{"id":1744,"date":"2018-12-11T13:34:51","date_gmt":"2018-12-11T18:34:51","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/chapter\/graph-linear-equations-in-two-variables\/"},"modified":"2018-12-11T13:34:51","modified_gmt":"2018-12-11T18:34:51","slug":"graph-linear-equations-in-two-variables","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/chapter\/graph-linear-equations-in-two-variables\/","title":{"raw":"Graph Linear Equations in Two Variables","rendered":"Graph Linear Equations in 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>Plot points in a rectangular coordinate system<\/li><li>Graph a linear equation by plotting points<\/li><li>Graph vertical and horizontal lines<\/li><li>Find the x- and y-intercepts<\/li><li>Graph a line using the intercepts<\/li><\/ul><\/div><div data-type=\"note\" id=\"fs-id1167835512168\" class=\"be-prepared\"><p id=\"fs-id1167834301088\">Before you get started, take this readiness quiz.<\/p><ol id=\"fs-id1167826978613\" type=\"1\"><li>Evaluate \\(5x-4\\) when \\(x=-1.\\)<div data-type=\"newline\"><br><\/div> If you missed this problem, review <a href=\"\/contents\/d3697553-3900-453f-8c08-8f74d55711ba#fs-id1167836530265\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li><li>Evaluate \\(3x-2y\\) when \\(x=4,y=-3.\\)<div data-type=\"newline\"><br><\/div> If you missed this problem, review <a href=\"\/contents\/05eab039-6d1c-4d80-8c8c-94469164a52c#fs-id1167832053133\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li><li>Solve for <em data-effect=\"italics\">y<\/em>: \\(8-3y=20.\\)<div data-type=\"newline\"><br><\/div> If you missed this problem, review <a href=\"\/contents\/9f100e8f-6d15-4cae-bc22-c306e9d7d55c#fs-id1167836432956\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li><\/ol><\/div><div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167835237326\"><h3 data-type=\"title\">Plot Points on a Rectangular Coordinate System<\/h3><p id=\"fs-id1167834423120\">Just like maps use a grid system to identify locations, a grid system is used in algebra to show a relationship between two variables in a rectangular coordinate system. The rectangular coordinate system is also called the <em data-effect=\"italics\">xy<\/em>-plane or the \u201ccoordinate plane.\u201d<\/p><p id=\"fs-id1167832052908\">The rectangular coordinate system is formed by two intersecting number lines, one horizontal and one vertical. The horizontal number line is called the <em data-effect=\"italics\">x<\/em>-axis. The vertical number line is called the <em data-effect=\"italics\">y<\/em>-axis. These axes divide a plane into four regions, called quadrants. The quadrants are identified by Roman numerals, beginning on the upper right and proceeding counterclockwise. See <a href=\"#CNX_IntAlg_Figure_03_01_001\" class=\"autogenerated-content\">(Figure)<\/a>.<\/p><div class=\"bc-figure figure\" id=\"CNX_IntAlg_Figure_03_01_001\"><span data-type=\"media\" id=\"fs-id1167835381166\" data-alt=\"This figure shows a square grid. A horizontal number line in the middle is labeled x. A vertical number line in the middle is labeled y. The number lines intersect at zero and together divide the square grid into 4 equally sized smaller squares. The square in the top right is labeled I. The square in the top left is labeled II. The square in the bottom left is labeled III. The square in the bottom right is labeled IV.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_001_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a square grid. A horizontal number line in the middle is labeled x. A vertical number line in the middle is labeled y. The number lines intersect at zero and together divide the square grid into 4 equally sized smaller squares. The square in the top right is labeled I. The square in the top left is labeled II. The square in the bottom left is labeled III. The square in the bottom right is labeled IV.\"><\/span><\/div><p id=\"fs-id1167832053394\">In the rectangular coordinate system, every point is represented by an <span data-type=\"term\">ordered pair<\/span>. The first number in the ordered pair is the <em data-effect=\"italics\">x<\/em>-coordinate of the point, and the second number is the <em data-effect=\"italics\">y<\/em>-coordinate of the point. The phrase \u201cordered pair\u201d means that the order is important.<\/p><div data-type=\"note\" id=\"fs-id1167834538411\"><div data-type=\"title\">Ordered Pair<\/div><p id=\"fs-id1167835240319\">An <strong data-effect=\"bold\">ordered pair<\/strong>, \\(\\left(x,y\\right)\\) gives the coordinates of a point in a rectangular coordinate system. The first number is the <em data-effect=\"italics\">x<\/em>-coordinate. The second number is the <em data-effect=\"italics\">y<\/em>-coordinate.<\/p><span data-type=\"media\" id=\"fs-id1167835478964\" data-alt=\"This figure shows the expression (x, y). The variable x is labeled x-coordinate. The variable y is labeled y-coordinate.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_002_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows the expression (x, y). The variable x is labeled x-coordinate. The variable y is labeled y-coordinate.\"><\/span><\/div><p id=\"fs-id1167834397450\">What is the ordered pair of the point where the axes cross? At that point both coordinates are zero, so its ordered pair is \\(\\left(0,0\\right).\\) The point \\(\\left(0,0\\right)\\) has a special name. It is called the <span data-type=\"term\">origin<\/span>.<\/p><div data-type=\"note\" id=\"fs-id1167835194673\"><div data-type=\"title\">The Origin<\/div><p id=\"fs-id1167835327055\">The point \\(\\left(0,0\\right)\\) is called the <strong data-effect=\"bold\">origin<\/strong>. It is the point where the <em data-effect=\"italics\">x<\/em>-axis and <em data-effect=\"italics\">y<\/em>-axis intersect.<\/p><\/div><p id=\"fs-id1167834377125\">We use the coordinates to locate a point on the <em data-effect=\"italics\">xy<\/em>-plane. Let\u2019s plot the point \\(\\left(1,3\\right)\\) as an example. First, locate 1 on the <em data-effect=\"italics\">x<\/em>-axis and lightly sketch a vertical line through \\(x=1.\\) Then, locate 3 on the <em data-effect=\"italics\">y<\/em>-axis and sketch a horizontal line through \\(y=3.\\) Now, find the point where these two lines meet\u2014that is the point with coordinates \\(\\left(1,3\\right).\\) See <a href=\"#CNX_IntAlg_Figure_03_01_003\" class=\"autogenerated-content\">(Figure)<\/a>.<\/p><div class=\"bc-figure figure\" id=\"CNX_IntAlg_Figure_03_01_003\"><span data-type=\"media\" id=\"fs-id1167835376254\" data-alt=\"This figure shows a point plotted on the x y-coordinate plane. The x and y axes run from negative 6 to 6. The point (1, 3) is labeled. A dashed vertical line goes through the point and intersects the x-axis at xplus1. A dashed horizontal line goes through the point and intersects the y-axis at yplus3.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_003_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a point plotted on the x y-coordinate plane. The x and y axes run from negative 6 to 6. The point (1, 3) is labeled. A dashed vertical line goes through the point and intersects the x-axis at xplus1. A dashed horizontal line goes through the point and intersects the y-axis at yplus3.\"><\/span><\/div><p id=\"fs-id1167835374762\">Notice that the vertical line through \\(x=1\\) and the horizontal line through \\(y=3\\) are not part of the graph. We just used them to help us locate the point \\(\\left(1,3\\right).\\)<\/p><p id=\"fs-id1167832059451\">When one of the coordinate is zero, the point lies on one of the axes. In <a href=\"#CNX_IntAlg_Figure_03_01_004\" class=\"autogenerated-content\">(Figure)<\/a> the point \\(\\left(0,4\\right)\\) is on the <em data-effect=\"italics\">y<\/em>-axis and the point \\(\\left(-2,0\\right)\\) is on the <em data-effect=\"italics\">x<\/em>-axis.<\/p><div class=\"bc-figure figure\" id=\"CNX_IntAlg_Figure_03_01_004\"><span data-type=\"media\" id=\"fs-id1167835512862\" data-alt=\"This figure shows points plotted on the x y-coordinate plane. The x and y axes run from negative 6 to 6. The point (negative 2, 0) is labeled and lies on the x-axis. The point (0, 4) is labeled and lies on the y-axis.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_004_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows points plotted on the x y-coordinate plane. The x and y axes run from negative 6 to 6. The point (negative 2, 0) is labeled and lies on the x-axis. The point (0, 4) is labeled and lies on the y-axis.\"><\/span><\/div><div data-type=\"note\" id=\"fs-id1167835330675\"><div data-type=\"title\">Points on the Axes<\/div><p id=\"fs-id1167834403364\">Points with a <em data-effect=\"italics\">y<\/em>-coordinate equal to 0 are on the <em data-effect=\"italics\">x<\/em>-axis, and have coordinates \\(\\left(a,0\\right).\\)<\/p><p id=\"fs-id1167835167375\">Points with an <em data-effect=\"italics\">x<\/em>-coordinate equal to 0 are on the <em data-effect=\"italics\">y<\/em>-axis, and have coordinates \\(\\left(0,b\\right).\\)<\/p><\/div><div data-type=\"example\" id=\"fs-id1167834190090\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167834423551\"><div data-type=\"problem\" id=\"fs-id1167835416428\"><p id=\"fs-id1167831893573\">Plot each point in the rectangular coordinate system and identify the quadrant in which the point is located:<\/p><p id=\"fs-id1167831103917\"><span class=\"token\">\u24d0<\/span>\\(\\left(-5,4\\right)\\)<span class=\"token\">\u24d1<\/span>\\(\\left(-3,-4\\right)\\)<span class=\"token\">\u24d2<\/span>\\(\\left(2,-3\\right)\\)<span class=\"token\">\u24d3<\/span>\\(\\left(0,-1\\right)\\)<span class=\"token\">\u24d4<\/span>\\(\\left(3,\\frac{5}{2}\\right).\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835336204\"><p id=\"fs-id1167835610006\">The first number of the coordinate pair is the <em data-effect=\"italics\">x<\/em>-coordinate, and the second number is the <em data-effect=\"italics\">y<\/em>-coordinate. To plot each point, sketch a vertical line through the <em data-effect=\"italics\">x<\/em>-coordinate and a horizontal line through the <em data-effect=\"italics\">y<\/em>-coordinate. Their intersection is the point.<\/p><div data-type=\"newline\"><br><\/div> <span class=\"token\">\u24d0<\/span> Since \\(x=-5,\\) the point is to the left of the <em data-effect=\"italics\">y<\/em>-axis. Also, since \\(y=4,\\) the point is above the <em data-effect=\"italics\">x<\/em>-axis. The point \\(\\left(-5,4\\right)\\) is in Quadrant II.<div data-type=\"newline\"><br><\/div> <span class=\"token\">\u24d1<\/span> Since \\(x=-3,\\) the point is to the left of the <em data-effect=\"italics\">y<\/em>-axis. Also, since \\(y=-4,\\) the point is below the <em data-effect=\"italics\">x<\/em>-axis. The point \\(\\left(-3,-4\\right)\\) is in Quadrant III.<div data-type=\"newline\"><br><\/div> <span class=\"token\">\u24d2<\/span> Since \\(x=2,\\) the point is to the right of the <em data-effect=\"italics\">y<\/em>-axis. Since \\(y=-3,\\) the point is below the <em data-effect=\"italics\">x<\/em>-axis. The point \\(\\left(2,-3\\right)\\) is in Quadrant IV.<div data-type=\"newline\"><br><\/div> <span class=\"token\">\u24d3<\/span> Since \\(x=0,\\) the point whose coordinates are \\(\\left(0,-1\\right)\\) is on the <em data-effect=\"italics\">y<\/em>-axis.<div data-type=\"newline\"><br><\/div> <span class=\"token\">\u24d4<\/span> Since \\(x=3,\\) the point is to the right of the <em data-effect=\"italics\">y<\/em>-axis. Since \\(y=\\frac{5}{2},\\) the point is above the <em data-effect=\"italics\">x<\/em>-axis. (It may be helpful to write \\(\\frac{5}{2}\\) as a mixed number or decimal.) The point \\(\\left(3,\\frac{5}{2}\\right)\\) is in Quadrant I.<div data-type=\"newline\"><br><\/div> <span data-type=\"media\" id=\"fs-id1167835356754\" data-alt=\"This figure shows points plotted on the x y-coordinate plane. The x and y axes run from negative 6 to 6. The following points are labeled: (3, 5 divided by 2), (negative 2, 3), negative 5, 4), (negative 3, negative 4), and (2, negative 3).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_005_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows points plotted on the x y-coordinate plane. The x and y axes run from negative 6 to 6. The following points are labeled: (3, 5 divided by 2), (negative 2, 3), negative 5, 4), (negative 3, negative 4), and (2, negative 3).\"><\/span><\/div><\/div><\/div><div data-type=\"note\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167834495427\"><div data-type=\"problem\" id=\"fs-id1167835207148\"><p id=\"fs-id1167834402753\">Plot each point in a rectangular coordinate system and identify the quadrant in which the point is located:<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d0<\/span> \\(\\left(-2,1\\right)\\) <span class=\"token\">\u24d1<\/span> \\(\\left(-3,-1\\right)\\) <span class=\"token\">\u24d2<\/span> \\(\\left(4,-4\\right)\\) <span class=\"token\">\u24d3<\/span> \\(\\left(-4,4\\right)\\) <span class=\"token\">\u24d4<\/span> \\(\\left(-4,\\frac{3}{2}\\right)\\)<\/div><div data-type=\"solution\" id=\"fs-id1167835369019\"><span data-type=\"media\" id=\"fs-id1167831920613\" data-alt=\"This figure shows points plotted on the x y-coordinate plane. The x and y axes run from negative 6 to 6. The point labeled a is 2 units to the left of the origin and 1 unit above the origin and is located in quadrant II. The point labeled b is 3 units to the left of the origin and 1 unit below the origin and is located in quadrant III. The point labeled c is 4 units to the right of the origin and 4 units below the origin and is located in quadrant IV. The point labeled d is 4 units to the left of the origin and 4 units above the origin and is located in quadrant II. The point labeled e is 4 units to the left of the origin and 1 and a half units above the origin and is located in quadrant II.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_301_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows points plotted on the x y-coordinate plane. The x and y axes run from negative 6 to 6. The point labeled a is 2 units to the left of the origin and 1 unit above the origin and is located in quadrant II. The point labeled b is 3 units to the left of the origin and 1 unit below the origin and is located in quadrant III. The point labeled c is 4 units to the right of the origin and 4 units below the origin and is located in quadrant IV. The point labeled d is 4 units to the left of the origin and 4 units above the origin and is located in quadrant II. The point labeled e is 4 units to the left of the origin and 1 and a half units above the origin and is located in quadrant II.\"><\/span><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167834161888\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167831920690\"><div data-type=\"problem\" id=\"fs-id1167832068391\"><p id=\"fs-id1167831823636\">Plot each point in a rectangular coordinate system and identify the quadrant in which the point is located:<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d0<\/span> \\(\\left(-4,1\\right)\\) <span class=\"token\">\u24d1<\/span> \\(\\left(-2,3\\right)\\) <span class=\"token\">\u24d2<\/span> \\(\\left(2,-5\\right)\\) <span class=\"token\">\u24d3<\/span> \\(\\left(-2,5\\right)\\) <span class=\"token\">\u24d4<\/span> \\(\\left(-3,\\frac{5}{2}\\right)\\)<\/div><div data-type=\"solution\"><span data-type=\"media\" data-alt=\"This figure shows points plotted on the x y-coordinate plane. The x and y axes run from negative 6 to 6. The point labeled a is 4 units to the left of the origin and 1 unit above the origin and is located in quadrant II. The point labeled b is 2 units to the left of the origin and 3 units above the origin and is located in quadrant II. The point labeled c is 2 units to the right of the origin and 5 units below the origin and is located in quadrant IV. The point labeled d is 2 units to the left of the origin and 5 units above the origin and is located in quadrant II. The point labeled e is 3 units to the left of the origin and 2 and a half units above the origin and is located in quadrant II.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_302_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows points plotted on the x y-coordinate plane. The x and y axes run from negative 6 to 6. The point labeled a is 4 units to the left of the origin and 1 unit above the origin and is located in quadrant II. The point labeled b is 2 units to the left of the origin and 3 units above the origin and is located in quadrant II. The point labeled c is 2 units to the right of the origin and 5 units below the origin and is located in quadrant IV. The point labeled d is 2 units to the left of the origin and 5 units above the origin and is located in quadrant II. The point labeled e is 3 units to the left of the origin and 2 and a half units above the origin and is located in quadrant II.\"><\/span><\/div><\/div><\/div><p>The signs of the <em data-effect=\"italics\">x<\/em>-coordinate and <em data-effect=\"italics\">y<\/em>-coordinate affect the location of the points. You may have noticed some patterns as you graphed the points in the previous example. We can summarize sign patterns of the quadrants in this way:<\/p><div data-type=\"note\" id=\"fs-id1167832067452\"><div data-type=\"title\">Quadrants<\/div><div data-type=\"equation\" id=\"fs-id1167835328203\" class=\"unnumbered\" data-label=\"\">\\(\\begin{array}{cccc}\\hfill \\mathbf{\\text{Quadrant I}}\\hfill &amp; \\hfill \\phantom{\\rule{2em}{0ex}}\\mathbf{\\text{Quadrant II}}\\hfill &amp; \\hfill \\phantom{\\rule{2em}{0ex}}\\mathbf{\\text{Quadrant III}}\\hfill &amp; \\hfill \\phantom{\\rule{2em}{0ex}}\\mathbf{\\text{Quadrant IV}}\\hfill \\\\ \\hfill \\left(x,y\\right)\\hfill &amp; \\hfill \\phantom{\\rule{2em}{0ex}}\\left(x,y\\right)\\hfill &amp; \\hfill \\phantom{\\rule{2em}{0ex}}\\left(x,y\\right)\\hfill &amp; \\hfill \\phantom{\\rule{2em}{0ex}}\\left(x,y\\right)\\hfill \\\\ \\hfill \\left(+,+\\right)\\hfill &amp; \\hfill \\phantom{\\rule{2em}{0ex}}\\left(-,+\\right)\\hfill &amp; \\hfill \\phantom{\\rule{2em}{0ex}}\\left(-,-\\right)\\hfill &amp; \\hfill \\phantom{\\rule{2em}{0ex}}\\left(+,-\\right)\\hfill \\end{array}\\)<\/div><span data-type=\"media\" id=\"fs-id1167832066567\" data-alt=\"This figure shows the x y-coordinate plane with the four quadrants labeled. In the top right of the plane is quadrant I labeled (plus, plus). In the top left of the plane is quadrant II labeled (minus, plus). In the bottom left of the plane is quadrant III labeled (minus, minus). In the bottom right of the plane is quadrant IV labeled (plus, minus).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_006_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows the x y-coordinate plane with the four quadrants labeled. In the top right of the plane is quadrant I labeled (plus, plus). In the top left of the plane is quadrant II labeled (minus, plus). In the bottom left of the plane is quadrant III labeled (minus, minus). In the bottom right of the plane is quadrant IV labeled (plus, minus).\"><\/span><\/div><p id=\"fs-id1167835534159\">Up to now, all the equations you have solved were equations with just one variable. In almost every case, when you solved the equation you got exactly one solution. But equations can have more than one variable. Equations with two variables may be of the form \\(Ax+By=C.\\) An equation of this form is called a <span data-type=\"term\">linear equation<\/span> in two variables.<\/p><div data-type=\"note\" id=\"fs-id1167835355041\"><div data-type=\"title\">Linear Equation<\/div><p id=\"fs-id1167835305763\">An equation of the form \\(Ax+By=C,\\) where <em data-effect=\"italics\">A<\/em> and <em data-effect=\"italics\">B<\/em> are not both zero, is called a <strong data-effect=\"bold\">linear equation<\/strong> in two variables.<\/p><\/div><p id=\"fs-id1167835233457\">Here is an example of a linear equation in two variables, <em data-effect=\"italics\">x<\/em> and <em data-effect=\"italics\">y<\/em>.<\/p><span data-type=\"media\" id=\"fs-id1167831869016\" data-alt=\"This figure shows the equation A x plus B y plus C. Below this is the equation x plus 4 y plus 8. Below this are the equations A plus 1, B plus 4, C plus 8. B and 4 are the same color in all the equations. C and 8 are the same color in all the equations.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_007_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows the equation A x plus B y plus C. Below this is the equation x plus 4 y plus 8. Below this are the equations A plus 1, B plus 4, C plus 8. B and 4 are the same color in all the equations. C and 8 are the same color in all the equations.\"><\/span><p id=\"fs-id1167835308773\">The equation \\(y=-3x+5\\) is also a linear equation. But it does not appear to be in the form \\(Ax+By=C.\\) We can use the Addition Property of Equality and rewrite it in \\(Ax+By=C\\) form.<\/p><div data-type=\"equation\" id=\"fs-id1167835510205\" class=\"unnumbered\" data-label=\"\">\\(\\begin{array}{cccccc}&amp; &amp; &amp; \\hfill y&amp; =\\hfill &amp; -3x+5\\hfill \\\\ \\text{Add to both sides.}\\hfill &amp; &amp; &amp; \\hfill y+3x&amp; =\\hfill &amp; 3x+5+3x\\hfill \\\\ \\text{Simplify.}\\hfill &amp; &amp; &amp; \\hfill y+3x&amp; =\\hfill &amp; 5\\hfill \\\\ \\text{Use the Commutative Property to put it in}\\hfill &amp; &amp; &amp; \\\\ Ax+By=C\\phantom{\\rule{0.2em}{0ex}}\\text{form.}\\hfill &amp; &amp; &amp; 3x+y\\hfill &amp; =\\hfill &amp; 5\\hfill \\end{array}\\)<\/div><p id=\"fs-id1167834430875\">By rewriting \\(y=-3x+5\\) as \\(3x+y=5,\\) we can easily see that it is a linear equation in two variables because it is of the form \\(Ax+By=C.\\) When an equation is in the form \\(Ax+By=C,\\) we say it is in <span data-type=\"term\">standard form of a linear equation<\/span>.<\/p><div data-type=\"note\" id=\"fs-id1167835214784\"><div data-type=\"title\">Standard Form of Linear Equation<\/div><p id=\"fs-id1167835237077\">A linear equation is in <strong data-effect=\"bold\">standard form<\/strong> when it is written \\(Ax+By=C.\\)<\/p><\/div><p id=\"fs-id1167835346035\">Most people prefer to have <em data-effect=\"italics\">A<\/em>, <em data-effect=\"italics\">B<\/em>, and <em data-effect=\"italics\">C<\/em> be integers and \\(A\\ge 0\\) when writing a linear equation in standard form, although it is not strictly necessary.<\/p><p id=\"fs-id1167834423852\">Linear equations have infinitely many solutions. For every number that is substituted for <em data-effect=\"italics\">x<\/em> there is a corresponding <em data-effect=\"italics\">y<\/em> value. This pair of values is a <span data-type=\"term\">solution<\/span> to the linear equation and is represented by the ordered pair \\(\\left(x,y\\right).\\) When we substitute these values of <em data-effect=\"italics\">x<\/em> and <em data-effect=\"italics\">y<\/em> into the equation, the result is a true statement, because the value on the left side is equal to the value on the right side.<\/p><div data-type=\"note\" id=\"fs-id1167835311076\"><div data-type=\"title\">Solution of a Linear Equation in Two Variables<\/div><p id=\"fs-id1167834377117\">An ordered pair \\(\\left(x,y\\right)\\) is a <strong data-effect=\"bold\">solution<\/strong> of the linear equation\\(Ax+By=C,\\) if the equation is a true statement when the <em data-effect=\"italics\">x<\/em>- and <em data-effect=\"italics\">y<\/em>-values of the ordered pair are substituted into the equation.<\/p><\/div><p id=\"fs-id1167834190603\">Linear equations have infinitely many solutions. We can plot these solutions in the rectangular coordinate system. The points will line up perfectly in a straight line. We connect the points with a straight line to get the graph of the equation. We put arrows on the ends of each side of the line to indicate that the line continues in both directions.<\/p><p id=\"fs-id1167835362190\">A graph is a visual representation of all the solutions of the equation. It is an example of the saying, \u201cA picture is worth a thousand words.\u201d The line shows you <em data-effect=\"italics\">all<\/em> the solutions to that equation. Every point on the line is a solution of the equation. And, every solution of this equation is on this line. This line is called the graph of the equation. Points <em data-effect=\"italics\">not<\/em> on the line are not solutions!<\/p><div data-type=\"note\" id=\"fs-id1167835309099\"><div data-type=\"title\">Graph of a Linear Equation<\/div><p id=\"fs-id1167834279380\">The graph of a linear equation \\(Ax+By=C\\) is a straight line.<\/p><ul id=\"fs-id1167830898857\" data-bullet-style=\"bullet\"><li>Every point on the line is a solution of the equation.<\/li><li>Every solution of this equation is a point on this line.<\/li><\/ul><\/div><div data-type=\"example\" id=\"fs-id1167835400321\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167835190713\"><div data-type=\"problem\" id=\"fs-id1167835304682\"><p id=\"fs-id1167835274549\">The graph of \\(y=2x-3\\) is shown.<\/p><span data-type=\"media\" id=\"fs-id1167835421585\" data-alt=\"This figure shows a straight line graphed on the x y-coordinate plane. The x and y-axes run from negative 10 to 10. The line has arrows on both ends and goes through the points (negative 3, negative 9), (negative 2, negative 7), (negative 1, negative 5), (0, negative 3), (1, negative 1), (2, 1), (3, 3), (4, 5), (5, 7), and (6, 9). The line is labeled y plus 2 x minus 3.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_008_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a straight line graphed on the x y-coordinate plane. The x and y-axes run from negative 10 to 10. The line has arrows on both ends and goes through the points (negative 3, negative 9), (negative 2, negative 7), (negative 1, negative 5), (0, negative 3), (1, negative 1), (2, 1), (3, 3), (4, 5), (5, 7), and (6, 9). The line is labeled y plus 2 x minus 3.\"><\/span><p id=\"fs-id1167835379641\">For each ordered pair, decide:<\/p><p id=\"fs-id1167831888160\"><span class=\"token\">\u24d0<\/span> Is the ordered pair a solution to the equation?<\/p><p id=\"fs-id1167835213648\"><span class=\"token\">\u24d1<\/span> Is the point on the line?<\/p><p id=\"fs-id1167831909258\">A: \\(\\left(0,-3\\right)\\) B: \\(\\left(3,3\\right)\\) C: \\(\\left(2,-3\\right)\\) D: \\(\\left(-1,-5\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834188708\"><p id=\"fs-id1167831893488\">Substitute the <em data-effect=\"italics\">x<\/em>- and <em data-effect=\"italics\">y<\/em>-values into the equation to check if the ordered pair is a solution to the equation.<\/p><div data-type=\"newline\"><br><\/div><div data-type=\"newline\"><br><\/div> <span class=\"token\">\u24d0<\/span><div data-type=\"newline\"><br><\/div> <span data-type=\"media\" id=\"fs-id1167835210264\" data-alt=\"Example A shows the ordered pair (0, negative 3). Under this is the equation y plus 2 x minus 3. Under this is the equation negative 3 equals 2 times 0 minus 3. The negative 3 and 0 are colored the same as the negative 3 and 0 in the ordered pair at the top. There is a question mark above the plus sign. Below this is the equation negative 3 plus negative 3. Below this is the statement (0, negative 3) is a solution. Example B shows the ordered pair (3, 3). Under this is the equation y plus 2 x minus 3. Under this is the equation 3 equals 2 times 3 minus 3. The 3 and 3 are colored the same as the 3 and 3 in the ordered pair at the top. There is a question mark above the plus sign. Below this is the equation 3 plus 3. Below this is the statement (3, 3) is a solution. Example C shows the ordered pair (2, negative 3). Under this is the equation y plus 2 x minus 3. Under this is the equation negative 3 equals 2 times 2 minus 3. The negative 3 and 2 are colored the same as the negative 3 and 2 in the ordered pair at the top. There is a question mark above the plus sign. Below this is the inequality negative 3 is not equal to 1. Below this is the statement (2, negative 3) is not a solution. Example D shows the ordered pair (negative 1, negative 5). Under this is the equation y plus 2 x minus 3. Under this is the equation negative 5 equals 2 times negative 1 minus 3. The negative 1 and negative 5 are colored the same as the negative 1 and negative 5 in the ordered pair at the top. There is a question mark above the plus sign. Below this is the equation negative 5 plus negative 5. Below this is the statement (negative 1, negative 5) is a solution.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_009_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Example A shows the ordered pair (0, negative 3). Under this is the equation y plus 2 x minus 3. Under this is the equation negative 3 equals 2 times 0 minus 3. The negative 3 and 0 are colored the same as the negative 3 and 0 in the ordered pair at the top. There is a question mark above the plus sign. Below this is the equation negative 3 plus negative 3. Below this is the statement (0, negative 3) is a solution. Example B shows the ordered pair (3, 3). Under this is the equation y plus 2 x minus 3. Under this is the equation 3 equals 2 times 3 minus 3. The 3 and 3 are colored the same as the 3 and 3 in the ordered pair at the top. There is a question mark above the plus sign. Below this is the equation 3 plus 3. Below this is the statement (3, 3) is a solution. Example C shows the ordered pair (2, negative 3). Under this is the equation y plus 2 x minus 3. Under this is the equation negative 3 equals 2 times 2 minus 3. The negative 3 and 2 are colored the same as the negative 3 and 2 in the ordered pair at the top. There is a question mark above the plus sign. Below this is the inequality negative 3 is not equal to 1. Below this is the statement (2, negative 3) is not a solution. Example D shows the ordered pair (negative 1, negative 5). Under this is the equation y plus 2 x minus 3. Under this is the equation negative 5 equals 2 times negative 1 minus 3. The negative 1 and negative 5 are colored the same as the negative 1 and negative 5 in the ordered pair at the top. There is a question mark above the plus sign. Below this is the equation negative 5 plus negative 5. Below this is the statement (negative 1, negative 5) is a solution.\"><\/span><p><\/p><div data-type=\"newline\"><br><\/div><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span> Plot the points \\(\\left(0,-3\\right),\\)\\(\\left(3,3\\right),\\)\\(\\left(2,-3\\right),\\) and \\(\\left(-1,-5\\right).\\)<div data-type=\"newline\"><br><\/div> <span data-type=\"media\" id=\"fs-id1167834061635\" data-alt=\"This figure shows the graph of the linear equation y plus 2 x minus 3 and some points graphed on the x y-coordinate plane. The x and y-axes run from negative 10 to 10. The line has arrows on both ends and goes through the points (negative 1, negative 5), (0, negative 3), and (3, 3). The point (2, negative 3) is also plotted but not on the line.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_010_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows the graph of the linear equation y plus 2 x minus 3 and some points graphed on the x y-coordinate plane. The x and y-axes run from negative 10 to 10. The line has arrows on both ends and goes through the points (negative 1, negative 5), (0, negative 3), and (3, 3). The point (2, negative 3) is also plotted but not on the line.\"><\/span><p><\/p><div data-type=\"newline\"><br><\/div> The points \\(\\left(0,3\\right),\\)\\(\\left(3,-3\\right),\\) and \\(\\left(-1,-5\\right)\\) are on the line \\(y=2x-3,\\) and the point \\(\\left(2,-3\\right)\\) is not on the line.<div data-type=\"newline\"><br><\/div> The points that are solutions to \\(y=2x-3\\) are on the line, but the point that is not a solution is not on the line.<\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167835353963\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167835529924\"><div data-type=\"problem\"><p id=\"fs-id1167835337195\">Use graph of \\(y=3x-1.\\) For each ordered pair, decide:<\/p><p id=\"fs-id1167834314579\"><span class=\"token\">\u24d0<\/span> Is the ordered pair a solution to the equation?<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span> Is the point on the line?<p id=\"fs-id1167835320293\">A \\(\\left(0,-1\\right)\\) B \\(\\left(2,5\\right)\\)<\/p><span data-type=\"media\" id=\"fs-id1167834156869\" data-alt=\"This figure shows a straight line graphed on the x y-coordinate plane. The x and y-axes run from negative 10 to 10. The line has arrows on both ends and goes through the points (negative 3, negative 10), (negative 2, negative 7), (negative 1, negative 4), (0, negative 1), (1, 2), (2, 5), and (3, 8). The line is labeled y plus 3 x minus 1.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_011_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a straight line graphed on the x y-coordinate plane. The x and y-axes run from negative 10 to 10. The line has arrows on both ends and goes through the points (negative 3, negative 10), (negative 2, negative 7), (negative 1, negative 4), (0, negative 1), (1, 2), (2, 5), and (3, 8). The line is labeled y plus 3 x minus 1.\"><\/span><\/div><div data-type=\"solution\" id=\"fs-id1167835239222\"><p id=\"fs-id1167835203046\"><span class=\"token\">\u24d0<\/span> yes, yes <span class=\"token\">\u24d1<\/span> yes, yes<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167828447116\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167835364020\"><div data-type=\"problem\" id=\"fs-id1167835253741\"><p id=\"fs-id1167835191408\">Use graph of \\(y=3x-1.\\) For each ordered pair, decide:<\/p><p><span class=\"token\">\u24d0<\/span> Is the ordered pair a solution to the equation?<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span> Is the point on the line?<p id=\"fs-id1167830962082\">A\\(\\left(3,-1\\right)\\) B\\(\\left(-1,-4\\right)\\)<\/p><span data-type=\"media\" id=\"fs-id1167835174029\" data-alt=\"This figure shows a straight line graphed on the x y-coordinate plane. The x and y-axes run from negative 10 to 10. The line has arrows on both ends and goes through the points (negative 3, negative 10), (negative 2, negative 7), (negative 1, negative 4), (0, negative 1), (1, 2), (2, 5), and (3, 8). The line is labeled y plus 3 x minus 1.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_012_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a straight line graphed on the x y-coordinate plane. The x and y-axes run from negative 10 to 10. The line has arrows on both ends and goes through the points (negative 3, negative 10), (negative 2, negative 7), (negative 1, negative 4), (0, negative 1), (1, 2), (2, 5), and (3, 8). The line is labeled y plus 3 x minus 1.\"><\/span><\/div><div data-type=\"solution\" id=\"fs-id1167831910982\"><p id=\"fs-id1167832051834\"><span class=\"token\">\u24d0<\/span> no, no <span class=\"token\">\u24d1<\/span> yes, yes<\/p><\/div><\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167835417087\"><h3 data-type=\"title\">Graph a Linear Equation by Plotting Points<\/h3><p id=\"fs-id1167835421131\">There are several methods that can be used to graph a linear equation. The first method we will use is called plotting points, or the Point-Plotting Method. We find three points whose coordinates are solutions to the equation and then plot them in a rectangular coordinate system. By connecting these points in a line, we have the graph of the linear equation.<\/p><div data-type=\"example\" id=\"fs-id1167832065594\" class=\"textbox textbox--examples\"><div data-type=\"title\">How to Graph a Linear Equation by Plotting Points<\/div><div data-type=\"exercise\"><div data-type=\"problem\" id=\"fs-id1167827957268\"><p id=\"fs-id1167834422792\">Graph the equation \\(y=2x+1\\) by plotting points.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835229769\"><span data-type=\"media\" id=\"fs-id1167830960982\" data-alt=\"Step 1 is to Find three points whose coordinates are solutions to the equation. You can choose any values for x or y. In this case since y is isolated on the left side of the equations, it is easier to choose values for x. Choosing x plus 0. We substitute this into the equation y plus 2 x plus 1 to get y plus 2 times 0 plus 1. This simplifies to y plus 0 plus 1. So y plus 1. Choosing x plus 1. We substitute this into the equation y plus 2 x plus 1 to get y plus 2 times 1 plus 1. This simplifies to y plus 2 plus 1. So y plus 3. Choosing x plus negative 2. We substitute this into the equation y plus 2 x plus 1 to get y plus 2 times negative 2 plus 1. This simplifies to y plus negative 4 plus 1. The y plus negative 3. Next we want to organize the solutions in a table. For this problem we will put the three solutions we just found in a table. The table has 5 rows and 3 columns. The first row is a title row with the equation y plus 2 x plus 1. The second row is a header row with the headers x, y, and (x, y). The third row has the numbers 0, 1, and (0, 1). The fourth row has the numbers 1, 3, and (1, 3). The fifth row has the numbers negative 2, negative 3, and (negative 2, negative 3).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_013a_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 1 is to Find three points whose coordinates are solutions to the equation. You can choose any values for x or y. In this case since y is isolated on the left side of the equations, it is easier to choose values for x. Choosing x plus 0. We substitute this into the equation y plus 2 x plus 1 to get y plus 2 times 0 plus 1. This simplifies to y plus 0 plus 1. So y plus 1. Choosing x plus 1. We substitute this into the equation y plus 2 x plus 1 to get y plus 2 times 1 plus 1. This simplifies to y plus 2 plus 1. So y plus 3. Choosing x plus negative 2. We substitute this into the equation y plus 2 x plus 1 to get y plus 2 times negative 2 plus 1. This simplifies to y plus negative 4 plus 1. The y plus negative 3. Next we want to organize the solutions in a table. For this problem we will put the three solutions we just found in a table. The table has 5 rows and 3 columns. The first row is a title row with the equation y plus 2 x plus 1. The second row is a header row with the headers x, y, and (x, y). The third row has the numbers 0, 1, and (0, 1). The fourth row has the numbers 1, 3, and (1, 3). The fifth row has the numbers negative 2, negative 3, and (negative 2, negative 3).\"><\/span><span data-type=\"media\" id=\"fs-id1167830698562\" data-alt=\"Step 2 is to plot the points in a rectangular coordinate system. Plot: (0, 1), (1, 3), (negative 2, negative 3). The figure then shows a graph of some points plotted on the x y-coordinate plane. The x and y axes run from negative 6 to 6. The points (0, 1), (1, 3), and (negative 2, negative 3) are plotted. Check that the points line up. If they do not, carefully check your work! Do the point line up? Yes, the points in this example line up.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_013b_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 2 is to plot the points in a rectangular coordinate system. Plot: (0, 1), (1, 3), (negative 2, negative 3). The figure then shows a graph of some points plotted on the x y-coordinate plane. The x and y axes run from negative 6 to 6. The points (0, 1), (1, 3), and (negative 2, negative 3) are plotted. Check that the points line up. If they do not, carefully check your work! Do the point line up? Yes, the points in this example line up.\"><\/span><span data-type=\"media\" id=\"fs-id1167835180910\" data-alt=\"Step 3 is to draw the line through the three points. Extend the line to fill the grid and put arrows on both ends of the line. This line is the graph of y plus 2 x plus 1. The figure shows the graph of a straight line on the x y-coordinate plane. The x and y axes run from negative 6 to 6. The points (negative 2, negative 3), (0, 1), and (1, 3) are plotted. The straight line goes through the three points and has arrows on both ends.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_013c_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 3 is to draw the line through the three points. Extend the line to fill the grid and put arrows on both ends of the line. This line is the graph of y plus 2 x plus 1. The figure shows the graph of a straight line on the x y-coordinate plane. The x and y axes run from negative 6 to 6. The points (negative 2, negative 3), (0, 1), and (1, 3) are plotted. The straight line goes through the three points and has arrows on both ends.\"><\/span><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167835623532\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167835348900\"><div data-type=\"problem\" id=\"fs-id1167834479634\"><p>Graph the equation by plotting points: \\(y=2x-3.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167830699920\"><p id=\"fs-id1167835279909\"><\/p><div data-type=\"newline\"><br><\/div><span data-type=\"media\" id=\"fs-id1167835420948\" data-alt=\"This figure shows a straight line graphed on the x y-coordinate plane. The x and y-axes run from negative 8 to 8. The line goes through the points (negative 2, negative 7), (negative 1, negative 5), (0, negative 3), (1, negative 1), (2, 1), (3, 3), (4, 5), and (5, 7).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_303_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a straight line graphed on the x y-coordinate plane. The x and y-axes run from negative 8 to 8. The line goes through the points (negative 2, negative 7), (negative 1, negative 5), (0, negative 3), (1, negative 1), (2, 1), (3, 3), (4, 5), and (5, 7).\"><\/span><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167835170817\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167831847014\"><div data-type=\"problem\" id=\"fs-id1167832059861\"><p id=\"fs-id1167835350848\">Graph the equation by plotting points: \\(y=-2x+4.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835358649\"><p id=\"fs-id1167835193486\"><\/p><div data-type=\"newline\"><br><\/div><span data-type=\"media\" id=\"fs-id1167835329515\" data-alt=\"This figure shows a straight line graphed on the x y-coordinate plane. The x and y-axes run from negative 8 to 8. The line goes through the points (negative 2, 8), (negative 1, 6), (0, 4), (1, 2), (2, 0), (3, negative 2), (4, negative 4), (5, negative 6) and (6, negative 8).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_304_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a straight line graphed on the x y-coordinate plane. The x and y-axes run from negative 8 to 8. The line goes through the points (negative 2, 8), (negative 1, 6), (0, 4), (1, 2), (2, 0), (3, negative 2), (4, negative 4), (5, negative 6) and (6, negative 8).\"><\/span><\/div><\/div><\/div><p id=\"fs-id1167835382110\">The steps to take when graphing a linear equation by plotting points are summarized here.<\/p><div data-type=\"note\" id=\"fs-id1167835350108\" class=\"howto\"><div data-type=\"title\">Graph a linear equation by plotting points.<\/div><ol id=\"fs-id1167835305295\" type=\"1\" class=\"stepwise\"><li>Find three points whose coordinates are solutions to the equation. Organize them in a table.<\/li><li>Plot the points in a rectangular coordinate system. Check that the points line up. If they do not, carefully check your work.<\/li><li>Draw the line through the three points. Extend the line to fill the grid and put arrows on both ends of the line.<\/li><\/ol><\/div><p>It is true that it only takes two points to determine a line, but it is a good habit to use three points. If you only plot two points and one of them is incorrect, you can still draw a line but it will not represent the solutions to the equation. It will be the wrong line.<\/p><p id=\"fs-id1167834531459\">If you use three points, and one is incorrect, the points will not line up. This tells you something is wrong and you need to check your work. Look at the difference between these illustrations.<\/p><span data-type=\"media\" id=\"fs-id1167834111513\" data-alt=\"The figure shows two images. In the first image there are three points with a straight line going through all three. In the second image there are three points that do not all lie on a straight line.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_014_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows two images. In the first image there are three points with a straight line going through all three. In the second image there are three points that do not all lie on a straight line.\"><\/span><p id=\"fs-id1167826828398\">When an equation includes a fraction as the coefficient of \\(x\\), we can still substitute any numbers for <em data-effect=\"italics\">x<\/em>. But the arithmetic is easier if we make \u201cgood\u201d choices for the values of <em data-effect=\"italics\">x<\/em>. This way we will avoid fractional answers, which are hard to graph precisely.<\/p><div data-type=\"example\" id=\"fs-id1167834408393\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167835232887\"><div data-type=\"problem\" id=\"fs-id1167835214095\"><p id=\"fs-id1167830914798\">Graph the equation: \\(y=\\frac{1}{2}x+3.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167832128684\"><p id=\"fs-id1167832055099\">Find three points that are solutions to the equation. Since this equation has the fraction \\(\\frac{1}{2}\\) as a coefficient of <em data-effect=\"italics\">x<\/em>, we will choose values of <em data-effect=\"italics\">x<\/em> carefully. We will use zero as one choice and multiples of 2 for the other choices. Why are multiples of two a good choice for values of <em data-effect=\"italics\">x<\/em>? By choosing multiples of 2 the multiplication by \\(\\frac{1}{2}\\) simplifies to a whole number<\/p><div data-type=\"newline\"><br><\/div> <span data-type=\"media\" id=\"fs-id1167831023926\" data-alt=\"The first set of equations starts with x plus 0. Under this is the equation y plus 1 half x plus 3. Under this is the equation y plus 1 half times 0 plus 3. Below this is the equation y plus 0 plus 3. Below this is the equation y plus 3. The second set of equations starts with x plus 2. Under this is the equation y plus 1 half x plus 3. Under this is the equation y plus 1 half times 2 plus 3. Below this is the equation y plus 1 plus 3. Below this is the equation y plus 4. The third set of equations starts with x plus 4. Under this is the equation y plus 1 half x plus 3. Under this is the equation y plus 1 half times 4 plus 3. Below this is the equation y plus 2 plus 3. Below this is the equation y plus 5.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_015_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The first set of equations starts with x plus 0. Under this is the equation y plus 1 half x plus 3. Under this is the equation y plus 1 half times 0 plus 3. Below this is the equation y plus 0 plus 3. Below this is the equation y plus 3. The second set of equations starts with x plus 2. Under this is the equation y plus 1 half x plus 3. Under this is the equation y plus 1 half times 2 plus 3. Below this is the equation y plus 1 plus 3. Below this is the equation y plus 4. The third set of equations starts with x plus 4. Under this is the equation y plus 1 half x plus 3. Under this is the equation y plus 1 half times 4 plus 3. Below this is the equation y plus 2 plus 3. Below this is the equation y plus 5.\"><\/span><p><\/p><div data-type=\"newline\"><br><\/div> The points are shown in <a href=\"#fs-id1167835202852\" class=\"autogenerated-content\">(Figure)<\/a>.<div data-type=\"newline\"><br><\/div> <table id=\"fs-id1167835202852\" summary=\"The table has 5 rows and 3 columns. The first row is a title row with the equation y plus 1 half x plus 3. The second row is a header row with the headers x, y, and (x, y). The third row has the numbers 0, 3, and (0, 3). The fourth row has the numbers 2, 4, and (2, 4). The fifth row has the numbers 4, 5, and (4, 5).\"><tbody><tr valign=\"top\"><td colspan=\"3\" data-valign=\"middle\" data-align=\"center\"><strong data-effect=\"bold\">\\(y=\\frac{1}{2}x+3\\)<\/strong><\/td><\/tr><tr valign=\"top\"><td data-valign=\"middle\" data-align=\"center\"><strong data-effect=\"bold\"><em data-effect=\"italics\">x<\/em><\/strong><\/td><td data-valign=\"middle\" data-align=\"center\"><strong data-effect=\"bold\"><em data-effect=\"italics\">y<\/em><\/strong><\/td><td data-valign=\"middle\" data-align=\"center\"><strong data-effect=\"bold\">\\(\\left(x,y\\right)\\)<\/strong><\/td><\/tr><tr valign=\"top\"><td data-valign=\"middle\" data-align=\"center\">0<\/td><td data-valign=\"middle\" data-align=\"center\">3<\/td><td data-valign=\"middle\" data-align=\"center\">\\(\\left(0,3\\right)\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"middle\" data-align=\"center\">2<\/td><td data-valign=\"middle\" data-align=\"center\">4<\/td><td data-valign=\"middle\" data-align=\"center\">\\(\\left(2,4\\right)\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"middle\" data-align=\"center\">4<\/td><td data-valign=\"middle\" data-align=\"center\">5<\/td><td data-valign=\"middle\" data-align=\"center\">\\(\\left(4,5\\right)\\)<\/td><\/tr><\/tbody><\/table><p><\/p><div data-type=\"newline\"><br><\/div> Plot the points, check that they line up, and draw the line.<div data-type=\"newline\"><br><\/div> <span data-type=\"media\" id=\"fs-id1167835489362\" data-alt=\"The figure shows the graph of a straight line on the x y-coordinate plane. The x and y axes run from negative 7 to 7. The points (0, 3), (2, 4), and (4, 5) are plotted. The straight line goes through the three points and has arrows on both ends. The line is labeled y plus 1 divided by 2 times x plus 3.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_016_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows the graph of a straight line on the x y-coordinate plane. The x and y axes run from negative 7 to 7. The points (0, 3), (2, 4), and (4, 5) are plotted. The straight line goes through the three points and has arrows on both ends. The line is labeled y plus 1 divided by 2 times x plus 3.\"><\/span><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167834194734\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167830837122\"><div data-type=\"problem\" id=\"fs-id1167826778512\"><p id=\"fs-id1167835595642\">Graph the equation: \\(y=\\frac{1}{3}x-1.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167830757662\"><p id=\"fs-id1167834111766\"><\/p><div data-type=\"newline\"><br><\/div><span data-type=\"media\" id=\"fs-id1167835350606\" data-alt=\"This figure shows a straight line graphed on the x y-coordinate plane. The x and y-axes run from negative 12 to 12. The line goes through the points (negative 12, negative 5), (negative 9, negative 4), (negative 6, negative 3), (negative 3, negative 2), (0, negative 1), (3, 0), (6, 1), (9, 2), and (12, 3).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_305_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a straight line graphed on the x y-coordinate plane. The x and y-axes run from negative 12 to 12. The line goes through the points (negative 12, negative 5), (negative 9, negative 4), (negative 6, negative 3), (negative 3, negative 2), (0, negative 1), (3, 0), (6, 1), (9, 2), and (12, 3).\"><\/span><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167834459155\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167826994459\"><div data-type=\"problem\" id=\"fs-id1167835345869\"><p id=\"fs-id1167835259316\">Graph the equation: \\(y=\\frac{1}{4}x+2.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834439103\"><p id=\"fs-id1167832074220\"><\/p><div data-type=\"newline\"><br><\/div><span data-type=\"media\" data-alt=\"This figure shows a straight line graphed on the x y-coordinate plane. The x and y-axes run from negative 12 to 12. The line goes through the points (negative 12, negative 1), (negative 8, 0), (negative 4, 1), (0, 2), (4, 3), (8, 4), and (12, 5).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_306_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a straight line graphed on the x y-coordinate plane. The x and y-axes run from negative 12 to 12. The line goes through the points (negative 12, negative 1), (negative 8, 0), (negative 4, 1), (0, 2), (4, 3), (8, 4), and (12, 5).