#45;&#49;&#53;&#121;&#61;&#45;&#51;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#54;&#120;&#43;&#51;&#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=\"142\" style=\"vertical-align: -17px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2f7b5a2c671c59d629bdc2113be4ed0b_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;&#52;&#121;&#45;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#52;&#120;&#45;&#50;&#121;&#61;&#45;&#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=\"125\" style=\"vertical-align: -17px;\" \/><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831239296\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167831239299\">\n<p id=\"fs-id1167826807915\">\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-dc60a182f9b3f5fc2cdd13f351005bb6_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;&#55;&#120;&#45;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#51;&#120;&#45;&#50;&#121;&#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=\"120\" style=\"vertical-align: -17px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f4ba0ccec14e5d20afc7ec09e310ad19_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;&#49;&#50;&#120;&#45;&#53;&#121;&#61;&#45;&#52;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#51;&#120;&#43;&#55;&#121;&#61;&#45;&#49;&#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=\"142\" style=\"vertical-align: -17px;\" \/><\/div>\n<div data-type=\"solution\" id=\"fs-id1167835370118\">\n<p id=\"fs-id1167835370121\"><span class=\"token\">\u24d0<\/span> substitution <span class=\"token\">\u24d1<\/span> elimination<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831920498\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167831920500\">\n<p id=\"fs-id1167831920502\">\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7b5711d3fe2c1abce97492d5f1a56d87_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;&#120;&#43;&#57;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#53;&#120;&#45;&#50;&#121;&#61;&#45;&#50;&#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=\"133\" style=\"vertical-align: -17px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b3da1863cc4b90390bdcb246fde9876e_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;&#120;&#45;&#52;&#121;&#61;&#50;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#51;&#120;&#43;&#53;&#121;&#61;&#45;&#49;&#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=\"134\" style=\"vertical-align: -17px;\" \/><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834394773\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167834394775\">\n<p id=\"fs-id1167830963602\">\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e818a3dfbeef49d6aee226b966c760bd_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;&#49;&#52;&#120;&#45;&#49;&#53;&#121;&#61;&#45;&#51;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#55;&#120;&#43;&#50;&#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=\"152\" style=\"vertical-align: -17px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b158f5661df0f640d0ebcaa2d6eae2d6_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;&#57;&#121;&#45;&#49;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#50;&#120;&#45;&#55;&#121;&#61;&#45;&#50;&#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=\"43\" width=\"134\" style=\"vertical-align: -17px;\" \/><\/div>\n<div data-type=\"solution\" id=\"fs-id1167835379819\">\n<p id=\"fs-id1167835379821\"><span class=\"token\">\u24d0<\/span> elimination <span class=\"token\">\u24d1<\/span> substituion<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"writing\" data-depth=\"2\" id=\"fs-id1167830984904\">\n<h4 data-type=\"title\">Writing Exercises<\/h4>\n<div data-type=\"exercise\" id=\"fs-id1167830984912\">\n<div data-type=\"problem\" id=\"fs-id1167830984914\">\n<p id=\"fs-id1167828426612\">In a system of linear equations, the two equations have the same intercepts. Describe the possible solutions to the system.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167830964362\">\n<div data-type=\"problem\" id=\"fs-id1167830964364\">\n<p>Solve the system of equations by substitution and explain all your steps in words: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0f02580527de2982a879c2479fd464f0_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;&#43;&#121;&#61;&#49;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#120;&#61;&#121;&#45;&#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=\"123\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834397152\">\n<p id=\"fs-id1167834397154\">Answers will vary.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167832153527\">\n<div data-type=\"problem\">\n<p>Solve the system of equations by elimination and explain all your steps in words: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ea99b4efaa875e3df03ca6477a318965_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;&#53;&#120;&#43;&#52;&#121;&#61;&#49;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#50;&#120;&#61;&#51;&#121;&#43;&#50;&#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=\"132\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1167835433652\">\n<p id=\"fs-id1167835433654\">Solve the system of equations <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f8ad708fe8b845433e2833c8447d52e5_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;&#49;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#120;&#45;&#121;&#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=\"102\" style=\"vertical-align: -17px;\" \/><\/p>\n<p id=\"fs-id1167834494857\"><span class=\"token\">\u24d0<\/span> by graphing <span class=\"token\">\u24d1<\/span> by substitution<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span> Which method do you prefer? Why?