{"id":84,"date":"2023-06-21T13:22:31","date_gmt":"2023-06-21T13:22:31","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/chapter\/graph-linear-equations\/"},"modified":"2023-08-24T05:00:42","modified_gmt":"2023-08-24T05:00:42","slug":"graph-linear-equations","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/chapter\/graph-linear-equations\/","title":{"raw":"\u25aa   Graphing Linear Equations (with two variables)","rendered":"\u25aa   Graphing Linear Equations (with two variables)"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Plot linear equations in two variables on the coordinate plane.<\/li>\r\n \t<li>Use intercepts to plot lines.<\/li>\r\n \t<li>Use a graphing utility to graph a linear equation on a coordinate plane.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div>\r\n\r\nWe can plot a set of points to represent an equation. When such an equation contains both an <em>x <\/em>variable and a <em>y <\/em>variable, it is called an <strong>equation in two variables<\/strong>. Its graph is called a <strong>graph in two variables<\/strong>. Any graph on a two-dimensional plane is a graph in two variables.\r\n\r\nSuppose we want to graph the equation [latex]y=2x - 1[\/latex]. We can begin by substituting a value for <em>x<\/em> into the equation and determining the resulting value of <em>y<\/em>. Each pair of <em>x\u00a0<\/em>and <em>y-<\/em>values is an ordered pair that can be plotted. The table below\u00a0lists values of <em>x<\/em> from \u20133 to 3 and the resulting values for <em>y<\/em>.\r\n<table summary=\"This is a table with 8 rows and 3 columns. The first row has columns labeled: x, y = 2x-1, (x, y). The entries in the second row are: negative 3; y = 2 times negative 3 minus 1 = negative 7; (-3, -7). The entries in the third row are: negative 2; y = 2 times negative 2 minus 1 = negative 5; (-2, -5). The entries in the fourth row are: negative1; y = 2 times negative 1 minus 1 = negative 3; (-1, -3). The entries in the fifth row are: 0; y = 2 times 0 minus 1 = negative 1; (0, -1). The entries in the sixth row are: 1; y = 2 times 1 minus 1 = 1; (1, 1). The entries in the seventh row are: 2; y = 2 times 2 minus 1 = 3; (2, 3). The entries in the eight row are: 3, y = 2 times 3 minus 1 = 5; (3,5)\">\r\n<tbody>\r\n<tr>\r\n<td>[latex]x[\/latex]<\/td>\r\n<td>[latex]y=2x - 1[\/latex]<\/td>\r\n<td>[latex]\\left(x,y\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]-3[\/latex]<\/td>\r\n<td>[latex]y=2\\left(-3\\right)-1=-7[\/latex]<\/td>\r\n<td>[latex]\\left(-3,-7\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]-2[\/latex]<\/td>\r\n<td>[latex]y=2\\left(-2\\right)-1=-5[\/latex]<\/td>\r\n<td>[latex]\\left(-2,-5\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]-1[\/latex]<\/td>\r\n<td>[latex]y=2\\left(-1\\right)-1=-3[\/latex]<\/td>\r\n<td>[latex]\\left(-1,-3\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]0[\/latex]<\/td>\r\n<td>[latex]y=2\\left(0\\right)-1=-1[\/latex]<\/td>\r\n<td>[latex]\\left(0,-1\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]1[\/latex]<\/td>\r\n<td>[latex]y=2\\left(1\\right)-1=1[\/latex]<\/td>\r\n<td>[latex]\\left(1,1\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]2[\/latex]<\/td>\r\n<td>[latex]y=2\\left(2\\right)-1=3[\/latex]<\/td>\r\n<td>[latex]\\left(2,3\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]3[\/latex]<\/td>\r\n<td>[latex]y=2\\left(3\\right)-1=5[\/latex]<\/td>\r\n<td>[latex]\\left(3,5\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nWe can plot these points from the table. The points for this particular equation form a line, so we can connect them.<strong>\u00a0<\/strong>This is not true for all equations.