\"><\/span><\/div><\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"1\"><h3 data-type=\"title\">Graph Vertical and Horizontal Lines<\/h3><p id=\"fs-id1167835340147\">Some linear equations have only one variable. They may have just <em data-effect=\"italics\">x<\/em> and no <em data-effect=\"italics\">y<\/em>, or just <em data-effect=\"italics\">y<\/em> without an <em data-effect=\"italics\">x<\/em>. This changes how we make a table of values to get the points to plot.<\/p><p id=\"fs-id1167835519235\">Let\u2019s consider the equation \\(x=-3.\\) This equation has only one variable, <em data-effect=\"italics\">x<\/em>. The equation says that <em data-effect=\"italics\">x<\/em> is <em data-effect=\"italics\">always<\/em> equal to\\(-3,\\) so its value does not depend on <em data-effect=\"italics\">y<\/em>. No matter what is the value of <em data-effect=\"italics\">y<\/em>, the value of <em data-effect=\"italics\">x<\/em> is always \\(-3.\\)<\/p><p id=\"fs-id1167831191426\">So to make a table of values, write \\(-3\\) in for all the <em data-effect=\"italics\">x<\/em>-values. Then choose any values for <em data-effect=\"italics\">y<\/em>. Since <em data-effect=\"italics\">x<\/em> does not depend on <em data-effect=\"italics\">y<\/em>, you can choose any numbers you like. But to fit the points on our coordinate graph, we\u2019ll use 1, 2, and 3 for the <em data-effect=\"italics\">y<\/em>-coordinates. See <a href=\"#fs-id1167831922183\" class=\"autogenerated-content\">(Figure)<\/a>.<\/p><table id=\"fs-id1167831922183\" summary=\"The table has 5 rows and 3 columns. The first row is a title row with the equation x plus negative 3. The second row is a header row with the headers x, y, and (x, y). The third row has the numbers negative 3, 1, and (negative 3, 1). The fourth row has the numbers negative 3, 2, and (negative 3, 2). The fifth row has the numbers negative 3, 3, and (negative 3, 3).\"><tbody><tr valign=\"top\"><td colspan=\"3\" data-valign=\"middle\" data-align=\"center\"><strong data-effect=\"bold\">\\(x=-3\\)<\/strong><\/td><\/tr><tr valign=\"top\"><td data-valign=\"middle\" data-align=\"center\"><strong data-effect=\"bold\"><em data-effect=\"italics\">x<\/em><\/strong><\/td><td data-valign=\"middle\" data-align=\"center\"><strong data-effect=\"bold\"><em data-effect=\"italics\">y<\/em><\/strong><\/td><td data-valign=\"middle\" data-align=\"center\"><strong data-effect=\"bold\">\\(\\left(x,y\\right)\\)<\/strong><\/td><\/tr><tr valign=\"top\"><td data-valign=\"middle\" data-align=\"center\">\\(-3\\)<\/td><td data-valign=\"middle\" data-align=\"center\">1<\/td><td data-valign=\"middle\" data-align=\"center\">\\(\\left(-3,1\\right)\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"middle\" data-align=\"center\">\\(-3\\)<\/td><td data-valign=\"middle\" data-align=\"center\">2<\/td><td data-valign=\"middle\" data-align=\"center\">\\(\\left(-3,2\\right)\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"middle\" data-align=\"center\">\\(-3\\)<\/td><td data-valign=\"middle\" data-align=\"center\">3<\/td><td data-valign=\"middle\" data-align=\"center\">\\(\\left(-3,3\\right)\\)<\/td><\/tr><\/tbody><\/table><p id=\"fs-id1167826996487\">Plot the points from the table and connect them with a straight line. Notice that we have graphed a <span data-type=\"term\">vertical line<\/span>.<\/p><span data-type=\"media\" id=\"fs-id1167834327439\" data-alt=\"The figure shows the graph of a straight vertical line on the x y-coordinate plane. The x and y axes run from negative 7 to 7. The points (negative 3, 1), (negative 3, 2), and (negative 3, 3) are plotted. The line goes through the three points and has arrows on both ends. The line is labeled x plus negative 3.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_017_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows the graph of a straight vertical line on the x y-coordinate plane. The x and y axes run from negative 7 to 7. The points (negative 3, 1), (negative 3, 2), and (negative 3, 3) are plotted. The line goes through the three points and has arrows on both ends. The line is labeled x plus negative 3.\"><\/span><p id=\"fs-id1167834377262\">What if the equation has <em data-effect=\"italics\">y<\/em> but no <em data-effect=\"italics\">x<\/em>? Let\u2019s graph the equation \\(y=4.\\) This time the <em data-effect=\"italics\">y-<\/em>value is a constant, so in this equation, <em data-effect=\"italics\">y<\/em> does not depend on <em data-effect=\"italics\">x<\/em>. Fill in 4 for all the <em data-effect=\"italics\">y<\/em>\u2019s in <a href=\"#fs-id1167835382105\" class=\"autogenerated-content\">(Figure)<\/a> and then choose any values for <em data-effect=\"italics\">x<\/em>. We\u2019ll use 0, 2, and 4 for the <em data-effect=\"italics\">x<\/em>-coordinates.<\/p><table id=\"fs-id1167835382105\" summary=\"The table has 5 rows and 3 columns. The first row is a title row with the equation y plus 4. The second row is a header row with the headers x, y, and (x, y). The third row has the numbers 0, 4, and (0, 4). The fourth row has the numbers 2, 4, and (2, 4). The fifth row has the numbers 4, 4, and (4, 4).\"><tbody><tr valign=\"top\"><td colspan=\"3\" data-valign=\"middle\" data-align=\"center\"><strong data-effect=\"bold\">\\(y=4\\)<\/strong><\/td><\/tr><tr valign=\"top\"><td data-valign=\"middle\" data-align=\"center\"><strong data-effect=\"bold\"><em data-effect=\"italics\">x<\/em><\/strong><\/td><td data-valign=\"middle\" data-align=\"center\"><strong data-effect=\"bold\"><em data-effect=\"italics\">y<\/em><\/strong><\/td><td data-valign=\"middle\" data-align=\"center\"><strong data-effect=\"bold\">\\(\\left(x,y\\right)\\)<\/strong><\/td><\/tr><tr valign=\"top\"><td data-valign=\"middle\" data-align=\"center\">0<\/td><td data-valign=\"middle\" data-align=\"center\">4<\/td><td data-valign=\"middle\" data-align=\"center\">\\(\\left(0,4\\right)\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"middle\" data-align=\"center\">2<\/td><td data-valign=\"middle\" data-align=\"center\">4<\/td><td data-valign=\"middle\" data-align=\"center\">\\(\\left(2,4\\right)\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"middle\" data-align=\"center\">4<\/td><td data-valign=\"middle\" data-align=\"center\">4<\/td><td data-valign=\"middle\" data-align=\"center\">\\(\\left(4,4\\right)\\)<\/td><\/tr><\/tbody><\/table><p id=\"fs-id1167831970006\">In this figure, we have graphed a <span data-type=\"term\">horizontal line<\/span> passing through the <em data-effect=\"italics\">y<\/em>-axis at 4.<\/p><span data-type=\"media\" id=\"fs-id1167832066046\" data-alt=\"The figure shows the graph of a straight horizontal line on the x y-coordinate plane. The x and y axes run from negative 7 to 7. The points (0, 4), (2, 4), and (4, 4) are plotted. The line goes through the three points and has arrows on both ends. The line is labeled y plus 4.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_018_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows the graph of a straight horizontal line on the x y-coordinate plane. The x and y axes run from negative 7 to 7. The points (0, 4), (2, 4), and (4, 4) are plotted. The line goes through the three points and has arrows on both ends. The line is labeled y plus 4.\"><\/span><div data-type=\"note\" id=\"fs-id1167832075518\"><div data-type=\"title\">Vertical and Horizontal Lines<\/div><p id=\"fs-id1167832152868\">A <strong data-effect=\"bold\">vertical line<\/strong> is the graph of an equation of the form \\(x=a.\\)<\/p><p id=\"fs-id1167834395223\">\\(\\phantom{\\rule{12em}{0ex}}\\)The line passes through the <em data-effect=\"italics\">x<\/em>-axis at \\(\\left(a,0\\right).\\)<\/p><p id=\"fs-id1167835421545\">A <strong data-effect=\"bold\">horizontal line<\/strong> is the graph of an equation of the form \\(y=b.\\)<\/p><p id=\"fs-id1167835367829\">\\(\\phantom{\\rule{12em}{0ex}}\\)The line passes through the <em data-effect=\"italics\">y<\/em>-axis at \\(\\left(0,b\\right).\\)<\/p><\/div><div data-type=\"example\" id=\"fs-id1167831872404\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167831872406\"><div data-type=\"problem\" id=\"fs-id1167835622210\"><p id=\"fs-id1167835622212\">Graph: <span class=\"token\">\u24d0<\/span> \\(x=2\\) <span class=\"token\">\u24d1<\/span> \\(y=-1.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167826802100\"><p id=\"fs-id1167835240910\"><span class=\"token\">\u24d0<\/span> The equation has only one variable, <em data-effect=\"italics\">x<\/em>, and <em data-effect=\"italics\">x<\/em> is always equal to 2. We create a table where <em data-effect=\"italics\">x<\/em> is always 2 and then put in any values for <em data-effect=\"italics\">y<\/em>. The graph is a vertical line passing through the <em data-effect=\"italics\">x<\/em>-axis at 2.<\/p><div data-type=\"newline\"><br><\/div> <table id=\"fs-id1167834195786\" class=\"unnumbered\" summary=\"The table has 5 rows and 3 columns. The first row is a title row with the equation x plus 2. The second row is a header row with the headers x, y, and (x, y). The third row has the numbers 2, 1, and (2, 1). The fourth row has the numbers 2, 2, and (2, 2). The fifth row has the numbers 2, 3, and (2, 3).\" data-label=\"\"><tbody><tr valign=\"top\"><td colspan=\"3\" data-valign=\"middle\" data-align=\"center\"><strong data-effect=\"bold\">\\(x=2\\)<\/strong><\/td><\/tr><tr valign=\"top\"><td data-valign=\"middle\" data-align=\"center\"><strong data-effect=\"bold\"><em data-effect=\"italics\">x<\/em><\/strong><\/td><td data-valign=\"middle\" data-align=\"center\"><strong data-effect=\"bold\"><em data-effect=\"italics\">y<\/em><\/strong><\/td><td data-valign=\"middle\" data-align=\"center\"><strong data-effect=\"bold\">\\(\\left(x,y\\right)\\)<\/strong><\/td><\/tr><tr valign=\"top\"><td data-valign=\"middle\" data-align=\"center\">2<\/td><td data-valign=\"middle\" data-align=\"center\">1<\/td><td data-valign=\"middle\" data-align=\"center\">\\(\\left(2,1\\right)\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"middle\" data-align=\"center\">2<\/td><td data-valign=\"middle\" data-align=\"center\">2<\/td><td data-valign=\"middle\" data-align=\"center\">\\(\\left(2,2\\right)\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"middle\" data-align=\"center\">2<\/td><td data-valign=\"middle\" data-align=\"center\">3<\/td><td data-valign=\"middle\" data-align=\"center\">\\(\\left(2,3\\right)\\)<\/td><\/tr><\/tbody><\/table><p><\/p><div data-type=\"newline\"><br><\/div><span data-type=\"media\" id=\"fs-id1167831919492\" data-alt=\"The figure shows the graph of a straight vertical line on the x y-coordinate plane. The x and y axes run from negative 7 to 7. The points (2, 1), (2, 2), and (2, 3) are plotted. The line goes through the three points and has arrows on both ends. The line is labeled x plus 2.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_019_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows the graph of a straight vertical line on the x y-coordinate plane. The x and y axes run from negative 7 to 7. The points (2, 1), (2, 2), and (2, 3) are plotted. The line goes through the three points and has arrows on both ends. The line is labeled x plus 2.\"><\/span><p><\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span> Similarly, the equation \\(y=-1\\) has only one variable, <em data-effect=\"italics\">y<\/em>. The value of <em data-effect=\"italics\">y<\/em> is constant. All the ordered pairs in the next table have the same <em data-effect=\"italics\">y<\/em>-coordinate. The graph is a horizontal line passing through the <em data-effect=\"italics\">y<\/em>-axis at \\(-1.\\)<div data-type=\"newline\"><br><\/div> <table id=\"fs-id1167835361018\" class=\"unnumbered\" summary=\"The table has 5 rows and 3 columns. The first row is a title row with the equation y plus negative 1. The second row is a header row with the headers x, y, and (x, y). The third row has the numbers 0, negative 1, and (0, negative 1). The fourth row has the numbers 3, negative 1, and (3, negative 1). The fifth row has the numbers negative 3, negative 1, and (negative 3, negative 1).\" data-label=\"\"><tbody><tr valign=\"top\"><td colspan=\"3\" data-valign=\"middle\" data-align=\"center\"><strong data-effect=\"bold\">\\(y=-1\\)<\/strong><\/td><\/tr><tr valign=\"top\"><td data-valign=\"middle\" data-align=\"center\"><strong data-effect=\"bold\"><em data-effect=\"italics\">x<\/em><\/strong><\/td><td data-valign=\"middle\" data-align=\"center\"><strong data-effect=\"bold\"><em data-effect=\"italics\">y<\/em><\/strong><\/td><td data-valign=\"middle\" data-align=\"center\"><strong data-effect=\"bold\">\\(\\left(x,y\\right)\\)<\/strong><\/td><\/tr><tr valign=\"top\"><td data-valign=\"middle\" data-align=\"center\">0<\/td><td data-valign=\"middle\" data-align=\"center\">\\(-1\\)<\/td><td data-valign=\"middle\" data-align=\"center\">\\(\\left(0,-1\\right)\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"middle\" data-align=\"center\">3<\/td><td data-valign=\"middle\" data-align=\"center\">\\(-1\\)<\/td><td data-valign=\"middle\" data-align=\"center\">\\(\\left(3,-1\\right)\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"middle\" data-align=\"center\">\\(-3\\)<\/td><td data-valign=\"middle\" data-align=\"center\">\\(-1\\)<\/td><td data-valign=\"middle\" data-align=\"center\">\\(\\left(-3,-1\\right)\\)<\/td><\/tr><\/tbody><\/table><p><\/p><div data-type=\"newline\"><br><\/div><span data-type=\"media\" id=\"fs-id1167834430851\" data-alt=\"The figure shows the graph of a straight horizontal line on the x y-coordinate plane. The x and y axes run from negative 7 to 7. The points (negative 3, negative 1), (0, negative 1), and (3, negative 1) are plotted. The line goes through the three points and has arrows on both ends. The line is labeled y plus negative 1.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_020_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows the graph of a straight horizontal line on the x y-coordinate plane. The x and y axes run from negative 7 to 7. The points (negative 3, negative 1), (0, negative 1), and (3, negative 1) are plotted. The line goes through the three points and has arrows on both ends. The line is labeled y plus negative 1.\"><\/span><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167835254336\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167835254340\"><div data-type=\"problem\" id=\"fs-id1167835254342\"><p id=\"fs-id1167835180556\">Graph the equations: <span class=\"token\">\u24d0<\/span> \\(x=5\\) <span class=\"token\">\u24d1<\/span> \\(y=-4.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167828435630\"><p id=\"fs-id1167828435632\"><span class=\"token\">\u24d0<\/span><\/p><div data-type=\"newline\"><br><\/div><span data-type=\"media\" id=\"fs-id1167835215981\" data-alt=\"The figure shows the graph of a straight vertical line on the x y-coordinate plane. The x and y axes run from negative 12 to 12. The line goes through the points (5, negative 3), (5, negative 2), (5, negative 1), (5, 0), (5, 1), (5, 2), and (5, 3).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_307_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows the graph of a straight vertical line on the x y-coordinate plane. The x and y axes run from negative 12 to 12. The line goes through the points (5, negative 3), (5, negative 2), (5, negative 1), (5, 0), (5, 1), (5, 2), and (5, 3).\"><\/span><p><\/p><div data-type=\"newline\"><br><\/div><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span><div data-type=\"newline\"><br><\/div><span data-type=\"media\" id=\"fs-id1167834062206\" data-alt=\"The figure shows the graph of a straight horizontal line on the x y-coordinate plane. The x and y axes run from negative 12 to 12. The line goes through the points (negative 3, negative 4), (negative 2, negative 4), (negative 1, negative 4), (0, negative 4), (1, negative 4), (2, negative 4), and (3, negative 4).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_308_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows the graph of a straight horizontal line on the x y-coordinate plane. The x and y axes run from negative 12 to 12. The line goes through the points (negative 3, negative 4), (negative 2, negative 4), (negative 1, negative 4), (0, negative 4), (1, negative 4), (2, negative 4), and (3, negative 4).\"><\/span><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167834061463\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167834061467\"><div data-type=\"problem\" id=\"fs-id1167834061469\"><p id=\"fs-id1167834061471\">Graph the equations: <span class=\"token\">\u24d0<\/span> \\(x=-2\\) <span class=\"token\">\u24d1<\/span> \\(y=3.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167831086624\"><p id=\"fs-id1167831086626\"><span class=\"token\">\u24d0<\/span><\/p><div data-type=\"newline\"><br><\/div><span data-type=\"media\" id=\"fs-id1167834376341\" data-alt=\"The figure shows the graph of a straight vertical line on the x y-coordinate plane. The x and y axes run from negative 12 to 12. The line goes through the points (negative 2, negative 3), (negative 2, negative 2), (negative 2, negative 1), (negative 2, 0), (negative 2, 1), (negative 2, 2), and (negative 2, 3).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_309_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows the graph of a straight vertical line on the x y-coordinate plane. The x and y axes run from negative 12 to 12. The line goes through the points (negative 2, negative 3), (negative 2, negative 2), (negative 2, negative 1), (negative 2, 0), (negative 2, 1), (negative 2, 2), and (negative 2, 3).\"><\/span><p><\/p><div data-type=\"newline\"><br><\/div><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span><div data-type=\"newline\"><br><\/div><span data-type=\"media\" data-alt=\"The figure shows the graph of a straight horizontal line on the x y-coordinate plane. The x and y axes run from negative 12 to 12. The line goes through the points (negative 3, 3), (negative 2, 3), (negative 1, 3), (0, 3), (1, 3), (2, 3), and (3, 3).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_310_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows the graph of a straight horizontal line on the x y-coordinate plane. The x and y axes run from negative 12 to 12. The line goes through the points (negative 3, 3), (negative 2, 3), (negative 1, 3), (0, 3), (1, 3), (2, 3), and (3, 3).\"><\/span><\/div><\/div><\/div><p id=\"fs-id1167834099415\">What is the difference between the equations \\(y=4x\\) and \\(y=4?\\)<\/p><p id=\"fs-id1167835337692\">The equation \\(y=4x\\) has both <em data-effect=\"italics\">x<\/em> and <em data-effect=\"italics\">y<\/em>. The value of <em data-effect=\"italics\">y<\/em> depends on the value of <em data-effect=\"italics\">x<\/em>, so the <em data-effect=\"italics\">y<\/em> -coordinate changes according to the value of <em data-effect=\"italics\">x<\/em>. The equation \\(y=4\\) has only one variable. The value of <em data-effect=\"italics\">y<\/em> is constant, it does not depend on the value of <em data-effect=\"italics\">x<\/em>, so the <em data-effect=\"italics\">y<\/em>-coordinate is always 4.<\/p><span data-type=\"media\" id=\"fs-id1167835544872\" data-alt=\"This figure has two tables. The first table has 5 rows and 3 columns. The first row is a title row with the equation y plus 4 x. The second row is a header row with the headers x, y, and (x, y). The third row has the numbers 0, 0, and (0, 0). The fourth row has the numbers 1, 4, and (1, 4). The fifth row has the numbers 2, 8, and (2, 8). The second table has 5 rows and 3 columns. The first row is a title row with the equation y plus 4. The second row is a header row with the headers x, y, and (x, y). The third row has the numbers 0, 4, and (0, 4). The fourth row has the numbers 1, 4, and (1, 4). The fifth row has the numbers 2, 4, and (2, 4).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_039_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure has two tables. The first table has 5 rows and 3 columns. The first row is a title row with the equation y plus 4 x. The second row is a header row with the headers x, y, and (x, y). The third row has the numbers 0, 0, and (0, 0). The fourth row has the numbers 1, 4, and (1, 4). The fifth row has the numbers 2, 8, and (2, 8). The second table has 5 rows and 3 columns. The first row is a title row with the equation y plus 4. The second row is a header row with the headers x, y, and (x, y). The third row has the numbers 0, 4, and (0, 4). The fourth row has the numbers 1, 4, and (1, 4). The fifth row has the numbers 2, 4, and (2, 4).\"><\/span><span data-type=\"media\" id=\"fs-id1167835368651\" data-alt=\"The figure shows the graphs of a straight horizontal line and a straight slanted line on the same x y-coordinate plane. The x and y axes run from negative 7 to 7. The horizontal line goes through the points (0, 4), (1, 4), and (2,4) and is labeled y plus 4. The slanted line goes through the points (0, 0), (1, 4), and (2, 8) and is labeled y plus 4 x.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_021_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows the graphs of a straight horizontal line and a straight slanted line on the same x y-coordinate plane. The x and y axes run from negative 7 to 7. The horizontal line goes through the points (0, 4), (1, 4), and (2,4) and is labeled y plus 4. The slanted line goes through the points (0, 0), (1, 4), and (2, 8) and is labeled y plus 4 x.\"><\/span><p id=\"fs-id1167835368655\">Notice, in the graph, the equation \\(y=4x\\) gives a slanted line, while \\(y=4\\) gives a horizontal line.<\/p><div data-type=\"example\" id=\"fs-id1167826972423\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167826972426\"><div data-type=\"problem\" id=\"fs-id1167826972428\"><p id=\"fs-id1167826972430\">Graph \\(y=-3x\\) and \\(y=-3\\) in the same rectangular coordinate system.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167831954216\"><p id=\"fs-id1167831954218\">We notice that the first equation has the variable <em data-effect=\"italics\">x<\/em>, while the second does not. We make a table of points for each equation and then graph the lines. The two graphs are shown.<\/p><span data-type=\"media\" id=\"fs-id1167835512135\" data-alt=\"This figure has two tables. The first table has 5 rows and 3 columns. The first row is a title row with the equation y plus negative 3 x. The second row is a header row with the headers x, y, and (x, y). The third row has the numbers 0, 0, and (0, 0). The fourth row has the numbers 1, negative 3, and (1, negative 3). The fifth row has the numbers 2, negative 6, and (2, neg ative 6). The second table has 5 rows and 3 columns. The first row is a title row with the equation y plus negative 3. The second row is a header row with the headers x, y, and (x, y). The third row has the numbers 0, negative 3, and (0, negative 3). The fourth row has the numbers 1, negative 3, and (1, negative 3). The fifth row has the numbers 2, negative 3, and (2, negative 3).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_040_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure has two tables. The first table has 5 rows and 3 columns. The first row is a title row with the equation y plus negative 3 x. The second row is a header row with the headers x, y, and (x, y). The third row has the numbers 0, 0, and (0, 0). The fourth row has the numbers 1, negative 3, and (1, negative 3). The fifth row has the numbers 2, negative 6, and (2, neg ative 6). The second table has 5 rows and 3 columns. The first row is a title row with the equation y plus negative 3. The second row is a header row with the headers x, y, and (x, y). The third row has the numbers 0, negative 3, and (0, negative 3). The fourth row has the numbers 1, negative 3, and (1, negative 3). The fifth row has the numbers 2, negative 3, and (2, negative 3).\"><\/span><p id=\"fs-id1165926641234\"><\/p><div data-type=\"newline\"><br><\/div><span data-type=\"media\" id=\"fs-id1167834094570\" data-alt=\"The figure shows the graphs of a straight horizontal line and a straight slanted line on the same x y-coordinate plane. The x and y axes run from negative 7 to 7. The horizontal line goes through the points (0, negative 3), (1, negative 3), and (2, negative 3) and is labeled y plus negative 3. The slanted line goes through the points (0, 0), (1, negative 3), and (2, negative 6) and is labeled y plus negative 3 x.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_022_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows the graphs of a straight horizontal line and a straight slanted line on the same x y-coordinate plane. The x and y axes run from negative 7 to 7. The horizontal line goes through the points (0, negative 3), (1, negative 3), and (2, negative 3) and is labeled y plus negative 3. The slanted line goes through the points (0, 0), (1, negative 3), and (2, negative 6) and is labeled y plus negative 3 x.\"><\/span><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167834395000\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167831954167\"><div data-type=\"problem\" id=\"fs-id1167831954169\"><p id=\"fs-id1167831954171\">Graph the equations in the same rectangular coordinate system: \\(y=-4x\\) and \\(y=-4.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835322119\"><p id=\"fs-id1167831872346\"><\/p><div data-type=\"newline\"><br><\/div><span data-type=\"media\" id=\"fs-id1167832066671\" data-alt=\"The figure shows the graphs of a straight horizontal line and a straight slanted line on the same x y-coordinate plane. The x and y axes run from negative 12 to 12. The horizontal line goes through the points (0, negative 4), (1, negative 4), and (2, negative 4). The slanted line goes through the points (0, 0), (1, negative 4), and (2, negative 8).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_311_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows the graphs of a straight horizontal line and a straight slanted line on the same x y-coordinate plane. The x and y axes run from negative 12 to 12. The horizontal line goes through the points (0, negative 4), (1, negative 4), and (2, negative 4). The slanted line goes through the points (0, 0), (1, negative 4), and (2, negative 8).\"><\/span><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167832066929\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167834252684\"><div data-type=\"problem\" id=\"fs-id1167834252686\"><p id=\"fs-id1167834252688\">Graph the equations in the same rectangular coordinate system: \\(y=3\\) and \\(y=3x.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835304740\"><p id=\"fs-id1167830963408\"><\/p><div data-type=\"newline\"><br><\/div><span data-type=\"media\" id=\"fs-id1167835304742\" data-alt=\"The figure shows the graphs of a straight horizontal line and a straight slanted line on the same x y-coordinate plane. The x and y axes run from negative 12 to 12. The horizontal line goes through the points (0, 3), (1, 3), and (2, 3). The slanted line goes through the points (0, 0), (1, 3), and (2, 6).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_312_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows the graphs of a straight horizontal line and a straight slanted line on the same x y-coordinate plane. The x and y axes run from negative 12 to 12. The horizontal line goes through the points (0, 3), (1, 3), and (2, 3). The slanted line goes through the points (0, 0), (1, 3), and (2, 6).\"><\/span><\/div><\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167835400371\"><h3 data-type=\"title\">Find <em data-effect=\"italics\">x<\/em>- and <em data-effect=\"italics\">y<\/em>-intercepts<\/h3><p id=\"fs-id1167835331638\">Every linear equation can be represented by a unique line that shows all the solutions of the equation. We have seen that when graphing a line by plotting points, you can use any three solutions to graph. This means that two people graphing the line might use different sets of three points.<\/p><p id=\"fs-id1167835331644\">At first glance, their two lines might not appear to be the same, since they would have different points labeled. But if all the work was done correctly, the lines should be exactly the same. One way to recognize that they are indeed the same line is to look at where the line crosses the <em data-effect=\"italics\">x<\/em>-axis and the <em data-effect=\"italics\">y<\/em>-axis. These points are called the <span data-type=\"term\">intercepts of a line<\/span>.<\/p><div data-type=\"note\" id=\"fs-id1167831871999\"><div data-type=\"title\">Intercepts of a Line<\/div><p id=\"fs-id1167834138160\">The points where a line crosses the <em data-effect=\"italics\">x<\/em>-axis and the <em data-effect=\"italics\">y<\/em>-axis are called the <strong data-effect=\"bold\">intercepts of the line<\/strong>.<\/p><\/div><p id=\"fs-id1167834194637\">Let\u2019s look at the graphs of the lines.<\/p><span data-type=\"media\" id=\"fs-id1167835343912\" data-alt=\"The figure shows four graphs of different equations. In example a the graph of 2 x plus y plus 6 is graphed on the x y-coordinate plane. The x and y axes run from negative 8 to 8. The points (0, 6) and (3, 0) are plotted and labeled. A straight line goes through both points and has arrows on both ends. In example b the graph of 3 x minus 4 y plus 12 is graphed on the x y-coordinate plane. The x and y axes run from negative 8 to 8. The points (0, negative 3) and (4, 0) are plotted and labeled. A straight line goes through both points and has arrows on both ends. In example c the graph of x minus y plus 5 is graphed on the x y-coordinate plane. The x and y axes run from negative 8 to 8. The points (0, negative 5) and (5, 0) are plotted and labeled. A straight line goes through both points and has arrows on both ends. In example d the graph of y plus negative 2 x is graphed on the x y-coordinate plane. The x and y axes run from negative 8 to 8. The point (0, 0) is plotted and labeled. A straight line goes through this point and the points (negative 1, 2) and (1, negative 2) and has arrows on both ends.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_023_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows four graphs of different equations. In example a the graph of 2 x plus y plus 6 is graphed on the x y-coordinate plane. The x and y axes run from negative 8 to 8. The points (0, 6) and (3, 0) are plotted and labeled. A straight line goes through both points and has arrows on both ends. In example b the graph of 3 x minus 4 y plus 12 is graphed on the x y-coordinate plane. The x and y axes run from negative 8 to 8. The points (0, negative 3) and (4, 0) are plotted and labeled. A straight line goes through both points and has arrows on both ends. In example c the graph of x minus y plus 5 is graphed on the x y-coordinate plane. The x and y axes run from negative 8 to 8. The points (0, negative 5) and (5, 0) are plotted and labeled. A straight line goes through both points and has arrows on both ends. In example d the graph of y plus negative 2 x is graphed on the x y-coordinate plane. The x and y axes run from negative 8 to 8. The point (0, 0) is plotted and labeled. A straight line goes through this point and the points (negative 1, 2) and (1, negative 2) and has arrows on both ends.\"><\/span><p id=\"fs-id1167831887736\">First, notice where each of these lines crosses the <em data-effect=\"italics\">x<\/em>-axis. See <a href=\"#fs-id1167835367746\" class=\"autogenerated-content\">(Figure)<\/a>.<\/p><p id=\"fs-id1167835345975\">Now, let\u2019s look at the points where these lines cross the <em data-effect=\"italics\">y<\/em>-axis.<\/p><table id=\"fs-id1167835367746\" summary=\"The table has 6 rows and 5 columns. The first row is a header row with the headers \u201cFigure\u201d, \u201cThe line crosses the x-axis at:\u201d, \u201cOrdered pair for this point\u201d, \u201cThe line crosses the y-axis at:\u201d, and \u201cOrdered pair for this point\u201d. The second row contains \u201cFigure a\u201d, 3, (3, 0), 6, (0, 6). The third row contains \u201cFigure b\u201d, 4, (4, 0), negative 3, (0, negative 3). The fourth row contains \u201cFigure c\u201d, 5, (5, 0), negative 5, (0, negative 5). The fifth row contains \u201cFigure d\u201d, 0, (0, 0), 0, (0, 0). The sixth row contains \u201cGeneral Figure\u201d, a, (a, 0), b, (0, b).\"><thead><tr valign=\"top\"><th data-valign=\"middle\" data-align=\"left\"><strong data-effect=\"bold\">Figure<\/strong><\/th><th data-valign=\"top\" data-align=\"center\"><strong data-effect=\"bold\">The line crosses<br><\/strong><\/th><\/tr><\/thead><\/table><\/div>the <em data-effect=\"italics\">x<\/em>-axis at:<strong data-effect=\"bold\">Ordered pair<br>for this point<\/strong><strong data-effect=\"bold\">The line crosses<br>the <em data-effect=\"italics\">y-<\/em>axis at:<\/strong><strong data-effect=\"bold\">Ordered pair<br>for this point<\/strong>Figure (a)3\\(\\left(3,0\\right)\\)6\\(\\left(0,6\\right)\\)Figure (b)4\\(\\left(4,0\\right)\\)\\(-3\\)\\(\\left(0,-3\\right)\\)Figure (c)5\\(\\left(5,0\\right)\\)\\(-5\\)\\(\\left(0,5\\right)\\)Figure (d)0\\(\\left(0,0\\right)\\)0\\(\\left(0,0\\right)\\)General Figure<em data-effect=\"italics\">a<\/em>\\(\\left(a,0\\right)\\)<em data-effect=\"italics\">b<\/em>\\(\\left(0,b\\right)\\)<p id=\"fs-id1167834535861\">Do you see a pattern?<\/p><p id=\"fs-id1167834535864\">For each line, the <em data-effect=\"italics\">y<\/em>-coordinate of the point where the line crosses the <em data-effect=\"italics\">x<\/em>-axis is zero. The point where the line crosses the <em data-effect=\"italics\">x<\/em>-axis has the form \\(\\left(a,0\\right)\\) and is called the <em data-effect=\"italics\">x-intercept<\/em> of the line. The <em data-effect=\"italics\">x<\/em>-intercept occurs when <em data-effect=\"italics\">y<\/em> is zero.<\/p><p id=\"fs-id1167835257786\">In each line, the <em data-effect=\"italics\">x<\/em><strong data-effect=\"bold\">-<\/strong>coordinate of the point where the line crosses the <em data-effect=\"italics\">y<\/em>-axis is zero. The point where the line crosses the <em data-effect=\"italics\">y<\/em>-axis has the form \\(\\left(0,b\\right)\\) and is called the <em data-effect=\"italics\">y-intercept<\/em> of the line. The <em data-effect=\"italics\">y<\/em>-intercept occurs when <em data-effect=\"italics\">x<\/em> is zero.<\/p><div data-type=\"note\"><div data-type=\"title\"><em data-effect=\"italics\">x<\/em>-intercept and <em data-effect=\"italics\">y<\/em>-intercept of a Line<\/div><p id=\"fs-id1167835309776\">The <em data-effect=\"italics\">x<\/em>-intercept is the point \\(\\left(a,0\\right)\\) where the line crosses the <em data-effect=\"italics\">x<\/em>-axis.<\/p><p id=\"fs-id1167831238979\">The <em data-effect=\"italics\">y<\/em>-intercept is the point \\(\\left(0,b\\right)\\) where the line crosses the <em data-effect=\"italics\">y<\/em>-axis.<\/p><span data-type=\"media\" id=\"fs-id1167834448946\" data-alt=\"The table has 3 rows and 2 columns. The first row is a header row with the headers x and y. The second row contains a and 0. The third row contains 0 and b.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_038_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The table has 3 rows and 2 columns. The first row is a header row with the headers x and y. The second row contains a and 0. The third row contains 0 and b.\"><\/span><\/div><div data-type=\"example\" id=\"fs-id1167834300811\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167834300813\"><div data-type=\"problem\" id=\"fs-id1167834300815\"><p id=\"fs-id1167835376064\">Find the <em data-effect=\"italics\">x<\/em>- and <em data-effect=\"italics\">y<\/em>-intercepts on each graph shown.<\/p><span data-type=\"media\" id=\"fs-id1167834279734\" data-alt=\"The figure has three graphs. Figure a shows a straight line graphed on the x y-coordinate plane. The x and y axes run from negative 8 to 8. The line goes through the points (negative 8, 6), (negative 4, 4), (0, 2), (4, 0), (8, negative 2). Figure b shows a straight line graphed on the x y-coordinate plane. The x and y axes run from negative 8 to 8. The line goes through the points (0, negative 6), (2, 0), and (4, 6). Figure c shows a straight line graphed on the x y-coordinate plane. The x and y axes run from negative 8 to 8. The line goes through the points (negative 5, 0), (negative 3, negative 3), (0, negative 5), (1, negative 6), and (2, negative 7).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_024_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure has three graphs. Figure a shows a straight line graphed on the x y-coordinate plane. The x and y axes run from negative 8 to 8. The line goes through the points (negative 8, 6), (negative 4, 4), (0, 2), (4, 0), (8, negative 2). Figure b shows a straight line graphed on the x y-coordinate plane. The x and y axes run from negative 8 to 8. The line goes through the points (0, negative 6), (2, 0), and (4, 6). Figure c shows a straight line graphed on the x y-coordinate plane. The x and y axes run from negative 8 to 8. The line goes through the points (negative 5, 0), (negative 3, negative 3), (0, negative 5), (1, negative 6), and (2, negative 7).\"><\/span><\/div><div data-type=\"solution\" id=\"fs-id1167835595936\"><p id=\"fs-id1167835360226\"><span class=\"token\">\u24d0<\/span> The graph crosses the <em data-effect=\"italics\">x<\/em>-axis at the point \\(\\left(4,0\\right).\\) The <em data-effect=\"italics\">x-<\/em>intercept is \\(\\left(4,0\\right).\\)<\/p><div data-type=\"newline\"><br><\/div> The graph crosses the <em data-effect=\"italics\">y<\/em>-axis at the point \\(\\left(0,2\\right).\\) The <em data-effect=\"italics\">y<\/em>-intercept is \\(\\left(0,2\\right).\\)<div data-type=\"newline\"><br><\/div><div data-type=\"newline\"><br><\/div> <span class=\"token\">\u24d1<\/span> The graph crosses the <em data-effect=\"italics\">x<\/em>-axis at the point \\(\\left(2,0\\right).\\) The <em data-effect=\"italics\">x<\/em>-intercept is \\(\\left(2,0\\right).\\)<div data-type=\"newline\"><br><\/div> The graph crosses the <em data-effect=\"italics\">y<\/em>-axis at the point \\(\\left(0,-6\\right).\\) The <em data-effect=\"italics\">y<\/em>-intercept is \\(\\left(0,-6\\right).\\)<div data-type=\"newline\"><br><\/div><div data-type=\"newline\"><br><\/div> <span class=\"token\">\u24d2<\/span> The graph crosses the <em data-effect=\"italics\">x<\/em>-axis at the point \\(\\left(-5,0\\right).\\) The <em data-effect=\"italics\">x<\/em>-intercept is \\(\\left(-5,0\\right).\\)<div data-type=\"newline\"><br><\/div> The graph crosses the <em data-effect=\"italics\">y<\/em>-axis at the point \\(\\left(0,-5\\right).\\) The <em data-effect=\"italics\">y<\/em>-intercept is \\(\\left(0,-5\\right).\\)<\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167826828292\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167826986762\"><div data-type=\"problem\" id=\"fs-id1167826986764\"><p id=\"fs-id1167826986766\">Find the <em data-effect=\"italics\">x<\/em>- and <em data-effect=\"italics\">y<\/em>-intercepts on the graph.<\/p><span data-type=\"media\" id=\"fs-id1167830697612\" data-alt=\"This figure a shows a straight line graphed on the x y-coordinate plane. The x and y axes run from negative 10 to 10. The line goes through the points (negative 6, negative 8), (negative 4, negative 6), (negative 2, negative 4), (0, negative 2), (2, 0), (4, 2), (6, 4), (8, 6).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_025_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure a shows a straight line graphed on the x y-coordinate plane. The x and y axes run from negative 10 to 10. The line goes through the points (negative 6, negative 8), (negative 4, negative 6), (negative 2, negative 4), (0, negative 2), (2, 0), (4, 2), (6, 4), (8, 6).\"><\/span><\/div><div data-type=\"solution\" id=\"fs-id1167834219644\"><p><em data-effect=\"italics\">x<\/em>-intercept: \\(\\left(2,0\\right),\\)<\/p><div data-type=\"newline\"><br><\/div><em data-effect=\"italics\">y<\/em>-intercept: \\(\\left(0,-2\\right)\\)<\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167832153754\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167832153758\"><div data-type=\"problem\" id=\"fs-id1167832153760\"><p id=\"fs-id1167832153762\">Find the <em data-effect=\"italics\">x<\/em>- and <em data-effect=\"italics\">y<\/em>-intercepts on the graph.<\/p><span data-type=\"media\" id=\"fs-id1167835326401\" data-alt=\"This figure a shows a straight line graphed on the x y-coordinate plane. The x and y axes run from negative 10 to 10. The line goes through the points (negative 6, 6), (negative 3, 4), (0, 2), (3, 0), (6, negative 2), and (9, negative 4).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_026_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure a shows a straight line graphed on the x y-coordinate plane. The x and y axes run from negative 10 to 10. The line goes through the points (negative 6, 6), (negative 3, 4), (0, 2), (3, 0), (6, negative 2), and (9, negative 4).\"><\/span><\/div><div data-type=\"solution\" id=\"fs-id1167835346249\"><p id=\"fs-id1167835346251\"><em data-effect=\"italics\">x<\/em>-intercept: \\(\\left(3,0\\right),\\)<\/p><div data-type=\"newline\"><br><\/div><em data-effect=\"italics\">y<\/em>-intercept: \\(\\left(0,2\\right)\\)<\/div><\/div><\/div><p id=\"fs-id1167835410917\">Recognizing that the <em data-effect=\"italics\">x<\/em>-intercept occurs when <em data-effect=\"italics\">y<\/em> is zero and that the <em data-effect=\"italics\">y<\/em>-intercept occurs when <em data-effect=\"italics\">x<\/em> is zero, gives us a method to find the intercepts of a line from its equation. To find the <em data-effect=\"italics\">x<\/em>-intercept, let \\(y=0\\) and solve for <em data-effect=\"italics\">x<\/em>. To find the <em data-effect=\"italics\">y<\/em>-intercept, let \\(x=0\\) and solve for <em data-effect=\"italics\">y<\/em>.<\/p><div data-type=\"note\" id=\"fs-id1167835410566\"><div data-type=\"title\">Find the <em data-effect=\"italics\">x<\/em>- and <em data-effect=\"italics\">y<\/em>-intercepts from the Equation of a Line<\/div><p id=\"fs-id1167835345789\">Use the equation of the line. To find:<\/p><ul id=\"fs-id1167834246665\" data-bullet-style=\"bullet\"><li>the <em data-effect=\"italics\">x<\/em>-intercept of the line, let \\(y=0\\) and solve for <em data-effect=\"italics\">x<\/em>.<\/li><li>the <em data-effect=\"italics\">y<\/em>-intercept of the line, let \\(x=0\\) and solve for <em data-effect=\"italics\">y<\/em>.<\/li><\/ul><\/div><div data-type=\"example\" id=\"fs-id1167827987818\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167827987820\"><div data-type=\"problem\" id=\"fs-id1167827987822\"><p id=\"fs-id1167827987825\">Find the intercepts of \\(2x+y=8.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835345155\"><p id=\"fs-id1167835345157\">We will let \\(y=0\\) to find the <em data-effect=\"italics\">x<\/em>-intercept, and let \\(x=0\\) to find the <em data-effect=\"italics\">y<\/em>-intercept. We will fill in a table, which reminds us of what we need to find.<\/p><div data-type=\"newline\"><br><\/div> <span data-type=\"media\" id=\"fs-id1167834433642\" data-alt=\"The figure has a table with 4 rows and 2 columns. The first row is a title row with the equation 2 x plus y plus 8. The second row is a header row with the headers x and y. The third row is labeled x-intercept and has the first column blank and a 0 in the second column. The fourth row is labeled y-intercept and has a 0 in the first column and the second column blank.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_027_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure has a table with 4 rows and 2 columns. The first row is a title row with the equation 2 x plus y plus 8. The second row is a header row with the headers x and y. The third row is labeled x-intercept and has the first column blank and a 0 in the second column. The fourth row is labeled y-intercept and has a 0 in the first column and the second column blank.\"><\/span><p><\/p><div data-type=\"newline\"><br><\/div><table id=\"fs-id1167834376970\" class=\"unnumbered unstyled\" summary=\"The figure has the equation 2 x plus y plus 8. Let y plus 0. The next equation is 2 x plus 0 plus 8, where the 0 is emphasized. Simplifying we get 2 x plus 8. Then y plus 4. The x-intercept is (4, 0). To find the y-intercept, let x plus 0. Again we start with the equation 2 x plus y plus 8. Let x plus 0. The next equation is 2 times 0 plus y plus 8, where the 0 is emphasized. Simplifying we get 0 plus y plus 8. Then y plus 8. The y-intercept is (0, 8).\" data-label=\"\"><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">To find the <em data-effect=\"italics\">x<\/em>-intercept, let \\(y=0.\\)<\/td><td><\/td><\/tr><tr valign=\"top\"><td><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834191753\" data-alt=\".\"><img src=\"CNX_IntAlg_Figure_03_01_028a_img_new.jpg#fixme#fixme\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Let \\(y=0.\\)<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167832138837\" data-alt=\".\"><img src=\"CNX_IntAlg_Figure_03_01_028b_img_new.jpg#fixme#fixme\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Simplify.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167828377143\" data-alt=\".\"><img src=\"CNX_IntAlg_Figure_03_01_028c_img_new.jpg#fixme#fixme\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835390567\" data-alt=\".\"><img src=\"CNX_IntAlg_Figure_03_01_028d_img_new.jpg#fixme#fixme\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">The <em data-effect=\"italics\">x<\/em>-intercept is:<\/td><td data-valign=\"top\" data-align=\"left\">\\(\\phantom{\\rule{2em}{0ex}}\\left(4,0\\right)\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">To find the <em data-effect=\"italics\">y<\/em>-intercept, let \\(x=0.\\)<\/td><td><\/td><\/tr><tr valign=\"top\"><td><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835418854\" data-alt=\".\"><img src=\"CNX_IntAlg_Figure_03_01_028e_img_new.jpg#fixme#fixme\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Let \\(x=0.\\)<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834387657\" data-alt=\".\"><img src=\"CNX_IntAlg_Figure_03_01_028f_img_new.jpg#fixme#fixme\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Simplify.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835320774\" data-alt=\".\"><img src=\"CNX_IntAlg_Figure_03_01_028g_img_new.jpg#fixme#fixme\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835324543\" data-alt=\".\"><img src=\"CNX_IntAlg_Figure_03_01_028h_img_new.jpg#fixme#fixme\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">The <em data-effect=\"italics\">y<\/em>-intercept is:<\/td><td data-valign=\"top\" data-align=\"left\">\\(\\phantom{\\rule{2em}{0ex}}\\left(0,8\\right)\\)<\/td><\/tr><\/tbody><\/table><p><\/p><div data-type=\"newline\"><br><\/div> The intercepts are the points \\(\\left(4,0\\right)\\) and \\(\\left(0,8\\right)\\) as shown in the table.<div data-type=\"newline\"><br><\/div> <table id=\"fs-id1167835173826\" class=\"unnumbered\" summary=\"The table shows the equation 2x plus y equals 8. Below that are two columns. The left column is x and the right column is y. The first row shows that x is 4 and y is 0. The second column shows that the x is 0 and the y is 8.\" data-label=\"\"><tbody><tr valign=\"top\"><td colspan=\"2\" data-valign=\"middle\" data-align=\"center\"><strong data-effect=\"bold\">\\(2x+y=8\\)<\/strong><\/td><\/tr><tr valign=\"top\"><td data-valign=\"middle\" data-align=\"center\"><strong data-effect=\"bold\"><em data-effect=\"italics\">x<\/em><\/strong><\/td><td data-valign=\"middle\" data-align=\"center\"><strong data-effect=\"bold\"><em data-effect=\"italics\">y<\/em><\/strong><\/td><\/tr><tr valign=\"top\"><td data-valign=\"middle\" data-align=\"center\">4<\/td><td data-valign=\"middle\" data-align=\"center\">0<\/td><\/tr><tr valign=\"top\"><td data-valign=\"middle\" data-align=\"center\">0<\/td><td data-valign=\"middle\" data-align=\"center\">8<\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167835353165\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167835353170\"><div data-type=\"problem\" id=\"fs-id1167835353172\"><p id=\"fs-id1167835353174\">Find the intercepts: \\(3x+y=12.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167831872300\"><p id=\"fs-id1167827943100\"><em data-effect=\"italics\">x<\/em>-intercept: \\(\\left(4,0\\right),\\)<\/p><div data-type=\"newline\"><br><\/div><em data-effect=\"italics\">y<\/em>-intercept: \\(\\left(0,12\\right)\\)<\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167830865897\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167835348348\"><div data-type=\"problem\" id=\"fs-id1167835348350\"><p id=\"fs-id1167835348352\">Find the intercepts: \\(x+4y=8.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835333372\"><p id=\"fs-id1167835333374\"><em data-effect=\"italics\">x<\/em>-intercept: \\(\\left(8,0\\right),\\)<\/p><div data-type=\"newline\"><br><\/div><em data-effect=\"italics\">y<\/em>-intercept: \\(\\left(0,2\\right)\\)<\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167832051938\"><h3 data-type=\"title\">Graph a Line Using the Intercepts<\/h3><p id=\"fs-id1167832051944\">To graph a linear equation by plotting points, you need to find three points whose coordinates are solutions to the equation. You can use the <em data-effect=\"italics\">x-<\/em> and <em data-effect=\"italics\">y-<\/em> intercepts as two of your three points. Find the intercepts, and then find a third point to ensure accuracy. Make sure the points line up\u2014then draw the line. This method is often the quickest way to graph a line.<\/p><div data-type=\"example\" id=\"fs-id1167834345742\" class=\"textbox textbox--examples\"><div data-type=\"title\">How to Graph a Line Using the Intercepts<\/div><div data-type=\"exercise\" id=\"fs-id1167834345744\"><div data-type=\"problem\" id=\"fs-id1167834345746\"><p id=\"fs-id1167834431456\">Graph \\(\u2013x+2y=6\\) using the intercepts.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835305984\"><span data-type=\"media\" id=\"fs-id1167835305986\" data-alt=\"Step 1 is to find the x and y-intercepts of the line. To find the x-intercept let y plus 0 and solve for x. The equation negative x plus 2 y plus 6 becomes negative x plus 2 times 0 plus 6. This simplifies to negative x plus 6. This is equivalent to x plus negative 6. The x-intercept is (negative 6, 0). To find the y-intercept let x plus 0 and solve for y. The equation negative x plus 2 y plus 6 becomes negative 0 plus 2 y plus 6. This simplifies to negative 2 y plus 6. This is equivalent to y plus 3. The y-intercept is (0, 3).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_029a_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 1 is to find the x and y-intercepts of the line. To find the x-intercept let y plus 0 and solve for x. The equation negative x plus 2 y plus 6 becomes negative x plus 2 times 0 plus 6. This simplifies to negative x plus 6. This is equivalent to x plus negative 6. The x-intercept is (negative 6, 0). To find the y-intercept let x plus 0 and solve for y. The equation negative x plus 2 y plus 6 becomes negative 0 plus 2 y plus 6. This simplifies to negative 2 y plus 6. This is equivalent to y plus 3. The y-intercept is (0, 3).\"><\/span><span data-type=\"media\" id=\"fs-id1167835305991\" data-alt=\"Step 2 is to find another solution to the equation. We\u2019ll use x plus 2. The equation negative x plus 2 y plus 6 becomes negative 2 plus 2 y plus 6. This simplifies to 2 y plus 8. This is equivalent to y plus 4. The third point is (2, 4).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_029b_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 2 is to find another solution to the equation. We\u2019ll use x plus 2. The equation negative x plus 2 y plus 6 becomes negative 2 plus 2 y plus 6. This simplifies to 2 y plus 8. This is equivalent to y plus 4. The third point is (2, 4).