{"id":10683,"date":"2017-06-05T14:59:54","date_gmt":"2017-06-05T14:59:54","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/prealgebra\/?post_type=chapter&#038;p=10683"},"modified":"2024-04-30T23:16:38","modified_gmt":"2024-04-30T23:16:38","slug":"finding-and-using-the-intercepts-from-the-equation-of-a-line","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/chapter\/finding-and-using-the-intercepts-from-the-equation-of-a-line\/","title":{"raw":"Graphing Lines Using X- and Y- Intercepts","rendered":"Graphing Lines Using X- and Y- Intercepts"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Graph a linear equation using x- and y-intercepts<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h3>Graph a Line Using the Intercepts<\/h3>\r\nTo graph a linear equation by plotting points, you can use the intercepts as two of your three points. Find the two intercepts, and then a third point to ensure accuracy, and draw the line. This method is often the quickest way to graph a line.\r\n<div class=\"textbox shaded\">\r\n<h3>Graph a line using the intercepts<\/h3>\r\n<ol id=\"eip-id1168468755039\" class=\"stepwise\">\r\n \t<li>Find the [latex]x-[\/latex] and [latex]\\text{y-intercepts}[\/latex] of the line.\r\n<ul id=\"fs-id1365900\">\r\n \t<li>Let [latex]y=0[\/latex] and solve for [latex]x[\/latex]<\/li>\r\n \t<li>Let [latex]x=0[\/latex] and solve for [latex]y[\/latex].<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>Find a third solution to the equation.<\/li>\r\n \t<li>Plot the three points and then check that they line up.<\/li>\r\n \t<li>Draw the line.<\/li>\r\n<\/ol>\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nGraph [latex]-x+2y=6[\/latex] using intercepts.\r\n\r\nSolution\r\nFirst, find the [latex]x\\text{-intercept}[\/latex].\r\n<p style=\"text-align: center;\">Let [latex]y=0[\/latex],\r\n[latex]\\begin{array}{}\\\\ -x+2y=6\\\\ -x+2\\left(0\\right)=6\\\\ -x=6\\\\ x=-6\\end{array}[\/latex]\r\nThe [latex]x\\text{-intercept}[\/latex] is [latex]\\left(-6,0\\right)[\/latex].<\/p>\r\nNow find the [latex]y\\text{-intercept}[\/latex].\r\n<p style=\"text-align: center;\">Let [latex]x=0[\/latex].\r\n[latex]\\begin{array}{}\\\\ -x+2y=6\\\\ -0+2y=6\\\\ \\\\ \\\\ 2y=6\\\\ y=3\\end{array}[\/latex]\r\nThe [latex]y\\text{-intercept}[\/latex] is [latex]\\left(0,3\\right)[\/latex].<\/p>\r\nFind a third point.\r\n<p style=\"text-align: center;\">We\u2019ll use [latex]x=2[\/latex],\r\n[latex]\\begin{array}{}\\\\ -x+2y=6\\\\ -2+2y=6\\\\ \\\\ \\\\ 2y=8\\\\ y=4\\end{array}[\/latex]\r\nA third solution to the equation is [latex]\\left(2,4\\right)[\/latex].<\/p>\r\nSummarize the three points in a table and then plot them on a graph.\r\n<table id=\"fs-id1822511\" class=\"unnumbered\" summary=\"This table it titled - x + 2 y = 6. It has 4 rows and 3 columns. The first row is a header row and it labels each column \">\r\n<thead>\r\n<tr valign=\"top\">\r\n<th colspan=\"3\">[latex]-x+2y=6[\/latex]<\/th>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<th><em><strong>x<\/strong><\/em><\/th>\r\n<th><em><strong>y<\/strong><\/em><\/th>\r\n<th><em><strong>(x,y)<\/strong><\/em><\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td>[latex]-6[\/latex]<\/td>\r\n<td>[latex]0[\/latex]<\/td>\r\n<td>[latex]\\left(-6,0\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]0[\/latex]<\/td>\r\n<td>[latex]3[\/latex]<\/td>\r\n<td>[latex]\\left(0,3\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]2[\/latex]<\/td>\r\n<td>[latex]4[\/latex]<\/td>\r\n<td>[latex]\\left(2,4\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/25224431\/CNX_BMath_Figure_11_03_009.png\" alt=\"The graph shows the x y-coordinate plane. The x and y-axis each run from -10 to 10. Three labeled points are shown at \u201cordered pair -6, 0\u201d, \u201cordered pair 0, 3\u201d and \u201cordered pair 2, 4\u201d.\" width=\"264\" height=\"270\" \/>\r\nDo the points line up? Yes, so draw line through the points.\r\n\r\n<img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/25224432\/CNX_BMath_Figure_11_03_010.png\" alt=\"The graph shows the x y-coordinate plane. The x and y-axis each run from -10 to 10. Three labeled points are shown at \u201cordered pair -6, 0\u201d, \u201cordered pair 0, 3\u201d and \u201cordered pair 2, 4\u201d. A line passes through the three labeled points.\" width=\"263\" height=\"270\" \/>\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nGraph [latex]5y+3x=30[\/latex]\u00a0using the <em>x<\/em> and <em>y<\/em>-intercepts.\r\n\r\n[reveal-answer q=\"153435\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"153435\"]When an equation is in [latex]Ax+By=C[\/latex]\u00a0form, you can easily find the <i>x<\/i>- and <i>y<\/i>-intercepts and then graph.