\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/12042408\/CNX_CAT_Figure_02_01_006.jpg\" alt=\"This is a graph of a line on an x, y coordinate plane. The x- and y-axis range from negative 8 to 8. A line passes through the points (-3, -7); (-2, -5); (-1, -3); (0, -1); (1, 1); (2, 3); and (3, 5).\" width=\"731\" height=\"669\" \/>\r\n\r\nNote that the <em>x-<\/em>values chosen are arbitrary regardless of the type of equation we are graphing. Of course, some situations may require particular values of <em>x<\/em> to be plotted in order to see a particular result. Otherwise, it is logical to choose values that can be calculated easily, and it is always a good idea to choose values that are both negative and positive. There is no rule dictating how many points to plot, although we need at least two to graph a line. Keep in mind, however, that the more points we plot, the more accurately we can sketch the graph.\r\n<div class=\"textbox\">\r\n<h3>How To: Given an equation, graph by plotting points<\/h3>\r\n<ol>\r\n \t<li>Make a table with one column labeled <em>x<\/em>, a second column labeled with the equation, and a third column listing the resulting ordered pairs.<\/li>\r\n \t<li>Enter <em>x-<\/em>values down the first column using positive and negative values. Selecting the <em>x-<\/em>values in numerical order will make graphing easier.<\/li>\r\n \t<li>Select <em>x-<\/em>values that will yield <em>y-<\/em>values with little effort, preferably ones that can be calculated mentally.<\/li>\r\n \t<li>Plot the ordered pairs.<\/li>\r\n \t<li>Connect the points if they form a line.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Graphing an Equation in Two Variables by Plotting Points<\/h3>\r\nGraph the equation [latex]y=-x+2[\/latex] by plotting points.\r\n[reveal-answer q=\"792137\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"792137\"]\r\n\r\nFirst, we construct a table similar to the one below. Choose <em>x<\/em> values and calculate <em>y.<\/em>\r\n<table summary=\"The table shows 8 rows and 3 columns. The entries in the first row are: x; y = negative x plus 2; and (x, y). The entries in the second row are: negative 5; y = the opposite of negative 5 plus 2 = 7; (-5, 7). The entries in the third row are: negative 3; y = the opposite of negative 3 plus 2 = 5; (-3, 5). The entries in the fourth row are: -1; y = the opposite of negative 1 plus 2 = 3; (-1, 3). The entries in the fifth row are: 0; y = opposite of zero plus 2 = 2; (0, 2). The entries in the sixth row are: 1; y = the opposite of 1 plus 2 = 1; (1, 1). The entries in the seventh row are: 3; y = the opposite of 3 plus 2 = negative 1; (3, -1). The entries in the eighth row are: 5; y = the opposite of 5 plus 2 = negative 3; (5, -3).\">\r\n<tbody>\r\n<tr>\r\n<td>[latex]x[\/latex]<\/td>\r\n<td>[latex]y=-x+2[\/latex]<\/td>\r\n<td>[latex]\\left(x,y\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]-5[\/latex]<\/td>\r\n<td>[latex]y=-\\left(-5\\right)+2=7[\/latex]<\/td>\r\n<td>[latex]\\left(-5,7\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]-3[\/latex]<\/td>\r\n<td>[latex]y=-\\left(-3\\right)+2=5[\/latex]<\/td>\r\n<td>[latex]\\left(-3,5\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]-1[\/latex]<\/td>\r\n<td>[latex]y=-\\left(-1\\right)+2=3[\/latex]<\/td>\r\n<td>[latex]\\left(-1,3\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]0[\/latex]<\/td>\r\n<td>[latex]y=-\\left(0\\right)+2=2[\/latex]<\/td>\r\n<td>[latex]\\left(0,2\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]1[\/latex]<\/td>\r\n<td>[latex]y=-\\left(1\\right)+2=1[\/latex]<\/td>\r\n<td>[latex]\\left(1,1\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]3[\/latex]<\/td>\r\n<td>[latex]y=-\\left(3\\right)+2=-1[\/latex]<\/td>\r\n<td>[latex]\\left(3,-1\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]5[\/latex]<\/td>\r\n<td>[latex]y=-\\left(5\\right)+2=-3[\/latex]<\/td>\r\n<td>[latex]\\left(5,-3\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nNow, plot the points. Connect them if they form a line.