\"><\/span><span data-type=\"media\" id=\"fs-id1167835375646\" data-alt=\"Step 3 is to plot the three points. The figure shows a table with 4 rows and 3 columns. The first row is a header row with the headers x, y, and (x, y). The second row contains negative 6, 0, and (negative 6, 0). The third row contains 0, 3, and (0, 3). The fourth row contains 2, 4, and (2, 4). The figure also has a graph of the three points on the x y-coordinate plane. The x and y-axes run from negative 6 to 6. The three points (negative 6, 0), (0, 3), and (2, 4) are plotted and labeled.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_029c_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 3 is to plot the three points. The figure shows a table with 4 rows and 3 columns. The first row is a header row with the headers x, y, and (x, y). The second row contains negative 6, 0, and (negative 6, 0). The third row contains 0, 3, and (0, 3). The fourth row contains 2, 4, and (2, 4). The figure also has a graph of the three points on the x y-coordinate plane. The x and y-axes run from negative 6 to 6. The three points (negative 6, 0), (0, 3), and (2, 4) are plotted and labeled.\"><\/span><span data-type=\"media\" id=\"fs-id1167834556100\" data-alt=\"Step 4 is to draw the line. The figure shows a graph of a straight line on the x y-coordinate plane. The x and y-axes run from negative 6 to 6. The straight line goes through the points (negative 6, 0), (0, 3), and (2, 4).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_029d_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 4 is to draw the line. The figure shows a graph of a straight line on the x y-coordinate plane. The x and y-axes run from negative 6 to 6. The straight line goes through the points (negative 6, 0), (0, 3), and (2, 4).\"><\/span><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167835337758\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167835337762\"><div data-type=\"problem\" id=\"fs-id1167835337764\"><p id=\"fs-id1167835337766\">Graph using the intercepts: \\(x\u20132y=4.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834501732\"><span data-type=\"media\" id=\"fs-id1167835312208\" data-alt=\"The figure shows a graph of a straight line on the x y-coordinate plane. The x and y-axes run from negative 12 to 12. The straight line goes through the points (negative 4, negative 4), (negative 2, negative 3), (0, negative 2), (2, negative 1), (4, 0), (6, 1), and (8, 2).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_313_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows a graph of a straight line on the x y-coordinate plane. The x and y-axes run from negative 12 to 12. The straight line goes through the points (negative 4, negative 4), (negative 2, negative 3), (0, negative 2), (2, negative 1), (4, 0), (6, 1), and (8, 2).\"><\/span><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167835621569\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167834448816\"><div data-type=\"problem\" id=\"fs-id1167834448818\"><p id=\"fs-id1167834448820\">Graph using the intercepts: \\(\u2013x+3y=6.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835503875\"><p id=\"fs-id1167832116054\"><\/p><div data-type=\"newline\"><br><\/div><span data-type=\"media\" id=\"fs-id1167835503877\" data-alt=\"The figure shows a graph of a straight line on the x y-coordinate plane. The x and y-axes run from negative 12 to 12. The straight line goes through the points (negative 9, negative 1), (negative 6, 0), (negative 3, 1), (0, 2), (3, 3), (6, 4), and (9, 5).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_314_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows a graph of a straight line on the x y-coordinate plane. The x and y-axes run from negative 12 to 12. The straight line goes through the points (negative 9, negative 1), (negative 6, 0), (negative 3, 1), (0, 2), (3, 3), (6, 4), and (9, 5).\"><\/span><\/div><\/div><\/div><p id=\"fs-id1167835345522\">The steps to graph a linear equation using the intercepts are summarized here.<\/p><div data-type=\"note\" id=\"fs-id1167835345525\" class=\"howto\"><div data-type=\"title\">Graph a linear equation using the intercepts.<\/div><ol id=\"fs-id1167835215399\" type=\"1\" class=\"stepwise\"><li>Find the <em data-effect=\"italics\">x<\/em>- and <em data-effect=\"italics\">y<\/em>-intercepts of the line. <ul id=\"fs-id1167835368530\" data-bullet-style=\"bullet\"><li>Let \\(y=0\\) and solve for <em data-effect=\"italics\">x<\/em>.<\/li><li>Let \\(x=0\\) and solve for <em data-effect=\"italics\">y<\/em>.<\/li><\/ul><\/li><li>Find a third solution to the equation.<\/li><li>Plot the three points and check that they line up.<\/li><li>Draw the line.<\/li><\/ol><\/div><div data-type=\"example\" class=\"textbox textbox--examples\"><div data-type=\"exercise\"><div data-type=\"problem\"><p id=\"fs-id1167831238959\">Graph \\(4x-3y=12\\) using the intercepts.<\/p><\/div><div data-type=\"solution\"><p>Find the intercepts and a third point.<\/p><div data-type=\"newline\"><br><\/div> <span data-type=\"media\" id=\"fs-id1167835319374\" data-alt=\"To find the x-intercept let y plus 0 and solve for x. The equation 4 x minus 3 y plus 12 becomes 4 x minus 3 times 0 plus 12. This simplifies to negative 4 x plus 12. This is equivalent to x plus 3. To find the y-intercept let x plus 0 and solve for y. The equation 4 x minus 3 y plus 12 becomes 4 times 0 minus 3 y plus 12. This simplifies to negative 3 y plus 12. This is equivalent to y plus negative 4. To find the third point let y plus 4 and solve for x. The equation 4 x minus 3 y plus 12 becomes 4 x minus 3 times 4 plus 12. This simplifies to negative 4 x plus 24. This is equivalent to x plus 6.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_030_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"To find the x-intercept let y plus 0 and solve for x. The equation 4 x minus 3 y plus 12 becomes 4 x minus 3 times 0 plus 12. This simplifies to negative 4 x plus 12. This is equivalent to x plus 3. To find the y-intercept let x plus 0 and solve for y. The equation 4 x minus 3 y plus 12 becomes 4 times 0 minus 3 y plus 12. This simplifies to negative 3 y plus 12. This is equivalent to y plus negative 4. To find the third point let y plus 4 and solve for x. The equation 4 x minus 3 y plus 12 becomes 4 x minus 3 times 4 plus 12. This simplifies to negative 4 x plus 24. This is equivalent to x plus 6.\"><\/span><p><\/p><div data-type=\"newline\"><br><\/div><div data-type=\"newline\"><br><\/div> We list the points in the table and show the graph.<div data-type=\"newline\"><br><\/div> <table id=\"fs-id1167834554869\" class=\"unnumbered\" summary=\"The table shows the equation 4 x minus 3 y plus 12. There are three columns below: x, y, and x y. The first row shows that x is 3, y is 0, and x y is 3, 0. The second row shows that x is 0, y is negative 4, and x y is 0, negative 4. The third column shows that x is 6, y is 4, and x y is 6, 4.\"><tbody><tr valign=\"top\"><td colspan=\"3\" data-valign=\"middle\" data-align=\"center\"><strong data-effect=\"bold\">\\(4x-3y=12\\)<\/strong><\/td><\/tr><tr valign=\"top\"><td data-valign=\"middle\" data-align=\"center\"><strong data-effect=\"bold\"><em data-effect=\"italics\">x<\/em><\/strong><\/td><td data-valign=\"middle\" data-align=\"center\"><strong data-effect=\"bold\"><em data-effect=\"italics\">y<\/em><\/strong><\/td><td data-valign=\"middle\" data-align=\"center\"><strong data-effect=\"bold\">\\(\\left(x,y\\right)\\)<\/strong><\/td><\/tr><tr valign=\"top\"><td data-valign=\"middle\" data-align=\"center\">3<\/td><td data-valign=\"middle\" data-align=\"center\">0<\/td><td data-valign=\"middle\" data-align=\"center\">\\(\\left(3,0\\right)\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"middle\" data-align=\"center\">0<\/td><td data-valign=\"middle\" data-align=\"center\">\\(-4\\)<\/td><td data-valign=\"middle\" data-align=\"center\">\\(\\left(0,-4\\right)\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"middle\" data-align=\"center\">6<\/td><td data-valign=\"middle\" data-align=\"center\">4<\/td><td data-valign=\"middle\" data-align=\"center\">\\(\\left(6,4\\right)\\)<\/td><\/tr><\/tbody><\/table><p><\/p><div data-type=\"newline\"><br><\/div><span data-type=\"media\" id=\"fs-id1167834583414\" data-alt=\"The figure shows a graph of the equation 4 x minus 3 y plus 12 on the x y-coordinate plane. The x and y-axes run from negative 7 to 7. The straight line goes through the points (0, negative 4), (3, 0), and (6, 4).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_031_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows a graph of the equation 4 x minus 3 y plus 12 on the x y-coordinate plane. The x and y-axes run from negative 7 to 7. The straight line goes through the points (0, negative 4), (3, 0), and (6, 4).\"><\/span><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167835343715\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167835280874\"><div data-type=\"problem\" id=\"fs-id1167835280877\"><p id=\"fs-id1167835280879\">Graph using the intercepts: \\(5x-2y=10.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835306373\"><p id=\"fs-id1167831891814\"><\/p><div data-type=\"newline\"><br><\/div><span data-type=\"media\" id=\"fs-id1167835423208\" data-alt=\"The figure shows a graph of a straight line on the x y-coordinate plane. The x and y-axes run from negative 8 to 8. The straight line goes through the points (0, negative 5), (2, 0), and (4, 5).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_315_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows a graph of a straight line on the x y-coordinate plane. The x and y-axes run from negative 8 to 8. The straight line goes through the points (0, negative 5), (2, 0), and (4, 5).\"><\/span><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167835363636\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167835363641\"><div data-type=\"problem\" id=\"fs-id1167834120979\"><p id=\"fs-id1167834120981\">Graph using the intercepts: \\(3x-4y=12.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835420632\"><p id=\"fs-id1167835318697\"><\/p><div data-type=\"newline\"><br><\/div><span data-type=\"media\" id=\"fs-id1167835420634\" data-alt=\"The figure shows a graph of a straight line on the x y-coordinate plane. The x and y-axes run from negative 8 to 8. The straight line goes through the points (negative 4, negative 6), (0, negative 3), (4, 0), and (8, 3).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_316_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows a graph of a straight line on the x y-coordinate plane. The x and y-axes run from negative 8 to 8. The straight line goes through the points (negative 4, negative 6), (0, negative 3), (4, 0), and (8, 3).\"><\/span><\/div><\/div><\/div><p id=\"fs-id1171792848414\">When the line passes through the origin, the <em data-effect=\"italics\">x<\/em>-intercept and the <em data-effect=\"italics\">y<\/em>-intercept are the same point.<\/p><div data-type=\"example\" id=\"fs-id1167835514433\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167835514436\"><div data-type=\"problem\" id=\"fs-id1167835514438\"><p id=\"fs-id1167835514440\">Graph \\(y=5x\\) using the intercepts.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834376079\"><span data-type=\"media\" id=\"fs-id1167834376081\" data-alt=\"To find the x-intercept let y plus 0 and solve for x. The equation y plus 5 x becomes 0 plus 5 x. This simplifies to 0 plus x. The x-intercept is (0, 0). To find the y-intercept let x plus 0 and solve for y. The equation y plus 5 x becomes y plus 5 times 0. This simplifies to y plus 0. The y-intercept is also (0, 0).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_032_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"To find the x-intercept let y plus 0 and solve for x. The equation y plus 5 x becomes 0 plus 5 x. This simplifies to 0 plus x. The x-intercept is (0, 0). To find the y-intercept let x plus 0 and solve for y. The equation y plus 5 x becomes y plus 5 times 0. This simplifies to y plus 0. The y-intercept is also (0, 0).\"><\/span><p id=\"fs-id1171792730711\"><\/p><div data-type=\"newline\"><br><\/div><p id=\"fs-id1167834185105\">This line has only one intercept. It is the point \\(\\left(0,0\\right).\\)<\/p><div data-type=\"newline\"><br><\/div> To ensure accuracy, we need to plot three points. Since the <em data-effect=\"italics\">x<\/em>- and <em data-effect=\"italics\">y<\/em>-intercepts are the same point, we need <em data-effect=\"italics\">two<\/em> more points to graph the line.<span data-type=\"media\" id=\"fs-id1167835364300\" data-alt=\"To find a second point let x plus 1 and solve for y. The equation y plus 5 x becomes y plus 5 times 1. This simplifies to y plus 5. To find a third point let x plus negative 1 and solve for y. The equation y plus 5 x becomes y plus 5 times negative 1. This simplifies to y plus negative 5\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_033_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"To find a second point let x plus 1 and solve for y. The equation y plus 5 x becomes y plus 5 times 1. This simplifies to y plus 5. To find a third point let x plus negative 1 and solve for y. The equation y plus 5 x becomes y plus 5 times negative 1. This simplifies to y plus negative 5\"><\/span><p id=\"fs-id1171791389259\">The resulting three points are summarized in the table.<\/p><table id=\"fs-id1167835367099\" class=\"unnumbered\" summary=\"The table has 5 rows and 3 columns. The first row is a title row with the equation y plus 5 x. The second row is a header row with the headers x, y, and (x, y). The third row contains negative 0, 0, and (0, 0). The fourth row contains 1, 5, and (1, 5). The fifth row contains negative 1, negative 5, and (negative 1, negative 5).\"><tbody><tr valign=\"top\"><td colspan=\"3\" data-valign=\"middle\" data-align=\"center\"><strong data-effect=\"bold\">\\(y=5x\\)<\/strong><\/td><\/tr><tr valign=\"top\"><td data-valign=\"middle\" data-align=\"center\"><strong data-effect=\"bold\"><em data-effect=\"italics\">x<\/em><\/strong><\/td><td data-valign=\"middle\" data-align=\"center\"><strong data-effect=\"bold\"><em data-effect=\"italics\">y<\/em><\/strong><\/td><td data-valign=\"middle\" data-align=\"center\"><strong data-effect=\"bold\">\\(\\left(x,y\\right)\\)<\/strong><\/td><\/tr><tr valign=\"top\"><td data-valign=\"middle\" data-align=\"center\">0<\/td><td data-valign=\"middle\" data-align=\"center\">0<\/td><td data-valign=\"middle\" data-align=\"center\">\\(\\left(0,0\\right)\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"middle\" data-align=\"center\">1<\/td><td data-valign=\"middle\" data-align=\"center\">5<\/td><td data-valign=\"middle\" data-align=\"center\">\\(\\left(1,5\\right)\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"middle\" data-align=\"center\">\\(-1\\)<\/td><td data-valign=\"middle\" data-align=\"center\">\\(-5\\)<\/td><td data-valign=\"middle\" data-align=\"center\">\\(\\left(-1,-5\\right)\\)<\/td><\/tr><\/tbody><\/table><p id=\"fs-id1167835335617\">Plot the three points, check that they line up, and draw the line.<\/p><span data-type=\"media\" id=\"fs-id1167835335620\" data-alt=\"The figure shows a graph of the equation y plus 5 x on the x y-coordinate plane. The x and y-axes run from negative 10 to 10. The straight line goes through the points (negative 1, negative 5), (0, 0), and (1, 5).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_034_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows a graph of the equation y plus 5 x on the x y-coordinate plane. The x and y-axes run from negative 10 to 10. The straight line goes through the points (negative 1, negative 5), (0, 0), and (1, 5).\"><\/span><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167835216609\" class=\"try\"><div data-type=\"exercise\"><div data-type=\"problem\" id=\"fs-id1167835377793\"><p id=\"fs-id1167835377795\">Graph using the intercepts: \\(y=4x.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167832151038\"><p id=\"fs-id1167834377227\"><\/p><div data-type=\"newline\"><br><\/div><span data-type=\"media\" id=\"fs-id1167832151040\" data-alt=\"The figure shows a graph of a straight line on the x y-coordinate plane. The x and y-axes run from negative 12 to 12. The straight line goes through the points (negative 1, negative 4), (0, 0), and (1, 4).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_317_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows a graph of a straight line on the x y-coordinate plane. The x and y-axes run from negative 12 to 12. The straight line goes through the points (negative 1, negative 4), (0, 0), and (1, 4).\"><\/span><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167826874357\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167834309127\"><div data-type=\"problem\" id=\"fs-id1167834309129\"><p id=\"fs-id1167834309131\">Graph the intercepts: \\(y=\\text{\u2212}x.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834433015\"><p id=\"fs-id1167832043239\"><\/p><div data-type=\"newline\"><br><\/div><span data-type=\"media\" id=\"fs-id1167834433018\" data-alt=\"The figure shows a graph of a straight line on the x y-coordinate plane. The x and y-axes run from negative 12 to 12. The straight line goes through the points (negative 1, 1), (0, 0), and (1, negative 1).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_318_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows a graph of a straight line on the x y-coordinate plane. The x and y-axes run from negative 12 to 12. The straight line goes through the points (negative 1, 1), (0, 0), and (1, negative 1).\"><\/span><\/div><\/div><\/div><\/div><div class=\"textbox\" data-depth=\"1\" id=\"fs-id1167832150989\"><h3 data-type=\"title\">Key Concepts<\/h3><ul id=\"fs-id1167835373759\" data-bullet-style=\"bullet\"><li><strong data-effect=\"bold\">Points on the Axes<\/strong><ul id=\"fs-id1167834162010\" data-bullet-style=\"bullet\"><li>Points with a <em data-effect=\"italics\">y<\/em>-coordinate equal to 0 are on the <em data-effect=\"italics\">x<\/em>-axis, and have coordinates \\(\\left(a,0\\right).\\)<\/li><li>Points with an <em data-effect=\"italics\">x<\/em>-coordinate equal to \\(0\\) are on the <em data-effect=\"italics\">y<\/em>-axis, and have coordinates \\(\\left(0,b\\right).\\)<\/li><\/ul><\/li><li><strong data-effect=\"bold\">Quadrant<\/strong><div data-type=\"equation\" id=\"fs-id1167831081622\" class=\"unnumbered\" data-label=\"\">\\(\\begin{array}{cccc}\\hfill \\mathbf{\\text{Quadrant I}}\\hfill &amp; \\hfill \\phantom{\\rule{2em}{0ex}}\\mathbf{\\text{Quadrant II}}\\hfill &amp; \\hfill \\phantom{\\rule{2em}{0ex}}\\mathbf{\\text{Quadrant III}}\\hfill &amp; \\hfill \\phantom{\\rule{2em}{0ex}}\\mathbf{\\text{Quadrant IV}}\\hfill \\\\ \\hfill \\left(x,y\\right)\\hfill &amp; \\hfill \\phantom{\\rule{2em}{0ex}}\\left(x,y\\right)\\hfill &amp; \\hfill \\phantom{\\rule{2em}{0ex}}\\left(x,y\\right)\\hfill &amp; \\hfill \\phantom{\\rule{2em}{0ex}}\\left(x,y\\right)\\hfill \\\\ \\hfill \\left(+,+\\right)\\hfill &amp; \\hfill \\phantom{\\rule{2em}{0ex}}\\left(-,+\\right)\\hfill &amp; \\hfill \\phantom{\\rule{2em}{0ex}}\\left(-,-\\right)\\hfill &amp; \\hfill \\phantom{\\rule{2em}{0ex}}\\left(+,-\\right)\\hfill \\end{array}\\)<\/div><span data-type=\"media\" id=\"fs-id1167835512761\" data-alt=\"This figure shows the x y-coordinate plane with the four quadrants labeled. In the top right of the plane is quadrant I labeled (plus, plus). In the top left of the plane is quadrant II labeled (minus, plus). In the bottom left of the plane is quadrant III labeled (minus, minus). In the bottom right of the plane is quadrant IV labeled (plus, minus).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_036_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows the x y-coordinate plane with the four quadrants labeled. In the top right of the plane is quadrant I labeled (plus, plus). In the top left of the plane is quadrant II labeled (minus, plus). In the bottom left of the plane is quadrant III labeled (minus, minus). In the bottom right of the plane is quadrant IV labeled (plus, minus).\"><\/span><\/li><li><strong data-effect=\"bold\">Graph of a Linear Equation:<\/strong> The graph of a linear equation \\(Ax+By=C\\) is a straight line.<div data-type=\"newline\"><br><\/div> Every point on the line is a solution of the equation.<div data-type=\"newline\"><br><\/div> Every solution of this equation is a point on this line.<\/li><li><strong data-effect=\"bold\">How to graph a linear equation by plotting points.<\/strong><ol id=\"fs-id1167830837149\" type=\"1\" class=\"stepwise\"><li>Find three points whose coordinates are solutions to the equation. Organize them in a table.<\/li><li>Plot the points in a rectangular coordinate system. Check that the points line up. If they do not, carefully check your work.<\/li><li>Draw the line through the three points. Extend the line to fill the grid and put arrows on both ends of the line.<\/li><\/ol><\/li><li><strong data-effect=\"bold\"><em data-effect=\"italics\">x<\/em>-intercept and <em data-effect=\"italics\">y<\/em>-intercept of a Line<\/strong><ul id=\"fs-id1167831076503\" data-bullet-style=\"bullet\"><li>The <em data-effect=\"italics\">x<\/em>-intercept is the point \\(\\left(a,0\\right)\\) where the line crosses the <em data-effect=\"italics\">x<\/em>-axis.<\/li><li>The <em data-effect=\"italics\">y<\/em>-intercept is the point \\(\\left(0,b\\right)\\) where the line crosses the <em data-effect=\"italics\">y<\/em>-axis. <span data-type=\"media\" id=\"fs-id1167835351058\" data-alt=\"The table has 3 rows and 2 columns. The first row is a header row with the headers x and y. The second row contains a and 0. The x-intercept occurs when y is zero. The third row contains 0 and b. The y-intercept occurs when x is zero.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_037_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The table has 3 rows and 2 columns. The first row is a header row with the headers x and y. The second row contains a and 0. The x-intercept occurs when y is zero. The third row contains 0 and b. The y-intercept occurs when x is zero.\"><\/span><\/li><\/ul><\/li><li><strong data-effect=\"bold\">Find the <em data-effect=\"italics\">x<\/em>- and <em data-effect=\"italics\">y<\/em>-intercepts from the Equation of a Line<\/strong><ul id=\"fs-id1167835326058\" data-bullet-style=\"bullet\"><li>Use the equation of the line. To find:<div data-type=\"newline\"><br><\/div> the <em data-effect=\"italics\">x<\/em>-intercept of the line, let \\(y=0\\) and solve for <em data-effect=\"italics\">x<\/em>.<div data-type=\"newline\"><br><\/div> the <em data-effect=\"italics\">y<\/em>-intercept of the line, let \\(x=0\\) and solve for <em data-effect=\"italics\">y<\/em>.<\/li><\/ul><\/li><li><strong data-effect=\"bold\">How to graph a linear equation using the intercepts.<\/strong><ol id=\"fs-id1167835514188\" type=\"1\" class=\"stepwise\"><li>Find the <em data-effect=\"italics\">x<\/em>- and <em data-effect=\"italics\">y<\/em>-intercepts of the line.<div data-type=\"newline\"><br><\/div> Let \\(y=0\\) and solve for <em data-effect=\"italics\">x.<\/em><div data-type=\"newline\"><br><\/div> Let \\(x=0\\) and solve for <em data-effect=\"italics\">y<\/em>.<\/li><li>Find a third solution to the equation.<\/li><li>Plot the three points and check that they line up.<\/li><li>Draw the line<\/li><\/ol><\/li><\/ul><\/div><div class=\"textbox\" data-depth=\"1\" id=\"fs-id1167835381776\"><div class=\"practice-perfect\" data-depth=\"2\" id=\"fs-id1167834060085a\"><h4 data-type=\"title\">Practice Makes Perfect<\/h4><p id=\"fs-id1167834060092\"><strong data-effect=\"bold\">Plot Points in a Rectangular Coordinate System<\/strong><\/p><p id=\"fs-id1167831837573\">In the following exercises, plot each point in a rectangular coordinate system and identify the quadrant in which the point is located.<\/p><div data-type=\"exercise\" class=\"material-set-2\"><div data-type=\"problem\"><p id=\"fs-id1167835384360\"><span class=\"token\">\u24d0<\/span>\\(\\left(-4,2\\right)\\)<span class=\"token\">\u24d1<\/span>\\(\\left(-1,-2\\right)\\)<span class=\"token\">\u24d2<\/span>\\(\\left(3,-5\\right)\\)<span class=\"token\">\u24d3<\/span>\\(\\left(-3,0\\right)\\)<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d4<\/span>\\(\\left(\\frac{5}{3},2\\right)\\)<\/div><div data-type=\"solution\" id=\"fs-id1167835381602\"><span data-type=\"media\" id=\"fs-id1167835381604\" data-alt=\"This figure shows points plotted on the x y-coordinate plane. The x and y axes run from negative 6 to 6. The point labeled a is 4 units to the left of the origin and 2 units above the origin and is located in quadrant II. The point labeled b is 1 unit to the left of the origin and 2 units below the origin and is located in quadrant III. The point labeled c is 3 units to the right of the origin and 5 units below the origin and is located in quadrant IV. The point labeled d is 3 units to the left of the origin and 5 units above the origin and is located in quadrant II. The point labeled e is 1 and a half units to the right of the origin and 2 units above the origin and is located in quadrant I.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_319_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows points plotted on the x y-coordinate plane. The x and y axes run from negative 6 to 6. The point labeled a is 4 units to the left of the origin and 2 units above the origin and is located in quadrant II. The point labeled b is 1 unit to the left of the origin and 2 units below the origin and is located in quadrant III. The point labeled c is 3 units to the right of the origin and 5 units below the origin and is located in quadrant IV. The point labeled d is 3 units to the left of the origin and 5 units above the origin and is located in quadrant II. The point labeled e is 1 and a half units to the right of the origin and 2 units above the origin and is located in quadrant I.\"><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834133113\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167834133115\"><p id=\"fs-id1167834133117\"><span class=\"token\">\u24d0<\/span>\\(\\left(-2,-3\\right)\\)<span class=\"token\">\u24d1<\/span>\\(\\left(3,-3\\right)\\)<span class=\"token\">\u24d2<\/span>\\(\\left(-4,1\\right)\\)<span class=\"token\">\u24d3<\/span>\\(\\left(4,-1\\right)\\)<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d4<\/span>\\(\\left(\\frac{3}{2},1\\right)\\)<\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835489339\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167835489342\"><p id=\"fs-id1167835351140\"><span class=\"token\">\u24d0<\/span>\\(\\left(3,-1\\right)\\)<span class=\"token\">\u24d1<\/span>\\(\\left(-3,1\\right)\\)<span class=\"token\">\u24d2<\/span>\\(\\left(-2,\\phantom{\\rule{0.2em}{0ex}}\\text{0}\\right)\\)<span class=\"token\">\u24d3<\/span>\\(\\left(-4,-3\\right)\\)<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d4<\/span>\\(\\left(1,\\frac{14}{5}\\right)\\)<\/div><div data-type=\"solution\" id=\"fs-id1167835420000\"><span data-type=\"media\" id=\"fs-id1167835420002\" data-alt=\"This figure shows points plotted on the x y-coordinate plane. The x and y axes run from negative 6 to 6. The point labeled a is 3 units to the right of the origin and 1 unit below the origin and is located in quadrant IV. The point labeled b is 3 units to the left of the origin and 1 unit above the origin and is located in quadrant II. The point labeled c is 2 units to the left of the origin and 2 units above the origin and is located in quadrant II. The point labeled d is 4 units to the left of the origin and 3 units below the origin and is located in quadrant III. The point labeled e is 1 unit to the right of the origin and 3 and 4 fifths units above the origin and is located in quadrant I.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_321_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows points plotted on the x y-coordinate plane. The x and y axes run from negative 6 to 6. The point labeled a is 3 units to the right of the origin and 1 unit below the origin and is located in quadrant IV. The point labeled b is 3 units to the left of the origin and 1 unit above the origin and is located in quadrant II. The point labeled c is 2 units to the left of the origin and 2 units above the origin and is located in quadrant II. The point labeled d is 4 units to the left of the origin and 3 units below the origin and is located in quadrant III. The point labeled e is 1 unit to the right of the origin and 3 and 4 fifths units above the origin and is located in quadrant I.\"><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831959139\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167831959141\"><p id=\"fs-id1167831920778\"><span class=\"token\">\u24d0<\/span>\\(\\left(-1,1\\right)\\)<span class=\"token\">\u24d1<\/span>\\(\\left(-2,-1\\right)\\)<span class=\"token\">\u24d2<\/span>\\(\\left(2,0\\right)\\)<span class=\"token\">\u24d3<\/span>\\(\\left(1,-4\\right)\\)<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d4<\/span>\\(\\left(3,\\frac{7}{2}\\right)\\)<\/div><\/div><p id=\"fs-id1167835390387\">In the following exercises, for each ordered pair, decide<\/p><p id=\"fs-id1167835309845\"><span class=\"token\">\u24d0<\/span> is the ordered pair a solution to the equation? <span class=\"token\">\u24d1<\/span> is the point on the line?<\/p><div data-type=\"exercise\" id=\"fs-id1167834516106\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167834516108\"><p id=\"fs-id1167834516111\">\\(y=x+2;\\)<\/p><div data-type=\"newline\"><br><\/div>A: \\(\\left(0,2\\right);\\) B: \\(\\left(1,2\\right);\\) C: \\(\\left(-1,1\\right);\\) D: \\(\\left(-3,-1\\right).\\)<span data-type=\"media\" id=\"fs-id1167832042287\" data-alt=\"This figure shows a straight line graphed on the x y-coordinate plane. The x and y-axes run from negative 8 to 8. The line goes through the points (negative 3, negative 1), (negative 2, 0), (negative 1, 1), (0, 2), (1, 3), (2, 4), and (3, 5).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_201_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a straight line graphed on the x y-coordinate plane. The x and y-axes run from negative 8 to 8. The line goes through the points (negative 3, negative 1), (negative 2, 0), (negative 1, 1), (0, 2), (1, 3), (2, 4), and (3, 5).\"><\/span><\/div><div data-type=\"solution\" id=\"fs-id1167831911373\"><p id=\"fs-id1167830964089\"><span class=\"token\">\u24d0<\/span> A: yes, B: no, C: yes, D: yes <span class=\"token\">\u24d1<\/span> A: yes, B: no, C: yes, D: yes<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167832015739\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167832015741\"><p id=\"fs-id1167832015743\">\\(y=x-4;\\)<\/p><div data-type=\"newline\"><br><\/div>A: \\(\\left(0,-4\\right);\\) B: \\(\\left(3,-1\\right);\\) C: \\(\\left(2,2\\right);\\) D: \\(\\left(1,-5\\right).\\)<span data-type=\"media\" id=\"fs-id1167831921371\" data-alt=\"This figure shows a straight line graphed on the x y-coordinate plane. The x and y-axes run from negative 8 to 8. The line goes through the points (negative 3, negative 7), (negative 2, negative 6), (negative 1, negative 5), (0, negative 4), (1, negative 3), (2, negative 2), and (3, negative 1).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_202_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a straight line graphed on the x y-coordinate plane. The x and y-axes run from negative 8 to 8. The line goes through the points (negative 3, negative 7), (negative 2, negative 6), (negative 1, negative 5), (0, negative 4), (1, negative 3), (2, negative 2), and (3, negative 1).\"><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835379705\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167831911279\"><p id=\"fs-id1167831911281\">\\(y=\\frac{1}{2}x-3;\\)<\/p><div data-type=\"newline\"><br><\/div>A: \\(\\left(0,-3\\right);\\) B: \\(\\left(2,-2\\right);\\) C: \\(\\left(-2,-4\\right);\\) D: \\(\\left(4,1\\right)\\)<span data-type=\"media\" id=\"fs-id1167828365419\" data-alt=\"This figure shows a straight line graphed on the x y-coordinate plane. The x and y-axes run from negative 8 to 8. The line goes through the points (negative 4, negative 5), (negative 2, negative 4), (0, negative 3), (2, negative 2), (4, negative 1), and (6, 0).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_203_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a straight line graphed on the x y-coordinate plane. The x and y-axes run from negative 8 to 8. The line goes through the points (negative 4, negative 5), (negative 2, negative 4), (0, negative 3), (2, negative 2), (4, negative 1), and (6, 0).\"><\/span><\/div><div data-type=\"solution\" id=\"fs-id1167826783927\"><p id=\"fs-id1167826783929\"><span class=\"token\">\u24d0<\/span> A: yes, B: yes, C: yes, D: no <span class=\"token\">\u24d1<\/span> A: yes, B: yes, C: yes, D: no<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167832043531\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167832043533\"><p id=\"fs-id1167832043535\">\\(y=\\frac{1}{3}x+2;\\)<\/p><div data-type=\"newline\"><br><\/div>A: \\(\\left(0,2\\right);\\) B: \\(\\left(3,3\\right);\\) C: \\(\\left(-3,2\\right);\\) D: \\(\\left(-6,0\\right).\\)<span data-type=\"media\" id=\"fs-id1167835173842\" data-alt=\"This figure shows a straight line graphed on the x y-coordinate plane. The x and y-axes run from negative 8 to 8. The line goes through the points (negative 6, 0), (negative 3, 1), (0, 2), (3, 3), (6, 4), and (9, 5).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_204_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a straight line graphed on the x y-coordinate plane. The x and y-axes run from negative 8 to 8. The line goes through the points (negative 6, 0), (negative 3, 1), (0, 2), (3, 3), (6, 4), and (9, 5).\"><\/span><\/div><\/div><p id=\"fs-id1167834534357\"><strong data-effect=\"bold\">Graph a Linear Equation by Plotting Points<\/strong><\/p><p id=\"fs-id1167834227331\">In the following exercises, graph by plotting points.<\/p><div data-type=\"exercise\" id=\"fs-id1167834133322\"><div data-type=\"problem\" id=\"fs-id1167834133324\"><p id=\"fs-id1167834133326\">\\(y=x+2\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834062901\"><span data-type=\"media\" id=\"fs-id1167834062904\" data-alt=\"This figure shows a straight line graphed on the x y-coordinate plane. The x and y-axes run from negative 12 to 12. The line goes through the points (negative 3, negative 1), (negative 2, 0), (negative 1, 1), (0, 2), (1, 3), (2, 4), and (3, 5).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_325_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a straight line graphed on the x y-coordinate plane. The x and y-axes run from negative 12 to 12. The line goes through the points (negative 3, negative 1), (negative 2, 0), (negative 1, 1), (0, 2), (1, 3), (2, 4), and (3, 5).\"><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834049043\"><div data-type=\"problem\" id=\"fs-id1167834049046\"><p id=\"fs-id1167834402981\">\\(y=x-3\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834227334\"><div data-type=\"problem\" id=\"fs-id1167834227337\"><p id=\"fs-id1167834227339\">\\(y=3x-1\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167827987486\"><span data-type=\"media\" data-alt=\"This figure shows a straight line graphed on the x y-coordinate plane. The x and y-axes run from negative 12 to 12. The line goes through the points (negative 3, negative 10), (negative 2, negative 7), (negative 1, negative 4), (0, negative 1), (1, 2), (2, 5), (3, 8), and (4, 11).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_323_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a straight line graphed on the x y-coordinate plane. The x and y-axes run from negative 12 to 12. The line goes through the points (negative 3, negative 10), (negative 2, negative 7), (negative 1, negative 4), (0, negative 1), (1, 2), (2, 5), (3, 8), and (4, 11).\"><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167826781720\"><div data-type=\"problem\" id=\"fs-id1167831883533\"><p id=\"fs-id1167831883535\">\\(y=-2x+2\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835640011\"><div data-type=\"problem\" id=\"fs-id1167835640013\"><p id=\"fs-id1167835640015\">\\(y=\\text{\u2212}x-3\\)<\/p><\/div><div data-type=\"solution\"><span data-type=\"media\" id=\"fs-id1167835379779\" data-alt=\"This figure shows a straight line graphed on the x y-coordinate plane. The x and y-axes run from negative 12 to 12. The line goes through the points (negative 3, 0), (negative 2, negative 1), (negative 1, negative 2), (0, negative 3), (1, negative 4), (2, negative 5), (3, negative 6), and (4, negative 7).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_327_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a straight line graphed on the x y-coordinate plane. The x and y-axes run from negative 12 to 12. The line goes through the points (negative 3, 0), (negative 2, negative 1), (negative 1, negative 2), (0, negative 3), (1, negative 4), (2, negative 5), (3, negative 6), and (4, negative 7).\"><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834252346\"><div data-type=\"problem\" id=\"fs-id1167834252348\"><p id=\"fs-id1167834252350\">\\(y=\\text{\u2212}x-2\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167830868598\"><div data-type=\"problem\" id=\"fs-id1167831106804\"><p id=\"fs-id1167831106806\">\\(y=2x\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167831881337\"><span data-type=\"media\" id=\"fs-id1167831881340\" data-alt=\"This figure shows a straight line graphed on the x y-coordinate plane. The x and y-axes run from negative 12 to 12. The line goes through the points (negative 3, negative 6), (negative 2, negative 4), (negative 1, negative 2), (0, 0), (1, 2), (2, 4), and (3, 6).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_329_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a straight line graphed on the x y-coordinate plane. The x and y-axes run from negative 12 to 12. The line goes through the points (negative 3, negative 6), (negative 2, negative 4), (negative 1, negative 2), (0, 0), (1, 2), (2, 4), and (3, 6).\"><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834084968\"><div data-type=\"problem\" id=\"fs-id1167834084970\"><p id=\"fs-id1167834084972\">\\(y=-2x\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831228815\"><div data-type=\"problem\" id=\"fs-id1167831228817\"><p id=\"fs-id1167831228819\">\\(y=\\frac{1}{2}x+2\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167831116445\"><span data-type=\"media\" id=\"fs-id1167831116447\" data-alt=\"This figure shows a straight line graphed on the x y-coordinate plane. The x and y-axes run from negative 12 to 12. The line goes through the points (negative 6, negative 2), (negative 4, 0), (negative 2, 1), (0, 2), (2, 3), (4, 4), and (6, 5).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_331_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a straight line graphed on the x y-coordinate plane. The x and y-axes run from negative 12 to 12. The line goes through the points (negative 6, negative 2), (negative 4, 0), (negative 2, 1), (0, 2), (2, 3), (4, 4), and (6, 5).\"><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835201088\"><div data-type=\"problem\" id=\"fs-id1167834426344\"><p id=\"fs-id1167834426346\">\\(y=\\frac{1}{3}x-1\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834161715\"><div data-type=\"problem\" id=\"fs-id1167834161718\"><p id=\"fs-id1167834161720\">\\(y=\\frac{4}{3}x-5\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167830963879\"><span data-type=\"media\" id=\"fs-id1167830963881\" data-alt=\"This figure shows a straight line graphed on the x y-coordinate plane. The x and y-axes run from negative 12 to 12. The line goes through the points (negative 3, negative 9), (0, negative 5), (3, negative 1), (6, 3), and (9, 7).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_333_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a straight line graphed on the x y-coordinate plane. The x and y-axes run from negative 12 to 12. The line goes through the points (negative 3, negative 9), (0, negative 5), (3, negative 1), (6, 3), and (9, 7).\"><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167832214462\"><div data-type=\"problem\" id=\"fs-id1167832214465\"><p id=\"fs-id1167832214467\">\\(y=\\frac{3}{2}x-3\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835510781\"><div data-type=\"problem\" id=\"fs-id1167835510783\"><p id=\"fs-id1167835510785\">\\(y=-\\frac{2}{5}x+1\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167831884860\"><span data-type=\"media\" id=\"fs-id1167831884862\" data-alt=\"This figure shows a straight line graphed on the x y-coordinate plane. The x and y-axes run from negative 12 to 12. The line goes through the points (negative 10, 5), (negative 5, 3), (0, 1), (5, negative 1), and (10, negative 3).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_335_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a straight line graphed on the x y-coordinate plane. The x and y-axes run from negative 12 to 12. The line goes through the points (negative 10, 5), (negative 5, 3), (0, 1), (5, negative 1), and (10, negative 3).\"><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167827966773\"><div data-type=\"problem\" id=\"fs-id1167831112408\"><p id=\"fs-id1167831112410\">\\(y=-\\frac{4}{5}x-1\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167830693579\"><div data-type=\"problem\" id=\"fs-id1167834120600\"><p id=\"fs-id1167834120602\">\\(y=-\\frac{3}{2}x+2\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167832117891\"><span data-type=\"media\" id=\"fs-id1167832117893\" data-alt=\"This figure shows a straight line graphed on the x y-coordinate plane. The x and y-axes run from negative 12 to 12. The line goes through the points (negative 4, 8), (negative 2, 5), (0, 2), (2, negative 1), (4, negative 4), and (6, negative 7).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_337_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a straight line graphed on the x y-coordinate plane. The x and y-axes run from negative 12 to 12. The line goes through the points (negative 4, 8), (negative 2, 5), (0, 2), (2, negative 1), (4, negative 4), and (6, negative 7).\"><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835319192\"><div data-type=\"problem\" id=\"fs-id1167835319194\"><p id=\"fs-id1167830963603\">\\(y=-\\frac{5}{3}x+4\\)<\/p><\/div><\/div><p id=\"fs-id1167835366967\"><strong data-effect=\"bold\">Graph Vertical and Horizontal lines<\/strong><\/p><p id=\"fs-id1167835267904\">In the following exercises, graph each equation.<\/p><div data-type=\"exercise\" id=\"fs-id1167835267907\"><div data-type=\"problem\" id=\"fs-id1167835267909\"><p id=\"fs-id1167835267911\"><span class=\"token\">\u24d0<\/span>\\(x=4\\)<span class=\"token\">\u24d1<\/span>\\(y=3\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835231508\"><p id=\"fs-id1167835231510\"><span class=\"token\">\u24d0<\/span><\/p><div data-type=\"newline\"><br><\/div><span data-type=\"media\" id=\"fs-id1167835231516\" data-alt=\"This figure shows a vertical straight line graphed on the x y-coordinate plane. The x and y-axes run from negative 12 to 12. The line goes through the points (4, negative 1), (4, 0), and (4, 1).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_339_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a vertical straight line graphed on the x y-coordinate plane. The x and y-axes run from negative 12 to 12. The line goes through the points (4, negative 1), (4, 0), and (4, 1).\"><\/span><p><\/p><div data-type=\"newline\"><br><\/div><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span><div data-type=\"newline\"><br><\/div><span data-type=\"media\" id=\"fs-id1167834403045\" data-alt=\"This figure shows a horizontal straight line graphed on the x y-coordinate plane. The x and y-axes run from negative 12 to 12. The line goes through the points (negative 1, 3), (0, 3), and (1, 3).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_340_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a horizontal straight line graphed on the x y-coordinate plane. The x and y-axes run from negative 12 to 12. The line goes through the points (negative 1, 3), (0, 3), and (1, 3).\"><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834195667\"><div data-type=\"problem\" id=\"fs-id1167835231662\"><p id=\"fs-id1167835231664\"><span class=\"token\">\u24d0<\/span>\\(x=3\\)<span class=\"token\">\u24d1<\/span>\\(y=1\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167826937848\"><div data-type=\"problem\" id=\"fs-id1167826937850\"><p id=\"fs-id1167835519107\"><span class=\"token\">\u24d0<\/span>\\(x=-2\\)<span class=\"token\">\u24d1<\/span>\\(y=-5\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835254717\"><p id=\"fs-id1167831949111\"><span class=\"token\">\u24d0<\/span><\/p><div data-type=\"newline\"><br><\/div><span data-type=\"media\" id=\"fs-id1167831949117\" data-alt=\"This figure shows a vertical straight line graphed on the x y-coordinate plane. The x and y-axes run from negative 12 to 12. The line goes through the points (negative 2, negative 1), (negative 2, 0), and (negative 2, 1).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_343_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a vertical straight line graphed on the x y-coordinate plane. The x and y-axes run from negative 12 to 12. The line goes through the points (negative 2, negative 1), (negative 2, 0), and (negative 2, 1).\"><\/span><p><\/p><div data-type=\"newline\"><br><\/div><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span><div data-type=\"newline\"><br><\/div><span data-type=\"media\" id=\"fs-id1167835389720\" data-alt=\"This figure shows a horizontal straight line graphed on the x y-coordinate plane. The x and y-axes run from negative 12 to 12. The line goes through the points (negative 1, negative 5), (0, negative 5), and (1, negative 5).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_344_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a horizontal straight line graphed on the x y-coordinate plane. The x and y-axes run from negative 12 to 12. The line goes through the points (negative 1, negative 5), (0, negative 5), and (1, negative 5).\"><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835511308\"><div data-type=\"problem\" id=\"fs-id1167835511310\"><p id=\"fs-id1167835511312\"><span class=\"token\">\u24d0<\/span>\\(x=-5\\)<span class=\"token\">\u24d1<\/span>\\(y=-2\\)<\/p><\/div><\/div><p id=\"fs-id1167832153425\">In the following exercises, graph each pair of equations in the same rectangular coordinate system.<\/p><div data-type=\"exercise\" id=\"fs-id1167832153429\"><div data-type=\"problem\" id=\"fs-id1167832153431\"><p id=\"fs-id1167835362062\">\\(y=2x\\) and \\(y=2\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834448640\"><span data-type=\"media\" id=\"fs-id1167834448642\" data-alt=\"The figure shows the graphs of a straight horizontal line and a straight slanted line on the same x y-coordinate plane. The x and y axes run from negative 12 to 12. The horizontal line goes through the points (0, 2), (1, 2), and (2, 2). The slanted line goes through the points (0, 0), (1, 2), and (2, 4).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_347_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows the graphs of a straight horizontal line and a straight slanted line on the same x y-coordinate plane. The x and y axes run from negative 12 to 12. The horizontal line goes through the points (0, 2), (1, 2), and (2, 2). The slanted line goes through the points (0, 0), (1, 2), and (2, 4).\"><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835376055\"><div data-type=\"problem\" id=\"fs-id1167835376057\"><p id=\"fs-id1167834246714\">\\(y=5x\\) and \\(y=5\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831031010\"><div data-type=\"problem\" id=\"fs-id1167831031012\"><p id=\"fs-id1167831031014\">\\(y=-\\frac{1}{2}x\\) and \\(y=-\\frac{1}{2}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167831880532\"><span data-type=\"media\" id=\"fs-id1167832125741\" data-alt=\"The figure shows the graphs of a straight horizontal line and a straight slanted line on the same x y-coordinate plane. The x and y axes run from negative 12 to 12. The horizontal line goes through the points (0, negative 1 divided 2), (1, negative 1 divided 2), and (2, negative 1 divided 2). The slanted line goes through the points (0, 0), (1, negative 1 divided 2), and (2, negative 1).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_349_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows the graphs of a straight horizontal line and a straight slanted line on the same x y-coordinate plane. The x and y axes run from negative 12 to 12. The horizontal line goes through the points (0, negative 1 divided 2), (1, negative 1 divided 2), and (2, negative 1 divided 2). The slanted line goes through the points (0, 0), (1, negative 1 divided 2), and (2, negative 1).\"><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835301347\"><div data-type=\"problem\" id=\"fs-id1167835301350\"><p id=\"fs-id1167835301352\">\\(y=-\\frac{1}{3}x\\) and \\(y=-\\frac{1}{3}\\)<\/p><\/div><\/div><p id=\"fs-id1167835512622\"><strong data-effect=\"bold\">Find <em data-effect=\"italics\">x-<\/em> and <em data-effect=\"italics\">y-<\/em>Intercepts<\/strong><\/p><p id=\"fs-id1167834219350\">In the following exercises, find the <em data-effect=\"italics\">x<\/em>- and <em data-effect=\"italics\">y<\/em>-intercepts on each graph.<\/p><div data-type=\"exercise\" id=\"fs-id1167831911460\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167835596622\"><p id=\"fs-id1167835596624\"><\/p><div data-type=\"newline\"><br><\/div><span data-type=\"media\" id=\"fs-id1167835596625\" data-alt=\"The figure shows a straight line graphed on the x y-coordinate plane. The x and y axes run from negative 8 to 8. The line goes through the points (negative 6, 9), (negative 3, 6), (0, 3), (3, 0), and (6, negative 3).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_205_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows a straight line graphed on the x y-coordinate plane. The x and y axes run from negative 8 to 8. The line goes through the points (negative 6, 9), (negative 3, 6), (0, 3), (3, 0), and (6, negative 3).\"><\/span><\/div><div data-type=\"solution\" id=\"fs-id1167834501585\"><p id=\"fs-id1167834501587\">\\(\\left(3,0\\right),\\left(0,3\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167826967394\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167826967396\"><p id=\"fs-id1167826967398\"><\/p><div data-type=\"newline\"><br><\/div><span data-type=\"media\" id=\"fs-id1167832076530\" data-alt=\"The figure shows a straight line graphed on the x y-coordinate plane. The x and y axes run from negative 8 to 8. The line goes through the points (negative 6, 4), (negative 4, 2), (negative 2, 0), (0, negative 2), (2, negative 4), and (4, negative 6).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_206_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows a straight line graphed on the x y-coordinate plane. The x and y axes run from negative 8 to 8. The line goes through the points (negative 6, 4), (negative 4, 2), (negative 2, 0), (0, negative 2), (2, negative 4), and (4, negative 6).\"><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167827943612\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167827943614\"><p id=\"fs-id1167827943616\"><\/p><div data-type=\"newline\"><br><\/div><span data-type=\"media\" id=\"fs-id1167827943618\" data-alt=\"The figure shows a straight line graphed on the x y-coordinate plane. The x and y axes run from negative 8 to 8. The line goes through the points (negative 1, negative 6), (0, negative 5), (2, negative 3), (5, 0), and (7, 2).