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}5y+3x=30\\\\5y+3\\left(0\\right)=30\\\\5y+0=30\\\\5y=30\\\\y=\\,\\,\\,6\\\\y\\text{-intercept}\\,\\left(0,6\\right)\\end{array}[\/latex]<\/p>\r\nTo find the <i>y<\/i>-intercept, set [latex]x=0[\/latex]\u00a0and solve for <i>y<\/i>.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}5y+3x=30\\\\5\\left(0\\right)+3x=30\\\\0+3x=30\\\\3x=30\\\\x=10\\\\x\\text{-intercept}\\left(10,0\\right)\\end{array}[\/latex]<\/p>\r\nTo find the <i>x<\/i>-intercept, set [latex]y=0[\/latex] and solve for <i>x<\/i>.\r\n<h4>Answer<\/h4>\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064251\/image020-1.jpg\" alt=\"The graph shows the x y-coordinate plane. The x and y-axis are labeled -2 to 12 and go up by increments of one. Two labeled points are shown at \u201cordered pair 10, 0\u201d and \u201cordered pair 0, 6\u201d. A line passes through the two labeled points. The equation 5 y plus 3 x equals 30 is also written next to the line.\" width=\"425\" height=\"430\" \/>[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146999[\/ohm_question]\r\n\r\n<\/div>\r\nWatch the following video for more on how to graph a line using the intercepts.\r\n\r\nhttps:\/\/youtu.be\/k8r-q_T6UFk\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>example<\/h3>\r\nGraph [latex]4x - 3y=12[\/latex] using intercepts.\r\n[reveal-answer q=\"293852\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"293852\"]\r\n\r\nSolution\r\nFind the intercepts and a third point.\r\n\r\n<img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/25224438\/CNX_BMath_Figure_11_03_016_img.png\" alt=\"The figure shows 3 solutions to the equation 4 x - 3 y = 12. The first is titled \u201cx-intercept, let y = 0\u201d. The first line is 4 x - 3 y = 12. The second line shows 0 in red substituted for y, reading 4 x - 3 open parentheses 0 closed parentheses = 12. The third line is 4 x = 12. The last line is x = 3. The second solution is titled \u201cy-intercept, let x = 0\u201d. The first line is 4 x - 3 y = 12. The second line shows 0 in red substituted for x, reading 4 open parentheses 0 closed parentheses - 3 y = 12. The third line is -3 y = 12. The last line is y = -4. The third solution is titled \u201cthird point, let y = 4\u201d. The first line is 4 x - 3 y = 12. The second line shows 4 in red substituted for y, reading 4 x - 3 open parentheses 4 closed parentheses = 12. The third line is 4 x - 12 = 12. The last line is x = 6.\" width=\"524\" height=\"174\" \/>\r\nWe list the points and show the graph.\r\n<table id=\"fs-id1715615\" class=\"unnumbered\" summary=\"This table it titled 4 x - 3 y = 12. It has 4 rows and 3 columns. The first row is a header row and it labels each column \">\r\n<thead>\r\n<tr valign=\"top\">\r\n<th colspan=\"3\">[latex]4x - 3y=12[\/latex]<\/th>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<th>[latex]x[\/latex]<\/th>\r\n<th>[latex]y[\/latex]<\/th>\r\n<th>[latex]\\left(x,y\\right)[\/latex]<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td>[latex]3[\/latex]<\/td>\r\n<td>[latex]0[\/latex]<\/td>\r\n<td>[latex]\\left(3,0\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]0[\/latex]<\/td>\r\n<td>[latex]-4[\/latex]<\/td>\r\n<td>[latex]\\left(0,-4\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]6[\/latex]<\/td>\r\n<td>[latex]4[\/latex]<\/td>\r\n<td>[latex]\\left(6,4\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/25224440\/CNX_BMath_Figure_11_03_017_img.png\" alt=\"The graph shows the x y-coordinate plane. Both axes run from -7 to 7. Three unlabeled points are drawn at \u201cordered pair 0, -4\u201d, \u201cordered pair 3, 0\u201d and \u201cordered pair 6, 4\u201d. A line passes through the points.\" width=\"219\" height=\"225\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146998[\/ohm_question]\r\n\r\n<\/div>\r\nIn the next example, there is only one intercept because the line passes through the point (0,0).\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nGraph [latex]y=5x[\/latex] using the intercepts.\r\n[reveal-answer q=\"821667\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"821667\"]\r\n\r\nSolution\r\n\r\n<img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/25224444\/CNX_BMath_Figure_11_03_020_img.png\" alt=\"The figure shows 2 solutions to y = 5 x. The first solution is titled \u201cx-intercept; Let y = 0.\u201d The first line is y = 5 x. The second line is 0, shown in red, = 5 x. The third line is 0 = x. The fourth line is x = 0. The last line is \u201cThe x-intercept is \u201cordered pair 0, 0\u201d. The second solution is titled \u201cy-intercept; Let x = 0.