\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/12042411\/CNX_CAT_Figure_02_01_007.jpg\" alt=\"This image is a graph of a line on an x, y coordinate plane. The x-axis includes numbers that range from negative 7 to 7. The y-axis includes numbers that range from negative 5 to 8. A line passes through the points: (-5, 7); (-3, 5); (-1, 3); (0, 2); (1, 1); (3, -1); and (5, -3).\" width=\"731\" height=\"556\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nConstruct a table and graph the equation by plotting points: [latex]y=\\frac{1}{2}x+2[\/latex].\r\n[reveal-answer q=\"811886\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"811886\"]\r\n<table summary=\"The table shows 6 rows and 3 columns. The entries in the first row are: x; y = x divided by 2 plus 2, (x,y). The entries in the second row are: negative 2; y = (negative 2) divided by 2 plus 2 = 1; (-2, 1). The entries in the third row are: negative 1; y = (negative 1) divided by 2 plus 2 = 3\/2; (-1,3\/2). The entries in the fourth row are: 0; y = (0)\/2 + 2 = 2; (0,2). The entries in the fifth row are: 1; y = (1)\/2 + 2 = 5\/2; (1,5\/2). The entries in the sixth row are: 2; y = (2)\/2 + 2 = 3; (2,3).\">\r\n<tbody>\r\n<tr>\r\n<td>[latex]x[\/latex]<\/td>\r\n<td>[latex]y=\\frac{1}{2}x+2[\/latex]<\/td>\r\n<td>[latex]\\left(x,y\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]-2[\/latex]<\/td>\r\n<td>[latex]y=\\frac{1}{2}\\left(-2\\right)+2=1[\/latex]<\/td>\r\n<td>[latex]\\left(-2,1\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]-1[\/latex]<\/td>\r\n<td>[latex]y=\\frac{1}{2}\\left(-1\\right)+2=\\frac{3}{2}[\/latex]<\/td>\r\n<td>[latex]\\left(-1,\\frac{3}{2}\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]0[\/latex]<\/td>\r\n<td>[latex]y=\\frac{1}{2}\\left(0\\right)+2=2[\/latex]<\/td>\r\n<td>[latex]\\left(0,2\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]1[\/latex]<\/td>\r\n<td>[latex]y=\\frac{1}{2}\\left(1\\right)+2=\\frac{5}{2}[\/latex]<\/td>\r\n<td>[latex]\\left(1,\\frac{5}{2}\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]2[\/latex]<\/td>\r\n<td>[latex]y=\\frac{1}{2}\\left(2\\right)+2=3[\/latex]<\/td>\r\n<td>[latex]\\left(2,3\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/12042413\/CNX_CAT_Figure_02_01_008.jpg\" alt=\"This is an image of a graph on an x, y coordinate plane. The x and y-axis range from negative 5 to 5. A line passes through the points (-2, 1); (-1, 3\/2); (0, 2); (1, 5\/2); and (2, 3).\" width=\"487\" height=\"442\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<iframe id=\"mom3\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=110939&amp;theme=oea&amp;iframe_resize_id=mom3\" width=\"100%\" height=\"350\"><\/iframe>\r\n\r\n<\/div>\r\n<h2>Using Intercepts to Plot Lines in the Coordinate Plane<\/h2>\r\nThe <strong>intercepts<\/strong> of a graph are points where the graph crosses the axes. The <strong><em>x-<\/em>intercept<\/strong> is the point where the graph crosses the <em>x-<\/em>axis. At this point, the <em>y-<\/em>coordinate is zero. The <strong><em>y-<\/em>intercept<\/strong> is the point where the graph crosses the <em>y-<\/em>axis. At this point, the <em>x-<\/em>coordinate is zero.\r\n\r\nTo determine the <em>x-<\/em>intercept, we set <em>y <\/em>equal to zero and solve for <em>x<\/em>. Similarly, to determine the <em>y-<\/em>intercept, we set <em>x <\/em>equal to zero and solve for <em>y<\/em>. For example, lets find the intercepts of the equation [latex]y=3x - 1[\/latex].\r\n\r\nTo find the <em>x-<\/em>intercept, set [latex]y=0[\/latex].