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_207_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows a straight line graphed on the x y-coordinate plane. The x and y axes run from negative 8 to 8. The line goes through the points (negative 1, negative 6), (0, negative 5), (2, negative 3), (5, 0), and (7, 2).\"><\/span><\/div><div data-type=\"solution\" id=\"fs-id1167834060239\"><p id=\"fs-id1167834494858\">\\(\\left(5,0\\right),\\left(0,-5\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835545442\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167835545444\"><p id=\"fs-id1167835545446\"><\/p><div data-type=\"newline\"><br><\/div><span data-type=\"media\" id=\"fs-id1167835545447\" data-alt=\"The figure shows a straight line graphed on the x y-coordinate plane. The x and y axes run from negative 8 to 8. The line goes through the points (negative 2, negative 4), (negative 1, negative 2), (0, 0), (1, 2), and (2, 4).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_208_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows a straight line graphed on the x y-coordinate plane. The x and y axes run from negative 8 to 8. The line goes through the points (negative 2, negative 4), (negative 1, negative 2), (0, 0), (1, 2), and (2, 4).\"><\/span><\/div><\/div><p id=\"fs-id1167831922487\">In the following exercises, find the intercepts for each equation.<\/p><div data-type=\"exercise\" id=\"fs-id1167831922490\"><div data-type=\"problem\" id=\"fs-id1167831922492\"><p id=\"fs-id1167835353850\">\\(x-y=5\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834079438\"><p id=\"fs-id1167834079440\">\\(\\left(5,0\\right),\\left(0,-5\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831920143\"><div data-type=\"problem\" id=\"fs-id1167831920146\"><p id=\"fs-id1167831920148\">\\(x-y=-4\\)<\/p><\/div><\/div><div data-type=\"exercise\"><div data-type=\"problem\" id=\"fs-id1167834376169\"><p id=\"fs-id1167834376171\">\\(3x+y=6\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167831872331\"><p id=\"fs-id1167831872333\">\\(\\left(2,0\\right),\\left(0,6\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831071499\"><div data-type=\"problem\" id=\"fs-id1167831071501\"><p id=\"fs-id1167831071504\">\\(x-2y=8\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167832081988\"><div data-type=\"problem\" id=\"fs-id1167830699485\"><p id=\"fs-id1167830699487\">\\(4x-y=8\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835415278\"><p id=\"fs-id1167835415280\">\\(\\left(2,0\\right),\\left(0,-8\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835306018\"><div data-type=\"problem\" id=\"fs-id1167835306020\"><p id=\"fs-id1167835306022\">\\(5x-y=5\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167830705955\"><div data-type=\"problem\" id=\"fs-id1167830705957\"><p id=\"fs-id1167830705959\">\\(2x+5y=10\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167830704516\"><p id=\"fs-id1167830704518\">\\(\\left(5,0\\right),\\left(0,2\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835326514\"><div data-type=\"problem\" id=\"fs-id1167835344492\"><p id=\"fs-id1167835344494\">\\(3x-2y=12\\)<\/p><\/div><\/div><p id=\"fs-id1167834192141\"><strong data-effect=\"bold\">Graph a Line Using the Intercepts<\/strong><\/p><p id=\"fs-id1167832053315\">In the following exercises, graph using the intercepts.<\/p><div data-type=\"exercise\" id=\"fs-id1167832053318\"><div data-type=\"problem\" id=\"fs-id1167832053320\"><p id=\"fs-id1167832053323\">\\(-x+4y=8\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835321857\"><span data-type=\"media\" id=\"fs-id1167835321859\" data-alt=\"The figure shows a straight line graphed on the x y-coordinate plane. The x and y axes run from negative 12 to 12. The line goes through the points (negative 8, 0), (0, 2), (4, 3), and (8, 4).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_351_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows a straight line graphed on the x y-coordinate plane. The x and y axes run from negative 12 to 12. The line goes through the points (negative 8, 0), (0, 2), (4, 3), and (8, 4).\"><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834506047\"><div data-type=\"problem\" id=\"fs-id1167834506050\"><p id=\"fs-id1167834506052\">\\(x+2y=4\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835217987\"><div data-type=\"problem\" id=\"fs-id1167835217989\"><p id=\"fs-id1167835217992\">\\(x+y=-3\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835483756\"><span data-type=\"media\" data-alt=\"The figure shows a straight line graphed on the x y-coordinate plane. The x and y axes run from negative 12 to 12. The line goes through the points (negative 3, 0), (0, negative 3), and (3, negative 6).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_353_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows a straight line graphed on the x y-coordinate plane. The x and y axes run from negative 12 to 12. The line goes through the points (negative 3, 0), (0, negative 3), and (3, negative 6).\"><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831239741\"><div data-type=\"problem\" id=\"fs-id1167831239743\"><p id=\"fs-id1167831239746\">\\(x-y=-4\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831239409\"><div data-type=\"problem\" id=\"fs-id1167835376276\"><p id=\"fs-id1167827958241\">\\(4x+y=4\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835524772\"><span data-type=\"media\" id=\"fs-id1167835524774\" data-alt=\"The figure shows a straight line graphed on the x y-coordinate plane. The x and y axes run from negative 12 to 12. The line goes through the points (0, 4), (1, 0), and (2, negative 4).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_355_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows a straight line graphed on the x y-coordinate plane. The x and y axes run from negative 12 to 12. The line goes through the points (0, 4), (1, 0), and (2, negative 4).\"><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834063389\"><div data-type=\"problem\" id=\"fs-id1167834063392\"><p id=\"fs-id1167834495437\">\\(3x+y=3\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834182324\"><div data-type=\"problem\" id=\"fs-id1167831065894\"><p id=\"fs-id1167831065896\">\\(3x-y=-6\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835595573\"><span data-type=\"media\" id=\"fs-id1167835595575\" data-alt=\"The figure shows a straight line graphed on the x y-coordinate plane. The x and y axes run from negative 12 to 12. The line goes through the points (negative 2, 0), (negative 1, 3), and (0, 6).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_357_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows a straight line graphed on the x y-coordinate plane. The x and y axes run from negative 12 to 12. The line goes through the points (negative 2, 0), (negative 1, 3), and (0, 6).\"><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835512205\"><div data-type=\"problem\"><p>\\(2x-y=-8\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167826967127\"><div data-type=\"problem\" id=\"fs-id1167826967129\"><p id=\"fs-id1167826967131\">\\(2x+4y=12\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835229406\"><span data-type=\"media\" id=\"fs-id1167835229408\" data-alt=\"The figure shows a straight line graphed on the x y-coordinate plane. The x and y axes run from negative 8 to 8. The line goes through the points (0, 3), (2, 2), and (6, 0).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_359_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows a straight line graphed on the x y-coordinate plane. The x and y axes run from negative 8 to 8. The line goes through the points (0, 3), (2, 2), and (6, 0).\"><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167826967305\"><div data-type=\"problem\" id=\"fs-id1167826967307\"><p id=\"fs-id1167826967309\">\\(3x-2y=6\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167826987293\"><div data-type=\"problem\" id=\"fs-id1167826987295\"><p id=\"fs-id1167826987297\">\\(2x-5y=-20\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167830705513\"><span data-type=\"media\" id=\"fs-id1167830705515\" data-alt=\"The figure shows a straight line graphed on the x y-coordinate plane. The x and y axes run from negative 12 to 12. The line goes through the points (negative 10, 0), (0, 4), and (10, 8).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_361_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows a straight line graphed on the x y-coordinate plane. The x and y axes run from negative 12 to 12. The line goes through the points (negative 10, 0), (0, 4), and (10, 8).\"><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831891905\"><div data-type=\"problem\" id=\"fs-id1167835347628\"><p id=\"fs-id1167835347630\">\\(3x-4y=-12\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835356423\"><div data-type=\"problem\" id=\"fs-id1167835356425\"><p id=\"fs-id1167835356427\">\\(y=-2x\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835327678\"><span data-type=\"media\" id=\"fs-id1167835327680\" data-alt=\"The figure shows a straight line graphed on the x y-coordinate plane. The x and y axes run from negative 12 to 12. The line goes through the points (negative 1, 2), (0, 0), and (1, negative 2).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_363_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows a straight line graphed on the x y-coordinate plane. The x and y axes run from negative 12 to 12. The line goes through the points (negative 1, 2), (0, 0), and (1, negative 2).\"><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831923233\"><div data-type=\"problem\" id=\"fs-id1167831923235\"><p id=\"fs-id1167831923238\">\\(y=5x\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834156928\"><div data-type=\"problem\" id=\"fs-id1167834156930\"><p id=\"fs-id1167835349567\">\\(y=x\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835349579\"><span data-type=\"media\" id=\"fs-id1167826799424\" data-alt=\"The figure shows a straight line graphed on the x y-coordinate plane. The x and y axes run from negative 12 to 12. The line goes through the points (negative 1, negative 1), (0, 0), and (1, 1).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_365_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows a straight line graphed on the x y-coordinate plane. The x and y axes run from negative 12 to 12. The line goes through the points (negative 1, negative 1), (0, 0), and (1, 1).\"><\/span><\/div><\/div><div data-type=\"exercise\"><div data-type=\"problem\" id=\"fs-id1167834098231\"><p id=\"fs-id1167834098233\">\\(y=\\text{\u2212}x\\)<\/p><\/div><\/div><p id=\"fs-id1167835217776\"><strong data-effect=\"bold\">Mixed Practice<\/strong><\/p><p>In the following exercises, graph each equation.<\/p><div data-type=\"exercise\" id=\"fs-id1167835217785\"><div data-type=\"problem\" id=\"fs-id1167835217787\"><p id=\"fs-id1167834593552\">\\(y=\\frac{3}{2}x\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834061645\"><span data-type=\"media\" id=\"fs-id1167834061647\" data-alt=\"The figure shows a straight line graphed on the x y-coordinate plane. The x and y axes run from negative 8 to 8. The line goes through the points (negative 2, negative 3), (0, 0), and (2, 3).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_367_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows a straight line graphed on the x y-coordinate plane. The x and y axes run from negative 8 to 8. The line goes through the points (negative 2, negative 3), (0, 0), and (2, 3).\"><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167827966889\"><div data-type=\"problem\" id=\"fs-id1167831040610\"><p id=\"fs-id1167831040612\">\\(y=-\\frac{2}{3}x\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167832065980\"><div data-type=\"problem\" id=\"fs-id1167832065982\"><p id=\"fs-id1167832065984\">\\(y=-\\frac{1}{2}x+3\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835512468\"><span data-type=\"media\" id=\"fs-id1167835512470\" data-alt=\"The figure shows a straight line graphed on the x y-coordinate plane. The x and y axes run from negative 8 to 8. The line goes through the points (negative 2, 4), (0, 3), (2, 2), (4, 1), and (6, 0).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_369_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows a straight line graphed on the x y-coordinate plane. The x and y axes run from negative 8 to 8. The line goes through the points (negative 2, 4), (0, 3), (2, 2), (4, 1), and (6, 0).\"><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167827956693\"><div data-type=\"problem\" id=\"fs-id1167827956695\"><p id=\"fs-id1167827956697\">\\(y=\\frac{1}{4}x-2\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834063745\"><div data-type=\"problem\" id=\"fs-id1167834063747\"><p id=\"fs-id1167834063750\">\\(4x+y=2\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167831922540\"><span data-type=\"media\" id=\"fs-id1167831922543\" data-alt=\"The figure shows a straight line graphed on the x y-coordinate plane. The x and y axes run from negative 8 to 8. The line goes through the points (negative 1, 6), (0, 2), (1, negative 2), and (2, negative 4).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_371_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows a straight line graphed on the x y-coordinate plane. The x and y axes run from negative 8 to 8. The line goes through the points (negative 1, 6), (0, 2), (1, negative 2), and (2, negative 4).\"><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834124817\"><div data-type=\"problem\"><p id=\"fs-id1167834124822\">\\(5x+2y=10\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834396194\"><div data-type=\"problem\" id=\"fs-id1167834396197\"><p id=\"fs-id1167834396199\">\\(y=-1\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834195818\"><span data-type=\"media\" id=\"fs-id1167834195821\" data-alt=\"The figure shows a horizontal straight line graphed on the x y-coordinate plane. The x and y axes run from negative 8 to 8. The line goes through the points (negative 2, negative 1), (0, negative 1), and (1, negative 1).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_373_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows a horizontal straight line graphed on the x y-coordinate plane. The x and y axes run from negative 8 to 8. The line goes through the points (negative 2, negative 1), (0, negative 1), and (1, negative 1).\"><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835321946\"><div data-type=\"problem\" id=\"fs-id1167835321948\"><p id=\"fs-id1167835321950\">\\(x=3\\)<\/p><\/div><\/div><\/div><div class=\"writing\" data-depth=\"2\" id=\"fs-id1167831911726\"><h4 data-type=\"title\">Writing Exercises<\/h4><div data-type=\"exercise\" id=\"fs-id1167835376744\"><div data-type=\"problem\" id=\"fs-id1167835376746\"><p id=\"fs-id1167835376748\">Explain how you would choose three <em data-effect=\"italics\">x<\/em>-values to make a table to graph the line \\(y=\\frac{1}{5}x-2.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834584361\"><p id=\"fs-id1167834584364\">Answers will vary.<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834584369\"><div data-type=\"problem\" id=\"fs-id1167834584371\"><p id=\"fs-id1167834584373\">What is the difference between the equations of a vertical and a horizontal line?<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831824603\"><div data-type=\"problem\" id=\"fs-id1167826828710\"><p id=\"fs-id1167826828712\">Do you prefer to use the method of plotting points or the method using the intercepts to graph the equation \\(4x+y=-4?\\) Why?<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835322017\"><p id=\"fs-id1167835322019\">Answers will vary.<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835322025\"><div data-type=\"problem\" id=\"fs-id1167835322027\"><p>Do you prefer to use the method of plotting points or the method using the intercepts to graph the equation \\(y=\\frac{2}{3}x-2?\\) Why?<\/p><\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1167835218155\"><h4 data-type=\"title\">Self Check<\/h4><p id=\"fs-id1167835344533\"><span class=\"token\">\u24d0<\/span> After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.<\/p><span data-type=\"media\" id=\"fs-id1167835344541\" data-alt=\"This table has 6 rows and 4 columns. The first row is a header row and it labels each column. The first column header is \u201cI can\u2026\u201d, the second is \u201cConfidently\u201d, the third is \u201cWith some help\u201d, and the fourth is \u201cNo, I don\u2019t get it\u201d. Under the first column are the phrases \u201cplot points on a rectangular coordinate system\u201d, \u201cgraph a linear equation by plotting points\u201d, \u201cgraph vertical and horizontal lines\u201d, \u201cfind x and y intercepts\u201d, and \u201cgraph a line using intercepts\u201d. The other columns are left blank so that the learner may indicate their mastery level for each topic.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_211_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This table has 6 rows and 4 columns. The first row is a header row and it labels each column. The first column header is \u201cI can\u2026\u201d, the second is \u201cConfidently\u201d, the third is \u201cWith some help\u201d, and the fourth is \u201cNo, I don\u2019t get it\u201d. Under the first column are the phrases \u201cplot points on a rectangular coordinate system\u201d, \u201cgraph a linear equation by plotting points\u201d, \u201cgraph vertical and horizontal lines\u201d, \u201cfind x and y intercepts\u201d, and \u201cgraph a line using intercepts\u201d. The other columns are left blank so that the learner may indicate their mastery level for each topic.\"><\/span><p id=\"fs-id1167835340496\"><span class=\"token\">\u24d1<\/span> If most of your checks were:<\/p><p id=\"fs-id1167835340504\">Confidently. 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-id1167835374300\">With some help. 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-id1167835374307\">No, I don\u2019t get it. This is a warning sign and you must address 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-id1167831076624\"><dt>horizontal line<\/dt><dd id=\"fs-id1167826814098\">A horizontal line is the graph of an equation of the form \\(y=b.\\) The line passes through the <em data-effect=\"italics\">y<\/em>-axis at \\(\\left(0,b\\right).\\)<\/dd><\/dl><dl id=\"fs-id1167835355558\"><dt>intercepts of a line<\/dt><dd id=\"fs-id1167835355563\">The points where a line crosses the <em data-effect=\"italics\">x<\/em>-axis and the <em data-effect=\"italics\">y<\/em>-axis are called the intercepts of the line.<\/dd><\/dl><dl id=\"fs-id1167834064606\"><dt>linear equation<\/dt><dd id=\"fs-id1167834357135\">An equation of the form \\(Ax+By=C,\\) where <em data-effect=\"italics\">A<\/em> and <em data-effect=\"italics\">B<\/em> are not both zero, is called a linear equation in two variables.<\/dd><\/dl><dl id=\"fs-id1167834156774\"><dt>ordered pair<\/dt><dd id=\"fs-id1167835533863\">An ordered pair, \\(\\left(x,y\\right)\\) gives the coordinates of a point in a rectangular coordinate system. The first number is the <em data-effect=\"italics\">x<\/em>-coordinate. The second number is the <em data-effect=\"italics\">y<\/em>-coordinate.<\/dd><\/dl><dl id=\"fs-id1167834053603\"><dt>origin<\/dt><dd id=\"fs-id1167831239720\">The point \\(\\left(0,0\\right)\\) is called the origin. It is the point where the <em data-effect=\"italics\">x<\/em>-axis and <em data-effect=\"italics\">y<\/em>-axis intersect.<\/dd><\/dl><dl id=\"fs-id1167834505886\"><dt>solution of a linear equation in two variables<\/dt><dd id=\"fs-id1167831944043\">An ordered pair \\(\\left(x,y\\right)\\) is a solution of the linear equation\\(Ax+By=C,\\) if the equation is a true statement when the <em data-effect=\"italics\">x<\/em>- and <em data-effect=\"italics\">y<\/em>-values of the ordered pair are substituted into the equation.<\/dd><\/dl><dl id=\"fs-id1167835363804\"><dt>standard form of a linear equation<\/dt><dd id=\"fs-id1167835363810\">A linear equation is in standard form when it is written \\(Ax+By=C.\\)<\/dd><\/dl><dl id=\"fs-id1167835231613\"><dt>vertical line<\/dt><dd id=\"fs-id1167835231619\">A vertical line is the graph of an equation of the form \\(x=a.\\) The line passes through the <em data-effect=\"italics\">x<\/em>-axis at \\(\\left(a,0\\right).\\)<\/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>Plot points in a rectangular coordinate system<\/li>\n<li>Graph a linear equation by plotting points<\/li>\n<li>Graph vertical and horizontal lines<\/li>\n<li>Find the x- and y-intercepts<\/li>\n<li>Graph a line using the intercepts<\/li>\n<\/ul>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167835512168\" class=\"be-prepared\">\n<p id=\"fs-id1167834301088\">Before you get started, take this readiness quiz.<\/p>\n<ol id=\"fs-id1167826978613\" type=\"1\">\n<li>Evaluate <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b5a3b55311fd6ae3f6f9300e92bf8eab_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#120;&#45;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"50\" style=\"vertical-align: -1px;\" \/> when <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6de4e73609a66312d9714a253f9ae3a2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#45;&#49;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"61\" style=\"vertical-align: -1px;\" \/>\n<div data-type=\"newline\"><\/div>\n<p> If you missed this problem, review <a href=\"\/contents\/d3697553-3900-453f-8c08-8f74d55711ba#fs-id1167836530265\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li>\n<li>Evaluate <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-658375cbfd2d22228ce1a10678682490_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#120;&#45;&#50;&#121;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"59\" style=\"vertical-align: -4px;\" \/> when <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d9d1f0b412cc88cb332d84b773584664_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#52;&#44;&#121;&#61;&#45;&#51;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"110\" style=\"vertical-align: -4px;\" \/>\n<div data-type=\"newline\"><\/div>\n<p> If you missed this problem, review <a href=\"\/contents\/05eab039-6d1c-4d80-8c8c-94469164a52c#fs-id1167832053133\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li>\n<li>Solve for <em data-effect=\"italics\">y<\/em>: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f0b6bcafcdfbc00a26bc154fd6a47894_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;&#45;&#51;&#121;&#61;&#50;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"94\" style=\"vertical-align: -4px;\" \/>\n<div data-type=\"newline\"><\/div>\n<p> If you missed this problem, review <a href=\"\/contents\/9f100e8f-6d15-4cae-bc22-c306e9d7d55c#fs-id1167836432956\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li>\n<\/ol>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167835237326\">\n<h3 data-type=\"title\">Plot Points on a Rectangular Coordinate System<\/h3>\n<p id=\"fs-id1167834423120\">Just like maps use a grid system to identify locations, a grid system is used in algebra to show a relationship between two variables in a rectangular coordinate system. The rectangular coordinate system is also called the <em data-effect=\"italics\">xy<\/em>-plane or the \u201ccoordinate plane.\u201d<\/p>\n<p id=\"fs-id1167832052908\">The rectangular coordinate system is formed by two intersecting number lines, one horizontal and one vertical. The horizontal number line is called the <em data-effect=\"italics\">x<\/em>-axis. The vertical number line is called the <em data-effect=\"italics\">y<\/em>-axis. These axes divide a plane into four regions, called quadrants. The quadrants are identified by Roman numerals, beginning on the upper right and proceeding counterclockwise. See <a href=\"#CNX_IntAlg_Figure_03_01_001\" class=\"autogenerated-content\">(Figure)<\/a>.<\/p>\n<div class=\"bc-figure figure\" id=\"CNX_IntAlg_Figure_03_01_001\"><span data-type=\"media\" id=\"fs-id1167835381166\" data-alt=\"This figure shows a square grid. A horizontal number line in the middle is labeled x. A vertical number line in the middle is labeled y. The number lines intersect at zero and together divide the square grid into 4 equally sized smaller squares. The square in the top right is labeled I. The square in the top left is labeled II. The square in the bottom left is labeled III. The square in the bottom right is labeled IV.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_001_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a square grid. A horizontal number line in the middle is labeled x. A vertical number line in the middle is labeled y. The number lines intersect at zero and together divide the square grid into 4 equally sized smaller squares. The square in the top right is labeled I. The square in the top left is labeled II. The square in the bottom left is labeled III. The square in the bottom right is labeled IV.\" \/><\/span><\/div>\n<p id=\"fs-id1167832053394\">In the rectangular coordinate system, every point is represented by an <span data-type=\"term\">ordered pair<\/span>. The first number in the ordered pair is the <em data-effect=\"italics\">x<\/em>-coordinate of the point, and the second number is the <em data-effect=\"italics\">y<\/em>-coordinate of the point. The phrase \u201cordered pair\u201d means that the order is important.<\/p>\n<div data-type=\"note\" id=\"fs-id1167834538411\">\n<div data-type=\"title\">Ordered Pair<\/div>\n<p id=\"fs-id1167835240319\">An <strong data-effect=\"bold\">ordered pair<\/strong>, <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;\" \/> gives the coordinates of a point in a rectangular coordinate system. The first number is the <em data-effect=\"italics\">x<\/em>-coordinate. The second number is the <em data-effect=\"italics\">y<\/em>-coordinate.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167835478964\" data-alt=\"This figure shows the expression (x, y). The variable x is labeled x-coordinate. The variable y is labeled y-coordinate.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_002_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows the expression (x, y). The variable x is labeled x-coordinate. The variable y is labeled y-coordinate.\" \/><\/span><\/div>\n<p id=\"fs-id1167834397450\">What is the ordered pair of the point where the axes cross? At that point both coordinates are zero, so its ordered pair is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e9b349d335879ab45d8b79d5850b0860_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;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"45\" style=\"vertical-align: -4px;\" \/> The point <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;\" \/> has a special name. It is called the <span data-type=\"term\">origin<\/span>.<\/p>\n<div data-type=\"note\" id=\"fs-id1167835194673\">\n<div data-type=\"title\">The Origin<\/div>\n<p id=\"fs-id1167835327055\">The point <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;\" \/> is called the <strong data-effect=\"bold\">origin<\/strong>. It is the point where the <em data-effect=\"italics\">x<\/em>-axis and <em data-effect=\"italics\">y<\/em>-axis intersect.<\/p>\n<\/div>\n<p id=\"fs-id1167834377125\">We use the coordinates to locate a point on the <em data-effect=\"italics\">xy<\/em>-plane. Let\u2019s plot the point <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;\" \/> as an example. First, locate 1 on the <em data-effect=\"italics\">x<\/em>-axis and lightly sketch a vertical line through <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-29163feacef7bfd88b9b5d136f8fef91_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#49;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"47\" style=\"vertical-align: -1px;\" \/> Then, locate 3 on the <em data-effect=\"italics\">y<\/em>-axis and sketch a horizontal line through <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-34f0eb02c9be9d3f22890308095c7a3a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#51;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"46\" style=\"vertical-align: -4px;\" \/> Now, find the point where these two lines meet\u2014that is the point with coordinates <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8abf8911b9dbc811d668c10a487f5e2e_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;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"45\" style=\"vertical-align: -4px;\" \/> See <a href=\"#CNX_IntAlg_Figure_03_01_003\" class=\"autogenerated-content\">(Figure)<\/a>.<\/p>\n<div class=\"bc-figure figure\" id=\"CNX_IntAlg_Figure_03_01_003\"><span data-type=\"media\" id=\"fs-id1167835376254\" data-alt=\"This figure shows a point plotted on the x y-coordinate plane. The x and y axes run from negative 6 to 6. The point (1, 3) is labeled. A dashed vertical line goes through the point and intersects the x-axis at xplus1. A dashed horizontal line goes through the point and intersects the y-axis at yplus3.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_003_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a point plotted on the x y-coordinate plane. The x and y axes run from negative 6 to 6. The point (1, 3) is labeled. A dashed vertical line goes through the point and intersects the x-axis at xplus1. A dashed horizontal line goes through the point and intersects the y-axis at yplus3.\" \/><\/span><\/div>\n<p id=\"fs-id1167835374762\">Notice that the vertical line through <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3330a01aa4d7d81947b71297d8623d3b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"42\" style=\"vertical-align: -1px;\" \/> and the horizontal line through <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8e36d35d8563f5053efd9935e88634f9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"42\" style=\"vertical-align: -4px;\" \/> are not part of the graph. We just used them to help us locate the point <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8abf8911b9dbc811d668c10a487f5e2e_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;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"45\" style=\"vertical-align: -4px;\" \/><\/p>\n<p id=\"fs-id1167832059451\">When one of the coordinate is zero, the point lies on one of the axes. In <a href=\"#CNX_IntAlg_Figure_03_01_004\" class=\"autogenerated-content\">(Figure)<\/a> the point <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e2c3a69d33f9737210f9c4f1551f4b9e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/> is on the <em data-effect=\"italics\">y<\/em>-axis and the point <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-dd6f8f312ab67ad0422a4959540654f3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"51\" style=\"vertical-align: -4px;\" \/> is on the <em data-effect=\"italics\">x<\/em>-axis.<\/p>\n<div class=\"bc-figure figure\" id=\"CNX_IntAlg_Figure_03_01_004\"><span data-type=\"media\" id=\"fs-id1167835512862\" data-alt=\"This figure shows points plotted on the x y-coordinate plane. The x and y axes run from negative 6 to 6. The point (negative 2, 0) is labeled and lies on the x-axis. The point (0, 4) is labeled and lies on the y-axis.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_004_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows points plotted on the x y-coordinate plane. The x and y axes run from negative 6 to 6. The point (negative 2, 0) is labeled and lies on the x-axis. The point (0, 4) is labeled and lies on the y-axis.\" \/><\/span><\/div>\n<div data-type=\"note\" id=\"fs-id1167835330675\">\n<div data-type=\"title\">Points on the Axes<\/div>\n<p id=\"fs-id1167834403364\">Points with a <em data-effect=\"italics\">y<\/em>-coordinate equal to 0 are on the <em data-effect=\"italics\">x<\/em>-axis, and have coordinates <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-efddd49bc1c2dc39c3fad2e3635247c3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"46\" style=\"vertical-align: -4px;\" \/><\/p>\n<p id=\"fs-id1167835167375\">Points with an <em data-effect=\"italics\">x<\/em>-coordinate equal to 0 are on the <em data-effect=\"italics\">y<\/em>-axis, and have coordinates <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5bac775607ac99b49b72fd1654a94604_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"44\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"example\" id=\"fs-id1167834190090\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167834423551\">\n<div data-type=\"problem\" id=\"fs-id1167835416428\">\n<p id=\"fs-id1167831893573\">Plot each point in the rectangular coordinate system and identify the quadrant in which the point is located:<\/p>\n<p id=\"fs-id1167831103917\"><span class=\"token\">\u24d0<\/span><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;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9523da542e5bfc81814e047c926984c5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#44;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"65\" style=\"vertical-align: -4px;\" \/><span class=\"token\">\u24d2<\/span><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;\" \/><span class=\"token\">\u24d3<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ec6084fa217f87f8fc5df481ca60ccf0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"52\" style=\"vertical-align: -4px;\" \/><span class=\"token\">\u24d4<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b149aa5c71585b037d14a3db2ed122d7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"49\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835336204\">\n<p id=\"fs-id1167835610006\">The first number of the coordinate pair is the <em data-effect=\"italics\">x<\/em>-coordinate, and the second number is the <em data-effect=\"italics\">y<\/em>-coordinate. To plot each point, sketch a vertical line through the <em data-effect=\"italics\">x<\/em>-coordinate and a horizontal line through the <em data-effect=\"italics\">y<\/em>-coordinate. Their intersection is the point.<\/p>\n<div data-type=\"newline\"><\/div>\n<p> <span class=\"token\">\u24d0<\/span> Since <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f0fdc1717d47916064f25e11eb18b433_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#45;&#53;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"61\" style=\"vertical-align: -4px;\" \/> the point is to the left of the <em data-effect=\"italics\">y<\/em>-axis. Also, since <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bbd8a7270526868629f3c91c3b0b40d6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#52;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"46\" style=\"vertical-align: -4px;\" \/> the point is above the <em data-effect=\"italics\">x<\/em>-axis. The point <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;\" \/> is in Quadrant II.<\/p>\n<div data-type=\"newline\"><\/div>\n<p> <span class=\"token\">\u24d1<\/span> Since <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-09bee320bf56e1f1abb3aeb986256f0a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#45;&#51;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"61\" style=\"vertical-align: -4px;\" \/> the point is to the left of the <em data-effect=\"italics\">y<\/em>-axis. Also, since <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1d5cc0600fefd9fdcd926f46d9373f93_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#45;&#52;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"60\" style=\"vertical-align: -4px;\" \/> the point is below the <em data-effect=\"italics\">x<\/em>-axis. The point <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9523da542e5bfc81814e047c926984c5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#44;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"65\" style=\"vertical-align: -4px;\" \/> is in Quadrant III.<\/p>\n<div data-type=\"newline\"><\/div>\n<p> <span class=\"token\">\u24d2<\/span> Since <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1e211b3d19a6898a4c9192f117c1fe08_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#50;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"47\" style=\"vertical-align: -4px;\" \/> the point is to the right of the <em data-effect=\"italics\">y<\/em>-axis. Since <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a6687e095156a09b747a5f30eb2beef8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#45;&#51;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"60\" style=\"vertical-align: -4px;\" \/> the point is below the <em data-effect=\"italics\">x<\/em>-axis. The point <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;\" \/> is in Quadrant IV.<\/p>\n<div data-type=\"newline\"><\/div>\n<p> <span class=\"token\">\u24d3<\/span> Since <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-534b43efb5ec72c9aa9f5eaccec09e9c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#48;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"47\" style=\"vertical-align: -4px;\" \/> the point whose coordinates are <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ec6084fa217f87f8fc5df481ca60ccf0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"52\" style=\"vertical-align: -4px;\" \/> is on the <em data-effect=\"italics\">y<\/em>-axis.<\/p>\n<div data-type=\"newline\"><\/div>\n<p> <span class=\"token\">\u24d4<\/span> Since <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c7d62a78a97a33b2bfba1ddcfe9dbb8c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#51;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"47\" style=\"vertical-align: -4px;\" \/> the point is to the right of the <em data-effect=\"italics\">y<\/em>-axis. Since <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-45f1a0aee972754b8b8e25cfd30e3cc7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#50;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"48\" style=\"vertical-align: -6px;\" \/> the point is above the <em data-effect=\"italics\">x<\/em>-axis. (It may be helpful to write <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3c8a34714cd9e14f96438eaca16625df_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"7\" style=\"vertical-align: -6px;\" \/> as a mixed number or decimal.) The point <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1c98a0c2caafdf0d2bb0cfb73e36a2a0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"40\" style=\"vertical-align: -7px;\" \/> is in Quadrant I.<\/p>\n<div data-type=\"newline\"><\/div>\n<p> <span data-type=\"media\" id=\"fs-id1167835356754\" data-alt=\"This figure shows points plotted on the x y-coordinate plane. The x and y axes run from negative 6 to 6. The following points are labeled: (3, 5 divided by 2), (negative 2, 3), negative 5, 4), (negative 3, negative 4), and (2, negative 3).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_005_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows points plotted on the x y-coordinate plane. The x and y axes run from negative 6 to 6. The following points are labeled: (3, 5 divided by 2), (negative 2, 3), negative 5, 4), (negative 3, negative 4), and (2, negative 3).\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167834495427\">\n<div data-type=\"problem\" id=\"fs-id1167835207148\">\n<p id=\"fs-id1167834402753\">Plot each point in a rectangular coordinate system and identify the quadrant in which the point is located:<\/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-e86393e45c0f6cf9bb7fcf130d3db9da_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#44;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"51\" 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-d29d505cec09336aa569e1cca8670699_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#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\">\u24d2<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8680e4f49775c6a5702e20f9a2e105a0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#44;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"52\" style=\"vertical-align: -4px;\" \/> <span class=\"token\">\u24d3<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6bd03bb09a34e49dd9f3ebf548e3dab1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#52;&#44;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"51\" style=\"vertical-align: -4px;\" \/> <span class=\"token\">\u24d4<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6b3e4a390b9705038aecd2ab7d812a5c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#52;&#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=\"53\" style=\"vertical-align: -7px;\" \/><\/div>\n<div data-type=\"solution\" id=\"fs-id1167835369019\"><span data-type=\"media\" id=\"fs-id1167831920613\" data-alt=\"This figure shows points plotted on the x y-coordinate plane. The x and y axes run from negative 6 to 6. The point labeled a is 2 units to the left of the origin and 1 unit above the origin and is located in quadrant II. The point labeled b is 3 units to the left of the origin and 1 unit below the origin and is located in quadrant III. The point labeled c is 4 units to the right of the origin and 4 units below the origin and is located in quadrant IV. The point labeled d is 4 units to the left of the origin and 4 units above the origin and is located in quadrant II. The point labeled e is 4 units to the left of the origin and 1 and a half units above the origin and is located in quadrant II.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_301_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows points plotted on the x y-coordinate plane. The x and y axes run from negative 6 to 6. The point labeled a is 2 units to the left of the origin and 1 unit above the origin and is located in quadrant II. The point labeled b is 3 units to the left of the origin and 1 unit below the origin and is located in quadrant III. The point labeled c is 4 units to the right of the origin and 4 units below the origin and is located in quadrant IV. The point labeled d is 4 units to the left of the origin and 4 units above the origin and is located in quadrant II. The point labeled e is 4 units to the left of the origin and 1 and a half units above the origin and is located in quadrant II.\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167834161888\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167831920690\">\n<div data-type=\"problem\" id=\"fs-id1167832068391\">\n<p id=\"fs-id1167831823636\">Plot each point in a rectangular coordinate system and identify the quadrant in which the point is located:<\/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-cba4706a21fad1e77985563696d1bdae_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#52;&#44;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"51\" 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-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;\" \/> <span class=\"token\">\u24d2<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c02ba61efe13be423bc75f83a9846930_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#44;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"52\" style=\"vertical-align: -4px;\" \/> <span class=\"token\">\u24d3<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-edd8d8b57c2815309edbd447803c95fa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#44;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"51\" style=\"vertical-align: -4px;\" \/> <span class=\"token\">\u24d4<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b30fc963d933def9eed175726f096ea7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"53\" style=\"vertical-align: -7px;\" \/><\/div>\n<div data-type=\"solution\"><span data-type=\"media\" data-alt=\"This figure shows points plotted on the x y-coordinate plane. The x and y axes run from negative 6 to 6. The point labeled a is 4 units to the left of the origin and 1 unit above the origin and is located in quadrant II. The point labeled b is 2 units to the left of the origin and 3 units above the origin and is located in quadrant II. The point labeled c is 2 units to the right of the origin and 5 units below the origin and is located in quadrant IV. The point labeled d is 2 units to the left of the origin and 5 units above the origin and is located in quadrant II. The point labeled e is 3 units to the left of the origin and 2 and a half units above the origin and is located in quadrant II.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_302_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows points plotted on the x y-coordinate plane. The x and y axes run from negative 6 to 6. The point labeled a is 4 units to the left of the origin and 1 unit above the origin and is located in quadrant II. The point labeled b is 2 units to the left of the origin and 3 units above the origin and is located in quadrant II. The point labeled c is 2 units to the right of the origin and 5 units below the origin and is located in quadrant IV. The point labeled d is 2 units to the left of the origin and 5 units above the origin and is located in quadrant II. The point labeled e is 3 units to the left of the origin and 2 and a half units above the origin and is located in quadrant II.\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<p>The signs of the <em data-effect=\"italics\">x<\/em>-coordinate and <em data-effect=\"italics\">y<\/em>-coordinate affect the location of the points. You may have noticed some patterns as you graphed the points in the previous example. We can summarize sign patterns of the quadrants in this way:<\/p>\n<div data-type=\"note\" id=\"fs-id1167832067452\">\n<div data-type=\"title\">Quadrants<\/div>\n<div data-type=\"equation\" id=\"fs-id1167835328203\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-dc6169aa2426e841e122e51405b42edc_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;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#81;&#117;&#97;&#100;&#114;&#97;&#110;&#116;&#32;&#73;&#125;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#81;&#117;&#97;&#100;&#114;&#97;&#110;&#116;&#32;&#73;&#73;&#125;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#81;&#117;&#97;&#100;&#114;&#97;&#110;&#116;&#32;&#73;&#73;&#73;&#125;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#81;&#117;&#97;&#100;&#114;&#97;&#110;&#116;&#32;&#73;&#86;&#125;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#44;&#121;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#44;&#121;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#44;&#121;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#44;&#121;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#108;&#101;&#102;&#116;&#40;&#43;&#44;&#43;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#44;&#43;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#44;&#45;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#43;&#44;&#45;&#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;\" title=\"Rendered by QuickLaTeX.com\" height=\"61\" width=\"535\" style=\"vertical-align: -26px;\" \/><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167832066567\" data-alt=\"This figure shows the x y-coordinate plane with the four quadrants labeled. In the top right of the plane is quadrant I labeled (plus, plus). In the top left of the plane is quadrant II labeled (minus, plus). In the bottom left of the plane is quadrant III labeled (minus, minus). In the bottom right of the plane is quadrant IV labeled (plus, minus).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_006_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows the x y-coordinate plane with the four quadrants labeled. In the top right of the plane is quadrant I labeled (plus, plus). In the top left of the plane is quadrant II labeled (minus, plus). In the bottom left of the plane is quadrant III labeled (minus, minus). In the bottom right of the plane is quadrant IV labeled (plus, minus).\" \/><\/span><\/div>\n<p id=\"fs-id1167835534159\">Up to now, all the equations you have solved were equations with just one variable. In almost every case, when you solved the equation you got exactly one solution. But equations can have more than one variable. Equations with two variables may be of the form <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f8784c7e5aaf20bf04cc889b8ea7b8b0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#65;&#120;&#43;&#66;&#121;&#61;&#67;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"109\" style=\"vertical-align: -4px;\" \/> An equation of this form is called a <span data-type=\"term\">linear equation<\/span> in two variables.<\/p>\n<div data-type=\"note\" id=\"fs-id1167835355041\">\n<div data-type=\"title\">Linear Equation<\/div>\n<p id=\"fs-id1167835305763\">An equation of the form <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8c3fc394aedf0a1f1b44fa47faf55523_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#65;&#120;&#43;&#66;&#121;&#61;&#67;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"109\" style=\"vertical-align: -4px;\" \/> where <em data-effect=\"italics\">A<\/em> and <em data-effect=\"italics\">B<\/em> are not both zero, is called a <strong data-effect=\"bold\">linear equation<\/strong> in two variables.<\/p>\n<\/div>\n<p id=\"fs-id1167835233457\">Here is an example of a linear equation in two variables, <em data-effect=\"italics\">x<\/em> and <em data-effect=\"italics\">y<\/em>.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167831869016\" data-alt=\"This figure shows the equation A x plus B y plus C. Below this is the equation x plus 4 y plus 8. Below this are the equations A plus 1, B plus 4, C plus 8. B and 4 are the same color in all the equations. C and 8 are the same color in all the equations.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_007_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows the equation A x plus B y plus C. Below this is the equation x plus 4 y plus 8. Below this are the equations A plus 1, B plus 4, C plus 8. B and 4 are the same color in all the equations. C and 8 are the same color in all the equations.\" \/><\/span><\/p>\n<p id=\"fs-id1167835308773\">The equation <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9ca7e6824e1b3783daf117b3e7f97530_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#45;&#51;&#120;&#43;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"95\" style=\"vertical-align: -4px;\" \/> is also a linear equation. But it does not appear to be in the form <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f8784c7e5aaf20bf04cc889b8ea7b8b0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#65;&#120;&#43;&#66;&#121;&#61;&#67;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"109\" style=\"vertical-align: -4px;\" \/> We can use the Addition Property of Equality and rewrite it in <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0520a31036e9a951aea74693c8b23cb1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#65;&#120;&#43;&#66;&#121;&#61;&#67;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"106\" style=\"vertical-align: -4px;\" \/> form.