\u201d The first line is y = 5 x. The second line is y = 5 open parentheses 0, shown in red, closed parentheses. The third line is y = 0. The last line is \u201cThe y-intercept is \u201cordered pair 0, 0\u201d.\" width=\"392\" height=\"176\" \/>\r\nThis line has only one intercept! It is the point [latex]\\left(0,0\\right)[\/latex].\r\n\r\nTo ensure accuracy, we need to plot three points. Since the intercepts are the same point, we need two more points to graph the line. As always, we can choose any values for [latex]x[\/latex], so we\u2019ll let [latex]x[\/latex] be [latex]1[\/latex] and [latex]-1[\/latex].\r\n\r\n<img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/25224446\/CNX_BMath_Figure_11_03_021_img.png\" alt=\"The figure shows two substitutions in the equation y = 5 x. In the first substitution, the first line is y = 5 x. The second line is y = 5 open parentheses 1, shown in red, closed parentheses. The third line is y =5. The last line is \u201cordered pair 1, 5\u201d. In the second substitution, the first line is y = 5 x. The second line is y = 5 open parentheses -1, shown in red, closed parentheses. The third line is y = -5. The last line is \u201cordered pair -1, -5\u201d.\" width=\"143\" height=\"121\" \/>\r\nOrganize the points in a table.\r\n<table id=\"fs-id1363944\" class=\"unnumbered\" style=\"height: 60px;\" summary=\"This table it titled y = 5 x. It has 4 rows and 3 columns. The first row is a header row and it labels each column \">\r\n<thead>\r\n<tr style=\"height: 12px;\" valign=\"top\">\r\n<th style=\"height: 12px; width: 454.4px;\" colspan=\"3\">[latex]y=5x[\/latex]<\/th>\r\n<\/tr>\r\n<tr style=\"height: 12px;\" valign=\"top\">\r\n<th style=\"height: 12px; width: 112.875px;\">[latex]x[\/latex]<\/th>\r\n<th style=\"height: 12px; width: 112.875px;\">[latex]y[\/latex]<\/th>\r\n<th style=\"height: 12px; width: 205.95px;\">[latex]\\left(x,y\\right)[\/latex]<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr style=\"height: 12px;\" valign=\"top\">\r\n<td style=\"height: 12px; width: 112.875px;\">[latex]0[\/latex]<\/td>\r\n<td style=\"height: 12px; width: 112.875px;\">[latex]0[\/latex]<\/td>\r\n<td style=\"height: 12px; width: 205.95px;\">[latex]\\left(0,0\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\" valign=\"top\">\r\n<td style=\"height: 12px; width: 112.875px;\">[latex]1[\/latex]<\/td>\r\n<td style=\"height: 12px; width: 112.875px;\">[latex]5[\/latex]<\/td>\r\n<td style=\"height: 12px; width: 205.95px;\">[latex]\\left(1,5\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\" valign=\"top\">\r\n<td style=\"height: 12px; width: 112.875px;\">[latex]-1[\/latex]<\/td>\r\n<td style=\"height: 12px; width: 112.875px;\">[latex]-5[\/latex]<\/td>\r\n<td style=\"height: 12px; width: 205.95px;\">[latex]\\left(-1,-5\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nPlot the three points, check that they line up, and draw the line.\r\n\r\n<img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/25224447\/CNX_BMath_Figure_11_03_013_img.png\" alt=\"The graph shows the x y-coordinate plane. The x and y-axis each run from -10 to 10. A line passes through three labeled points, \u201cordered pair -1, -5\u201d, \u201cordered pair 0, 0\u201d, and ordered pair 1, 5\u201d.\" width=\"264\" height=\"270\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]147001[\/ohm_question]\r\n\r\n<\/div>\r\nIn the following video we show another example of how to plot a line using the intercepts of the line.\r\n\r\nhttps:\/\/youtu.be\/Wgpq0zO6z3o\r\n<h2>Choosing the Most Convenient Way to Graph a Line Given an Equation<\/h2>\r\nWhile we could graph any linear equation by plotting points, it may not always be the most convenient method. This table shows six of equations we\u2019ve graphed in this chapter, and the methods we used to graph them.\r\n<table id=\"eip-650\" class=\"unnumbered\" summary=\"...\">\r\n<thead>\r\n<tr>\r\n<th><strong>Equation<\/strong><\/th>\r\n<th><strong>Method<\/strong><\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>#1<\/td>\r\n<td>[latex]y=2x+1[\/latex]<\/td>\r\n<td>Plotting points<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>#2<\/td>\r\n<td>[latex]y=\\Large\\frac{1}{2}\\normalsize x+3[\/latex]<\/td>\r\n<td>Plotting points<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>#3<\/td>\r\n<td>[latex]x=-7[\/latex]<\/td>\r\n<td>Vertical line<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>#4<\/td>\r\n<td>[latex]y=4[\/latex]<\/td>\r\n<td>Horizontal line<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>#5<\/td>\r\n<td>[latex]2x+y=6[\/latex]<\/td>\r\n<td>Intercepts<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>#6<\/td>\r\n<td>[latex]4x - 3y=12[\/latex]<\/td>\r\n<td>Intercepts<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nWhat is it about the form of equation that can help us choose the most convenient method to graph its line?