<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834219257\">\n<p id=\"fs-id1167834219259\">Answers will vary.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1167834219266\">\n<h4 data-type=\"title\">Self Check<\/h4>\n<p id=\"fs-id1167834397580\">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-id1167832060286\" data-alt=\"This table has 4 columns 5 rows and a header row. The header row labels each column: I can, confidently, with some help and no, I don\u2019t get it. The first column has the following statements: determine whether an ordered pair is a solution of a system of equations, solve a system of linear equations by graphing, solve a system of equations by substitution, solve a system of equations by elimination, choose the most convenient method to solve a system of linear equations. The remaining columns are blank.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_01_201_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This table has 4 columns 5 rows and a header row. The header row labels each column: I can, confidently, with some help and no, I don\u2019t get it. The first column has the following statements: determine whether an ordered pair is a solution of a system of equations, solve a system of linear equations by graphing, solve a system of equations by substitution, solve a system of equations by elimination, choose the most convenient method to solve a system of linear equations. The remaining columns are blank.\" \/><\/span><\/p>\n<p id=\"fs-id1167832060283\">If most of your checks were:<\/p>\n<p id=\"fs-id1167832150935\"><strong data-effect=\"bold\">\u2026confidently.<\/strong> Congratulations! You have achieved the objectives in this section. Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific.<\/p>\n<p id=\"fs-id1167832150945\"><strong data-effect=\"bold\">\u2026with some help.<\/strong> This must be addressed quickly because topics you do not master become potholes in your road to success. In math every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Who can you ask for help? Your fellow classmates and instructor are good resources. Is there a place on campus where math tutors are available? Can your study skills be improved?<\/p>\n<p id=\"fs-id1167830699785\"><strong data-effect=\"bold\">\u2026no &#8211; I don\u2019t get it!<\/strong> This is a warning sign and you must not ignore it. You should get help right away or you will quickly be overwhelmed. See your instructor as soon as you can to discuss your situation. Together you can come up with a plan to get you the help you need.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"glossary\" class=\"textbox shaded\">\n<h3 data-type=\"glossary-title\">Glossary<\/h3>\n<dl id=\"fs-id1167835321158\">\n<dt>coincident lines<\/dt>\n<dd id=\"fs-id1167835321161\">Coincident lines have the same slope and same <em data-effect=\"italics\">y<\/em>-intercept.<\/dd>\n<\/dl>\n<dl id=\"fs-id1167831890606\">\n<dt>consistent and inconsistent systems<\/dt>\n<dd id=\"fs-id1167831890609\">Consistent system of equations is a system of equations with at least one solution; inconsistent system of equations is a system of equations with no solution.<\/dd>\n<\/dl>\n<dl id=\"fs-id1167831890614\">\n<dt>solutions of a system of equations<\/dt>\n<dd id=\"fs-id1167834191858\">Solutions of a system of equations are the values of the variables that make <em data-effect=\"italics\">all<\/em> the equations true; solution is represented by an ordered pair <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0ab61ec2033ed5b69d0447eac5d6a4f8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#44;&#121;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"47\" style=\"vertical-align: -4px;\" \/><\/dd>\n<\/dl>\n<dl id=\"fs-id1167828182376\">\n<dt>system of linear equations<\/dt>\n<dd id=\"fs-id1167828182379\">When two or more linear equations are grouped together, they form a system of linear equations.<\/dd>\n<\/dl>\n<\/div>\n","protected":false},"author":103,"menu_order":2,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-2373","chapter","type-chapter","status-publish","hentry"],"part":2305,"_links":{"self":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters\/2373","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":1,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters\/2373\/revisions"}],"predecessor-version":[{"id":15155,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters\/2373\/revisions\/15155"}],"part":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/parts\/2305"}],"metadata":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters\/2373\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/media?parent=2373"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapter-type?post=2373"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/contributor?post=2373"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/license?post=2373"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}