<\/p>\n<div data-type=\"equation\" id=\"fs-id1167835510205\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a65d595b8da4676a5b2e42d6deebadb8_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;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#121;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#45;&#51;&#120;&#43;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#65;&#100;&#100;&#32;&#116;&#111;&#32;&#98;&#111;&#116;&#104;&#32;&#115;&#105;&#100;&#101;&#115;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#121;&#43;&#51;&#120;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#51;&#120;&#43;&#53;&#43;&#51;&#120;&#92;&#104;&#102;&#105;&#108;&#108;&#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;&#104;&#102;&#105;&#108;&#108;&#32;&#121;&#43;&#51;&#120;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#85;&#115;&#101;&#32;&#116;&#104;&#101;&#32;&#67;&#111;&#109;&#109;&#117;&#116;&#97;&#116;&#105;&#118;&#101;&#32;&#80;&#114;&#111;&#112;&#101;&#114;&#116;&#121;&#32;&#116;&#111;&#32;&#112;&#117;&#116;&#32;&#105;&#116;&#32;&#105;&#110;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#92;&#32;&#65;&#120;&#43;&#66;&#121;&#61;&#67;&#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;&#102;&#111;&#114;&#109;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#51;&#120;&#43;&#121;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#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;\" title=\"Rendered by QuickLaTeX.com\" height=\"105\" width=\"572\" style=\"vertical-align: -48px;\" \/><\/div>\n<p id=\"fs-id1167834430875\">By rewriting <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9ca7e6824e1b3783daf117b3e7f97530_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#45;&#51;&#120;&#43;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"95\" style=\"vertical-align: -4px;\" \/> as <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-883e18013e6eb1ace8ecb1889da9a56b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#120;&#43;&#121;&#61;&#53;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"87\" style=\"vertical-align: -4px;\" \/> we can easily see that it is a linear equation in two variables because it is of the form <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f8784c7e5aaf20bf04cc889b8ea7b8b0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#65;&#120;&#43;&#66;&#121;&#61;&#67;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"109\" style=\"vertical-align: -4px;\" \/> When an equation is in the form <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8c3fc394aedf0a1f1b44fa47faf55523_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#65;&#120;&#43;&#66;&#121;&#61;&#67;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"109\" style=\"vertical-align: -4px;\" \/> we say it is in <span data-type=\"term\">standard form of a linear equation<\/span>.<\/p>\n<div data-type=\"note\" id=\"fs-id1167835214784\">\n<div data-type=\"title\">Standard Form of Linear Equation<\/div>\n<p id=\"fs-id1167835237077\">A linear equation is in <strong data-effect=\"bold\">standard form<\/strong> when it is written <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f8784c7e5aaf20bf04cc889b8ea7b8b0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#65;&#120;&#43;&#66;&#121;&#61;&#67;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"109\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<p id=\"fs-id1167835346035\">Most people prefer to have <em data-effect=\"italics\">A<\/em>, <em data-effect=\"italics\">B<\/em>, and <em data-effect=\"italics\">C<\/em> be integers and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c7c76516e02e800674d79e291782f797_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#65;&#92;&#103;&#101;&#32;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"46\" style=\"vertical-align: -3px;\" \/> when writing a linear equation in standard form, although it is not strictly necessary.<\/p>\n<p id=\"fs-id1167834423852\">Linear equations have infinitely many solutions. For every number that is substituted for <em data-effect=\"italics\">x<\/em> there is a corresponding <em data-effect=\"italics\">y<\/em> value. This pair of values is a <span data-type=\"term\">solution<\/span> to the linear equation and is represented by the 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;\" \/> When we substitute these values of <em data-effect=\"italics\">x<\/em> and <em data-effect=\"italics\">y<\/em> into the equation, the result is a true statement, because the value on the left side is equal to the value on the right side.<\/p>\n<div data-type=\"note\" id=\"fs-id1167835311076\">\n<div data-type=\"title\">Solution of a Linear Equation in Two Variables<\/div>\n<p id=\"fs-id1167834377117\">An ordered pair <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;\" \/> is a <strong data-effect=\"bold\">solution<\/strong> of the linear equation<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8c3fc394aedf0a1f1b44fa47faf55523_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#65;&#120;&#43;&#66;&#121;&#61;&#67;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"109\" style=\"vertical-align: -4px;\" \/> if the equation is a true statement when the <em data-effect=\"italics\">x<\/em>&#8211; and <em data-effect=\"italics\">y<\/em>-values of the ordered pair are substituted into the equation.<\/p>\n<\/div>\n<p id=\"fs-id1167834190603\">Linear equations have infinitely many solutions. We can plot these solutions in the rectangular coordinate system. The points will line up perfectly in a straight line. We connect the points with a straight line to get the graph of the equation. We put arrows on the ends of each side of the line to indicate that the line continues in both directions.<\/p>\n<p id=\"fs-id1167835362190\">A graph is a visual representation of all the solutions of the equation. It is an example of the saying, \u201cA picture is worth a thousand words.\u201d The line shows you <em data-effect=\"italics\">all<\/em> the solutions to that equation. Every point on the line is a solution of the equation. And, every solution of this equation is on this line. This line is called the graph of the equation. Points <em data-effect=\"italics\">not<\/em> on the line are not solutions!<\/p>\n<div data-type=\"note\" id=\"fs-id1167835309099\">\n<div data-type=\"title\">Graph of a Linear Equation<\/div>\n<p id=\"fs-id1167834279380\">The graph of a linear equation <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0520a31036e9a951aea74693c8b23cb1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#65;&#120;&#43;&#66;&#121;&#61;&#67;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"106\" style=\"vertical-align: -4px;\" \/> is a straight line.<\/p>\n<ul id=\"fs-id1167830898857\" data-bullet-style=\"bullet\">\n<li>Every point on the line is a solution of the equation.<\/li>\n<li>Every solution of this equation is a point on this line.<\/li>\n<\/ul>\n<\/div>\n<div data-type=\"example\" id=\"fs-id1167835400321\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167835190713\">\n<div data-type=\"problem\" id=\"fs-id1167835304682\">\n<p id=\"fs-id1167835274549\">The graph of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e09b4ae6d312635fcb8ec259a322e66d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#50;&#120;&#45;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"83\" style=\"vertical-align: -4px;\" \/> is shown.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167835421585\" data-alt=\"This figure shows a straight line graphed on the x y-coordinate plane. The x and y-axes run from negative 10 to 10. The line has arrows on both ends and goes through the points (negative 3, negative 9), (negative 2, negative 7), (negative 1, negative 5), (0, negative 3), (1, negative 1), (2, 1), (3, 3), (4, 5), (5, 7), and (6, 9). The line is labeled y plus 2 x minus 3.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_008_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a straight line graphed on the x y-coordinate plane. The x and y-axes run from negative 10 to 10. The line has arrows on both ends and goes through the points (negative 3, negative 9), (negative 2, negative 7), (negative 1, negative 5), (0, negative 3), (1, negative 1), (2, 1), (3, 3), (4, 5), (5, 7), and (6, 9). The line is labeled y plus 2 x minus 3.\" \/><\/span><\/p>\n<p id=\"fs-id1167835379641\">For each ordered pair, decide:<\/p>\n<p id=\"fs-id1167831888160\"><span class=\"token\">\u24d0<\/span> Is the ordered pair a solution to the equation?<\/p>\n<p id=\"fs-id1167835213648\"><span class=\"token\">\u24d1<\/span> Is the point on the line?<\/p>\n<p id=\"fs-id1167831909258\">A: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-017f20ea0c6fda3470cedb20ea0b5537_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"52\" style=\"vertical-align: -4px;\" \/> B: <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;\" \/> C: <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;\" \/> D: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d66960a9b4adfa7702388a061d743cbd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#44;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"65\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834188708\">\n<p id=\"fs-id1167831893488\">Substitute the <em data-effect=\"italics\">x<\/em>&#8211; and <em data-effect=\"italics\">y<\/em>-values into the equation to check if the ordered pair is a solution to the equation.<\/p>\n<div data-type=\"newline\"><\/div>\n<div data-type=\"newline\"><\/div>\n<p> <span class=\"token\">\u24d0<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p> <span data-type=\"media\" id=\"fs-id1167835210264\" data-alt=\"Example A shows the ordered pair (0, negative 3). Under this is the equation y plus 2 x minus 3. Under this is the equation negative 3 equals 2 times 0 minus 3. The negative 3 and 0 are colored the same as the negative 3 and 0 in the ordered pair at the top. There is a question mark above the plus sign. Below this is the equation negative 3 plus negative 3. Below this is the statement (0, negative 3) is a solution. Example B shows the ordered pair (3, 3). Under this is the equation y plus 2 x minus 3. Under this is the equation 3 equals 2 times 3 minus 3. The 3 and 3 are colored the same as the 3 and 3 in the ordered pair at the top. There is a question mark above the plus sign. Below this is the equation 3 plus 3. Below this is the statement (3, 3) is a solution. Example C shows the ordered pair (2, negative 3). Under this is the equation y plus 2 x minus 3. Under this is the equation negative 3 equals 2 times 2 minus 3. The negative 3 and 2 are colored the same as the negative 3 and 2 in the ordered pair at the top. There is a question mark above the plus sign. Below this is the inequality negative 3 is not equal to 1. Below this is the statement (2, negative 3) is not a solution. Example D shows the ordered pair (negative 1, negative 5). Under this is the equation y plus 2 x minus 3. Under this is the equation negative 5 equals 2 times negative 1 minus 3. The negative 1 and negative 5 are colored the same as the negative 1 and negative 5 in the ordered pair at the top. There is a question mark above the plus sign. Below this is the equation negative 5 plus negative 5. Below this is the statement (negative 1, negative 5) is a solution.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_009_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Example A shows the ordered pair (0, negative 3). Under this is the equation y plus 2 x minus 3. Under this is the equation negative 3 equals 2 times 0 minus 3. The negative 3 and 0 are colored the same as the negative 3 and 0 in the ordered pair at the top. There is a question mark above the plus sign. Below this is the equation negative 3 plus negative 3. Below this is the statement (0, negative 3) is a solution. Example B shows the ordered pair (3, 3). Under this is the equation y plus 2 x minus 3. Under this is the equation 3 equals 2 times 3 minus 3. The 3 and 3 are colored the same as the 3 and 3 in the ordered pair at the top. There is a question mark above the plus sign. Below this is the equation 3 plus 3. Below this is the statement (3, 3) is a solution. Example C shows the ordered pair (2, negative 3). Under this is the equation y plus 2 x minus 3. Under this is the equation negative 3 equals 2 times 2 minus 3. The negative 3 and 2 are colored the same as the negative 3 and 2 in the ordered pair at the top. There is a question mark above the plus sign. Below this is the inequality negative 3 is not equal to 1. Below this is the statement (2, negative 3) is not a solution. Example D shows the ordered pair (negative 1, negative 5). Under this is the equation y plus 2 x minus 3. Under this is the equation negative 5 equals 2 times negative 1 minus 3. The negative 1 and negative 5 are colored the same as the negative 1 and negative 5 in the ordered pair at the top. There is a question mark above the plus sign. Below this is the equation negative 5 plus negative 5. Below this is the statement (negative 1, negative 5) is a solution.\" \/><\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span> Plot the points <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3228e365cc5c34a8d1ac63acea4d0408_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"59\" style=\"vertical-align: -4px;\" \/><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-25d3073b0181d08c030a89f785a94281_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;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"45\" style=\"vertical-align: -4px;\" \/><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e9814b89722ed25612f3d4f22a746753_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;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"59\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-af443c2890d3eeb2456fc94734bd46cc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#44;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"73\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p> <span data-type=\"media\" id=\"fs-id1167834061635\" data-alt=\"This figure shows the graph of the linear equation y plus 2 x minus 3 and some points graphed on the x y-coordinate plane. The x and y-axes run from negative 10 to 10. The line has arrows on both ends and goes through the points (negative 1, negative 5), (0, negative 3), and (3, 3). The point (2, negative 3) is also plotted but not on the line.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_010_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows the graph of the linear equation y plus 2 x minus 3 and some points graphed on the x y-coordinate plane. The x and y-axes run from negative 10 to 10. The line has arrows on both ends and goes through the points (negative 1, negative 5), (0, negative 3), and (3, 3). The point (2, negative 3) is also plotted but not on the line.\" \/><\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p> The points <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-38bcc27d53dc5dea57f6bfec48ff1d7a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"45\" style=\"vertical-align: -4px;\" \/><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ef38aa0a60a619389bfd175e0ea2083c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#44;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"59\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d66960a9b4adfa7702388a061d743cbd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#44;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"65\" style=\"vertical-align: -4px;\" \/> are on the line <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fa9ece86fa22640223cce98a0c3eb364_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#50;&#120;&#45;&#51;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"87\" style=\"vertical-align: -4px;\" \/> and the point <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;\" \/> is not on the line.<\/p>\n<div data-type=\"newline\"><\/div>\n<p> The points that are solutions to <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e09b4ae6d312635fcb8ec259a322e66d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#50;&#120;&#45;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"83\" style=\"vertical-align: -4px;\" \/> are on the line, but the point that is not a solution is not on the line.<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167835353963\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167835529924\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167835337195\">Use graph of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4f5d4f96fca8517ea0cc90e71270f2ef_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#51;&#120;&#45;&#49;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"87\" style=\"vertical-align: -4px;\" \/> For each ordered pair, decide:<\/p>\n<p id=\"fs-id1167834314579\"><span class=\"token\">\u24d0<\/span> Is the ordered pair a solution to the equation?<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span> Is the point on the line?<\/p>\n<p id=\"fs-id1167835320293\">A <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ec6084fa217f87f8fc5df481ca60ccf0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"52\" style=\"vertical-align: -4px;\" \/> B <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2da1eb750fc283f55cb9396d5536b47a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#44;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/><\/p>\n<p><span data-type=\"media\" id=\"fs-id1167834156869\" data-alt=\"This figure shows a straight line graphed on the x y-coordinate plane. The x and y-axes run from negative 10 to 10. The line has arrows on both ends and goes through the points (negative 3, negative 10), (negative 2, negative 7), (negative 1, negative 4), (0, negative 1), (1, 2), (2, 5), and (3, 8). The line is labeled y plus 3 x minus 1.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_011_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a straight line graphed on the x y-coordinate plane. The x and y-axes run from negative 10 to 10. The line has arrows on both ends and goes through the points (negative 3, negative 10), (negative 2, negative 7), (negative 1, negative 4), (0, negative 1), (1, 2), (2, 5), and (3, 8). The line is labeled y plus 3 x minus 1.\" \/><\/span><\/div>\n<div data-type=\"solution\" id=\"fs-id1167835239222\">\n<p id=\"fs-id1167835203046\"><span class=\"token\">\u24d0<\/span> yes, yes <span class=\"token\">\u24d1<\/span> yes, yes<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167828447116\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167835364020\">\n<div data-type=\"problem\" id=\"fs-id1167835253741\">\n<p id=\"fs-id1167835191408\">Use graph of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4f5d4f96fca8517ea0cc90e71270f2ef_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#51;&#120;&#45;&#49;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"87\" style=\"vertical-align: -4px;\" \/> For each ordered pair, decide:<\/p>\n<p><span class=\"token\">\u24d0<\/span> Is the ordered pair a solution to the equation?<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span> Is the point on the line?<\/p>\n<p id=\"fs-id1167830962082\">A<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0d66a71b8940b998e4f29f8cccda06d4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#44;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"52\" style=\"vertical-align: -4px;\" \/> B<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8235f4c83db17d3a3454713a44752b19_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#44;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"65\" style=\"vertical-align: -4px;\" \/><\/p>\n<p><span data-type=\"media\" id=\"fs-id1167835174029\" data-alt=\"This figure shows a straight line graphed on the x y-coordinate plane. The x and y-axes run from negative 10 to 10. The line has arrows on both ends and goes through the points (negative 3, negative 10), (negative 2, negative 7), (negative 1, negative 4), (0, negative 1), (1, 2), (2, 5), and (3, 8). The line is labeled y plus 3 x minus 1.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_012_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a straight line graphed on the x y-coordinate plane. The x and y-axes run from negative 10 to 10. The line has arrows on both ends and goes through the points (negative 3, negative 10), (negative 2, negative 7), (negative 1, negative 4), (0, negative 1), (1, 2), (2, 5), and (3, 8). The line is labeled y plus 3 x minus 1.\" \/><\/span><\/div>\n<div data-type=\"solution\" id=\"fs-id1167831910982\">\n<p id=\"fs-id1167832051834\"><span class=\"token\">\u24d0<\/span> no, no <span class=\"token\">\u24d1<\/span> yes, yes<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167835417087\">\n<h3 data-type=\"title\">Graph a Linear Equation by Plotting Points<\/h3>\n<p id=\"fs-id1167835421131\">There are several methods that can be used to graph a linear equation. The first method we will use is called plotting points, or the Point-Plotting Method. We find three points whose coordinates are solutions to the equation and then plot them in a rectangular coordinate system. By connecting these points in a line, we have the graph of the linear equation.<\/p>\n<div data-type=\"example\" id=\"fs-id1167832065594\" class=\"textbox textbox--examples\">\n<div data-type=\"title\">How to Graph a Linear Equation by Plotting Points<\/div>\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1167827957268\">\n<p id=\"fs-id1167834422792\">Graph the equation <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8af503cd473a72e30e30c2d46c7575ff_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#50;&#120;&#43;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"82\" style=\"vertical-align: -4px;\" \/> by plotting points.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835229769\"><span data-type=\"media\" id=\"fs-id1167830960982\" data-alt=\"Step 1 is to Find three points whose coordinates are solutions to the equation. You can choose any values for x or y. In this case since y is isolated on the left side of the equations, it is easier to choose values for x. Choosing x plus 0. We substitute this into the equation y plus 2 x plus 1 to get y plus 2 times 0 plus 1. This simplifies to y plus 0 plus 1. So y plus 1. Choosing x plus 1. We substitute this into the equation y plus 2 x plus 1 to get y plus 2 times 1 plus 1. This simplifies to y plus 2 plus 1. So y plus 3. Choosing x plus negative 2. We substitute this into the equation y plus 2 x plus 1 to get y plus 2 times negative 2 plus 1. This simplifies to y plus negative 4 plus 1. The y plus negative 3. Next we want to organize the solutions in a table. For this problem we will put the three solutions we just found in a table. The table has 5 rows and 3 columns. The first row is a title row with the equation y plus 2 x plus 1. The second row is a header row with the headers x, y, and (x, y). The third row has the numbers 0, 1, and (0, 1). The fourth row has the numbers 1, 3, and (1, 3). The fifth row has the numbers negative 2, negative 3, and (negative 2, negative 3).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_013a_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 1 is to Find three points whose coordinates are solutions to the equation. You can choose any values for x or y. In this case since y is isolated on the left side of the equations, it is easier to choose values for x. Choosing x plus 0. We substitute this into the equation y plus 2 x plus 1 to get y plus 2 times 0 plus 1. This simplifies to y plus 0 plus 1. So y plus 1. Choosing x plus 1. We substitute this into the equation y plus 2 x plus 1 to get y plus 2 times 1 plus 1. This simplifies to y plus 2 plus 1. So y plus 3. Choosing x plus negative 2. We substitute this into the equation y plus 2 x plus 1 to get y plus 2 times negative 2 plus 1. This simplifies to y plus negative 4 plus 1. The y plus negative 3. Next we want to organize the solutions in a table. For this problem we will put the three solutions we just found in a table. The table has 5 rows and 3 columns. The first row is a title row with the equation y plus 2 x plus 1. The second row is a header row with the headers x, y, and (x, y). The third row has the numbers 0, 1, and (0, 1). The fourth row has the numbers 1, 3, and (1, 3). The fifth row has the numbers negative 2, negative 3, and (negative 2, negative 3).\" \/><\/span><span data-type=\"media\" id=\"fs-id1167830698562\" data-alt=\"Step 2 is to plot the points in a rectangular coordinate system. Plot: (0, 1), (1, 3), (negative 2, negative 3). The figure then shows a graph of some points plotted on the x y-coordinate plane. The x and y axes run from negative 6 to 6. The points (0, 1), (1, 3), and (negative 2, negative 3) are plotted. Check that the points line up. If they do not, carefully check your work! Do the point line up? Yes, the points in this example line up.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_013b_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 2 is to plot the points in a rectangular coordinate system. Plot: (0, 1), (1, 3), (negative 2, negative 3). The figure then shows a graph of some points plotted on the x y-coordinate plane. The x and y axes run from negative 6 to 6. The points (0, 1), (1, 3), and (negative 2, negative 3) are plotted. Check that the points line up. If they do not, carefully check your work! Do the point line up? Yes, the points in this example line up.\" \/><\/span><span data-type=\"media\" id=\"fs-id1167835180910\" data-alt=\"Step 3 is to draw the line through the three points. Extend the line to fill the grid and put arrows on both ends of the line. This line is the graph of y plus 2 x plus 1. The figure shows the graph of a straight line on the x y-coordinate plane. The x and y axes run from negative 6 to 6. The points (negative 2, negative 3), (0, 1), and (1, 3) are plotted. The straight line goes through the three points and has arrows on both ends.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_013c_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 3 is to draw the line through the three points. Extend the line to fill the grid and put arrows on both ends of the line. This line is the graph of y plus 2 x plus 1. The figure shows the graph of a straight line on the x y-coordinate plane. The x and y axes run from negative 6 to 6. The points (negative 2, negative 3), (0, 1), and (1, 3) are plotted. The straight line goes through the three points and has arrows on both ends.\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167835623532\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167835348900\">\n<div data-type=\"problem\" id=\"fs-id1167834479634\">\n<p>Graph the equation by plotting points: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fc617a1e7c2cc45ad6eac438f484cab7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#50;&#120;&#45;&#51;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"87\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167830699920\">\n<p id=\"fs-id1167835279909\">\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167835420948\" data-alt=\"This figure shows a straight line graphed on the x y-coordinate plane. The x and y-axes run from negative 8 to 8. The line goes through the points (negative 2, negative 7), (negative 1, negative 5), (0, negative 3), (1, negative 1), (2, 1), (3, 3), (4, 5), and (5, 7).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_303_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a straight line graphed on the x y-coordinate plane. The x and y-axes run from negative 8 to 8. The line goes through the points (negative 2, negative 7), (negative 1, negative 5), (0, negative 3), (1, negative 1), (2, 1), (3, 3), (4, 5), and (5, 7).\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167835170817\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167831847014\">\n<div data-type=\"problem\" id=\"fs-id1167832059861\">\n<p id=\"fs-id1167835350848\">Graph the equation by plotting points: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2f424cf5ea35bb809040c0ecf2943693_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#45;&#50;&#120;&#43;&#52;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"100\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835358649\">\n<p id=\"fs-id1167835193486\">\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167835329515\" data-alt=\"This figure shows a straight line graphed on the x y-coordinate plane. The x and y-axes run from negative 8 to 8. The line goes through the points (negative 2, 8), (negative 1, 6), (0, 4), (1, 2), (2, 0), (3, negative 2), (4, negative 4), (5, negative 6) and (6, negative 8).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_304_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a straight line graphed on the x y-coordinate plane. The x and y-axes run from negative 8 to 8. The line goes through the points (negative 2, 8), (negative 1, 6), (0, 4), (1, 2), (2, 0), (3, negative 2), (4, negative 4), (5, negative 6) and (6, negative 8).\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167835382110\">The steps to take when graphing a linear equation by plotting points are summarized here.<\/p>\n<div data-type=\"note\" id=\"fs-id1167835350108\" class=\"howto\">\n<div data-type=\"title\">Graph a linear equation by plotting points.<\/div>\n<ol id=\"fs-id1167835305295\" type=\"1\" class=\"stepwise\">\n<li>Find three points whose coordinates are solutions to the equation. Organize them in a table.<\/li>\n<li>Plot the points in a rectangular coordinate system. Check that the points line up. If they do not, carefully check your work.<\/li>\n<li>Draw the line through the three points. Extend the line to fill the grid and put arrows on both ends of the line.<\/li>\n<\/ol>\n<\/div>\n<p>It is true that it only takes two points to determine a line, but it is a good habit to use three points. If you only plot two points and one of them is incorrect, you can still draw a line but it will not represent the solutions to the equation. It will be the wrong line.<\/p>\n<p id=\"fs-id1167834531459\">If you use three points, and one is incorrect, the points will not line up. This tells you something is wrong and you need to check your work. Look at the difference between these illustrations.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167834111513\" data-alt=\"The figure shows two images. In the first image there are three points with a straight line going through all three. In the second image there are three points that do not all lie on a straight line.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_014_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows two images. In the first image there are three points with a straight line going through all three. In the second image there are three points that do not all lie on a straight line.\" \/><\/span><\/p>\n<p id=\"fs-id1167826828398\">When an equation includes a fraction as the coefficient of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ede05c264bba0eda080918aaa09c4658_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\" \/>, we can still substitute any numbers for <em data-effect=\"italics\">x<\/em>. But the arithmetic is easier if we make \u201cgood\u201d choices for the values of <em data-effect=\"italics\">x<\/em>. This way we will avoid fractional answers, which are hard to graph precisely.<\/p>\n<div data-type=\"example\" id=\"fs-id1167834408393\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167835232887\">\n<div data-type=\"problem\" id=\"fs-id1167835214095\">\n<p id=\"fs-id1167830914798\">Graph the equation: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9a7f08fe0dbb847539e1af2a481be0aa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#120;&#43;&#51;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"89\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167832128684\">\n<p id=\"fs-id1167832055099\">Find three points that are solutions to the equation. Since this equation has the fraction <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b8fa03e1b526c6d07ec843385490ca4c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"7\" style=\"vertical-align: -6px;\" \/> as a coefficient of <em data-effect=\"italics\">x<\/em>, we will choose values of <em data-effect=\"italics\">x<\/em> carefully. We will use zero as one choice and multiples of 2 for the other choices. Why are multiples of two a good choice for values of <em data-effect=\"italics\">x<\/em>? By choosing multiples of 2 the multiplication by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b8fa03e1b526c6d07ec843385490ca4c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"7\" style=\"vertical-align: -6px;\" \/> simplifies to a whole number<\/p>\n<div data-type=\"newline\"><\/div>\n<p> <span data-type=\"media\" id=\"fs-id1167831023926\" data-alt=\"The first set of equations starts with x plus 0. Under this is the equation y plus 1 half x plus 3. Under this is the equation y plus 1 half times 0 plus 3. Below this is the equation y plus 0 plus 3. Below this is the equation y plus 3. The second set of equations starts with x plus 2. Under this is the equation y plus 1 half x plus 3. Under this is the equation y plus 1 half times 2 plus 3. Below this is the equation y plus 1 plus 3. Below this is the equation y plus 4. The third set of equations starts with x plus 4. Under this is the equation y plus 1 half x plus 3. Under this is the equation y plus 1 half times 4 plus 3. Below this is the equation y plus 2 plus 3. Below this is the equation y plus 5.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_015_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The first set of equations starts with x plus 0. Under this is the equation y plus 1 half x plus 3. Under this is the equation y plus 1 half times 0 plus 3. Below this is the equation y plus 0 plus 3. Below this is the equation y plus 3. The second set of equations starts with x plus 2. Under this is the equation y plus 1 half x plus 3. Under this is the equation y plus 1 half times 2 plus 3. Below this is the equation y plus 1 plus 3. Below this is the equation y plus 4. The third set of equations starts with x plus 4. Under this is the equation y plus 1 half x plus 3. Under this is the equation y plus 1 half times 4 plus 3. Below this is the equation y plus 2 plus 3. Below this is the equation y plus 5.\" \/><\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p> The points are shown in <a href=\"#fs-id1167835202852\" class=\"autogenerated-content\">(Figure)<\/a>.<\/p>\n<div data-type=\"newline\"><\/div>\n<table id=\"fs-id1167835202852\" summary=\"The table has 5 rows and 3 columns. The first row is a title row with the equation y plus 1 half x plus 3. The second row is a header row with the headers x, y, and (x, y). The third row has the numbers 0, 3, and (0, 3). The fourth row has the numbers 2, 4, and (2, 4). The fifth row has the numbers 4, 5, and (4, 5).\">\n<tbody>\n<tr valign=\"top\">\n<td colspan=\"3\" data-valign=\"middle\" data-align=\"center\"><strong data-effect=\"bold\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4e2ae707ab5eccd1febc4c89a755aa4a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#120;&#43;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"85\" style=\"vertical-align: -6px;\" \/><\/strong><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"middle\" data-align=\"center\"><strong data-effect=\"bold\"><em data-effect=\"italics\">x<\/em><\/strong><\/td>\n<td data-valign=\"middle\" data-align=\"center\"><strong data-effect=\"bold\"><em data-effect=\"italics\">y<\/em><\/strong><\/td>\n<td data-valign=\"middle\" data-align=\"center\"><strong data-effect=\"bold\"><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;\" \/><\/strong><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"middle\" data-align=\"center\">0<\/td>\n<td data-valign=\"middle\" data-align=\"center\">3<\/td>\n<td data-valign=\"middle\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ef24d382319a3ce81f280194edba003a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"middle\" data-align=\"center\">2<\/td>\n<td data-valign=\"middle\" data-align=\"center\">4<\/td>\n<td data-valign=\"middle\" data-align=\"center\"><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;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"middle\" data-align=\"center\">4<\/td>\n<td data-valign=\"middle\" data-align=\"center\">5<\/td>\n<td data-valign=\"middle\" data-align=\"center\"><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;\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div data-type=\"newline\"><\/div>\n<p> Plot the points, check that they line up, and draw the line.<\/p>\n<div data-type=\"newline\"><\/div>\n<p> <span data-type=\"media\" id=\"fs-id1167835489362\" data-alt=\"The figure shows the graph of a straight line on the x y-coordinate plane. The x and y axes run from negative 7 to 7. The points (0, 3), (2, 4), and (4, 5) are plotted. The straight line goes through the three points and has arrows on both ends. The line is labeled y plus 1 divided by 2 times x plus 3.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_016_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows the graph of a straight line on the x y-coordinate plane. The x and y axes run from negative 7 to 7. The points (0, 3), (2, 4), and (4, 5) are plotted. The straight line goes through the three points and has arrows on both ends. The line is labeled y plus 1 divided by 2 times x plus 3.\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167834194734\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167830837122\">\n<div data-type=\"problem\" id=\"fs-id1167826778512\">\n<p id=\"fs-id1167835595642\">Graph the equation: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-129bc9b21302a72cefb62e5d1ea9a056_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#120;&#45;&#49;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"89\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167830757662\">\n<p id=\"fs-id1167834111766\">\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167835350606\" data-alt=\"This figure shows a straight line graphed on the x y-coordinate plane. The x and y-axes run from negative 12 to 12. The line goes through the points (negative 12, negative 5), (negative 9, negative 4), (negative 6, negative 3), (negative 3, negative 2), (0, negative 1), (3, 0), (6, 1), (9, 2), and (12, 3).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_305_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a straight line graphed on the x y-coordinate plane. The x and y-axes run from negative 12 to 12. The line goes through the points (negative 12, negative 5), (negative 9, negative 4), (negative 6, negative 3), (negative 3, negative 2), (0, negative 1), (3, 0), (6, 1), (9, 2), and (12, 3).\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167834459155\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167826994459\">\n<div data-type=\"problem\" id=\"fs-id1167835345869\">\n<p id=\"fs-id1167835259316\">Graph the equation: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7607c36d0c87d0ff77197c1bc14700ee_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#52;&#125;&#120;&#43;&#50;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"89\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834439103\">\n<p id=\"fs-id1167832074220\">\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" data-alt=\"This figure shows a straight line graphed on the x y-coordinate plane. The x and y-axes run from negative 12 to 12. The line goes through the points (negative 12, negative 1), (negative 8, 0), (negative 4, 1), (0, 2), (4, 3), (8, 4), and (12, 5).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_306_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a straight line graphed on the x y-coordinate plane. The x and y-axes run from negative 12 to 12. The line goes through the points (negative 12, negative 1), (negative 8, 0), (negative 4, 1), (0, 2), (4, 3), (8, 4), and (12, 5).\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\">\n<h3 data-type=\"title\">Graph Vertical and Horizontal Lines<\/h3>\n<p id=\"fs-id1167835340147\">Some linear equations have only one variable. They may have just <em data-effect=\"italics\">x<\/em> and no <em data-effect=\"italics\">y<\/em>, or just <em data-effect=\"italics\">y<\/em> without an <em data-effect=\"italics\">x<\/em>. This changes how we make a table of values to get the points to plot.<\/p>\n<p id=\"fs-id1167835519235\">Let\u2019s consider the equation <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-18062540cd799901f80ebaea09891a13_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#45;&#51;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"61\" style=\"vertical-align: 0px;\" \/> This equation has only one variable, <em data-effect=\"italics\">x<\/em>. The equation says that <em data-effect=\"italics\">x<\/em> is <em data-effect=\"italics\">always<\/em> equal to<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8f54336ab5f34f79c0d075fa22f60ec5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#51;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"26\" style=\"vertical-align: -4px;\" \/> so its value does not depend on <em data-effect=\"italics\">y<\/em>. No matter what is the value of <em data-effect=\"italics\">y<\/em>, the value of <em data-effect=\"italics\">x<\/em> is always <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cfa36400e576c82fd4847d2da37b1d56_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#51;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"26\" style=\"vertical-align: 0px;\" \/><\/p>\n<p id=\"fs-id1167831191426\">So to make a table of values, write <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-470cb162cf92c55d139f4f69216225e5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"22\" style=\"vertical-align: 0px;\" \/> in for all the <em data-effect=\"italics\">x<\/em>-values. Then choose any values for <em data-effect=\"italics\">y<\/em>. Since <em data-effect=\"italics\">x<\/em> does not depend on <em data-effect=\"italics\">y<\/em>, you can choose any numbers you like. But to fit the points on our coordinate graph, we\u2019ll use 1, 2, and 3 for the <em data-effect=\"italics\">y<\/em>-coordinates. See <a href=\"#fs-id1167831922183\" class=\"autogenerated-content\">(Figure)<\/a>.<\/p>\n<table id=\"fs-id1167831922183\" summary=\"The table has 5 rows and 3 columns. The first row is a title row with the equation x plus negative 3. The second row is a header row with the headers x, y, and (x, y). The third row has the numbers negative 3, 1, and (negative 3, 1). The fourth row has the numbers negative 3, 2, and (negative 3, 2). The fifth row has the numbers negative 3, 3, and (negative 3, 3).\">\n<tbody>\n<tr valign=\"top\">\n<td colspan=\"3\" data-valign=\"middle\" data-align=\"center\"><strong data-effect=\"bold\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7e135cd6350a4c21195c621240f7aee7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#45;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"57\" style=\"vertical-align: 0px;\" \/><\/strong><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"middle\" data-align=\"center\"><strong data-effect=\"bold\"><em data-effect=\"italics\">x<\/em><\/strong><\/td>\n<td data-valign=\"middle\" data-align=\"center\"><strong data-effect=\"bold\"><em data-effect=\"italics\">y<\/em><\/strong><\/td>\n<td data-valign=\"middle\" data-align=\"center\"><strong data-effect=\"bold\"><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;\" \/><\/strong><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"middle\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-470cb162cf92c55d139f4f69216225e5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"22\" style=\"vertical-align: 0px;\" \/><\/td>\n<td data-valign=\"middle\" data-align=\"center\">1<\/td>\n<td data-valign=\"middle\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0e0819cfb987fa92d333add15bb2e864_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#44;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"51\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"middle\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-470cb162cf92c55d139f4f69216225e5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"22\" style=\"vertical-align: 0px;\" \/><\/td>\n<td data-valign=\"middle\" data-align=\"center\">2<\/td>\n<td data-valign=\"middle\" data-align=\"center\"><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;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"middle\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-470cb162cf92c55d139f4f69216225e5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"22\" style=\"vertical-align: 0px;\" \/><\/td>\n<td data-valign=\"middle\" data-align=\"center\">3<\/td>\n<td data-valign=\"middle\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-03125d47f1e456b2a5c3479daa28f59c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#44;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"51\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1167826996487\">Plot the points from the table and connect them with a straight line. Notice that we have graphed a <span data-type=\"term\">vertical line<\/span>.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167834327439\" data-alt=\"The figure shows the graph of a straight vertical line on the x y-coordinate plane. The x and y axes run from negative 7 to 7. The points (negative 3, 1), (negative 3, 2), and (negative 3, 3) are plotted. The line goes through the three points and has arrows on both ends. The line is labeled x plus negative 3.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_017_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows the graph of a straight vertical line on the x y-coordinate plane. The x and y axes run from negative 7 to 7. The points (negative 3, 1), (negative 3, 2), and (negative 3, 3) are plotted. The line goes through the three points and has arrows on both ends. The line is labeled x plus negative 3.\" \/><\/span><\/p>\n<p id=\"fs-id1167834377262\">What if the equation has <em data-effect=\"italics\">y<\/em> but no <em data-effect=\"italics\">x<\/em>? Let\u2019s graph the equation <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-270d2ebaae94f65bccc2c78eddebf555_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#52;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"46\" style=\"vertical-align: -4px;\" \/> This time the <em data-effect=\"italics\">y-<\/em>value is a constant, so in this equation, <em data-effect=\"italics\">y<\/em> does not depend on <em data-effect=\"italics\">x<\/em>. Fill in 4 for all the <em data-effect=\"italics\">y<\/em>\u2019s in <a href=\"#fs-id1167835382105\" class=\"autogenerated-content\">(Figure)<\/a> and then choose any values for <em data-effect=\"italics\">x<\/em>. We\u2019ll use 0, 2, and 4 for the <em data-effect=\"italics\">x<\/em>-coordinates.<\/p>\n<table id=\"fs-id1167835382105\" summary=\"The table has 5 rows and 3 columns. The first row is a title row with the equation y plus 4. The second row is a header row with the headers x, y, and (x, y). The third row has the numbers 0, 4, and (0, 4). The fourth row has the numbers 2, 4, and (2, 4). The fifth row has the numbers 4, 4, and (4, 4).\">\n<tbody>\n<tr valign=\"top\">\n<td colspan=\"3\" data-valign=\"middle\" data-align=\"center\"><strong data-effect=\"bold\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0621c4f761e7864714642fcc62d4c42f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"42\" style=\"vertical-align: -4px;\" \/><\/strong><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"middle\" data-align=\"center\"><strong data-effect=\"bold\"><em data-effect=\"italics\">x<\/em><\/strong><\/td>\n<td data-valign=\"middle\" data-align=\"center\"><strong data-effect=\"bold\"><em data-effect=\"italics\">y<\/em><\/strong><\/td>\n<td data-valign=\"middle\" data-align=\"center\"><strong data-effect=\"bold\"><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;\" \/><\/strong><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"middle\" data-align=\"center\">0<\/td>\n<td data-valign=\"middle\" data-align=\"center\">4<\/td>\n<td data-valign=\"middle\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e2c3a69d33f9737210f9c4f1551f4b9e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"middle\" data-align=\"center\">2<\/td>\n<td data-valign=\"middle\" data-align=\"center\">4<\/td>\n<td data-valign=\"middle\" data-align=\"center\"><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;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"middle\" data-align=\"center\">4<\/td>\n<td data-valign=\"middle\" data-align=\"center\">4<\/td>\n<td data-valign=\"middle\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9f071f7020da53c225631b96b8f9875e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#44;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1167831970006\">In this figure, we have graphed a <span data-type=\"term\">horizontal line<\/span> passing through the <em data-effect=\"italics\">y<\/em>-axis at 4.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167832066046\" data-alt=\"The figure shows the graph of a straight horizontal line on the x y-coordinate plane. The x and y axes run from negative 7 to 7. The points (0, 4), (2, 4), and (4, 4) are plotted. The line goes through the three points and has arrows on both ends. The line is labeled y plus 4.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_018_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows the graph of a straight horizontal line on the x y-coordinate plane. The x and y axes run from negative 7 to 7. The points (0, 4), (2, 4), and (4, 4) are plotted. The line goes through the three points and has arrows on both ends. The line is labeled y plus 4.\" \/><\/span><\/p>\n<div data-type=\"note\" id=\"fs-id1167832075518\">\n<div data-type=\"title\">Vertical and Horizontal Lines<\/div>\n<p id=\"fs-id1167832152868\">A <strong data-effect=\"bold\">vertical line<\/strong> is the graph of an equation of the form <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5c2ae728b9a161e044125a0dc6c913c3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#97;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"47\" style=\"vertical-align: 0px;\" \/><\/p>\n<p id=\"fs-id1167834395223\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9ae8e085e3a4db2ba316ebcda0c17dea_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"1\" width=\"1\" style=\"vertical-align: 0px;\" \/>The line passes through the <em data-effect=\"italics\">x<\/em>-axis at <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-efddd49bc1c2dc39c3fad2e3635247c3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"46\" style=\"vertical-align: -4px;\" \/><\/p>\n<p id=\"fs-id1167835421545\">A <strong data-effect=\"bold\">horizontal line<\/strong> is the graph of an equation of the form <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-27334f3a0485b8c07e8c9ebc0be4df6b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#98;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"45\" style=\"vertical-align: -4px;\" \/><\/p>\n<p id=\"fs-id1167835367829\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9ae8e085e3a4db2ba316ebcda0c17dea_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"1\" width=\"1\" style=\"vertical-align: 0px;\" \/>The line passes through the <em data-effect=\"italics\">y<\/em>-axis at <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5bac775607ac99b49b72fd1654a94604_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"44\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"example\" id=\"fs-id1167831872404\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167831872406\">\n<div data-type=\"problem\" id=\"fs-id1167835622210\">\n<p id=\"fs-id1167835622212\">Graph: <span class=\"token\">\u24d0<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c657687cbbf5ea9a7545edb42190e592_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"42\" style=\"vertical-align: 0px;\" \/> <span class=\"token\">\u24d1<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1144e182f26c1fd5166b5411a3ed3cd4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#45;&#49;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"60\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167826802100\">\n<p id=\"fs-id1167835240910\"><span class=\"token\">\u24d0<\/span> The equation has only one variable, <em data-effect=\"italics\">x<\/em>, and <em data-effect=\"italics\">x<\/em> is always equal to 2. We create a table where <em data-effect=\"italics\">x<\/em> is always 2 and then put in any values for <em data-effect=\"italics\">y<\/em>. The graph is a vertical line passing through the <em data-effect=\"italics\">x<\/em>-axis at 2.<\/p>\n<div data-type=\"newline\"><\/div>\n<table id=\"fs-id1167834195786\" class=\"unnumbered\" summary=\"The table has 5 rows and 3 columns. The first row is a title row with the equation x plus 2. The second row is a header row with the headers x, y, and (x, y). The third row has the numbers 2, 1, and (2, 1). The fourth row has the numbers 2, 2, and (2, 2). The fifth row has the numbers 2, 3, and (2, 3).