\r\n\r\nNotice that in equations #1 and #2, <em>y<\/em> is isolated on one side of the equation, and its coefficient is 1. We found points by substituting values for <em>x<\/em> on the right side of the equation and then simplifying to get the corresponding <em>y-<\/em> values.\r\n\r\nEquations #3 and #4 each have just one variable. Remember, in this kind of equation the value of that one variable is constant; it does not depend on the value of the other variable. Equations of this form have graphs that are vertical or horizontal lines.\r\n\r\nIn equations #5 and #6, both <em>x<\/em> and <em>y<\/em> are on the same side of the equation. These two equations are of the form [latex]Ax+By=C[\/latex] . We substituted [latex]y=0[\/latex] and [latex]x=0[\/latex] to find the <em>x-<\/em> and <em>y-<\/em> intercepts, and then found a third point by choosing a value for <em>x<\/em> or <em>y<\/em>.\r\n\r\nThis leads to the following strategy for choosing the most convenient method to graph a line.\r\n<div class=\"textbox shaded\">\r\n<h3>Choose the most convenient method to graph a line<\/h3>\r\n<ol id=\"eip-id1168467125546\" class=\"stepwise\">\r\n \t<li>If the equation has only one variable. It is a vertical or horizontal line.\r\n<ul id=\"fs-id1715836\">\r\n \t<li>[latex]x=a[\/latex] is a vertical line passing through the [latex]x\\text{-axis}[\/latex] at [latex]a[\/latex]<\/li>\r\n \t<li>[latex]y=b[\/latex] is a horizontal line passing through the [latex]y\\text{-axis}[\/latex] at [latex]b[\/latex].<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>If [latex]y[\/latex] is isolated on one side of the equation. Graph by plotting points.\r\n<ul id=\"fs-id1715895\">\r\n \t<li>Choose any three values for [latex]x[\/latex] and then solve for the corresponding [latex]y\\text{-}[\/latex] values.<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>If the equation is of the form [latex]Ax+By=C[\/latex], find the intercepts.\r\n<ul id=\"fs-id1715124\">\r\n \t<li>Find the [latex]x\\text{-}[\/latex] and [latex]y\\text{-}[\/latex] intercepts and then a third point.<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ol>\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nIdentify the most convenient method to graph each line:\r\n\r\n1. [latex]y=-3[\/latex]\r\n2. [latex]4x - 6y=12[\/latex]\r\n3. [latex]x=2[\/latex]\r\n4. [latex]y=\\frac{2}{5}x - 1[\/latex]\r\n\r\nSolution\r\n1. [latex]y=-3[\/latex]\r\nThis equation has only one variable, [latex]y[\/latex]. Its graph is a horizontal line crossing the [latex]y\\text{-axis}[\/latex] at [latex]-3[\/latex].\r\n2. [latex]4x - 6y=12[\/latex]\r\nThis equation is of the form [latex]Ax+By=C[\/latex]. Find the intercepts and one more point.\r\n3. [latex]x=2[\/latex]\r\nThere is only one variable, [latex]x[\/latex]. The graph is a vertical line crossing the [latex]x\\text{-axis}[\/latex] at [latex]2[\/latex].\r\n4. [latex]y=\\Large\\frac{2}{5}\\normalsize x - 1[\/latex]\r\nSince [latex]y[\/latex] is isolated on the left side of the equation, it will be easiest to graph this line by plotting three points.\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]147002[\/ohm_question]\r\n\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Graph a linear equation using x- and y-intercepts<\/li>\n<\/ul>\n<\/div>\n<h3>Graph a Line Using the Intercepts<\/h3>\n<p>To graph a linear equation by plotting points, you can use the intercepts as two of your three points. Find the two intercepts, and then a third point to ensure accuracy, and draw the line. This method is often the quickest way to graph a line.<\/p>\n<div class=\"textbox shaded\">\n<h3>Graph a line using the intercepts<\/h3>\n<ol id=\"eip-id1168468755039\" class=\"stepwise\">\n<li>Find the [latex]x-[\/latex] and [latex]\\text{y-intercepts}[\/latex] of the line.\n<ul id=\"fs-id1365900\">\n<li>Let [latex]y=0[\/latex] and solve for [latex]x[\/latex]<\/li>\n<li>Let [latex]x=0[\/latex] and solve for [latex]y[\/latex].<\/li>\n<\/ul>\n<\/li>\n<li>Find a third solution to the equation.<\/li>\n<li>Plot the three points and then check that they line up.<\/li>\n<li>Draw the line.<\/li>\n<\/ol>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Graph [latex]-x+2y=6[\/latex] using intercepts.