{"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":"2018-01-15T18:40:07","modified_gmt":"2018-01-15T18:40:07","slug":"finding-and-using-the-intercepts-from-the-equation-of-a-line","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/csn-prealgebra\/chapter\/finding-and-using-the-intercepts-from-the-equation-of-a-line\/","title":{"raw":"Graphing Lines Using Points","rendered":"Graphing Lines Using Points"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Use intercepts to graph a line<\/li>\r\n \t<li>Use intercepts combined with a third point to graph a line<\/li>\r\n \t<li>Determine the most convenient way to graph a line given it's equation<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h3 data-type=\"title\">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 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 \" data-label=\"\">\r\n<thead>\r\n<tr valign=\"top\">\r\n<th colspan=\"3\" data-align=\"center\">[latex]-x+2y=6[\/latex]<\/th>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<th data-align=\"left\"><em data-effect=\"italics\"><strong>x<\/strong><\/em><\/th>\r\n<th data-align=\"left\"><em data-effect=\"italics\"><strong>y<\/strong><\/em><\/th>\r\n<th data-align=\"left\"><em data-effect=\"italics\"><strong>(x,y)<\/strong><\/em><\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td data-align=\"left\">[latex]-6[\/latex]<\/td>\r\n<td data-align=\"left\">[latex]0[\/latex]<\/td>\r\n<td data-align=\"left\">[latex]\\left(-6,0\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-align=\"left\">[latex]0[\/latex]<\/td>\r\n<td data-align=\"left\">[latex]3[\/latex]<\/td>\r\n<td data-align=\"left\">[latex]\\left(0,3\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-align=\"left\">[latex]2[\/latex]<\/td>\r\n<td data-align=\"left\">[latex]4[\/latex]<\/td>\r\n<td data-align=\"left\">[latex]\\left(2,4\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<img 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 \" data-media-type=\"image\/png\" \/>\r\nDo the points line up? Yes, so draw line through the points.\r\n\r\n<img 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 \" data-media-type=\"image\/png\" \/>\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\n&nbsp;\r\n<div class=\"textbox shaded\">\r\n<h3>Graph a line using the intercepts<\/h3>\r\n<ol id=\"eip-id1168468755039\" class=\"stepwise\" data-number-style=\"arabic\">\r\n \t<li>Find the [latex]x-[\/latex] and [latex]\\text{y-intercepts}[\/latex] of the line.\r\n<ul id=\"fs-id1365900\" data-bullet-style=\"open-circle\">\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]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 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 \" data-media-type=\"image\/png\" \/>\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 \" data-label=\"\">\r\n<thead>\r\n<tr valign=\"top\">\r\n<th colspan=\"3\" data-align=\"center\">[latex]4x - 3y=12[\/latex]<\/th>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<th data-align=\"left\">[latex]x[\/latex]<\/th>\r\n<th data-align=\"left\">[latex]y[\/latex]<\/th>\r\n<th data-align=\"left\">[latex]\\left(x,y\\right)[\/latex]<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td data-align=\"left\">[latex]3[\/latex]<\/td>\r\n<td data-align=\"left\">[latex]0[\/latex]<\/td>\r\n<td data-align=\"left\">[latex]\\left(3,0\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-align=\"left\">[latex]0[\/latex]<\/td>\r\n<td data-align=\"left\">[latex]-4[\/latex]<\/td>\r\n<td data-align=\"left\">[latex]\\left(0,-4\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-align=\"left\">[latex]6[\/latex]<\/td>\r\n<td data-align=\"left\">[latex]4[\/latex]<\/td>\r\n<td data-align=\"left\">[latex]\\left(6,4\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<img 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 \" data-media-type=\"image\/png\" \/>\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\n&nbsp;\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 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 \" data-media-type=\"image\/png\" \/>\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 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 \" data-media-type=\"image\/png\" \/>\r\nOrganize the points in a table.\r\n<table id=\"fs-id1363944\" class=\"unnumbered\" 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 \" data-label=\"\">\r\n<thead>\r\n<tr valign=\"top\">\r\n<th colspan=\"3\" data-align=\"center\">[latex]y=5x[\/latex]<\/th>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<th data-align=\"left\">[latex]x[\/latex]<\/th>\r\n<th data-align=\"left\">[latex]y[\/latex]<\/th>\r\n<th data-align=\"left\">[latex]\\left(x,y\\right)[\/latex]<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td data-align=\"left\">[latex]0[\/latex]<\/td>\r\n<td data-align=\"left\">[latex]0[\/latex]<\/td>\r\n<td data-align=\"left\">[latex]\\left(0,0\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-align=\"left\">[latex]1[\/latex]<\/td>\r\n<td data-align=\"left\">[latex]5[\/latex]<\/td>\r\n<td data-align=\"left\">[latex]\\left(1,5\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-align=\"left\">[latex]-1[\/latex]<\/td>\r\n<td data-align=\"left\">[latex]-5[\/latex]<\/td>\r\n<td data-align=\"left\">[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 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, \" data-media-type=\"image\/png\" \/>\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\n<p data-type=\"title\">In the following video we show another example of how to plot a line using the intercepts of the line.