\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td colspan=\"3\" data-valign=\"middle\" data-align=\"center\"><strong data-effect=\"bold\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c657687cbbf5ea9a7545edb42190e592_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"42\" style=\"vertical-align: 0px;\" \/><\/strong><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"middle\" data-align=\"center\"><strong data-effect=\"bold\"><em data-effect=\"italics\">x<\/em><\/strong><\/td>\n<td data-valign=\"middle\" data-align=\"center\"><strong data-effect=\"bold\"><em data-effect=\"italics\">y<\/em><\/strong><\/td>\n<td data-valign=\"middle\" data-align=\"center\"><strong data-effect=\"bold\"><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;\" \/><\/strong><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"middle\" data-align=\"center\">2<\/td>\n<td data-valign=\"middle\" data-align=\"center\">1<\/td>\n<td data-valign=\"middle\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bb160c5e6177bdd7a1d220c410258e5f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#44;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"middle\" data-align=\"center\">2<\/td>\n<td data-valign=\"middle\" data-align=\"center\">2<\/td>\n<td data-valign=\"middle\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f78f604644c2cdddbea2fc4d8ad49cca_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#44;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"middle\" data-align=\"center\">2<\/td>\n<td data-valign=\"middle\" data-align=\"center\">3<\/td>\n<td data-valign=\"middle\" data-align=\"center\"><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;\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167831919492\" data-alt=\"The figure shows the graph of a straight vertical line on the x y-coordinate plane. The x and y axes run from negative 7 to 7. The points (2, 1), (2, 2), and (2, 3) are plotted. The line goes through the three points and has arrows on both ends. The line is labeled x plus 2.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_019_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows the graph of a straight vertical line on the x y-coordinate plane. The x and y axes run from negative 7 to 7. The points (2, 1), (2, 2), and (2, 3) are plotted. The line goes through the three points and has arrows on both ends. The line is labeled x plus 2.\" \/><\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span> Similarly, the equation <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9f371b4e77462e28d9f6119571c92982_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"55\" style=\"vertical-align: -4px;\" \/> has only one variable, <em data-effect=\"italics\">y<\/em>. The value of <em data-effect=\"italics\">y<\/em> is constant. All the ordered pairs in the next table have the same <em data-effect=\"italics\">y<\/em>-coordinate. The graph is a horizontal line passing through the <em data-effect=\"italics\">y<\/em>-axis at <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fc343c70c6ffe28a6cd57dfb53f250b3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"26\" style=\"vertical-align: -1px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<table id=\"fs-id1167835361018\" class=\"unnumbered\" summary=\"The table has 5 rows and 3 columns. The first row is a title row with the equation y plus negative 1. The second row is a header row with the headers x, y, and (x, y). The third row has the numbers 0, negative 1, and (0, negative 1). The fourth row has the numbers 3, negative 1, and (3, negative 1). The fifth row has the numbers negative 3, negative 1, and (negative 3, negative 1).\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td colspan=\"3\" data-valign=\"middle\" data-align=\"center\"><strong data-effect=\"bold\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9f371b4e77462e28d9f6119571c92982_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"55\" style=\"vertical-align: -4px;\" \/><\/strong><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"middle\" data-align=\"center\"><strong data-effect=\"bold\"><em data-effect=\"italics\">x<\/em><\/strong><\/td>\n<td data-valign=\"middle\" data-align=\"center\"><strong data-effect=\"bold\"><em data-effect=\"italics\">y<\/em><\/strong><\/td>\n<td data-valign=\"middle\" data-align=\"center\"><strong data-effect=\"bold\"><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;\" \/><\/strong><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"middle\" data-align=\"center\">0<\/td>\n<td data-valign=\"middle\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7b34c01098c83fa602de54e9d74d63a9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"21\" style=\"vertical-align: -1px;\" \/><\/td>\n<td data-valign=\"middle\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ec6084fa217f87f8fc5df481ca60ccf0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"52\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"middle\" data-align=\"center\">3<\/td>\n<td data-valign=\"middle\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7b34c01098c83fa602de54e9d74d63a9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"21\" style=\"vertical-align: -1px;\" \/><\/td>\n<td data-valign=\"middle\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0d66a71b8940b998e4f29f8cccda06d4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#44;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"52\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"middle\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-470cb162cf92c55d139f4f69216225e5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"22\" style=\"vertical-align: 0px;\" \/><\/td>\n<td data-valign=\"middle\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7b34c01098c83fa602de54e9d74d63a9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"21\" style=\"vertical-align: -1px;\" \/><\/td>\n<td data-valign=\"middle\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d29d505cec09336aa569e1cca8670699_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#44;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"65\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167834430851\" data-alt=\"The figure shows the graph of a straight horizontal line on the x y-coordinate plane. The x and y axes run from negative 7 to 7. The points (negative 3, negative 1), (0, negative 1), and (3, negative 1) are plotted. The line goes through the three points and has arrows on both ends. The line is labeled y plus negative 1.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_020_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows the graph of a straight horizontal line on the x y-coordinate plane. The x and y axes run from negative 7 to 7. The points (negative 3, negative 1), (0, negative 1), and (3, negative 1) are plotted. The line goes through the three points and has arrows on both ends. The line is labeled y plus negative 1.\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167835254336\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167835254340\">\n<div data-type=\"problem\" id=\"fs-id1167835254342\">\n<p id=\"fs-id1167835180556\">Graph the equations: <span class=\"token\">\u24d0<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8ddab230605c435eb8b7408a736d3e77_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"42\" style=\"vertical-align: 0px;\" \/> <span class=\"token\">\u24d1<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8af9b026845f3fbfb2c441fcaba8be36_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#45;&#52;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"60\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167828435630\">\n<p id=\"fs-id1167828435632\"><span class=\"token\">\u24d0<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167835215981\" data-alt=\"The figure shows the graph of a straight vertical line on the x y-coordinate plane. The x and y axes run from negative 12 to 12. The line goes through the points (5, negative 3), (5, negative 2), (5, negative 1), (5, 0), (5, 1), (5, 2), and (5, 3).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_307_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows the graph of a straight vertical line on the x y-coordinate plane. The x and y axes run from negative 12 to 12. The line goes through the points (5, negative 3), (5, negative 2), (5, negative 1), (5, 0), (5, 1), (5, 2), and (5, 3).\" \/><\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167834062206\" data-alt=\"The figure shows the graph of a straight horizontal line on the x y-coordinate plane. The x and y axes run from negative 12 to 12. The line goes through the points (negative 3, negative 4), (negative 2, negative 4), (negative 1, negative 4), (0, negative 4), (1, negative 4), (2, negative 4), and (3, negative 4).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_308_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows the graph of a straight horizontal line on the x y-coordinate plane. The x and y axes run from negative 12 to 12. The line goes through the points (negative 3, negative 4), (negative 2, negative 4), (negative 1, negative 4), (0, negative 4), (1, negative 4), (2, negative 4), and (3, negative 4).\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167834061463\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167834061467\">\n<div data-type=\"problem\" id=\"fs-id1167834061469\">\n<p id=\"fs-id1167834061471\">Graph the equations: <span class=\"token\">\u24d0<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-01f282abd343bbe6b83c45e54b86c6ed_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#45;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"56\" style=\"vertical-align: 0px;\" \/> <span class=\"token\">\u24d1<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-34f0eb02c9be9d3f22890308095c7a3a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#51;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"46\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167831086624\">\n<p id=\"fs-id1167831086626\"><span class=\"token\">\u24d0<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167834376341\" data-alt=\"The figure shows the graph of a straight vertical line on the x y-coordinate plane. The x and y axes run from negative 12 to 12. The line goes through the points (negative 2, negative 3), (negative 2, negative 2), (negative 2, negative 1), (negative 2, 0), (negative 2, 1), (negative 2, 2), and (negative 2, 3).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_309_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows the graph of a straight vertical line on the x y-coordinate plane. The x and y axes run from negative 12 to 12. The line goes through the points (negative 2, negative 3), (negative 2, negative 2), (negative 2, negative 1), (negative 2, 0), (negative 2, 1), (negative 2, 2), and (negative 2, 3).\" \/><\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" data-alt=\"The figure shows the graph of a straight horizontal line on the x y-coordinate plane. The x and y axes run from negative 12 to 12. The line goes through the points (negative 3, 3), (negative 2, 3), (negative 1, 3), (0, 3), (1, 3), (2, 3), and (3, 3).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_310_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows the graph of a straight horizontal line on the x y-coordinate plane. The x and y axes run from negative 12 to 12. The line goes through the points (negative 3, 3), (negative 2, 3), (negative 1, 3), (0, 3), (1, 3), (2, 3), and (3, 3).\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167834099415\">What is the difference between the equations <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3830f1c4beca3ad0dd7e3a5dae581de2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#52;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"52\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8f9d33d92b853842ff676638161b437b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#52;&#63;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"50\" style=\"vertical-align: -4px;\" \/><\/p>\n<p id=\"fs-id1167835337692\">The equation <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3830f1c4beca3ad0dd7e3a5dae581de2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#52;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"52\" style=\"vertical-align: -4px;\" \/> has both <em data-effect=\"italics\">x<\/em> and <em data-effect=\"italics\">y<\/em>. The value of <em data-effect=\"italics\">y<\/em> depends on the value of <em data-effect=\"italics\">x<\/em>, so the <em data-effect=\"italics\">y<\/em> -coordinate changes according to the value of <em data-effect=\"italics\">x<\/em>. The equation <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0621c4f761e7864714642fcc62d4c42f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"42\" style=\"vertical-align: -4px;\" \/> has only one variable. The value of <em data-effect=\"italics\">y<\/em> is constant, it does not depend on the value of <em data-effect=\"italics\">x<\/em>, so the <em data-effect=\"italics\">y<\/em>-coordinate is always 4.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167835544872\" data-alt=\"This figure has two tables. The first table has 5 rows and 3 columns. The first row is a title row with the equation y plus 4 x. The second row is a header row with the headers x, y, and (x, y). The third row has the numbers 0, 0, and (0, 0). The fourth row has the numbers 1, 4, and (1, 4). The fifth row has the numbers 2, 8, and (2, 8). The second table has 5 rows and 3 columns. The first row is a title row with the equation y plus 4. The second row is a header row with the headers x, y, and (x, y). The third row has the numbers 0, 4, and (0, 4). The fourth row has the numbers 1, 4, and (1, 4). The fifth row has the numbers 2, 4, and (2, 4).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_039_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure has two tables. The first table has 5 rows and 3 columns. The first row is a title row with the equation y plus 4 x. The second row is a header row with the headers x, y, and (x, y). The third row has the numbers 0, 0, and (0, 0). The fourth row has the numbers 1, 4, and (1, 4). The fifth row has the numbers 2, 8, and (2, 8). The second table has 5 rows and 3 columns. The first row is a title row with the equation y plus 4. The second row is a header row with the headers x, y, and (x, y). The third row has the numbers 0, 4, and (0, 4). The fourth row has the numbers 1, 4, and (1, 4). The fifth row has the numbers 2, 4, and (2, 4).\" \/><\/span><span data-type=\"media\" id=\"fs-id1167835368651\" data-alt=\"The figure shows the graphs of a straight horizontal line and a straight slanted line on the same x y-coordinate plane. The x and y axes run from negative 7 to 7. The horizontal line goes through the points (0, 4), (1, 4), and (2,4) and is labeled y plus 4. The slanted line goes through the points (0, 0), (1, 4), and (2, 8) and is labeled y plus 4 x.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_021_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows the graphs of a straight horizontal line and a straight slanted line on the same x y-coordinate plane. The x and y axes run from negative 7 to 7. The horizontal line goes through the points (0, 4), (1, 4), and (2,4) and is labeled y plus 4. The slanted line goes through the points (0, 0), (1, 4), and (2, 8) and is labeled y plus 4 x.\" \/><\/span><\/p>\n<p id=\"fs-id1167835368655\">Notice, in the graph, the equation <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3830f1c4beca3ad0dd7e3a5dae581de2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#52;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"52\" style=\"vertical-align: -4px;\" \/> gives a slanted line, while <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0621c4f761e7864714642fcc62d4c42f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"42\" style=\"vertical-align: -4px;\" \/> gives a horizontal line.<\/p>\n<div data-type=\"example\" id=\"fs-id1167826972423\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167826972426\">\n<div data-type=\"problem\" id=\"fs-id1167826972428\">\n<p id=\"fs-id1167826972430\">Graph <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ae17fd29062052621010e587cc571ee5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#45;&#51;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"66\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-91f9ec631e44f3d108457c2f8adad27c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#45;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"56\" style=\"vertical-align: -4px;\" \/> in the same rectangular coordinate system.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167831954216\">\n<p id=\"fs-id1167831954218\">We notice that the first equation has the variable <em data-effect=\"italics\">x<\/em>, while the second does not. We make a table of points for each equation and then graph the lines. The two graphs are shown.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167835512135\" data-alt=\"This figure has two tables. The first table has 5 rows and 3 columns. The first row is a title row with the equation y plus negative 3 x. The second row is a header row with the headers x, y, and (x, y). The third row has the numbers 0, 0, and (0, 0). The fourth row has the numbers 1, negative 3, and (1, negative 3). The fifth row has the numbers 2, negative 6, and (2, neg ative 6). The second table has 5 rows and 3 columns. The first row is a title row with the equation y plus negative 3. The second row is a header row with the headers x, y, and (x, y). The third row has the numbers 0, negative 3, and (0, negative 3). The fourth row has the numbers 1, negative 3, and (1, negative 3). The fifth row has the numbers 2, negative 3, and (2, negative 3).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_040_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure has two tables. The first table has 5 rows and 3 columns. The first row is a title row with the equation y plus negative 3 x. The second row is a header row with the headers x, y, and (x, y). The third row has the numbers 0, 0, and (0, 0). The fourth row has the numbers 1, negative 3, and (1, negative 3). The fifth row has the numbers 2, negative 6, and (2, neg ative 6). The second table has 5 rows and 3 columns. The first row is a title row with the equation y plus negative 3. The second row is a header row with the headers x, y, and (x, y). The third row has the numbers 0, negative 3, and (0, negative 3). The fourth row has the numbers 1, negative 3, and (1, negative 3). The fifth row has the numbers 2, negative 3, and (2, negative 3).\" \/><\/span><\/p>\n<p id=\"fs-id1165926641234\">\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167834094570\" data-alt=\"The figure shows the graphs of a straight horizontal line and a straight slanted line on the same x y-coordinate plane. The x and y axes run from negative 7 to 7. The horizontal line goes through the points (0, negative 3), (1, negative 3), and (2, negative 3) and is labeled y plus negative 3. The slanted line goes through the points (0, 0), (1, negative 3), and (2, negative 6) and is labeled y plus negative 3 x.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_022_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows the graphs of a straight horizontal line and a straight slanted line on the same x y-coordinate plane. The x and y axes run from negative 7 to 7. The horizontal line goes through the points (0, negative 3), (1, negative 3), and (2, negative 3) and is labeled y plus negative 3. The slanted line goes through the points (0, 0), (1, negative 3), and (2, negative 6) and is labeled y plus negative 3 x.\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167834395000\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167831954167\">\n<div data-type=\"problem\" id=\"fs-id1167831954169\">\n<p id=\"fs-id1167831954171\">Graph the equations in the same rectangular coordinate system: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d6dd00142930f9fde84a1aaca1dedc3e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#45;&#52;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"66\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8af9b026845f3fbfb2c441fcaba8be36_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#45;&#52;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"60\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835322119\">\n<p id=\"fs-id1167831872346\">\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167832066671\" data-alt=\"The figure shows the graphs of a straight horizontal line and a straight slanted line on the same x y-coordinate plane. The x and y axes run from negative 12 to 12. The horizontal line goes through the points (0, negative 4), (1, negative 4), and (2, negative 4). The slanted line goes through the points (0, 0), (1, negative 4), and (2, negative 8).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_311_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows the graphs of a straight horizontal line and a straight slanted line on the same x y-coordinate plane. The x and y axes run from negative 12 to 12. The horizontal line goes through the points (0, negative 4), (1, negative 4), and (2, negative 4). The slanted line goes through the points (0, 0), (1, negative 4), and (2, negative 8).\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167832066929\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167834252684\">\n<div data-type=\"problem\" id=\"fs-id1167834252686\">\n<p id=\"fs-id1167834252688\">Graph the equations in the same rectangular coordinate system: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8e36d35d8563f5053efd9935e88634f9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"42\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-61eab8887b1816e55808a0611c382634_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#51;&#120;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"56\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835304740\">\n<p id=\"fs-id1167830963408\">\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167835304742\" data-alt=\"The figure shows the graphs of a straight horizontal line and a straight slanted line on the same x y-coordinate plane. The x and y axes run from negative 12 to 12. The horizontal line goes through the points (0, 3), (1, 3), and (2, 3). The slanted line goes through the points (0, 0), (1, 3), and (2, 6).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_312_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows the graphs of a straight horizontal line and a straight slanted line on the same x y-coordinate plane. The x and y axes run from negative 12 to 12. The horizontal line goes through the points (0, 3), (1, 3), and (2, 3). The slanted line goes through the points (0, 0), (1, 3), and (2, 6).\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167835400371\">\n<h3 data-type=\"title\">Find <em data-effect=\"italics\">x<\/em>&#8211; and <em data-effect=\"italics\">y<\/em>-intercepts<\/h3>\n<p id=\"fs-id1167835331638\">Every linear equation can be represented by a unique line that shows all the solutions of the equation. We have seen that when graphing a line by plotting points, you can use any three solutions to graph. This means that two people graphing the line might use different sets of three points.<\/p>\n<p id=\"fs-id1167835331644\">At first glance, their two lines might not appear to be the same, since they would have different points labeled. But if all the work was done correctly, the lines should be exactly the same. One way to recognize that they are indeed the same line is to look at where the line crosses the <em data-effect=\"italics\">x<\/em>-axis and the <em data-effect=\"italics\">y<\/em>-axis. These points are called the <span data-type=\"term\">intercepts of a line<\/span>.<\/p>\n<div data-type=\"note\" id=\"fs-id1167831871999\">\n<div data-type=\"title\">Intercepts of a Line<\/div>\n<p id=\"fs-id1167834138160\">The points where a line crosses the <em data-effect=\"italics\">x<\/em>-axis and the <em data-effect=\"italics\">y<\/em>-axis are called the <strong data-effect=\"bold\">intercepts of the line<\/strong>.<\/p>\n<\/div>\n<p id=\"fs-id1167834194637\">Let\u2019s look at the graphs of the lines.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167835343912\" data-alt=\"The figure shows four graphs of different equations. In example a the graph of 2 x plus y plus 6 is graphed on the x y-coordinate plane. The x and y axes run from negative 8 to 8. The points (0, 6) and (3, 0) are plotted and labeled. A straight line goes through both points and has arrows on both ends. In example b the graph of 3 x minus 4 y plus 12 is graphed on the x y-coordinate plane. The x and y axes run from negative 8 to 8. The points (0, negative 3) and (4, 0) are plotted and labeled. A straight line goes through both points and has arrows on both ends. In example c the graph of x minus y plus 5 is graphed on the x y-coordinate plane. The x and y axes run from negative 8 to 8. The points (0, negative 5) and (5, 0) are plotted and labeled. A straight line goes through both points and has arrows on both ends. In example d the graph of y plus negative 2 x is graphed on the x y-coordinate plane. The x and y axes run from negative 8 to 8. The point (0, 0) is plotted and labeled. A straight line goes through this point and the points (negative 1, 2) and (1, negative 2) and has arrows on both ends.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_023_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows four graphs of different equations. In example a the graph of 2 x plus y plus 6 is graphed on the x y-coordinate plane. The x and y axes run from negative 8 to 8. The points (0, 6) and (3, 0) are plotted and labeled. A straight line goes through both points and has arrows on both ends. In example b the graph of 3 x minus 4 y plus 12 is graphed on the x y-coordinate plane. The x and y axes run from negative 8 to 8. The points (0, negative 3) and (4, 0) are plotted and labeled. A straight line goes through both points and has arrows on both ends. In example c the graph of x minus y plus 5 is graphed on the x y-coordinate plane. The x and y axes run from negative 8 to 8. The points (0, negative 5) and (5, 0) are plotted and labeled. A straight line goes through both points and has arrows on both ends. In example d the graph of y plus negative 2 x is graphed on the x y-coordinate plane. The x and y axes run from negative 8 to 8. The point (0, 0) is plotted and labeled. A straight line goes through this point and the points (negative 1, 2) and (1, negative 2) and has arrows on both ends.\" \/><\/span><\/p>\n<p id=\"fs-id1167831887736\">First, notice where each of these lines crosses the <em data-effect=\"italics\">x<\/em>-axis. See <a href=\"#fs-id1167835367746\" class=\"autogenerated-content\">(Figure)<\/a>.<\/p>\n<p id=\"fs-id1167835345975\">Now, let\u2019s look at the points where these lines cross the <em data-effect=\"italics\">y<\/em>-axis.<\/p>\n<table id=\"fs-id1167835367746\" summary=\"The table has 6 rows and 5 columns. The first row is a header row with the headers \u201cFigure\u201d, \u201cThe line crosses the x-axis at:\u201d, \u201cOrdered pair for this point\u201d, \u201cThe line crosses the y-axis at:\u201d, and \u201cOrdered pair for this point\u201d. The second row contains \u201cFigure a\u201d, 3, (3, 0), 6, (0, 6). The third row contains \u201cFigure b\u201d, 4, (4, 0), negative 3, (0, negative 3). The fourth row contains \u201cFigure c\u201d, 5, (5, 0), negative 5, (0, negative 5). The fifth row contains \u201cFigure d\u201d, 0, (0, 0), 0, (0, 0). The sixth row contains \u201cGeneral Figure\u201d, a, (a, 0), b, (0, b).\">\n<thead>\n<tr valign=\"top\">\n<th data-valign=\"middle\" data-align=\"left\"><strong data-effect=\"bold\">Figure<\/strong><\/th>\n<th data-valign=\"top\" data-align=\"center\"><strong data-effect=\"bold\">The line crosses<br \/><\/strong><\/th>\n<\/tr>\n<\/thead>\n<\/table>\n<\/div>\n<p>the <em data-effect=\"italics\">x<\/em>-axis at:<strong data-effect=\"bold\">Ordered pair<br \/>for this point<\/strong><strong data-effect=\"bold\">The line crosses<br \/>the <em data-effect=\"italics\">y-<\/em>axis at:<\/strong><strong data-effect=\"bold\">Ordered pair<br \/>for this point<\/strong>Figure (a)3<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-28a186f2424d2e935d6aa6388441b6d2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/>6<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3be590e6084f04549fae6ac7c149a1d3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/>Figure (b)4<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;\" \/><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-470cb162cf92c55d139f4f69216225e5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"22\" style=\"vertical-align: 0px;\" \/><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-017f20ea0c6fda3470cedb20ea0b5537_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"52\" style=\"vertical-align: -4px;\" \/>Figure (c)5<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e3e1928d65786877c787a2d401d9e77e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7b5b9d9f382b11767d19f257afca0019_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"21\" style=\"vertical-align: 0px;\" \/><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6655aef23bcf82d48b1ff5bf888d5b2a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/>Figure (d)0<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;\" \/>0<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;\" \/>General Figure<em data-effect=\"italics\">a<\/em><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-53d4347201ab8f9c6f195eeec4b01f0c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/><em data-effect=\"italics\">b<\/em><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-05dbabd42fe84ba97e673be0628c5974_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"37\" style=\"vertical-align: -4px;\" \/><\/p>\n<p id=\"fs-id1167834535861\">Do you see a pattern?<\/p>\n<p id=\"fs-id1167834535864\">For each line, the <em data-effect=\"italics\">y<\/em>-coordinate of the point where the line crosses the <em data-effect=\"italics\">x<\/em>-axis is zero. The point where the line crosses the <em data-effect=\"italics\">x<\/em>-axis has the form <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-53d4347201ab8f9c6f195eeec4b01f0c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/> and is called the <em data-effect=\"italics\">x-intercept<\/em> of the line. The <em data-effect=\"italics\">x<\/em>-intercept occurs when <em data-effect=\"italics\">y<\/em> is zero.<\/p>\n<p id=\"fs-id1167835257786\">In each line, the <em data-effect=\"italics\">x<\/em><strong data-effect=\"bold\">&#8211;<\/strong>coordinate of the point where the line crosses the <em data-effect=\"italics\">y<\/em>-axis is zero. The point where the line crosses the <em data-effect=\"italics\">y<\/em>-axis has the form <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-05dbabd42fe84ba97e673be0628c5974_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"37\" style=\"vertical-align: -4px;\" \/> and is called the <em data-effect=\"italics\">y-intercept<\/em> of the line. The <em data-effect=\"italics\">y<\/em>-intercept occurs when <em data-effect=\"italics\">x<\/em> is zero.<\/p>\n<div data-type=\"note\">\n<div data-type=\"title\"><em data-effect=\"italics\">x<\/em>-intercept and <em data-effect=\"italics\">y<\/em>-intercept of a Line<\/div>\n<p id=\"fs-id1167835309776\">The <em data-effect=\"italics\">x<\/em>-intercept is the point <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-53d4347201ab8f9c6f195eeec4b01f0c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/> where the line crosses the <em data-effect=\"italics\">x<\/em>-axis.<\/p>\n<p id=\"fs-id1167831238979\">The <em data-effect=\"italics\">y<\/em>-intercept is the point <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-05dbabd42fe84ba97e673be0628c5974_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"37\" style=\"vertical-align: -4px;\" \/> where the line crosses the <em data-effect=\"italics\">y<\/em>-axis.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167834448946\" data-alt=\"The table has 3 rows and 2 columns. The first row is a header row with the headers x and y. The second row contains a and 0. The third row contains 0 and b.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_038_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The table has 3 rows and 2 columns. The first row is a header row with the headers x and y. The second row contains a and 0. The third row contains 0 and b.\" \/><\/span><\/div>\n<div data-type=\"example\" id=\"fs-id1167834300811\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167834300813\">\n<div data-type=\"problem\" id=\"fs-id1167834300815\">\n<p id=\"fs-id1167835376064\">Find the <em data-effect=\"italics\">x<\/em>&#8211; and <em data-effect=\"italics\">y<\/em>-intercepts on each graph shown.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167834279734\" data-alt=\"The figure has three graphs. Figure a shows a straight line graphed on the x y-coordinate plane. The x and y axes run from negative 8 to 8. The line goes through the points (negative 8, 6), (negative 4, 4), (0, 2), (4, 0), (8, negative 2). Figure b shows a straight line graphed on the x y-coordinate plane. The x and y axes run from negative 8 to 8. The line goes through the points (0, negative 6), (2, 0), and (4, 6). Figure c shows a straight line graphed on the x y-coordinate plane. The x and y axes run from negative 8 to 8. The line goes through the points (negative 5, 0), (negative 3, negative 3), (0, negative 5), (1, negative 6), and (2, negative 7).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_024_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure has three graphs. Figure a shows a straight line graphed on the x y-coordinate plane. The x and y axes run from negative 8 to 8. The line goes through the points (negative 8, 6), (negative 4, 4), (0, 2), (4, 0), (8, negative 2). Figure b shows a straight line graphed on the x y-coordinate plane. The x and y axes run from negative 8 to 8. The line goes through the points (0, negative 6), (2, 0), and (4, 6). Figure c shows a straight line graphed on the x y-coordinate plane. The x and y axes run from negative 8 to 8. The line goes through the points (negative 5, 0), (negative 3, negative 3), (0, negative 5), (1, negative 6), and (2, negative 7).\" \/><\/span><\/div>\n<div data-type=\"solution\" id=\"fs-id1167835595936\">\n<p id=\"fs-id1167835360226\"><span class=\"token\">\u24d0<\/span> The graph crosses the <em data-effect=\"italics\">x<\/em>-axis at the point <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b82d01b79621a013d9a24273324abbe9_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;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"45\" style=\"vertical-align: -4px;\" \/> The <em data-effect=\"italics\">x-<\/em>intercept is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b82d01b79621a013d9a24273324abbe9_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;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"45\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p> The graph crosses the <em data-effect=\"italics\">y<\/em>-axis at the point <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2787442576eb47fcba2bd3a113db823f_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;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"45\" style=\"vertical-align: -4px;\" \/> The <em data-effect=\"italics\">y<\/em>-intercept is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2787442576eb47fcba2bd3a113db823f_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;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"45\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<div data-type=\"newline\"><\/div>\n<p> <span class=\"token\">\u24d1<\/span> The graph crosses the <em data-effect=\"italics\">x<\/em>-axis at the point <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4d2abda359ee91996db7fe7bb27fa18d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"45\" style=\"vertical-align: -4px;\" \/> The <em data-effect=\"italics\">x<\/em>-intercept is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4d2abda359ee91996db7fe7bb27fa18d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"45\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p> The graph crosses the <em data-effect=\"italics\">y<\/em>-axis at the point <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-04c8c35675ca868fa2e02a300d5167b0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#45;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"59\" style=\"vertical-align: -4px;\" \/> The <em data-effect=\"italics\">y<\/em>-intercept is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-04c8c35675ca868fa2e02a300d5167b0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#45;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"59\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<div data-type=\"newline\"><\/div>\n<p> <span class=\"token\">\u24d2<\/span> The graph crosses the <em data-effect=\"italics\">x<\/em>-axis at the point <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-675cf805c5a8d766a987138da86670c2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#53;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"59\" style=\"vertical-align: -4px;\" \/> The <em data-effect=\"italics\">x<\/em>-intercept is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-675cf805c5a8d766a987138da86670c2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#53;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"59\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p> The graph crosses the <em data-effect=\"italics\">y<\/em>-axis at the point <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-07522d36f772a7e68964fbb659fdca09_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"59\" style=\"vertical-align: -4px;\" \/> The <em data-effect=\"italics\">y<\/em>-intercept is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-07522d36f772a7e68964fbb659fdca09_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"59\" style=\"vertical-align: -4px;\" \/><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167826828292\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167826986762\">\n<div data-type=\"problem\" id=\"fs-id1167826986764\">\n<p id=\"fs-id1167826986766\">Find the <em data-effect=\"italics\">x<\/em>&#8211; and <em data-effect=\"italics\">y<\/em>-intercepts on the graph.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167830697612\" data-alt=\"This figure a shows a straight line graphed on the x y-coordinate plane. The x and y axes run from negative 10 to 10. The line goes through the points (negative 6, negative 8), (negative 4, negative 6), (negative 2, negative 4), (0, negative 2), (2, 0), (4, 2), (6, 4), (8, 6).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_025_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure a shows a straight line graphed on the x y-coordinate plane. The x and y axes run from negative 10 to 10. The line goes through the points (negative 6, negative 8), (negative 4, negative 6), (negative 2, negative 4), (0, negative 2), (2, 0), (4, 2), (6, 4), (8, 6).\" \/><\/span><\/div>\n<div data-type=\"solution\" id=\"fs-id1167834219644\">\n<p><em data-effect=\"italics\">x<\/em>-intercept: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b9bfe5f0e85bf2ee8d0ce478041861f9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"45\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><em data-effect=\"italics\">y<\/em>-intercept: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3ecef9f206503704c74407265b403ee3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"52\" style=\"vertical-align: -4px;\" \/><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167832153754\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167832153758\">\n<div data-type=\"problem\" id=\"fs-id1167832153760\">\n<p id=\"fs-id1167832153762\">Find the <em data-effect=\"italics\">x<\/em>&#8211; and <em data-effect=\"italics\">y<\/em>-intercepts on the graph.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167835326401\" data-alt=\"This figure a shows a straight line graphed on the x y-coordinate plane. The x and y axes run from negative 10 to 10. The line goes through the points (negative 6, 6), (negative 3, 4), (0, 2), (3, 0), (6, negative 2), and (9, negative 4).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_026_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure a shows a straight line graphed on the x y-coordinate plane. The x and y axes run from negative 10 to 10. The line goes through the points (negative 6, 6), (negative 3, 4), (0, 2), (3, 0), (6, negative 2), and (9, negative 4).\" \/><\/span><\/div>\n<div data-type=\"solution\" id=\"fs-id1167835346249\">\n<p id=\"fs-id1167835346251\"><em data-effect=\"italics\">x<\/em>-intercept: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0b641bd134e4dda02d1c48080f8e069b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"45\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><em data-effect=\"italics\">y<\/em>-intercept: <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;\" \/><\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167835410917\">Recognizing that the <em data-effect=\"italics\">x<\/em>-intercept occurs when <em data-effect=\"italics\">y<\/em> is zero and that the <em data-effect=\"italics\">y<\/em>-intercept occurs when <em data-effect=\"italics\">x<\/em> is zero, gives us a method to find the intercepts of a line from its equation. To find the <em data-effect=\"italics\">x<\/em>-intercept, let <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5e8ef70615fdaee8588017ac1fdd2da0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"42\" style=\"vertical-align: -4px;\" \/> and solve for <em data-effect=\"italics\">x<\/em>. To find the <em data-effect=\"italics\">y<\/em>-intercept, let <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8203ced39e0cdafefa708857c7ec2264_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"43\" style=\"vertical-align: 0px;\" \/> and solve for <em data-effect=\"italics\">y<\/em>.<\/p>\n<div data-type=\"note\" id=\"fs-id1167835410566\">\n<div data-type=\"title\">Find the <em data-effect=\"italics\">x<\/em>&#8211; and <em data-effect=\"italics\">y<\/em>-intercepts from the Equation of a Line<\/div>\n<p id=\"fs-id1167835345789\">Use the equation of the line. To find:<\/p>\n<ul id=\"fs-id1167834246665\" data-bullet-style=\"bullet\">\n<li>the <em data-effect=\"italics\">x<\/em>-intercept of the line, let <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5e8ef70615fdaee8588017ac1fdd2da0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"42\" style=\"vertical-align: -4px;\" \/> and solve for <em data-effect=\"italics\">x<\/em>.<\/li>\n<li>the <em data-effect=\"italics\">y<\/em>-intercept of the line, let <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8203ced39e0cdafefa708857c7ec2264_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"43\" style=\"vertical-align: 0px;\" \/> and solve for <em data-effect=\"italics\">y<\/em>.<\/li>\n<\/ul>\n<\/div>\n<div data-type=\"example\" id=\"fs-id1167827987818\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167827987820\">\n<div data-type=\"problem\" id=\"fs-id1167827987822\">\n<p id=\"fs-id1167827987825\">Find the intercepts of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f2f6d8d28793d51b09ae30134c8fc29c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#120;&#43;&#121;&#61;&#56;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"87\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835345155\">\n<p id=\"fs-id1167835345157\">We will let <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5e8ef70615fdaee8588017ac1fdd2da0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"42\" style=\"vertical-align: -4px;\" \/> to find the <em data-effect=\"italics\">x<\/em>-intercept, and let <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;\" \/> to find the <em data-effect=\"italics\">y<\/em>-intercept. We will fill in a table, which reminds us of what we need to find.<\/p>\n<div data-type=\"newline\"><\/div>\n<p> <span data-type=\"media\" id=\"fs-id1167834433642\" data-alt=\"The figure has a table with 4 rows and 2 columns. The first row is a title row with the equation 2 x plus y plus 8. The second row is a header row with the headers x and y. The third row is labeled x-intercept and has the first column blank and a 0 in the second column. The fourth row is labeled y-intercept and has a 0 in the first column and the second column blank.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_027_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure has a table with 4 rows and 2 columns. The first row is a title row with the equation 2 x plus y plus 8. The second row is a header row with the headers x and y. The third row is labeled x-intercept and has the first column blank and a 0 in the second column. The fourth row is labeled y-intercept and has a 0 in the first column and the second column blank.\" \/><\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<table id=\"fs-id1167834376970\" class=\"unnumbered unstyled\" summary=\"The figure has the equation 2 x plus y plus 8. Let y plus 0. The next equation is 2 x plus 0 plus 8, where the 0 is emphasized. Simplifying we get 2 x plus 8. Then y plus 4. The x-intercept is (4, 0). To find the y-intercept, let x plus 0. Again we start with the equation 2 x plus y plus 8. Let x plus 0. The next equation is 2 times 0 plus y plus 8, where the 0 is emphasized. Simplifying we get 0 plus y plus 8. Then y plus 8. The y-intercept is (0, 8).\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">To find the <em data-effect=\"italics\">x<\/em>-intercept, let <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-03bd81b19337bd478ac4d869320205bf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"46\" style=\"vertical-align: -4px;\" \/><\/td>\n<td><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834191753\" data-alt=\".\"><img decoding=\"async\" src=\"CNX_IntAlg_Figure_03_01_028a_img_new.jpg#fixme#fixme\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Let <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-03bd81b19337bd478ac4d869320205bf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"46\" style=\"vertical-align: -4px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167832138837\" data-alt=\".\"><img decoding=\"async\" src=\"CNX_IntAlg_Figure_03_01_028b_img_new.jpg#fixme#fixme\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167828377143\" data-alt=\".\"><img decoding=\"async\" src=\"CNX_IntAlg_Figure_03_01_028c_img_new.jpg#fixme#fixme\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835390567\" data-alt=\".\"><img decoding=\"async\" src=\"CNX_IntAlg_Figure_03_01_028d_img_new.jpg#fixme#fixme\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">The <em data-effect=\"italics\">x<\/em>-intercept is:<\/td>\n<td data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8ae8cbb6a392f2b7d26808f2fd31f7f9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#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;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">To find the <em data-effect=\"italics\">y<\/em>-intercept, let <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d6889ee3f02f0af137641306363d2da7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"47\" style=\"vertical-align: 0px;\" \/><\/td>\n<td><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835418854\" data-alt=\".\"><img decoding=\"async\" src=\"CNX_IntAlg_Figure_03_01_028e_img_new.jpg#fixme#fixme\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Let <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d6889ee3f02f0af137641306363d2da7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"47\" style=\"vertical-align: 0px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834387657\" data-alt=\".\"><img decoding=\"async\" src=\"CNX_IntAlg_Figure_03_01_028f_img_new.jpg#fixme#fixme\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835320774\" data-alt=\".\"><img decoding=\"async\" src=\"CNX_IntAlg_Figure_03_01_028g_img_new.jpg#fixme#fixme\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835324543\" data-alt=\".\"><img decoding=\"async\" src=\"CNX_IntAlg_Figure_03_01_028h_img_new.jpg#fixme#fixme\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">The <em data-effect=\"italics\">y<\/em>-intercept is:<\/td>\n<td data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0cb385ca0585807f893bd3c2dffdc90b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div data-type=\"newline\"><\/div>\n<p> The intercepts are the points <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;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7ccd2ec7c9cce60dfe4df9c4dd1a3588_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/> as shown in the table.<\/p>\n<div data-type=\"newline\"><\/div>\n<table id=\"fs-id1167835173826\" class=\"unnumbered\" summary=\"The table shows the equation 2x plus y equals 8. Below that are two columns. The left column is x and the right column is y. The first row shows that x is 4 and y is 0. The second column shows that the x is 0 and the y is 8.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td colspan=\"2\" data-valign=\"middle\" data-align=\"center\"><strong data-effect=\"bold\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a5793bb7b8b037b12ce8c700c56480ba_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#120;&#43;&#121;&#61;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"83\" style=\"vertical-align: -4px;\" \/><\/strong><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"middle\" data-align=\"center\"><strong data-effect=\"bold\"><em data-effect=\"italics\">x<\/em><\/strong><\/td>\n<td data-valign=\"middle\" data-align=\"center\"><strong data-effect=\"bold\"><em data-effect=\"italics\">y<\/em><\/strong><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"middle\" data-align=\"center\">4<\/td>\n<td data-valign=\"middle\" data-align=\"center\">0<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"middle\" data-align=\"center\">0<\/td>\n<td data-valign=\"middle\" data-align=\"center\">8<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167835353165\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167835353170\">\n<div data-type=\"problem\" id=\"fs-id1167835353172\">\n<p id=\"fs-id1167835353174\">Find the intercepts: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ff43ca98dceafc074260265ee703e4ef_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#120;&#43;&#121;&#61;&#49;&#50;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"96\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167831872300\">\n<p id=\"fs-id1167827943100\"><em data-effect=\"italics\">x<\/em>-intercept: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-081727c272ed3219c0b60fdb30b9ce96_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;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"45\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><em data-effect=\"italics\">y<\/em>-intercept: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-977b0243afc1e93b6d551e14600e46a7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#49;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"47\" style=\"vertical-align: -4px;\" \/><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167830865897\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167835348348\">\n<div data-type=\"problem\" id=\"fs-id1167835348350\">\n<p id=\"fs-id1167835348352\">Find the intercepts: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b92f68643ac12760171214ba7983324b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#43;&#52;&#121;&#61;&#56;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"87\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835333372\">\n<p id=\"fs-id1167835333374\"><em data-effect=\"italics\">x<\/em>-intercept: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-07f8d1768c5429f80456c7cec386ed4b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#56;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"45\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><em data-effect=\"italics\">y<\/em>-intercept: <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;\" \/><\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167832051938\">\n<h3 data-type=\"title\">Graph a Line Using the Intercepts<\/h3>\n<p id=\"fs-id1167832051944\">To graph a linear equation by plotting points, you need to find three points whose coordinates are solutions to the equation. You can use the <em data-effect=\"italics\">x-<\/em> and <em data-effect=\"italics\">y-<\/em> intercepts as two of your three points. Find the intercepts, and then find a third point to ensure accuracy. Make sure the points line up\u2014then draw the line. This method is often the quickest way to graph a line.