<\/p>\n<p>Solution<br \/>\nFirst, find the [latex]x\\text{-intercept}[\/latex].<\/p>\n<p style=\"text-align: center;\">Let [latex]y=0[\/latex],<br \/>\n[latex]\\begin{array}{}\\\\ -x+2y=6\\\\ -x+2\\left(0\\right)=6\\\\ -x=6\\\\ x=-6\\end{array}[\/latex]<br \/>\nThe [latex]x\\text{-intercept}[\/latex] is [latex]\\left(-6,0\\right)[\/latex].<\/p>\n<p>Now find the [latex]y\\text{-intercept}[\/latex].<\/p>\n<p style=\"text-align: center;\">Let [latex]x=0[\/latex].<br \/>\n[latex]\\begin{array}{}\\\\ -x+2y=6\\\\ -0+2y=6\\\\ \\\\ \\\\ 2y=6\\\\ y=3\\end{array}[\/latex]<br \/>\nThe [latex]y\\text{-intercept}[\/latex] is [latex]\\left(0,3\\right)[\/latex].<\/p>\n<p>Find a third point.<\/p>\n<p style=\"text-align: center;\">We\u2019ll use [latex]x=2[\/latex],<br \/>\n[latex]\\begin{array}{}\\\\ -x+2y=6\\\\ -2+2y=6\\\\ \\\\ \\\\ 2y=8\\\\ y=4\\end{array}[\/latex]<br \/>\nA third solution to the equation is [latex]\\left(2,4\\right)[\/latex].<\/p>\n<p>Summarize the three points in a table and then plot them on a graph.<\/p>\n<table id=\"fs-id1822511\" class=\"unnumbered\" summary=\"This table it titled - x + 2 y = 6. It has 4 rows and 3 columns. The first row is a header row and it labels each column\">\n<thead>\n<tr valign=\"top\">\n<th colspan=\"3\">[latex]-x+2y=6[\/latex]<\/th>\n<\/tr>\n<tr valign=\"top\">\n<th><em><strong>x<\/strong><\/em><\/th>\n<th><em><strong>y<\/strong><\/em><\/th>\n<th><em><strong>(x,y)<\/strong><\/em><\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td>[latex]-6[\/latex]<\/td>\n<td>[latex]0[\/latex]<\/td>\n<td>[latex]\\left(-6,0\\right)[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]0[\/latex]<\/td>\n<td>[latex]3[\/latex]<\/td>\n<td>[latex]\\left(0,3\\right)[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]2[\/latex]<\/td>\n<td>[latex]4[\/latex]<\/td>\n<td>[latex]\\left(2,4\\right)[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/25224431\/CNX_BMath_Figure_11_03_009.png\" alt=\"The graph shows the x y-coordinate plane. The x and y-axis each run from -10 to 10. Three labeled points are shown at \u201cordered pair -6, 0\u201d, \u201cordered pair 0, 3\u201d and \u201cordered pair 2, 4\u201d.\" width=\"264\" height=\"270\" \/><br \/>\nDo the points line up? Yes, so draw line through the points.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/25224432\/CNX_BMath_Figure_11_03_010.png\" alt=\"The graph shows the x y-coordinate plane. The x and y-axis each run from -10 to 10. Three labeled points are shown at \u201cordered pair -6, 0\u201d, \u201cordered pair 0, 3\u201d and \u201cordered pair 2, 4\u201d. A line passes through the three labeled points.\" width=\"263\" height=\"270\" \/><\/p>\n<\/div>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Graph [latex]5y+3x=30[\/latex]\u00a0using the <em>x<\/em> and <em>y<\/em>-intercepts.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q153435\">Show Solution<\/span><\/p>\n<div id=\"q153435\" class=\"hidden-answer\" style=\"display: none\">When an equation is in [latex]Ax+By=C[\/latex]\u00a0form, you can easily find the <i>x<\/i>&#8211; and <i>y<\/i>-intercepts and then graph.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}5y+3x=30\\\\5y+3\\left(0\\right)=30\\\\5y+0=30\\\\5y=30\\\\y=\\,\\,\\,6\\\\y\\text{-intercept}\\,\\left(0,6\\right)\\end{array}[\/latex]<\/p>\n<p>To find the <i>y<\/i>-intercept, set [latex]x=0[\/latex]\u00a0and solve for <i>y<\/i>.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}5y+3x=30\\\\5\\left(0\\right)+3x=30\\\\0+3x=30\\\\3x=30\\\\x=10\\\\x\\text{-intercept}\\left(10,0\\right)\\end{array}[\/latex]<\/p>\n<p>To find the <i>x<\/i>-intercept, set [latex]y=0[\/latex] and solve for <i>x<\/i>.<\/p>\n<h4>Answer<\/h4>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064251\/image020-1.jpg\" alt=\"The graph shows the x y-coordinate plane. The x and y-axis are labeled -2 to 12 and go up by increments of one. Two labeled points are shown at \u201cordered pair 10, 0\u201d and \u201cordered pair 0, 6\u201d. A line passes through the two labeled points. The equation 5 y plus 3 x equals 30 is also written next to the line.\" width=\"425\" height=\"430\" \/><\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146999\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146999&theme=oea&iframe_resize_id=ohm146999&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>Watch the following video for more on how to graph a line using the intercepts.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Graph Linear Equations Using Intercepts\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/k8r-q_T6UFk?