<\/p>\r\nhttps:\/\/youtu.be\/Wgpq0zO6z3o\r\n<h2 data-type=\"title\">Choosing the Most Convenient Way to Graph a Line Given an Equation<\/h2>\r\n<p data-type=\"title\">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>\r\n\r\n<table id=\"eip-650\" class=\"unnumbered\" summary=\"...\" data-label=\"\">\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 data-effect=\"italics\">y<\/em> is isolated on one side of the equation, and its coefficient is 1. We found points by substituting values for <em data-effect=\"italics\">x<\/em> on the right side of the equation and then simplifying to get the corresponding <em data-effect=\"italics\">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 data-effect=\"italics\">x<\/em> and <em data-effect=\"italics\">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 data-effect=\"italics\">x-<\/em> and <em data-effect=\"italics\">y-<\/em> intercepts, and then found a third point by choosing a value for <em data-effect=\"italics\">x<\/em> or <em data-effect=\"italics\">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\" data-number-style=\"arabic\">\r\n \t<li>If the equation has only one variable. It is a vertical or horizontal line.\r\n<ul id=\"fs-id1715836\" data-bullet-style=\"open-circle\">\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\" data-bullet-style=\"open-circle\">\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\" data-bullet-style=\"open-circle\">\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>Use intercepts to graph a line<\/li>\n<li>Use intercepts combined with a third point to graph a line<\/li>\n<li>Determine the most convenient way to graph a line given it&#8217;s equation<\/li>\n<\/ul>\n<\/div>\n<h3 data-type=\"title\">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 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\" data-label=\"\">\n<thead>\n<tr valign=\"top\">\n<th colspan=\"3\" data-align=\"center\">[latex]-x+2y=6[\/latex]<\/th>\n<\/tr>\n<tr valign=\"top\">\n<th data-align=\"left\"><em data-effect=\"italics\"><strong>x<\/strong><\/em><\/th>\n<th data-align=\"left\"><em data-effect=\"italics\"><strong>y<\/strong><\/em><\/th>\n<th data-align=\"left\"><em data-effect=\"italics\"><strong>(x,y)<\/strong><\/em><\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td data-align=\"left\">[latex]-6[\/latex]<\/td>\n<td data-align=\"left\">[latex]0[\/latex]<\/td>\n<td data-align=\"left\">[latex]\\left(-6,0\\right)[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-align=\"left\">[latex]0[\/latex]<\/td>\n<td data-align=\"left\">[latex]3[\/latex]<\/td>\n<td data-align=\"left\">[latex]\\left(0,3\\right)[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-align=\"left\">[latex]2[\/latex]<\/td>\n<td data-align=\"left\">[latex]4[\/latex]<\/td>\n<td data-align=\"left\">[latex]\\left(2,4\\right)[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><img decoding=\"async\" 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\" data-media-type=\"image\/png\" \/><br \/>\nDo the points line up? Yes, so draw line through the points.<\/p>\n<p><img decoding=\"async\" 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\" data-media-type=\"image\/png\" \/><\/p>\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>&nbsp;<\/p>\n<div class=\"textbox shaded\">\n<h3>Graph a line using the intercepts<\/h3>\n<ol id=\"eip-id1168468755039\" class=\"stepwise\" data-number-style=\"arabic\">\n<li>Find the [latex]x-[\/latex] and [latex]\\text{y-intercepts}[\/latex] of the line.\n<ul id=\"fs-id1365900\" data-bullet-style=\"open-circle\">\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]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 decoding=\"async\" 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\" data-media-type=\"image\/png\" \/><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\" data-label=\"\">\n<thead>\n<tr valign=\"top\">\n<th colspan=\"3\" data-align=\"center\">[latex]4x - 3y=12[\/latex]<\/th>\n<\/tr>\n<tr valign=\"top\">\n<th data-align=\"left\">[latex]x[\/latex]<\/th>\n<th data-align=\"left\">[latex]y[\/latex]<\/th>\n<th data-align=\"left\">[latex]\\left(x,y\\right)[\/latex]<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td data-align=\"left\">[latex]3[\/latex]<\/td>\n<td data-align=\"left\">[latex]0[\/latex]<\/td>\n<td data-align=\"left\">[latex]\\left(3,0\\right)[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-align=\"left\">[latex]0[\/latex]<\/td>\n<td data-align=\"left\">[latex]-4[\/latex]<\/td>\n<td