<\/p>\n<div data-type=\"example\" id=\"fs-id1167834345742\" class=\"textbox textbox--examples\">\n<div data-type=\"title\">How to Graph a Line Using the Intercepts<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834345744\">\n<div data-type=\"problem\" id=\"fs-id1167834345746\">\n<p id=\"fs-id1167834431456\">Graph <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9d2fa5921b9545e5ff74ab908bb0d6ef_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#120;&#43;&#50;&#121;&#61;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"95\" style=\"vertical-align: -4px;\" \/> using the intercepts.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835305984\"><span data-type=\"media\" id=\"fs-id1167835305986\" data-alt=\"Step 1 is to find the x and y-intercepts of the line. To find the x-intercept let y plus 0 and solve for x. The equation negative x plus 2 y plus 6 becomes negative x plus 2 times 0 plus 6. This simplifies to negative x plus 6. This is equivalent to x plus negative 6. The x-intercept is (negative 6, 0). To find the y-intercept let x plus 0 and solve for y. The equation negative x plus 2 y plus 6 becomes negative 0 plus 2 y plus 6. This simplifies to negative 2 y plus 6. This is equivalent to y plus 3. The y-intercept is (0, 3).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_029a_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 1 is to find the x and y-intercepts of the line. To find the x-intercept let y plus 0 and solve for x. The equation negative x plus 2 y plus 6 becomes negative x plus 2 times 0 plus 6. This simplifies to negative x plus 6. This is equivalent to x plus negative 6. The x-intercept is (negative 6, 0). To find the y-intercept let x plus 0 and solve for y. The equation negative x plus 2 y plus 6 becomes negative 0 plus 2 y plus 6. This simplifies to negative 2 y plus 6. This is equivalent to y plus 3. The y-intercept is (0, 3).\" \/><\/span><span data-type=\"media\" id=\"fs-id1167835305991\" data-alt=\"Step 2 is to find another solution to the equation. We\u2019ll use x plus 2. The equation negative x plus 2 y plus 6 becomes negative 2 plus 2 y plus 6. This simplifies to 2 y plus 8. This is equivalent to y plus 4. The third point is (2, 4).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_029b_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 2 is to find another solution to the equation. We\u2019ll use x plus 2. The equation negative x plus 2 y plus 6 becomes negative 2 plus 2 y plus 6. This simplifies to 2 y plus 8. This is equivalent to y plus 4. The third point is (2, 4).\" \/><\/span><span data-type=\"media\" id=\"fs-id1167835375646\" data-alt=\"Step 3 is to plot the three points. The figure shows a table with 4 rows and 3 columns. The first row is a header row with the headers x, y, and (x, y). The second row contains negative 6, 0, and (negative 6, 0). The third row contains 0, 3, and (0, 3). The fourth row contains 2, 4, and (2, 4). The figure also has a graph of the three points on the x y-coordinate plane. The x and y-axes run from negative 6 to 6. The three points (negative 6, 0), (0, 3), and (2, 4) are plotted and labeled.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_029c_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 3 is to plot the three points. The figure shows a table with 4 rows and 3 columns. The first row is a header row with the headers x, y, and (x, y). The second row contains negative 6, 0, and (negative 6, 0). The third row contains 0, 3, and (0, 3). The fourth row contains 2, 4, and (2, 4). The figure also has a graph of the three points on the x y-coordinate plane. The x and y-axes run from negative 6 to 6. The three points (negative 6, 0), (0, 3), and (2, 4) are plotted and labeled.\" \/><\/span><span data-type=\"media\" id=\"fs-id1167834556100\" data-alt=\"Step 4 is to draw the line. The figure shows a graph of a straight line on the x y-coordinate plane. The x and y-axes run from negative 6 to 6. The straight line goes through the points (negative 6, 0), (0, 3), and (2, 4).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_029d_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 4 is to draw the line. The figure shows a graph of a straight line on the x y-coordinate plane. The x and y-axes run from negative 6 to 6. The straight line goes through the points (negative 6, 0), (0, 3), and (2, 4).\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167835337758\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167835337762\">\n<div data-type=\"problem\" id=\"fs-id1167835337764\">\n<p id=\"fs-id1167835337766\">Graph using the intercepts: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-818f942e04a0dc03655623304cb89058_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#45;&#50;&#121;&#61;&#52;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"87\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834501732\"><span data-type=\"media\" id=\"fs-id1167835312208\" data-alt=\"The figure shows a graph of a straight line on the x y-coordinate plane. The x and y-axes run from negative 12 to 12. The straight line goes through the points (negative 4, negative 4), (negative 2, negative 3), (0, negative 2), (2, negative 1), (4, 0), (6, 1), and (8, 2).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_313_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows a graph of a straight line on the x y-coordinate plane. The x and y-axes run from negative 12 to 12. The straight line goes through the points (negative 4, negative 4), (negative 2, negative 3), (0, negative 2), (2, negative 1), (4, 0), (6, 1), and (8, 2).\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167835621569\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167834448816\">\n<div data-type=\"problem\" id=\"fs-id1167834448818\">\n<p id=\"fs-id1167834448820\">Graph using the intercepts: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-786c6885543d1f49052772e36d12f63d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#120;&#43;&#51;&#121;&#61;&#54;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"99\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835503875\">\n<p id=\"fs-id1167832116054\">\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167835503877\" data-alt=\"The figure shows a graph of a straight line on the x y-coordinate plane. The x and y-axes run from negative 12 to 12. The straight line goes through the points (negative 9, negative 1), (negative 6, 0), (negative 3, 1), (0, 2), (3, 3), (6, 4), and (9, 5).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_314_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows a graph of a straight line on the x y-coordinate plane. The x and y-axes run from negative 12 to 12. The straight line goes through the points (negative 9, negative 1), (negative 6, 0), (negative 3, 1), (0, 2), (3, 3), (6, 4), and (9, 5).\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167835345522\">The steps to graph a linear equation using the intercepts are summarized here.<\/p>\n<div data-type=\"note\" id=\"fs-id1167835345525\" class=\"howto\">\n<div data-type=\"title\">Graph a linear equation using the intercepts.<\/div>\n<ol id=\"fs-id1167835215399\" type=\"1\" class=\"stepwise\">\n<li>Find the <em data-effect=\"italics\">x<\/em>&#8211; and <em data-effect=\"italics\">y<\/em>-intercepts of the line.\n<ul id=\"fs-id1167835368530\" data-bullet-style=\"bullet\">\n<li>Let <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5e8ef70615fdaee8588017ac1fdd2da0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"42\" style=\"vertical-align: -4px;\" \/> and solve for <em data-effect=\"italics\">x<\/em>.<\/li>\n<li>Let <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8203ced39e0cdafefa708857c7ec2264_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"43\" style=\"vertical-align: 0px;\" \/> and solve for <em data-effect=\"italics\">y<\/em>.<\/li>\n<\/ul>\n<\/li>\n<li>Find a third solution to the equation.<\/li>\n<li>Plot the three points and check that they line up.<\/li>\n<li>Draw the line.<\/li>\n<\/ol>\n<\/div>\n<div data-type=\"example\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167831238959\">Graph <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-efedfb1adc9ce250cfc862e4fb07e730_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#120;&#45;&#51;&#121;&#61;&#49;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"99\" style=\"vertical-align: -4px;\" \/> using the intercepts.<\/p>\n<\/div>\n<div data-type=\"solution\">\n<p>Find the intercepts and a third point.<\/p>\n<div data-type=\"newline\"><\/div>\n<p> <span data-type=\"media\" id=\"fs-id1167835319374\" data-alt=\"To find the x-intercept let y plus 0 and solve for x. The equation 4 x minus 3 y plus 12 becomes 4 x minus 3 times 0 plus 12. This simplifies to negative 4 x plus 12. This is equivalent to x plus 3. To find the y-intercept let x plus 0 and solve for y. The equation 4 x minus 3 y plus 12 becomes 4 times 0 minus 3 y plus 12. This simplifies to negative 3 y plus 12. This is equivalent to y plus negative 4. To find the third point let y plus 4 and solve for x. The equation 4 x minus 3 y plus 12 becomes 4 x minus 3 times 4 plus 12. This simplifies to negative 4 x plus 24. This is equivalent to x plus 6.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_030_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"To find the x-intercept let y plus 0 and solve for x. The equation 4 x minus 3 y plus 12 becomes 4 x minus 3 times 0 plus 12. This simplifies to negative 4 x plus 12. This is equivalent to x plus 3. To find the y-intercept let x plus 0 and solve for y. The equation 4 x minus 3 y plus 12 becomes 4 times 0 minus 3 y plus 12. This simplifies to negative 3 y plus 12. This is equivalent to y plus negative 4. To find the third point let y plus 4 and solve for x. The equation 4 x minus 3 y plus 12 becomes 4 x minus 3 times 4 plus 12. This simplifies to negative 4 x plus 24. This is equivalent to x plus 6.\" \/><\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<div data-type=\"newline\"><\/div>\n<p> We list the points in the table and show the graph.<\/p>\n<div data-type=\"newline\"><\/div>\n<table id=\"fs-id1167834554869\" class=\"unnumbered\" summary=\"The table shows the equation 4 x minus 3 y plus 12. There are three columns below: x, y, and x y. The first row shows that x is 3, y is 0, and x y is 3, 0. The second row shows that x is 0, y is negative 4, and x y is 0, negative 4. The third column shows that x is 6, y is 4, and x y is 6, 4.\">\n<tbody>\n<tr valign=\"top\">\n<td colspan=\"3\" data-valign=\"middle\" data-align=\"center\"><strong data-effect=\"bold\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-efedfb1adc9ce250cfc862e4fb07e730_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#120;&#45;&#51;&#121;&#61;&#49;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"99\" style=\"vertical-align: -4px;\" \/><\/strong><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"middle\" data-align=\"center\"><strong data-effect=\"bold\"><em data-effect=\"italics\">x<\/em><\/strong><\/td>\n<td data-valign=\"middle\" data-align=\"center\"><strong data-effect=\"bold\"><em data-effect=\"italics\">y<\/em><\/strong><\/td>\n<td data-valign=\"middle\" data-align=\"center\"><strong data-effect=\"bold\"><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;\" \/><\/strong><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"middle\" data-align=\"center\">3<\/td>\n<td data-valign=\"middle\" data-align=\"center\">0<\/td>\n<td data-valign=\"middle\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-28a186f2424d2e935d6aa6388441b6d2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"middle\" data-align=\"center\">0<\/td>\n<td data-valign=\"middle\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-00b9cce9021441b203ec0271d72e6ba2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"22\" style=\"vertical-align: -1px;\" \/><\/td>\n<td data-valign=\"middle\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-42f3c0bd5adc0ec0fa1707d7989e5c00_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"52\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"middle\" data-align=\"center\">6<\/td>\n<td data-valign=\"middle\" data-align=\"center\">4<\/td>\n<td data-valign=\"middle\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-86825c0bacfa777b8fe368820523b723_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#44;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167834583414\" data-alt=\"The figure shows a graph of the equation 4 x minus 3 y plus 12 on the x y-coordinate plane. The x and y-axes run from negative 7 to 7. The straight line goes through the points (0, negative 4), (3, 0), and (6, 4).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_031_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows a graph of the equation 4 x minus 3 y plus 12 on the x y-coordinate plane. The x and y-axes run from negative 7 to 7. The straight line goes through the points (0, negative 4), (3, 0), and (6, 4).\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167835343715\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167835280874\">\n<div data-type=\"problem\" id=\"fs-id1167835280877\">\n<p id=\"fs-id1167835280879\">Graph using the intercepts: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4904356b272a8dae0f397627a0568911_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#120;&#45;&#50;&#121;&#61;&#49;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"104\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835306373\">\n<p id=\"fs-id1167831891814\">\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167835423208\" data-alt=\"The figure shows a graph of a straight line on the x y-coordinate plane. The x and y-axes run from negative 8 to 8. The straight line goes through the points (0, negative 5), (2, 0), and (4, 5).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_315_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows a graph of a straight line on the x y-coordinate plane. The x and y-axes run from negative 8 to 8. The straight line goes through the points (0, negative 5), (2, 0), and (4, 5).\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167835363636\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167835363641\">\n<div data-type=\"problem\" id=\"fs-id1167834120979\">\n<p id=\"fs-id1167834120981\">Graph using the intercepts: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1afd50597e4a4545db7bcdd00a6a4597_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#120;&#45;&#52;&#121;&#61;&#49;&#50;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"104\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835420632\">\n<p id=\"fs-id1167835318697\">\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167835420634\" data-alt=\"The figure shows a graph of a straight line on the x y-coordinate plane. The x and y-axes run from negative 8 to 8. The straight line goes through the points (negative 4, negative 6), (0, negative 3), (4, 0), and (8, 3).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_316_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows a graph of a straight line on the x y-coordinate plane. The x and y-axes run from negative 8 to 8. The straight line goes through the points (negative 4, negative 6), (0, negative 3), (4, 0), and (8, 3).\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1171792848414\">When the line passes through the origin, the <em data-effect=\"italics\">x<\/em>-intercept and the <em data-effect=\"italics\">y<\/em>-intercept are the same point.<\/p>\n<div data-type=\"example\" id=\"fs-id1167835514433\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167835514436\">\n<div data-type=\"problem\" id=\"fs-id1167835514438\">\n<p id=\"fs-id1167835514440\">Graph <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-91c10e13707f47199b50bd3f85254528_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#53;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"52\" style=\"vertical-align: -4px;\" \/> using the intercepts.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834376079\"><span data-type=\"media\" id=\"fs-id1167834376081\" data-alt=\"To find the x-intercept let y plus 0 and solve for x. The equation y plus 5 x becomes 0 plus 5 x. This simplifies to 0 plus x. The x-intercept is (0, 0). To find the y-intercept let x plus 0 and solve for y. The equation y plus 5 x becomes y plus 5 times 0. This simplifies to y plus 0. The y-intercept is also (0, 0).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_032_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"To find the x-intercept let y plus 0 and solve for x. The equation y plus 5 x becomes 0 plus 5 x. This simplifies to 0 plus x. The x-intercept is (0, 0). To find the y-intercept let x plus 0 and solve for y. The equation y plus 5 x becomes y plus 5 times 0. This simplifies to y plus 0. The y-intercept is also (0, 0).\" \/><\/span><\/p>\n<p id=\"fs-id1171792730711\">\n<div data-type=\"newline\"><\/div>\n<p id=\"fs-id1167834185105\">This line has only one intercept. It is the point <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e9b349d335879ab45d8b79d5850b0860_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;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"45\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p> To ensure accuracy, we need to plot three points. Since the <em data-effect=\"italics\">x<\/em>&#8211; and <em data-effect=\"italics\">y<\/em>-intercepts are the same point, we need <em data-effect=\"italics\">two<\/em> more points to graph the line.<span data-type=\"media\" id=\"fs-id1167835364300\" data-alt=\"To find a second point let x plus 1 and solve for y. The equation y plus 5 x becomes y plus 5 times 1. This simplifies to y plus 5. To find a third point let x plus negative 1 and solve for y. The equation y plus 5 x becomes y plus 5 times negative 1. This simplifies to y plus negative 5\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_033_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"To find a second point let x plus 1 and solve for y. The equation y plus 5 x becomes y plus 5 times 1. This simplifies to y plus 5. To find a third point let x plus negative 1 and solve for y. The equation y plus 5 x becomes y plus 5 times negative 1. This simplifies to y plus negative 5\" \/><\/span><\/p>\n<p id=\"fs-id1171791389259\">The resulting three points are summarized in the table.<\/p>\n<table id=\"fs-id1167835367099\" class=\"unnumbered\" summary=\"The table has 5 rows and 3 columns. The first row is a title row with the equation y plus 5 x. The second row is a header row with the headers x, y, and (x, y). The third row contains negative 0, 0, and (0, 0). The fourth row contains 1, 5, and (1, 5). The fifth row contains negative 1, negative 5, and (negative 1, negative 5).\">\n<tbody>\n<tr valign=\"top\">\n<td colspan=\"3\" data-valign=\"middle\" data-align=\"center\"><strong data-effect=\"bold\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-91c10e13707f47199b50bd3f85254528_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#53;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"52\" style=\"vertical-align: -4px;\" \/><\/strong><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"middle\" data-align=\"center\"><strong data-effect=\"bold\"><em data-effect=\"italics\">x<\/em><\/strong><\/td>\n<td data-valign=\"middle\" data-align=\"center\"><strong data-effect=\"bold\"><em data-effect=\"italics\">y<\/em><\/strong><\/td>\n<td data-valign=\"middle\" data-align=\"center\"><strong data-effect=\"bold\"><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;\" \/><\/strong><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"middle\" data-align=\"center\">0<\/td>\n<td data-valign=\"middle\" data-align=\"center\">0<\/td>\n<td data-valign=\"middle\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c53627fd7039dcb62c54d86fe468e6e1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"middle\" data-align=\"center\">1<\/td>\n<td data-valign=\"middle\" data-align=\"center\">5<\/td>\n<td data-valign=\"middle\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fc88778f80c6fa1eb186cfd3741de73e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#44;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"middle\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7b34c01098c83fa602de54e9d74d63a9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"21\" style=\"vertical-align: -1px;\" \/><\/td>\n<td data-valign=\"middle\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7b5b9d9f382b11767d19f257afca0019_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"21\" style=\"vertical-align: 0px;\" \/><\/td>\n<td data-valign=\"middle\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d66960a9b4adfa7702388a061d743cbd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#44;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"65\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1167835335617\">Plot the three points, check that they line up, and draw the line.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167835335620\" data-alt=\"The figure shows a graph of the equation y plus 5 x on the x y-coordinate plane. The x and y-axes run from negative 10 to 10. The straight line goes through the points (negative 1, negative 5), (0, 0), and (1, 5).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_034_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows a graph of the equation y plus 5 x on the x y-coordinate plane. The x and y-axes run from negative 10 to 10. The straight line goes through the points (negative 1, negative 5), (0, 0), and (1, 5).\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167835216609\" class=\"try\">\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1167835377793\">\n<p id=\"fs-id1167835377795\">Graph using the intercepts: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-daf4fb55b4aeed3a4aa1689f9aa29cf9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#52;&#120;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"56\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167832151038\">\n<p id=\"fs-id1167834377227\">\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167832151040\" data-alt=\"The figure shows a graph of a straight line on the x y-coordinate plane. The x and y-axes run from negative 12 to 12. The straight line goes through the points (negative 1, negative 4), (0, 0), and (1, 4).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_317_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows a graph of a straight line on the x y-coordinate plane. The x and y-axes run from negative 12 to 12. The straight line goes through the points (negative 1, negative 4), (0, 0), and (1, 4).\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167826874357\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167834309127\">\n<div data-type=\"problem\" id=\"fs-id1167834309129\">\n<p id=\"fs-id1167834309131\">Graph the intercepts: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b4c77f70d6c2c1e49a4fd05268adcb50_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#120;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"47\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834433015\">\n<p id=\"fs-id1167832043239\">\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167834433018\" data-alt=\"The figure shows a graph of a straight line on the x y-coordinate plane. The x and y-axes run from negative 12 to 12. The straight line goes through the points (negative 1, 1), (0, 0), and (1, negative 1).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_318_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows a graph of a straight line on the x y-coordinate plane. The x and y-axes run from negative 12 to 12. The straight line goes through the points (negative 1, 1), (0, 0), and (1, negative 1).\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox\" data-depth=\"1\" id=\"fs-id1167832150989\">\n<h3 data-type=\"title\">Key Concepts<\/h3>\n<ul id=\"fs-id1167835373759\" data-bullet-style=\"bullet\">\n<li><strong data-effect=\"bold\">Points on the Axes<\/strong>\n<ul id=\"fs-id1167834162010\" data-bullet-style=\"bullet\">\n<li>Points with a <em data-effect=\"italics\">y<\/em>-coordinate equal to 0 are on the <em data-effect=\"italics\">x<\/em>-axis, and have coordinates <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-efddd49bc1c2dc39c3fad2e3635247c3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"46\" style=\"vertical-align: -4px;\" \/><\/li>\n<li>Points with an <em data-effect=\"italics\">x<\/em>-coordinate equal to <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a5e437be25f29374d30f66cd46adf81c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: 0px;\" \/> are on the <em data-effect=\"italics\">y<\/em>-axis, and have coordinates <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5bac775607ac99b49b72fd1654a94604_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"44\" style=\"vertical-align: -4px;\" \/><\/li>\n<\/ul>\n<\/li>\n<li><strong data-effect=\"bold\">Quadrant<\/strong>\n<div data-type=\"equation\" id=\"fs-id1167831081622\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-dc6169aa2426e841e122e51405b42edc_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;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#81;&#117;&#97;&#100;&#114;&#97;&#110;&#116;&#32;&#73;&#125;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#81;&#117;&#97;&#100;&#114;&#97;&#110;&#116;&#32;&#73;&#73;&#125;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#81;&#117;&#97;&#100;&#114;&#97;&#110;&#116;&#32;&#73;&#73;&#73;&#125;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#81;&#117;&#97;&#100;&#114;&#97;&#110;&#116;&#32;&#73;&#86;&#125;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#44;&#121;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#44;&#121;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#44;&#121;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#44;&#121;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#108;&#101;&#102;&#116;&#40;&#43;&#44;&#43;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#44;&#43;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#44;&#45;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#43;&#44;&#45;&#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;\" title=\"Rendered by QuickLaTeX.com\" height=\"61\" width=\"535\" style=\"vertical-align: -26px;\" \/><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167835512761\" data-alt=\"This figure shows the x y-coordinate plane with the four quadrants labeled. In the top right of the plane is quadrant I labeled (plus, plus). In the top left of the plane is quadrant II labeled (minus, plus). In the bottom left of the plane is quadrant III labeled (minus, minus). In the bottom right of the plane is quadrant IV labeled (plus, minus).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_036_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows the x y-coordinate plane with the four quadrants labeled. In the top right of the plane is quadrant I labeled (plus, plus). In the top left of the plane is quadrant II labeled (minus, plus). In the bottom left of the plane is quadrant III labeled (minus, minus). In the bottom right of the plane is quadrant IV labeled (plus, minus).\" \/><\/span><\/li>\n<li><strong data-effect=\"bold\">Graph of a Linear Equation:<\/strong> The graph of a linear equation <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0520a31036e9a951aea74693c8b23cb1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#65;&#120;&#43;&#66;&#121;&#61;&#67;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"106\" style=\"vertical-align: -4px;\" \/> is a straight line.\n<div data-type=\"newline\"><\/div>\n<p> Every point on the line is a solution of the equation.<\/p>\n<div data-type=\"newline\"><\/div>\n<p> Every solution of this equation is a point on this line.<\/li>\n<li><strong data-effect=\"bold\">How to graph a linear equation by plotting points.<\/strong>\n<ol id=\"fs-id1167830837149\" type=\"1\" class=\"stepwise\">\n<li>Find three points whose coordinates are solutions to the equation. Organize them in a table.<\/li>\n<li>Plot the points in a rectangular coordinate system. Check that the points line up. If they do not, carefully check your work.<\/li>\n<li>Draw the line through the three points. Extend the line to fill the grid and put arrows on both ends of the line.<\/li>\n<\/ol>\n<\/li>\n<li><strong data-effect=\"bold\"><em data-effect=\"italics\">x<\/em>-intercept and <em data-effect=\"italics\">y<\/em>-intercept of a Line<\/strong>\n<ul id=\"fs-id1167831076503\" data-bullet-style=\"bullet\">\n<li>The <em data-effect=\"italics\">x<\/em>-intercept is the point <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-53d4347201ab8f9c6f195eeec4b01f0c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/> where the line crosses the <em data-effect=\"italics\">x<\/em>-axis.<\/li>\n<li>The <em data-effect=\"italics\">y<\/em>-intercept is the point <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-05dbabd42fe84ba97e673be0628c5974_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"37\" style=\"vertical-align: -4px;\" \/> where the line crosses the <em data-effect=\"italics\">y<\/em>-axis. <span data-type=\"media\" id=\"fs-id1167835351058\" data-alt=\"The table has 3 rows and 2 columns. The first row is a header row with the headers x and y. The second row contains a and 0. The x-intercept occurs when y is zero. The third row contains 0 and b. The y-intercept occurs when x is zero.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_037_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The table has 3 rows and 2 columns. The first row is a header row with the headers x and y. The second row contains a and 0. The x-intercept occurs when y is zero. The third row contains 0 and b. The y-intercept occurs when x is zero.\" \/><\/span><\/li>\n<\/ul>\n<\/li>\n<li><strong data-effect=\"bold\">Find the <em data-effect=\"italics\">x<\/em>&#8211; and <em data-effect=\"italics\">y<\/em>-intercepts from the Equation of a Line<\/strong>\n<ul id=\"fs-id1167835326058\" data-bullet-style=\"bullet\">\n<li>Use the equation of the line. To find:\n<div data-type=\"newline\"><\/div>\n<p> the <em data-effect=\"italics\">x<\/em>-intercept of the line, let <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5e8ef70615fdaee8588017ac1fdd2da0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"42\" style=\"vertical-align: -4px;\" \/> and solve for <em data-effect=\"italics\">x<\/em>.<\/p>\n<div data-type=\"newline\"><\/div>\n<p> the <em data-effect=\"italics\">y<\/em>-intercept of the line, let <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8203ced39e0cdafefa708857c7ec2264_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"43\" style=\"vertical-align: 0px;\" \/> and solve for <em data-effect=\"italics\">y<\/em>.<\/li>\n<\/ul>\n<\/li>\n<li><strong data-effect=\"bold\">How to graph a linear equation using the intercepts.<\/strong>\n<ol id=\"fs-id1167835514188\" type=\"1\" class=\"stepwise\">\n<li>Find the <em data-effect=\"italics\">x<\/em>&#8211; and <em data-effect=\"italics\">y<\/em>-intercepts of the line.\n<div data-type=\"newline\"><\/div>\n<p> Let <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5e8ef70615fdaee8588017ac1fdd2da0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"42\" style=\"vertical-align: -4px;\" \/> and solve for <em data-effect=\"italics\">x.<\/em><\/p>\n<div data-type=\"newline\"><\/div>\n<p> Let <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8203ced39e0cdafefa708857c7ec2264_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"43\" style=\"vertical-align: 0px;\" \/> and solve for <em data-effect=\"italics\">y<\/em>.<\/li>\n<li>Find a third solution to the equation.<\/li>\n<li>Plot the three points and check that they line up.<\/li>\n<li>Draw the line<\/li>\n<\/ol>\n<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox\" data-depth=\"1\" id=\"fs-id1167835381776\">\n<div class=\"practice-perfect\" data-depth=\"2\" id=\"fs-id1167834060085a\">\n<h4 data-type=\"title\">Practice Makes Perfect<\/h4>\n<p id=\"fs-id1167834060092\"><strong data-effect=\"bold\">Plot Points in a Rectangular Coordinate System<\/strong><\/p>\n<p id=\"fs-id1167831837573\">In the following exercises, plot each point in a rectangular coordinate system and identify the quadrant in which the point is located.<\/p>\n<div data-type=\"exercise\" class=\"material-set-2\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167835384360\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ad41549099069a514e17dfbc399b6c68_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#52;&#44;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"51\" 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-ce38ace23dd1b8f15095a1aff4a73f36_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#44;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"65\" style=\"vertical-align: -4px;\" \/><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-afaa85d80f816c5a4c4a30f7848f7f78_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#44;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"52\" style=\"vertical-align: -4px;\" \/><span class=\"token\">\u24d3<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7ba9cc2a7f12e65a6b3de8f34bcc16e4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" 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\">\u24d4<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-64d8120241193ac47be39d83562a0dd1_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;&#51;&#125;&#44;&#50;&#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-id1167835381602\"><span data-type=\"media\" id=\"fs-id1167835381604\" data-alt=\"This figure shows points plotted on the x y-coordinate plane. The x and y axes run from negative 6 to 6. The point labeled a is 4 units to the left of the origin and 2 units above the origin and is located in quadrant II. The point labeled b is 1 unit to the left of the origin and 2 units below the origin and is located in quadrant III. The point labeled c is 3 units to the right of the origin and 5 units below the origin and is located in quadrant IV. The point labeled d is 3 units to the left of the origin and 5 units above the origin and is located in quadrant II. The point labeled e is 1 and a half units to the right of the origin and 2 units above the origin and is located in quadrant I.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_319_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows points plotted on the x y-coordinate plane. The x and y axes run from negative 6 to 6. The point labeled a is 4 units to the left of the origin and 2 units above the origin and is located in quadrant II. The point labeled b is 1 unit to the left of the origin and 2 units below the origin and is located in quadrant III. The point labeled c is 3 units to the right of the origin and 5 units below the origin and is located in quadrant IV. The point labeled d is 3 units to the left of the origin and 5 units above the origin and is located in quadrant II. The point labeled e is 1 and a half units to the right of the origin and 2 units above the origin and is located in quadrant I.\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834133113\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167834133115\">\n<p id=\"fs-id1167834133117\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e6b25c6279fe53efe7c78796a156eeed_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#44;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" 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-f054d79e93ab302f68adcad7d4c3ec21_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#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\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cba4706a21fad1e77985563696d1bdae_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#52;&#44;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"51\" style=\"vertical-align: -4px;\" \/><span class=\"token\">\u24d3<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6de38faced02869069c306c45a22c5e0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#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 data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d4<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8587ff653ad6b4047b73ccd5dbb63370_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#50;&#125;&#44;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"40\" style=\"vertical-align: -7px;\" \/><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835489339\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835489342\">\n<p id=\"fs-id1167835351140\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0d66a71b8940b998e4f29f8cccda06d4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#44;&#45;&#49;&#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-0e0819cfb987fa92d333add15bb2e864_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#44;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"51\" style=\"vertical-align: -4px;\" \/><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cb8b4631a3117b8d3055f32de28fef91_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#44;&#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;&#48;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"55\" style=\"vertical-align: -4px;\" \/><span class=\"token\">\u24d3<\/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 data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d4<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7b306914c9166160cb903d32a58e7a3d_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;&#49;&#52;&#125;&#123;&#53;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"47\" style=\"vertical-align: -7px;\" \/><\/div>\n<div data-type=\"solution\" id=\"fs-id1167835420000\"><span data-type=\"media\" id=\"fs-id1167835420002\" data-alt=\"This figure shows points plotted on the x y-coordinate plane. The x and y axes run from negative 6 to 6. The point labeled a is 3 units to the right of the origin and 1 unit below the origin and is located in quadrant IV. The point labeled b is 3 units to the left of the origin and 1 unit above the origin and is located in quadrant II. The point labeled c is 2 units to the left of the origin and 2 units above the origin and is located in quadrant II. The point labeled d is 4 units to the left of the origin and 3 units below the origin and is located in quadrant III. The point labeled e is 1 unit to the right of the origin and 3 and 4 fifths units above the origin and is located in quadrant I.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_321_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows points plotted on the x y-coordinate plane. The x and y axes run from negative 6 to 6. The point labeled a is 3 units to the right of the origin and 1 unit below the origin and is located in quadrant IV. The point labeled b is 3 units to the left of the origin and 1 unit above the origin and is located in quadrant II. The point labeled c is 2 units to the left of the origin and 2 units above the origin and is located in quadrant II. The point labeled d is 4 units to the left of the origin and 3 units below the origin and is located in quadrant III. The point labeled e is 1 unit to the right of the origin and 3 and 4 fifths units above the origin and is located in quadrant I.\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831959139\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167831959141\">\n<p id=\"fs-id1167831920778\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c1d1c22dc2eabfdd66670fe21fd738b8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#44;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"51\" style=\"vertical-align: -4px;\" \/><span class=\"token\">\u24d1<\/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\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-559928bd7c8949c8342dd73437aef05a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/><span class=\"token\">\u24d3<\/span><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 data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d4<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1bf9252343e7c78d63ebcdd2a4df8445_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#125;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"40\" style=\"vertical-align: -7px;\" \/><\/div>\n<\/div>\n<p id=\"fs-id1167835390387\">In the following exercises, for each ordered pair, decide<\/p>\n<p id=\"fs-id1167835309845\"><span class=\"token\">\u24d0<\/span> is the ordered pair a solution to the equation? <span class=\"token\">\u24d1<\/span> is the point on the line?<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167834516106\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167834516108\">\n<p id=\"fs-id1167834516111\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7ae0a2442b2303ccd57a555545fc5317_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#120;&#43;&#50;&#59;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"78\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p>A: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8bf1dcc5a62cf0d1f452068f53c0888a_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;&#59;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"45\" style=\"vertical-align: -4px;\" \/> B: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-beedb13754411c1a7e9a340d03724c80_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#44;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#59;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"45\" style=\"vertical-align: -4px;\" \/> C: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bd09beee40ffb8a60056d53f5341c880_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#44;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#59;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"59\" style=\"vertical-align: -4px;\" \/> D: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6f8be7d5c2fcc82848d6ac3b69a60254_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#44;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"73\" style=\"vertical-align: -4px;\" \/><span data-type=\"media\" id=\"fs-id1167832042287\" data-alt=\"This figure shows a straight line graphed on the x y-coordinate plane. The x and y-axes run from negative 8 to 8. The line goes through the points (negative 3, negative 1), (negative 2, 0), (negative 1, 1), (0, 2), (1, 3), (2, 4), and (3, 5).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_201_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a straight line graphed on the x y-coordinate plane. The x and y-axes run from negative 8 to 8. The line goes through the points (negative 3, negative 1), (negative 2, 0), (negative 1, 1), (0, 2), (1, 3), (2, 4), and (3, 5).\" \/><\/span><\/div>\n<div data-type=\"solution\" id=\"fs-id1167831911373\">\n<p id=\"fs-id1167830964089\"><span class=\"token\">\u24d0<\/span> A: yes, B: no, C: yes, D: yes <span class=\"token\">\u24d1<\/span> A: yes, B: no, C: yes, D: yes<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167832015739\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167832015741\">\n<p id=\"fs-id1167832015743\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1f2ce3d6f0f280d66f5104c00433b8ed_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#120;&#45;&#52;&#59;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"78\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p>A: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b8dd33deddac50fe17d4eaab40e9be3b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#59;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"59\" style=\"vertical-align: -4px;\" \/> B: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0ce1b3043bfc70090ad83d07fd550c49_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#44;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#59;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"59\" style=\"vertical-align: -4px;\" \/> C: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-11d2462786d528fa620cc9a81f88f04b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#44;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#59;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"45\" style=\"vertical-align: -4px;\" \/> D: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-65261358a1796b5039acb865fda2665a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#44;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"59\" style=\"vertical-align: -4px;\" \/><span data-type=\"media\" id=\"fs-id1167831921371\" data-alt=\"This figure shows a straight line graphed on the x y-coordinate plane. The x and y-axes run from negative 8 to 8. The line goes through the points (negative 3, negative 7), (negative 2, negative 6), (negative 1, negative 5), (0, negative 4), (1, negative 3), (2, negative 2), and (3, negative 1).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_202_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a straight line graphed on the x y-coordinate plane. The x and y-axes run from negative 8 to 8. The line goes through the points (negative 3, negative 7), (negative 2, negative 6), (negative 1, negative 5), (0, negative 4), (1, negative 3), (2, negative 2), and (3, negative 1).\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835379705\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167831911279\">\n<p id=\"fs-id1167831911281\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4c684a4c9bbb11c817fbeb07f2eb2103_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#120;&#45;&#51;&#59;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"89\" style=\"vertical-align: -6px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p>A: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8699c3ff4b8541caf17bd9b9ba1494e0_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;&#59;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"59\" style=\"vertical-align: -4px;\" \/> B: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3e2d4fab5e437f4c621efe487421c415_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;&#59;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"59\" style=\"vertical-align: -4px;\" \/> C: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-642c44a7ddc70c32477d4fad046d7e0f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#44;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#59;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"73\" style=\"vertical-align: -4px;\" \/> D: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-da4dc2348b880aa050a82db91856c8b2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#44;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/><span data-type=\"media\" id=\"fs-id1167828365419\" data-alt=\"This figure shows a straight line graphed on the x y-coordinate plane. The x and y-axes run from negative 8 to 8. The line goes through the points (negative 4, negative 5), (negative 2, negative 4), (0, negative 3), (2, negative 2), (4, negative 1), and (6, 0).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_203_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a straight line graphed on the x y-coordinate plane. The x and y-axes run from negative 8 to 8. The line goes through the points (negative 4, negative 5), (negative 2, negative 4), (0, negative 3), (2, negative 2), (4, negative 1), and (6, 0).\" \/><\/span><\/div>\n<div data-type=\"solution\" id=\"fs-id1167826783927\">\n<p id=\"fs-id1167826783929\"><span class=\"token\">\u24d0<\/span> A: yes, B: yes, C: yes, D: no <span class=\"token\">\u24d1<\/span> A: yes, B: yes, C: yes, D: no<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167832043531\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167832043533\">\n<p id=\"fs-id1167832043535\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-755a159d3917f77f24101a57cd8b331a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#120;&#43;&#50;&#59;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"89\" style=\"vertical-align: -6px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p>A: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8bf1dcc5a62cf0d1f452068f53c0888a_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;&#59;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"45\" style=\"vertical-align: -4px;\" \/> B: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-07417b84493fc43d4047a8cbb301d031_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;&#59;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"45\" style=\"vertical-align: -4px;\" \/> C: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-94440239e39db030b398f34d37fa0f4e_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;&#59;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"59\" style=\"vertical-align: -4px;\" \/> D: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6c313f19f44c0ee593d54242b770b7f5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#54;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"59\" style=\"vertical-align: -4px;\" \/><span data-type=\"media\" id=\"fs-id1167835173842\" data-alt=\"This figure shows a straight line graphed on the x y-coordinate plane. The x and y-axes run from negative 8 to 8. The line goes through the points (negative 6, 0), (negative 3, 1), (0, 2), (3, 3), (6, 4), and (9, 5).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_204_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a straight line graphed on the x y-coordinate plane. The x and y-axes run from negative 8 to 8. The line goes through the points (negative 6, 0), (negative 3, 1), (0, 2), (3, 3), (6, 4), and (9, 5).\" \/><\/span><\/div>\n<\/div>\n<p id=\"fs-id1167834534357\"><strong data-effect=\"bold\">Graph a Linear Equation by Plotting Points<\/strong><\/p>\n<p id=\"fs-id1167834227331\">In the following exercises, graph by plotting points.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167834133322\">\n<div data-type=\"problem\" id=\"fs-id1167834133324\">\n<p id=\"fs-id1167834133326\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9573bcdc8144ab85298374bd51ec886d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#120;&#43;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"73\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834062901\"><span data-type=\"media\" id=\"fs-id1167834062904\" data-alt=\"This figure shows a straight line graphed on the x y-coordinate plane. The x and y-axes run from negative 12 to 12. The line goes through the points (negative 3, negative 1), (negative 2, 0), (negative 1, 1), (0, 2), (1, 3), (2, 4), and (3, 5).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_325_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a straight line graphed on the x y-coordinate plane. The x and y-axes run from negative 12 to 12. The line goes through the points (negative 3, negative 1), (negative 2, 0), (negative 1, 1), (0, 2), (1, 3), (2, 4), and (3, 5).