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>example<\/h3>\n<p>Graph [latex]4x - 3y=12[\/latex] using intercepts.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q293852\">Show Solution<\/span><\/p>\n<div id=\"q293852\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<br \/>\nFind the intercepts and a third point.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/25224438\/CNX_BMath_Figure_11_03_016_img.png\" alt=\"The figure shows 3 solutions to the equation 4 x - 3 y = 12. The first is titled \u201cx-intercept, let y = 0\u201d. The first line is 4 x - 3 y = 12. The second line shows 0 in red substituted for y, reading 4 x - 3 open parentheses 0 closed parentheses = 12. The third line is 4 x = 12. The last line is x = 3. The second solution is titled \u201cy-intercept, let x = 0\u201d. The first line is 4 x - 3 y = 12. The second line shows 0 in red substituted for x, reading 4 open parentheses 0 closed parentheses - 3 y = 12. The third line is -3 y = 12. The last line is y = -4. The third solution is titled \u201cthird point, let y = 4\u201d. The first line is 4 x - 3 y = 12. The second line shows 4 in red substituted for y, reading 4 x - 3 open parentheses 4 closed parentheses = 12. The third line is 4 x - 12 = 12. The last line is x = 6.\" width=\"524\" height=\"174\" \/><br \/>\nWe list the points and show the graph.<\/p>\n<table id=\"fs-id1715615\" class=\"unnumbered\" summary=\"This table it titled 4 x - 3 y = 12. It has 4 rows and 3 columns. The first row is a header row and it labels each column\">\n<thead>\n<tr valign=\"top\">\n<th colspan=\"3\">[latex]4x - 3y=12[\/latex]<\/th>\n<\/tr>\n<tr valign=\"top\">\n<th>[latex]x[\/latex]<\/th>\n<th>[latex]y[\/latex]<\/th>\n<th>[latex]\\left(x,y\\right)[\/latex]<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td>[latex]3[\/latex]<\/td>\n<td>[latex]0[\/latex]<\/td>\n<td>[latex]\\left(3,0\\right)[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]0[\/latex]<\/td>\n<td>[latex]-4[\/latex]<\/td>\n<td>[latex]\\left(0,-4\\right)[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]6[\/latex]<\/td>\n<td>[latex]4[\/latex]<\/td>\n<td>[latex]\\left(6,4\\right)[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/25224440\/CNX_BMath_Figure_11_03_017_img.png\" alt=\"The graph shows the x y-coordinate plane. Both axes run from -7 to 7. Three unlabeled points are drawn at \u201cordered pair 0, -4\u201d, \u201cordered pair 3, 0\u201d and \u201cordered pair 6, 4\u201d. A line passes through the points.\" width=\"219\" height=\"225\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146998\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146998&theme=oea&iframe_resize_id=ohm146998&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>In the next example, there is only one intercept because the line passes through the point (0,0).<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Graph [latex]y=5x[\/latex] using the intercepts.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q821667\">Show Solution<\/span><\/p>\n<div id=\"q821667\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/25224444\/CNX_BMath_Figure_11_03_020_img.png\" alt=\"The figure shows 2 solutions to y = 5 x. The first solution is titled \u201cx-intercept; Let y = 0.\u201d The first line is y = 5 x. The second line is 0, shown in red, = 5 x. The third line is 0 = x. The fourth line is x = 0. The last line is \u201cThe x-intercept is \u201cordered pair 0, 0\u201d. The second solution is titled \u201cy-intercept; Let x = 0.\u201d The first line is y = 5 x. The second line is y = 5 open parentheses 0, shown in red, closed parentheses. The third line is y = 0. The last line is \u201cThe y-intercept is \u201cordered pair 0, 0\u201d.\" width=\"392\" height=\"176\" \/><br \/>\nThis line has only one intercept! It is the point [latex]\\left(0,0\\right)[\/latex].<\/p>\n<p>To ensure accuracy, we need to plot three points. Since the intercepts are the same point, we need two more points to graph the line. As always, we can choose any values for [latex]x[\/latex], so we\u2019ll let [latex]x[\/latex] be [latex]1[\/latex] and [latex]-1[\/latex].<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/25224446\/CNX_BMath_Figure_11_03_021_img.png\" alt=\"The figure shows two substitutions in the equation y = 5 x. In the first substitution, the first line is y = 5 x. The second line is y = 5 open parentheses 1, shown in red, closed parentheses. The third line is y =5. The last line is \u201cordered pair 1, 5\u201d. In the second substitution, the first line is y = 5 x. The second line is y = 5 open parentheses -1, shown in red, closed parentheses. The third line is y = -5. The last line is \u201cordered pair -1, -5\u201d.