data-align=\"left\">[latex]\\left(0,-4\\right)[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-align=\"left\">[latex]6[\/latex]<\/td>\n<td data-align=\"left\">[latex]4[\/latex]<\/td>\n<td data-align=\"left\">[latex]\\left(6,4\\right)[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><img decoding=\"async\" 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\" data-media-type=\"image\/png\" \/><\/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>&nbsp;<\/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 decoding=\"async\" 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\" data-media-type=\"image\/png\" \/><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 decoding=\"async\" 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\" data-media-type=\"image\/png\" \/><br \/>\nOrganize the points in a table.<\/p>\n<table id=\"fs-id1363944\" class=\"unnumbered\" 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\" data-label=\"\">\n<thead>\n<tr valign=\"top\">\n<th colspan=\"3\" data-align=\"center\">[latex]y=5x[\/latex]<\/th>\n<\/tr>\n<tr valign=\"top\">\n<th data-align=\"left\">[latex]x[\/latex]<\/th>\n<th data-align=\"left\">[latex]y[\/latex]<\/th>\n<th data-align=\"left\">[latex]\\left(x,y\\right)[\/latex]<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td data-align=\"left\">[latex]0[\/latex]<\/td>\n<td data-align=\"left\">[latex]0[\/latex]<\/td>\n<td data-align=\"left\">[latex]\\left(0,0\\right)[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-align=\"left\">[latex]1[\/latex]<\/td>\n<td data-align=\"left\">[latex]5[\/latex]<\/td>\n<td data-align=\"left\">[latex]\\left(1,5\\right)[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-align=\"left\">[latex]-1[\/latex]<\/td>\n<td data-align=\"left\">[latex]-5[\/latex]<\/td>\n<td data-align=\"left\">[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 decoding=\"async\" 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,\" data-media-type=\"image\/png\" \/><\/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 data-type=\"title\">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-1\" 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 data-type=\"title\">Choosing the Most Convenient Way to Graph a Line Given an Equation<\/h2>\n<p data-type=\"title\">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=\"...\" data-label=\"\">\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 data-effect=\"italics\">y<\/em> is isolated on one side of the equation, and its coefficient is 1. We found points by substituting values for <em data-effect=\"italics\">x<\/em> on the right side of the equation and then simplifying to get the corresponding <em data-effect=\"italics\">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 data-effect=\"italics\">x<\/em> and <em data-effect=\"italics\">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 data-effect=\"italics\">x-<\/em> and <em data-effect=\"italics\">y-<\/em> intercepts, and then found a third point by choosing a value for <em data-effect=\"italics\">x<\/em> or <em data-effect=\"italics\">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\" data-number-style=\"arabic\">\n<li>If the equation has only one variable. It is a vertical or horizontal line.\n<ul id=\"fs-id1715836\" data-bullet-style=\"open-circle\">\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\" data-bullet-style=\"open-circle\">\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\" data-bullet-style=\"open-circle\">\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":10,"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":"22e6b057-ede6-4386-bfc6-0fdf32a98dea","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\/csn-prealgebra\/wp-json\/pressbooks\/v2\/chapters\/10683","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/csn-prealgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/csn-prealgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/csn-prealgebra\/wp-json\/wp\/v2\/users\/17533"}],"version-history":[{"count":25,"href":"https:\/\/courses.lumenlearning.com\/csn-prealgebra\/wp-json\/pressbooks\/v2\/chapters\/10683\/revisions"}],"predecessor-version":[{"id":15695,"href":"https:\/\/courses.lumenlearning.com\/csn-prealgebra\/wp-json\/pressbooks\/v2\/chapters\/10683\/revisions\/15695"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/csn-prealgebra\/wp-json\/pressbooks\/v2\/parts\/8524"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/csn-prealgebra\/wp-json\/pressbooks\/v2\/chapters\/10683\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/csn-prealgebra\/wp-json\/wp\/v2\/media?parent=10683"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/csn-prealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=10683"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/csn-prealgebra\/wp-json\/wp\/v2\/contributor?post=10683"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/csn-prealgebra\/wp-json\/wp\/v2\/license?post=10683"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}