\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834049043\">\n<div data-type=\"problem\" id=\"fs-id1167834049046\">\n<p id=\"fs-id1167834402981\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7d6de0021f4a1d6e58bd3d8beec1c84a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#120;&#45;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"74\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834227334\">\n<div data-type=\"problem\" id=\"fs-id1167834227337\">\n<p id=\"fs-id1167834227339\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-09acfc77aa1f355947c21a5c0e345588_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#51;&#120;&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"82\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167827987486\"><span data-type=\"media\" data-alt=\"This figure shows a straight line graphed on the x y-coordinate plane. The x and y-axes run from negative 12 to 12. The line goes through the points (negative 3, negative 10), (negative 2, negative 7), (negative 1, negative 4), (0, negative 1), (1, 2), (2, 5), (3, 8), and (4, 11).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_323_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a straight line graphed on the x y-coordinate plane. The x and y-axes run from negative 12 to 12. The line goes through the points (negative 3, negative 10), (negative 2, negative 7), (negative 1, negative 4), (0, negative 1), (1, 2), (2, 5), (3, 8), and (4, 11).\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167826781720\">\n<div data-type=\"problem\" id=\"fs-id1167831883533\">\n<p id=\"fs-id1167831883535\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9ee9a398efc0528db8f6e51a774b2116_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#45;&#50;&#120;&#43;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"95\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835640011\">\n<div data-type=\"problem\" id=\"fs-id1167835640013\">\n<p id=\"fs-id1167835640015\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4597e9a5dd1ad5e5267c465d2e5539d2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#120;&#45;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"74\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\"><span data-type=\"media\" id=\"fs-id1167835379779\" data-alt=\"This figure shows a straight line graphed on the x y-coordinate plane. The x and y-axes run from negative 12 to 12. The line goes through the points (negative 3, 0), (negative 2, negative 1), (negative 1, negative 2), (0, negative 3), (1, negative 4), (2, negative 5), (3, negative 6), and (4, negative 7).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_327_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a straight line graphed on the x y-coordinate plane. The x and y-axes run from negative 12 to 12. The line goes through the points (negative 3, 0), (negative 2, negative 1), (negative 1, negative 2), (0, negative 3), (1, negative 4), (2, negative 5), (3, negative 6), and (4, negative 7).\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834252346\">\n<div data-type=\"problem\" id=\"fs-id1167834252348\">\n<p id=\"fs-id1167834252350\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0348a349216c4ef63f9122edbe7390e9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#120;&#45;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"73\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167830868598\">\n<div data-type=\"problem\" id=\"fs-id1167831106804\">\n<p id=\"fs-id1167831106806\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4ecf689c13800069b252cefb0f1d5b07_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#50;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"52\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167831881337\"><span data-type=\"media\" id=\"fs-id1167831881340\" data-alt=\"This figure shows a straight line graphed on the x y-coordinate plane. The x and y-axes run from negative 12 to 12. The line goes through the points (negative 3, negative 6), (negative 2, negative 4), (negative 1, negative 2), (0, 0), (1, 2), (2, 4), and (3, 6).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_329_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a straight line graphed on the x y-coordinate plane. The x and y-axes run from negative 12 to 12. The line goes through the points (negative 3, negative 6), (negative 2, negative 4), (negative 1, negative 2), (0, 0), (1, 2), (2, 4), and (3, 6).\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834084968\">\n<div data-type=\"problem\" id=\"fs-id1167834084970\">\n<p id=\"fs-id1167834084972\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b1860f3bc1742115ef9046a1238467de_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#45;&#50;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"66\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831228815\">\n<div data-type=\"problem\" id=\"fs-id1167831228817\">\n<p id=\"fs-id1167831228819\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-310709dccff6a1f76a02bb5e66ef0651_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#120;&#43;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"84\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167831116445\"><span data-type=\"media\" id=\"fs-id1167831116447\" data-alt=\"This figure shows a straight line graphed on the x y-coordinate plane. The x and y-axes run from negative 12 to 12. The line goes through the points (negative 6, negative 2), (negative 4, 0), (negative 2, 1), (0, 2), (2, 3), (4, 4), and (6, 5).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_331_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a straight line graphed on the x y-coordinate plane. The x and y-axes run from negative 12 to 12. The line goes through the points (negative 6, negative 2), (negative 4, 0), (negative 2, 1), (0, 2), (2, 3), (4, 4), and (6, 5).\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835201088\">\n<div data-type=\"problem\" id=\"fs-id1167834426344\">\n<p id=\"fs-id1167834426346\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5067836876049a69968419323279ddbf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#120;&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"84\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834161715\">\n<div data-type=\"problem\" id=\"fs-id1167834161718\">\n<p id=\"fs-id1167834161720\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-72b1ab9f028b3f33ac825a1027435960_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#51;&#125;&#120;&#45;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"84\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167830963879\"><span data-type=\"media\" id=\"fs-id1167830963881\" data-alt=\"This figure shows a straight line graphed on the x y-coordinate plane. The x and y-axes run from negative 12 to 12. The line goes through the points (negative 3, negative 9), (0, negative 5), (3, negative 1), (6, 3), and (9, 7).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_333_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a straight line graphed on the x y-coordinate plane. The x and y-axes run from negative 12 to 12. The line goes through the points (negative 3, negative 9), (0, negative 5), (3, negative 1), (6, 3), and (9, 7).\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167832214462\">\n<div data-type=\"problem\" id=\"fs-id1167832214465\">\n<p id=\"fs-id1167832214467\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-598897224fed3094e6102b49d8b7b0d9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#50;&#125;&#120;&#45;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"85\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835510781\">\n<div data-type=\"problem\" id=\"fs-id1167835510783\">\n<p id=\"fs-id1167835510785\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d457bf148676f064342f9c9c37f75810_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#53;&#125;&#120;&#43;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"97\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167831884860\"><span data-type=\"media\" id=\"fs-id1167831884862\" data-alt=\"This figure shows a straight line graphed on the x y-coordinate plane. The x and y-axes run from negative 12 to 12. The line goes through the points (negative 10, 5), (negative 5, 3), (0, 1), (5, negative 1), and (10, negative 3).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_335_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a straight line graphed on the x y-coordinate plane. The x and y-axes run from negative 12 to 12. The line goes through the points (negative 10, 5), (negative 5, 3), (0, 1), (5, negative 1), and (10, negative 3).\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167827966773\">\n<div data-type=\"problem\" id=\"fs-id1167831112408\">\n<p id=\"fs-id1167831112410\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6ea212d3edc781d4492da65c06ed06be_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#53;&#125;&#120;&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"97\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167830693579\">\n<div data-type=\"problem\" id=\"fs-id1167834120600\">\n<p id=\"fs-id1167834120602\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-aa79ba839bbcb41a00c9806e30191623_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#50;&#125;&#120;&#43;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"97\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167832117891\"><span data-type=\"media\" id=\"fs-id1167832117893\" data-alt=\"This figure shows a straight line graphed on the x y-coordinate plane. The x and y-axes run from negative 12 to 12. The line goes through the points (negative 4, 8), (negative 2, 5), (0, 2), (2, negative 1), (4, negative 4), and (6, negative 7).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_337_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a straight line graphed on the x y-coordinate plane. The x and y-axes run from negative 12 to 12. The line goes through the points (negative 4, 8), (negative 2, 5), (0, 2), (2, negative 1), (4, negative 4), and (6, negative 7).\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835319192\">\n<div data-type=\"problem\" id=\"fs-id1167835319194\">\n<p id=\"fs-id1167830963603\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bbff23cbf656646e7e192d7a96c1e527_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#51;&#125;&#120;&#43;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"98\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167835366967\"><strong data-effect=\"bold\">Graph Vertical and Horizontal lines<\/strong><\/p>\n<p id=\"fs-id1167835267904\">In the following exercises, graph each equation.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167835267907\">\n<div data-type=\"problem\" id=\"fs-id1167835267909\">\n<p id=\"fs-id1167835267911\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2145acc2878ed61214887e120f2485b7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"43\" style=\"vertical-align: -1px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8e36d35d8563f5053efd9935e88634f9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"42\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835231508\">\n<p id=\"fs-id1167835231510\"><span class=\"token\">\u24d0<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167835231516\" data-alt=\"This figure shows a vertical straight line graphed on the x y-coordinate plane. The x and y-axes run from negative 12 to 12. The line goes through the points (4, negative 1), (4, 0), and (4, 1).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_339_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a vertical straight line graphed on the x y-coordinate plane. The x and y-axes run from negative 12 to 12. The line goes through the points (4, negative 1), (4, 0), and (4, 1).\" \/><\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167834403045\" data-alt=\"This figure shows a horizontal straight line graphed on the x y-coordinate plane. The x and y-axes run from negative 12 to 12. The line goes through the points (negative 1, 3), (0, 3), and (1, 3).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_340_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a horizontal straight line graphed on the x y-coordinate plane. The x and y-axes run from negative 12 to 12. The line goes through the points (negative 1, 3), (0, 3), and (1, 3).\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834195667\">\n<div data-type=\"problem\" id=\"fs-id1167835231662\">\n<p id=\"fs-id1167835231664\"><span class=\"token\">\u24d0<\/span><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;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0e3ca3f6eb8810e090b4ceee7f6e129b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"41\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167826937848\">\n<div data-type=\"problem\" id=\"fs-id1167826937850\">\n<p id=\"fs-id1167835519107\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-01f282abd343bbe6b83c45e54b86c6ed_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#45;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"56\" style=\"vertical-align: 0px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ee7bf59b42611ad4534a6d8ca47648e2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#45;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"55\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835254717\">\n<p id=\"fs-id1167831949111\"><span class=\"token\">\u24d0<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167831949117\" data-alt=\"This figure shows a vertical straight line graphed on the x y-coordinate plane. The x and y-axes run from negative 12 to 12. The line goes through the points (negative 2, negative 1), (negative 2, 0), and (negative 2, 1).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_343_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a vertical straight line graphed on the x y-coordinate plane. The x and y-axes run from negative 12 to 12. The line goes through the points (negative 2, negative 1), (negative 2, 0), and (negative 2, 1).\" \/><\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167835389720\" data-alt=\"This figure shows a horizontal straight line graphed on the x y-coordinate plane. The x and y-axes run from negative 12 to 12. The line goes through the points (negative 1, negative 5), (0, negative 5), and (1, negative 5).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_344_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a horizontal straight line graphed on the x y-coordinate plane. The x and y-axes run from negative 12 to 12. The line goes through the points (negative 1, negative 5), (0, negative 5), and (1, negative 5).\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835511308\">\n<div data-type=\"problem\" id=\"fs-id1167835511310\">\n<p id=\"fs-id1167835511312\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d3e213d8d687e32831c24e16c432b60e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#45;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"56\" style=\"vertical-align: 0px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f986dbfac9d3f29a18cba91e9efa9d2c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#45;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"55\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167832153425\">In the following exercises, graph each pair of equations in the same rectangular coordinate system.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167832153429\">\n<div data-type=\"problem\" id=\"fs-id1167832153431\">\n<p id=\"fs-id1167835362062\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4ecf689c13800069b252cefb0f1d5b07_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#50;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"52\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-552d8ed773e160e229551b39aff39445_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"41\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834448640\"><span data-type=\"media\" id=\"fs-id1167834448642\" data-alt=\"The figure shows the graphs of a straight horizontal line and a straight slanted line on the same x y-coordinate plane. The x and y axes run from negative 12 to 12. The horizontal line goes through the points (0, 2), (1, 2), and (2, 2). The slanted line goes through the points (0, 0), (1, 2), and (2, 4).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_347_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows the graphs of a straight horizontal line and a straight slanted line on the same x y-coordinate plane. The x and y axes run from negative 12 to 12. The horizontal line goes through the points (0, 2), (1, 2), and (2, 2). The slanted line goes through the points (0, 0), (1, 2), and (2, 4).\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835376055\">\n<div data-type=\"problem\" id=\"fs-id1167835376057\">\n<p id=\"fs-id1167834246714\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-91c10e13707f47199b50bd3f85254528_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#53;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"52\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-39af43abd99adaf051fde7775af522c7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"41\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831031010\">\n<div data-type=\"problem\" id=\"fs-id1167831031012\">\n<p id=\"fs-id1167831031014\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1b298864de4217bd4810d7b724265182_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"68\" style=\"vertical-align: -6px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1d73f92d367c6005c529d143de2bb9ef_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"56\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167831880532\"><span data-type=\"media\" id=\"fs-id1167832125741\" data-alt=\"The figure shows the graphs of a straight horizontal line and a straight slanted line on the same x y-coordinate plane. The x and y axes run from negative 12 to 12. The horizontal line goes through the points (0, negative 1 divided 2), (1, negative 1 divided 2), and (2, negative 1 divided 2). The slanted line goes through the points (0, 0), (1, negative 1 divided 2), and (2, negative 1).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_349_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows the graphs of a straight horizontal line and a straight slanted line on the same x y-coordinate plane. The x and y axes run from negative 12 to 12. The horizontal line goes through the points (0, negative 1 divided 2), (1, negative 1 divided 2), and (2, negative 1 divided 2). The slanted line goes through the points (0, 0), (1, negative 1 divided 2), and (2, negative 1).\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835301347\">\n<div data-type=\"problem\" id=\"fs-id1167835301350\">\n<p id=\"fs-id1167835301352\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e74e44beeaa92662efcc22a46416c6ca_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"68\" style=\"vertical-align: -6px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9c6dfce7ea311b94f51717efc951c245_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"56\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167835512622\"><strong data-effect=\"bold\">Find <em data-effect=\"italics\">x-<\/em> and <em data-effect=\"italics\">y-<\/em>Intercepts<\/strong><\/p>\n<p id=\"fs-id1167834219350\">In the following exercises, find the <em data-effect=\"italics\">x<\/em>&#8211; and <em data-effect=\"italics\">y<\/em>-intercepts on each graph.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167831911460\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835596622\">\n<p id=\"fs-id1167835596624\">\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167835596625\" data-alt=\"The figure shows a straight line graphed on the x y-coordinate plane. The x and y axes run from negative 8 to 8. The line goes through the points (negative 6, 9), (negative 3, 6), (0, 3), (3, 0), and (6, negative 3).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_205_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows a straight line graphed on the x y-coordinate plane. The x and y axes run from negative 8 to 8. The line goes through the points (negative 6, 9), (negative 3, 6), (0, 3), (3, 0), and (6, negative 3).\" \/><\/span><\/div>\n<div data-type=\"solution\" id=\"fs-id1167834501585\">\n<p id=\"fs-id1167834501587\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-869b99518d49b8e3bee9ba43edd3c259_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"88\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167826967394\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167826967396\">\n<p id=\"fs-id1167826967398\">\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167832076530\" data-alt=\"The figure shows a straight line graphed on the x y-coordinate plane. The x and y axes run from negative 8 to 8. The line goes through the points (negative 6, 4), (negative 4, 2), (negative 2, 0), (0, negative 2), (2, negative 4), and (4, negative 6).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_206_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows a straight line graphed on the x y-coordinate plane. The x and y axes run from negative 8 to 8. The line goes through the points (negative 6, 4), (negative 4, 2), (negative 2, 0), (0, negative 2), (2, negative 4), and (4, negative 6).\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167827943612\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167827943614\">\n<p id=\"fs-id1167827943616\">\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167827943618\" data-alt=\"The figure shows a straight line graphed on the x y-coordinate plane. The x and y axes run from negative 8 to 8. The line goes through the points (negative 1, negative 6), (0, negative 5), (2, negative 3), (5, 0), and (7, 2).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_207_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows a straight line graphed on the x y-coordinate plane. The x and y axes run from negative 8 to 8. The line goes through the points (negative 1, negative 6), (0, negative 5), (2, negative 3), (5, 0), and (7, 2).\" \/><\/span><\/div>\n<div data-type=\"solution\" id=\"fs-id1167834060239\">\n<p id=\"fs-id1167834494858\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3c1a23c4a2ce518346a024cd569116ae_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"102\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835545442\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835545444\">\n<p id=\"fs-id1167835545446\">\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167835545447\" data-alt=\"The figure shows a straight line graphed on the x y-coordinate plane. The x and y axes run from negative 8 to 8. The line goes through the points (negative 2, negative 4), (negative 1, negative 2), (0, 0), (1, 2), and (2, 4).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_208_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows a straight line graphed on the x y-coordinate plane. The x and y axes run from negative 8 to 8. The line goes through the points (negative 2, negative 4), (negative 1, negative 2), (0, 0), (1, 2), and (2, 4).\" \/><\/span><\/div>\n<\/div>\n<p id=\"fs-id1167831922487\">In the following exercises, find the intercepts for each equation.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167831922490\">\n<div data-type=\"problem\" id=\"fs-id1167831922492\">\n<p id=\"fs-id1167835353850\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f246c694f3d64b7ca3b620ac0c2bac11_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#45;&#121;&#61;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"73\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834079438\">\n<p id=\"fs-id1167834079440\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3c1a23c4a2ce518346a024cd569116ae_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"102\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831920143\">\n<div data-type=\"problem\" id=\"fs-id1167831920146\">\n<p id=\"fs-id1167831920148\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4130f8c9dac07d3b719f961e72ec907e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#45;&#121;&#61;&#45;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"88\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1167834376169\">\n<p id=\"fs-id1167834376171\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-827cfd6c83e73df785c1f64d3c021e06_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#120;&#43;&#121;&#61;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"83\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167831872331\">\n<p id=\"fs-id1167831872333\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a47dd28ce431ce4f6fd961a8ee74b456_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"88\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831071499\">\n<div data-type=\"problem\" id=\"fs-id1167831071501\">\n<p id=\"fs-id1167831071504\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-629e8a8bc3f2d8f9ede90d5b0e034675_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#45;&#50;&#121;&#61;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"83\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167832081988\">\n<div data-type=\"problem\" id=\"fs-id1167830699485\">\n<p id=\"fs-id1167830699487\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6cb7584b6cb0af341c1c860d92c901d4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#120;&#45;&#121;&#61;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"83\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835415278\">\n<p id=\"fs-id1167835415280\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f5a18e540cbc13b5e869d3e58c0158f2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#45;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"102\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835306018\">\n<div data-type=\"problem\" id=\"fs-id1167835306020\">\n<p id=\"fs-id1167835306022\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-74e2108dc21fffcf5a654cd5aa1e0820_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#120;&#45;&#121;&#61;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"82\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167830705955\">\n<div data-type=\"problem\" id=\"fs-id1167830705957\">\n<p id=\"fs-id1167830705959\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9f3f68fd17add94c24081fb0b84fc759_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#120;&#43;&#53;&#121;&#61;&#49;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"100\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167830704516\">\n<p id=\"fs-id1167830704518\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-453f0fbbab3b56bd33135c06268ee28e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"88\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835326514\">\n<div data-type=\"problem\" id=\"fs-id1167835344492\">\n<p id=\"fs-id1167835344494\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4e9f65ece64d249676d42030d4afdc1f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#120;&#45;&#50;&#121;&#61;&#49;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"99\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167834192141\"><strong data-effect=\"bold\">Graph a Line Using the Intercepts<\/strong><\/p>\n<p id=\"fs-id1167832053315\">In the following exercises, graph using the intercepts.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167832053318\">\n<div data-type=\"problem\" id=\"fs-id1167832053320\">\n<p id=\"fs-id1167832053323\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-db4a8c081ae2904ee5a9a22a8da85067_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#120;&#43;&#52;&#121;&#61;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"95\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835321857\"><span data-type=\"media\" id=\"fs-id1167835321859\" data-alt=\"The figure shows a straight line graphed on the x y-coordinate plane. The x and y axes run from negative 12 to 12. The line goes through the points (negative 8, 0), (0, 2), (4, 3), and (8, 4).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_351_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows a straight line graphed on the x y-coordinate plane. The x and y axes run from negative 12 to 12. The line goes through the points (negative 8, 0), (0, 2), (4, 3), and (8, 4).\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834506047\">\n<div data-type=\"problem\" id=\"fs-id1167834506050\">\n<p id=\"fs-id1167834506052\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5d617c009105da612113637708d4f4b7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#43;&#50;&#121;&#61;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"83\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835217987\">\n<div data-type=\"problem\" id=\"fs-id1167835217989\">\n<p id=\"fs-id1167835217992\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c5ca20ad43730fc891b9d2a2757b968d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#43;&#121;&#61;&#45;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"88\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835483756\"><span data-type=\"media\" data-alt=\"The figure shows a straight line graphed on the x y-coordinate plane. The x and y axes run from negative 12 to 12. The line goes through the points (negative 3, 0), (0, negative 3), and (3, negative 6).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_353_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows a straight line graphed on the x y-coordinate plane. The x and y axes run from negative 12 to 12. The line goes through the points (negative 3, 0), (0, negative 3), and (3, negative 6).\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831239741\">\n<div data-type=\"problem\" id=\"fs-id1167831239743\">\n<p id=\"fs-id1167831239746\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4130f8c9dac07d3b719f961e72ec907e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#45;&#121;&#61;&#45;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"88\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831239409\">\n<div data-type=\"problem\" id=\"fs-id1167835376276\">\n<p id=\"fs-id1167827958241\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-664ee5b0cec38831f83aaeeff09b841d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#120;&#43;&#121;&#61;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"83\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835524772\"><span data-type=\"media\" id=\"fs-id1167835524774\" data-alt=\"The figure shows a straight line graphed on the x y-coordinate plane. The x and y axes run from negative 12 to 12. The line goes through the points (0, 4), (1, 0), and (2, negative 4).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_355_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows a straight line graphed on the x y-coordinate plane. The x and y axes run from negative 12 to 12. The line goes through the points (0, 4), (1, 0), and (2, negative 4).\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834063389\">\n<div data-type=\"problem\" id=\"fs-id1167834063392\">\n<p id=\"fs-id1167834495437\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-334c636933233ad8becae682c430de38_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#120;&#43;&#121;&#61;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"83\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834182324\">\n<div data-type=\"problem\" id=\"fs-id1167831065894\">\n<p id=\"fs-id1167831065896\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a07491e39cfbc2a57d7a1cc5c8266c4f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#120;&#45;&#121;&#61;&#45;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"97\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835595573\"><span data-type=\"media\" id=\"fs-id1167835595575\" data-alt=\"The figure shows a straight line graphed on the x y-coordinate plane. The x and y axes run from negative 12 to 12. The line goes through the points (negative 2, 0), (negative 1, 3), and (0, 6).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_357_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows a straight line graphed on the x y-coordinate plane. The x and y axes run from negative 12 to 12. The line goes through the points (negative 2, 0), (negative 1, 3), and (0, 6).\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835512205\">\n<div data-type=\"problem\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-15b35ead2f2f32cb07260d9582e86f2f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#120;&#45;&#121;&#61;&#45;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"97\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167826967127\">\n<div data-type=\"problem\" id=\"fs-id1167826967129\">\n<p id=\"fs-id1167826967131\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3b66882a8c748fed24695ac1bb43fff5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#120;&#43;&#52;&#121;&#61;&#49;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"99\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835229406\"><span data-type=\"media\" id=\"fs-id1167835229408\" data-alt=\"The figure shows a straight line graphed on the x y-coordinate plane. The x and y axes run from negative 8 to 8. The line goes through the points (0, 3), (2, 2), and (6, 0).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_359_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows a straight line graphed on the x y-coordinate plane. The x and y axes run from negative 8 to 8. The line goes through the points (0, 3), (2, 2), and (6, 0).\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167826967305\">\n<div data-type=\"problem\" id=\"fs-id1167826967307\">\n<p id=\"fs-id1167826967309\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4fa0b37b12e2ed9b6d126e5523cd6dbe_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#120;&#45;&#50;&#121;&#61;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"91\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167826987293\">\n<div data-type=\"problem\" id=\"fs-id1167826987295\">\n<p id=\"fs-id1167826987297\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-96aa1b1af0405002221ad03086502739_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#120;&#45;&#53;&#121;&#61;&#45;&#50;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"114\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167830705513\"><span data-type=\"media\" id=\"fs-id1167830705515\" data-alt=\"The figure shows a straight line graphed on the x y-coordinate plane. The x and y axes run from negative 12 to 12. The line goes through the points (negative 10, 0), (0, 4), and (10, 8).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_361_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows a straight line graphed on the x y-coordinate plane. The x and y axes run from negative 12 to 12. The line goes through the points (negative 10, 0), (0, 4), and (10, 8).\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831891905\">\n<div data-type=\"problem\" id=\"fs-id1167835347628\">\n<p id=\"fs-id1167835347630\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b6ee8f1156ed21d958f66549e5dcf9a3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#120;&#45;&#52;&#121;&#61;&#45;&#49;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"113\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835356423\">\n<div data-type=\"problem\" id=\"fs-id1167835356425\">\n<p id=\"fs-id1167835356427\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b1860f3bc1742115ef9046a1238467de_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#45;&#50;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"66\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835327678\"><span data-type=\"media\" id=\"fs-id1167835327680\" data-alt=\"The figure shows a straight line graphed on the x y-coordinate plane. The x and y axes run from negative 12 to 12. The line goes through the points (negative 1, 2), (0, 0), and (1, negative 2).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_363_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows a straight line graphed on the x y-coordinate plane. The x and y axes run from negative 12 to 12. The line goes through the points (negative 1, 2), (0, 0), and (1, negative 2).\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831923233\">\n<div data-type=\"problem\" id=\"fs-id1167831923235\">\n<p id=\"fs-id1167831923238\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-91c10e13707f47199b50bd3f85254528_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#53;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"52\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834156928\">\n<div data-type=\"problem\" id=\"fs-id1167834156930\">\n<p id=\"fs-id1167835349567\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0a99c26c04ea6299b78366ce136d5675_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"43\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835349579\"><span data-type=\"media\" id=\"fs-id1167826799424\" data-alt=\"The figure shows a straight line graphed on the x y-coordinate plane. The x and y axes run from negative 12 to 12. The line goes through the points (negative 1, negative 1), (0, 0), and (1, 1).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_365_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows a straight line graphed on the x y-coordinate plane. The x and y axes run from negative 12 to 12. The line goes through the points (negative 1, negative 1), (0, 0), and (1, 1).\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1167834098231\">\n<p id=\"fs-id1167834098233\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-64c381a25fe27d286e35d4c136a53cbc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"43\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167835217776\"><strong data-effect=\"bold\">Mixed Practice<\/strong><\/p>\n<p>In the following exercises, graph each equation.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167835217785\">\n<div data-type=\"problem\" id=\"fs-id1167835217787\">\n<p id=\"fs-id1167834593552\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c15a64bb1e0ff79129445d69f8291fba_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#50;&#125;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"54\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834061645\"><span data-type=\"media\" id=\"fs-id1167834061647\" data-alt=\"The figure shows a straight line graphed on the x y-coordinate plane. The x and y axes run from negative 8 to 8. The line goes through the points (negative 2, negative 3), (0, 0), and (2, 3).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_367_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows a straight line graphed on the x y-coordinate plane. The x and y axes run from negative 8 to 8. The line goes through the points (negative 2, negative 3), (0, 0), and (2, 3).\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167827966889\">\n<div data-type=\"problem\" id=\"fs-id1167831040610\">\n<p id=\"fs-id1167831040612\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b8c762b607852b3ed17ae06656bfd07a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"68\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167832065980\">\n<div data-type=\"problem\" id=\"fs-id1167832065982\">\n<p id=\"fs-id1167832065984\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-36804a7858d70313deacf2883a46e44e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#120;&#43;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"98\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835512468\"><span data-type=\"media\" id=\"fs-id1167835512470\" data-alt=\"The figure shows a straight line graphed on the x y-coordinate plane. The x and y axes run from negative 8 to 8. The line goes through the points (negative 2, 4), (0, 3), (2, 2), (4, 1), and (6, 0).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_369_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows a straight line graphed on the x y-coordinate plane. The x and y axes run from negative 8 to 8. The line goes through the points (negative 2, 4), (0, 3), (2, 2), (4, 1), and (6, 0).\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167827956693\">\n<div data-type=\"problem\" id=\"fs-id1167827956695\">\n<p id=\"fs-id1167827956697\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b486331756e7260a3c1bf72d928621f9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#52;&#125;&#120;&#45;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"84\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834063745\">\n<div data-type=\"problem\" id=\"fs-id1167834063747\">\n<p id=\"fs-id1167834063750\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a10adf4a4d4580cd65cca21e24b150a1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#120;&#43;&#121;&#61;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"82\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167831922540\"><span data-type=\"media\" id=\"fs-id1167831922543\" data-alt=\"The figure shows a straight line graphed on the x y-coordinate plane. The x and y axes run from negative 8 to 8. The line goes through the points (negative 1, 6), (0, 2), (1, negative 2), and (2, negative 4).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_371_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows a straight line graphed on the x y-coordinate plane. The x and y axes run from negative 8 to 8. The line goes through the points (negative 1, 6), (0, 2), (1, negative 2), and (2, negative 4).\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834124817\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167834124822\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-79b727dd3be137db9ceda850e7b18c09_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#120;&#43;&#50;&#121;&#61;&#49;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"100\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834396194\">\n<div data-type=\"problem\" id=\"fs-id1167834396197\">\n<p id=\"fs-id1167834396199\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9f371b4e77462e28d9f6119571c92982_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"55\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834195818\"><span data-type=\"media\" id=\"fs-id1167834195821\" data-alt=\"The figure shows a horizontal straight line graphed on the x y-coordinate plane. The x and y axes run from negative 8 to 8. The line goes through the points (negative 2, negative 1), (0, negative 1), and (1, negative 1).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_373_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows a horizontal straight line graphed on the x y-coordinate plane. The x and y axes run from negative 8 to 8. The line goes through the points (negative 2, negative 1), (0, negative 1), and (1, negative 1).\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835321946\">\n<div data-type=\"problem\" id=\"fs-id1167835321948\">\n<p id=\"fs-id1167835321950\"><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;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"writing\" data-depth=\"2\" id=\"fs-id1167831911726\">\n<h4 data-type=\"title\">Writing Exercises<\/h4>\n<div data-type=\"exercise\" id=\"fs-id1167835376744\">\n<div data-type=\"problem\" id=\"fs-id1167835376746\">\n<p id=\"fs-id1167835376748\">Explain how you would choose three <em data-effect=\"italics\">x<\/em>-values to make a table to graph the line <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e68099111280b9d450421404c2aedf66_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#53;&#125;&#120;&#45;&#50;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"89\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834584361\">\n<p id=\"fs-id1167834584364\">Answers will vary.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834584369\">\n<div data-type=\"problem\" id=\"fs-id1167834584371\">\n<p id=\"fs-id1167834584373\">What is the difference between the equations of a vertical and a horizontal line?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831824603\">\n<div data-type=\"problem\" id=\"fs-id1167826828710\">\n<p id=\"fs-id1167826828712\">Do you prefer to use the method of plotting points or the method using the intercepts to graph the equation <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-59cf9b8e07e4912b2a556b21e9e84456_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#120;&#43;&#121;&#61;&#45;&#52;&#63;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"105\" style=\"vertical-align: -4px;\" \/> Why?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835322017\">\n<p id=\"fs-id1167835322019\">Answers will vary.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835322025\">\n<div data-type=\"problem\" id=\"fs-id1167835322027\">\n<p>Do you prefer to use the method of plotting points or the method using the intercepts to graph the equation <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-69795281d32beb7b3acb9fde028203cd_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;&#50;&#63;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"93\" style=\"vertical-align: -6px;\" \/> Why?<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1167835218155\">\n<h4 data-type=\"title\">Self Check<\/h4>\n<p id=\"fs-id1167835344533\"><span class=\"token\">\u24d0<\/span> After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167835344541\" data-alt=\"This table has 6 rows and 4 columns. The first row is a header row and it labels each column. The first column header is \u201cI can\u2026\u201d, the second is \u201cConfidently\u201d, the third is \u201cWith some help\u201d, and the fourth is \u201cNo, I don\u2019t get it\u201d. Under the first column are the phrases \u201cplot points on a rectangular coordinate system\u201d, \u201cgraph a linear equation by plotting points\u201d, \u201cgraph vertical and horizontal lines\u201d, \u201cfind x and y intercepts\u201d, and \u201cgraph a line using intercepts\u201d. The other columns are left blank so that the learner may indicate their mastery level for each topic.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_03_01_211_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This table has 6 rows and 4 columns. The first row is a header row and it labels each column. The first column header is \u201cI can\u2026\u201d, the second is \u201cConfidently\u201d, the third is \u201cWith some help\u201d, and the fourth is \u201cNo, I don\u2019t get it\u201d. Under the first column are the phrases \u201cplot points on a rectangular coordinate system\u201d, \u201cgraph a linear equation by plotting points\u201d, \u201cgraph vertical and horizontal lines\u201d, \u201cfind x and y intercepts\u201d, and \u201cgraph a line using intercepts\u201d. The other columns are left blank so that the learner may indicate their mastery level for each topic.\" \/><\/span><\/p>\n<p id=\"fs-id1167835340496\"><span class=\"token\">\u24d1<\/span> If most of your checks were:<\/p>\n<p id=\"fs-id1167835340504\">Confidently. 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-id1167835374300\">With some help. 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-id1167835374307\">No, I don\u2019t get it. This is a warning sign and you must address 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-id1167831076624\">\n<dt>horizontal line<\/dt>\n<dd id=\"fs-id1167826814098\">A horizontal line is the graph of an equation of the form <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-27334f3a0485b8c07e8c9ebc0be4df6b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#98;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"45\" style=\"vertical-align: -4px;\" \/> The line passes through the <em data-effect=\"italics\">y<\/em>-axis at <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5bac775607ac99b49b72fd1654a94604_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"44\" style=\"vertical-align: -4px;\" \/><\/dd>\n<\/dl>\n<dl id=\"fs-id1167835355558\">\n<dt>intercepts of a line<\/dt>\n<dd id=\"fs-id1167835355563\">The points where a line crosses the <em data-effect=\"italics\">x<\/em>-axis and the <em data-effect=\"italics\">y<\/em>-axis are called the intercepts of the line.<\/dd>\n<\/dl>\n<dl id=\"fs-id1167834064606\">\n<dt>linear equation<\/dt>\n<dd id=\"fs-id1167834357135\">An equation of the form <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8c3fc394aedf0a1f1b44fa47faf55523_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#65;&#120;&#43;&#66;&#121;&#61;&#67;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"109\" style=\"vertical-align: -4px;\" \/> where <em data-effect=\"italics\">A<\/em> and <em data-effect=\"italics\">B<\/em> are not both zero, is called a linear equation in two variables.<\/dd>\n<\/dl>\n<dl id=\"fs-id1167834156774\">\n<dt>ordered pair<\/dt>\n<dd id=\"fs-id1167835533863\">An ordered pair, <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;\" \/> gives the coordinates of a point in a rectangular coordinate system. The first number is the <em data-effect=\"italics\">x<\/em>-coordinate. The second number is the <em data-effect=\"italics\">y<\/em>-coordinate.<\/dd>\n<\/dl>\n<dl id=\"fs-id1167834053603\">\n<dt>origin<\/dt>\n<dd id=\"fs-id1167831239720\">The point <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;\" \/> is called the origin. It is the point where the <em data-effect=\"italics\">x<\/em>-axis and <em data-effect=\"italics\">y<\/em>-axis intersect.<\/dd>\n<\/dl>\n<dl id=\"fs-id1167834505886\">\n<dt>solution of a linear equation in two variables<\/dt>\n<dd id=\"fs-id1167831944043\">An ordered pair <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;\" \/> is a solution of the linear equation<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8c3fc394aedf0a1f1b44fa47faf55523_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#65;&#120;&#43;&#66;&#121;&#61;&#67;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"109\" style=\"vertical-align: -4px;\" \/> if the equation is a true statement when the <em data-effect=\"italics\">x<\/em>&#8211; and <em data-effect=\"italics\">y<\/em>-values of the ordered pair are substituted into the equation.<\/dd>\n<\/dl>\n<dl id=\"fs-id1167835363804\">\n<dt>standard form of a linear equation<\/dt>\n<dd id=\"fs-id1167835363810\">A linear equation is in standard form when it is written <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f8784c7e5aaf20bf04cc889b8ea7b8b0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#65;&#120;&#43;&#66;&#121;&#61;&#67;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"109\" style=\"vertical-align: -4px;\" \/><\/dd>\n<\/dl>\n<dl id=\"fs-id1167835231613\">\n<dt>vertical line<\/dt>\n<dd id=\"fs-id1167835231619\">A vertical line is the graph of an equation of the form <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5c2ae728b9a161e044125a0dc6c913c3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#97;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"47\" style=\"vertical-align: 0px;\" \/> The line passes through the <em data-effect=\"italics\">x<\/em>-axis at <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-efddd49bc1c2dc39c3fad2e3635247c3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"46\" style=\"vertical-align: -4px;\" \/><\/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-1744","chapter","type-chapter","status-publish","hentry"],"part":1643,"_links":{"self":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters\/1744","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\/1744\/revisions"}],"predecessor-version":[{"id":15148,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters\/1744\/revisions\/15148"}],"part":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/parts\/1643"}],"metadata":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters\/1744\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/media?parent=1744"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapter-type?post=1744"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/contributor?post=1744"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/license?post=1744"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}