\" width=\"143\" height=\"121\" \/><br \/>\nOrganize the points in a table.<\/p>\n<table id=\"fs-id1363944\" class=\"unnumbered\" style=\"height: 60px;\" summary=\"This table it titled y = 5 x. It has 4 rows and 3 columns. The first row is a header row and it labels each column\">\n<thead>\n<tr style=\"height: 12px;\" valign=\"top\">\n<th style=\"height: 12px; width: 454.4px;\" colspan=\"3\">[latex]y=5x[\/latex]<\/th>\n<\/tr>\n<tr style=\"height: 12px;\" valign=\"top\">\n<th style=\"height: 12px; width: 112.875px;\">[latex]x[\/latex]<\/th>\n<th style=\"height: 12px; width: 112.875px;\">[latex]y[\/latex]<\/th>\n<th style=\"height: 12px; width: 205.95px;\">[latex]\\left(x,y\\right)[\/latex]<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr style=\"height: 12px;\" valign=\"top\">\n<td style=\"height: 12px; width: 112.875px;\">[latex]0[\/latex]<\/td>\n<td style=\"height: 12px; width: 112.875px;\">[latex]0[\/latex]<\/td>\n<td style=\"height: 12px; width: 205.95px;\">[latex]\\left(0,0\\right)[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 12px;\" valign=\"top\">\n<td style=\"height: 12px; width: 112.875px;\">[latex]1[\/latex]<\/td>\n<td style=\"height: 12px; width: 112.875px;\">[latex]5[\/latex]<\/td>\n<td style=\"height: 12px; width: 205.95px;\">[latex]\\left(1,5\\right)[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 12px;\" valign=\"top\">\n<td style=\"height: 12px; width: 112.875px;\">[latex]-1[\/latex]<\/td>\n<td style=\"height: 12px; width: 112.875px;\">[latex]-5[\/latex]<\/td>\n<td style=\"height: 12px; width: 205.95px;\">[latex]\\left(-1,-5\\right)[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Plot the three points, check that they line up, and draw the line.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/25224447\/CNX_BMath_Figure_11_03_013_img.png\" alt=\"The graph shows the x y-coordinate plane. The x and y-axis each run from -10 to 10. A line passes through three labeled points, \u201cordered pair -1, -5\u201d, \u201cordered pair 0, 0\u201d, and ordered pair 1, 5\u201d.\" width=\"264\" height=\"270\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm147001\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=147001&theme=oea&iframe_resize_id=ohm147001&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>In the following video we show another example of how to plot a line using the intercepts of the line.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Ex 1:  Graph a Linear Equation in Standard Form Using the Intercepts\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/Wgpq0zO6z3o?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Choosing the Most Convenient Way to Graph a Line Given an Equation<\/h2>\n<p>While we could graph any linear equation by plotting points, it may not always be the most convenient method. This table shows six of equations we\u2019ve graphed in this chapter, and the methods we used to graph them.<\/p>\n<table id=\"eip-650\" class=\"unnumbered\" summary=\"...\">\n<thead>\n<tr>\n<th><strong>Equation<\/strong><\/th>\n<th><strong>Method<\/strong><\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>#1<\/td>\n<td>[latex]y=2x+1[\/latex]<\/td>\n<td>Plotting points<\/td>\n<\/tr>\n<tr>\n<td>#2<\/td>\n<td>[latex]y=\\Large\\frac{1}{2}\\normalsize x+3[\/latex]<\/td>\n<td>Plotting points<\/td>\n<\/tr>\n<tr>\n<td>#3<\/td>\n<td>[latex]x=-7[\/latex]<\/td>\n<td>Vertical line<\/td>\n<\/tr>\n<tr>\n<td>#4<\/td>\n<td>[latex]y=4[\/latex]<\/td>\n<td>Horizontal line<\/td>\n<\/tr>\n<tr>\n<td>#5<\/td>\n<td>[latex]2x+y=6[\/latex]<\/td>\n<td>Intercepts<\/td>\n<\/tr>\n<tr>\n<td>#6<\/td>\n<td>[latex]4x - 3y=12[\/latex]<\/td>\n<td>Intercepts<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>What is it about the form of equation that can help us choose the most convenient method to graph its line?<\/p>\n<p>Notice that in equations #1 and #2, <em>y<\/em> is isolated on one side of the equation, and its coefficient is 1. We found points by substituting values for <em>x<\/em> on the right side of the equation and then simplifying to get the corresponding <em>y-<\/em> values.<\/p>\n<p>Equations #3 and #4 each have just one variable. Remember, in this kind of equation the value of that one variable is constant; it does not depend on the value of the other variable. Equations of this form have graphs that are vertical or horizontal lines.<\/p>\n<p>In equations #5 and #6, both <em>x<\/em> and <em>y<\/em> are on the same side of the equation. These two equations are of the form [latex]Ax+By=C[\/latex] . We substituted [latex]y=0[\/latex] and [latex]x=0[\/latex] to find the <em>x-<\/em> and <em>y-<\/em> intercepts, and then found a third point by choosing a value for <em>x<\/em> or <em>y<\/em>.<\/p>\n<p>This leads to the following strategy for choosing the most convenient method to graph a line.<\/p>\n<div class=\"textbox shaded\">\n<h3>Choose the most convenient method to graph a line<\/h3>\n<ol id=\"eip-id1168467125546\" class=\"stepwise\">\n<li>If the equation has only one variable. It is a vertical or horizontal line.\n<ul id=\"fs-id1715836\">\n<li>[latex]x=a[\/latex] is a vertical line passing through the [latex]x\\text{-axis}[\/latex] at [latex]a[\/latex]<\/li>\n<li>[latex]y=b[\/latex] is a horizontal line passing through the [latex]y\\text{-axis}[\/latex] at [latex]b[\/latex].<\/li>\n<\/ul>\n<\/li>\n<li>If [latex]y[\/latex] is isolated on one side of the equation. Graph by plotting points.\n<ul id=\"fs-id1715895\">\n<li>Choose any three values for [latex]x[\/latex] and then solve for the corresponding [latex]y\\text{-}[\/latex] values.<\/li>\n<\/ul>\n<\/li>\n<li>If the equation is of the form [latex]Ax+By=C[\/latex], find the intercepts.\n<ul id=\"fs-id1715124\">\n<li>Find the [latex]x\\text{-}[\/latex] and [latex]y\\text{-}[\/latex] intercepts and then a third point.<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Identify the most convenient method to graph each line:<\/p>\n<p>1. [latex]y=-3[\/latex]<br \/>\n2. [latex]4x - 6y=12[\/latex]<br \/>\n3. [latex]x=2[\/latex]<br \/>\n4. [latex]y=\\frac{2}{5}x - 1[\/latex]<\/p>\n<p>Solution<br \/>\n1. [latex]y=-3[\/latex]<br \/>\nThis equation has only one variable, [latex]y[\/latex]. Its graph is a horizontal line crossing the [latex]y\\text{-axis}[\/latex] at [latex]-3[\/latex].<br \/>\n2. [latex]4x - 6y=12[\/latex]<br \/>\nThis equation is of the form [latex]Ax+By=C[\/latex]. Find the intercepts and one more point.<br \/>\n3. [latex]x=2[\/latex]<br \/>\nThere is only one variable, [latex]x[\/latex]. The graph is a vertical line crossing the [latex]x\\text{-axis}[\/latex] at [latex]2[\/latex].<br \/>\n4. [latex]y=\\Large\\frac{2}{5}\\normalsize x - 1[\/latex]<br \/>\nSince [latex]y[\/latex] is isolated on the left side of the equation, it will be easiest to graph this line by plotting three points.<\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm147002\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=147002&theme=oea&iframe_resize_id=ohm147002&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-10683\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Question ID 147001, 146998, 146999, 147002. <strong>Authored by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Specific attribution<\/div><ul class=\"citation-list\"><li>Prealgebra. <strong>Provided by<\/strong>: OpenStax. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download for free at http:\/\/cnx.org\/contents\/caa57dab-41c7-455e-bd6f-f443cda5519c@9.757<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":17533,"menu_order":14,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"Prealgebra\",\"author\":\"\",\"organization\":\"OpenStax\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download for free at http:\/\/cnx.org\/contents\/caa57dab-41c7-455e-bd6f-f443cda5519c@9.757\"},{\"type\":\"original\",\"description\":\"Question ID 147001, 146998, 146999, 147002\",\"author\":\"Lumen Learning\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"0a4cf7b473d74cb5b51e82292e049f41, 6fd56c9af01f48bc98560f9913f061c8","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-10683","chapter","type-chapter","status-publish","hentry"],"part":8524,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/pressbooks\/v2\/chapters\/10683","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/wp\/v2\/users\/17533"}],"version-history":[{"count":36,"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/pressbooks\/v2\/chapters\/10683\/revisions"}],"predecessor-version":[{"id":20475,"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/pressbooks\/v2\/chapters\/10683\/revisions\/20475"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/pressbooks\/v2\/parts\/8524"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/pressbooks\/v2\/chapters\/10683\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/wp\/v2\/media?parent=10683"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/pressbooks\/v2\/chapter-type?post=10683"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/wp\/v2\/contributor?post=10683"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/wp\/v2\/license?post=10683"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}