\r\n<div style=\"text-align: center;\">[latex]\\begin{array}{llllll}y=3x - 1\\hfill &amp; \\hfill \\\\ 0=3x - 1\\hfill &amp; \\hfill \\\\ 1=3x\\hfill &amp; \\hfill \\\\ \\frac{1}{3}=x\\hfill &amp; \\hfill \\\\ \\left(\\frac{1}{3},0\\right)\\hfill &amp; x\\text{-intercept}\\hfill \\end{array}[\/latex]<\/div>\r\nTo find the <em>y-<\/em>intercept, set [latex]x=0[\/latex].\r\n<div style=\"text-align: center;\">[latex]\\begin{array}{lllll}y=3x - 1\\hfill &amp; \\hfill \\\\ y=3\\left(0\\right)-1\\hfill &amp; \\hfill \\\\ y=-1\\hfill &amp; \\hfill \\\\ \\left(0,-1\\right)\\hfill &amp; y\\text{-intercept}\\hfill \\end{array}[\/latex]<\/div>\r\nWe can confirm that our results make sense by observing a graph of the equation. Notice that the graph crosses the axes where we predicted it would.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/12042423\/CNX_CAT_Figure_02_01_012.jpg\" alt=\"This is an image of a line graph on an x, y coordinate plane. The x and y-axis range from negative 4 to 4. The function y = 3x \u2013 1 is plotted on the coordinate plane\" width=\"487\" height=\"366\" \/>\r\n<div class=\"textbox\">\r\n<h3>How To: Given an equation, find the intercepts<\/h3>\r\n<ol>\r\n \t<li>Find the <em>x<\/em>-intercept by setting [latex]y=0[\/latex] and solving for [latex]x[\/latex].<\/li>\r\n \t<li>Find the <em>y-<\/em>intercept by setting [latex]x=0[\/latex] and solving for [latex]y[\/latex].<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Finding the Intercepts of the Given Equation<\/h3>\r\nFind the intercepts of the equation [latex]y=-3x - 4[\/latex]. Then sketch the graph using only the intercepts.\r\n\r\n[reveal-answer q=\"814560\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"814560\"]\r\nSet [latex]y=0[\/latex] to find the <em>x-<\/em>intercept.\r\n<div style=\"text-align: center;\">[latex]\\begin{array}{l}y=-3x - 4\\hfill \\\\ 0=-3x - 4\\hfill \\\\ 4=-3x\\hfill \\\\ -\\frac{4}{3}=x\\hfill \\\\ \\left(-\\frac{4}{3},0\\right)x\\text{-intercept}\\hfill \\end{array}[\/latex]<\/div>\r\nSet [latex]x=0[\/latex] to find the <em>y-<\/em>intercept.\r\n<div style=\"text-align: center;\">[latex]\\begin{array}{l}y=-3x - 4\\hfill \\\\ y=-3\\left(0\\right)-4\\hfill \\\\ y=-4\\hfill \\\\ \\left(0,-4\\right)y\\text{-intercept}\\hfill \\end{array}[\/latex]<\/div>\r\nPlot both points and draw a line passing through them.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/12042425\/CNX_CAT_Figure_02_01_013.jpg\" alt=\"This is an image of a line graph on an x, y coordinate plane. The x-axis ranges from negative 5 to 5. The y-axis ranges from negative 6 to 3. The line passes through the points (-4\/3, 0) and (0, -4).\" width=\"487\" height=\"406\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nFind the intercepts of the equation and sketch the graph: [latex]y=-\\frac{3}{4}x+3[\/latex].\r\n\r\n[reveal-answer q=\"80464\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"80464\"]\r\n\r\n<em>x<\/em>-intercept is [latex]\\left(4,0\\right)[\/latex]; <em>y-<\/em>intercept is [latex]\\left(0,3\\right)[\/latex].\r\n\r\n<img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/09\/25200257\/CNX_CAT_Figure_02_01_014.jpg\" alt=\"This is an image of a line graph on an x, y coordinate plane. The x and y axes range from negative 4 to 6. The function y = -3x\/4 + 3 is plotted.\" width=\"487\" height=\"447\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<iframe id=\"mom4\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=92757&amp;theme=oea&amp;iframe_resize_id=mom4\" width=\"100%\" height=\"450\"><\/iframe>\r\n\r\n<\/div>\r\n<h2>Using a Graphing Utility to Plot Lines<\/h2>\r\nhttps:\/\/www.geogebra.org\/graphing\r\n\r\nhttps:\/\/www.desmos.com\/calculator\r\n\r\nhttps:\/\/www.mathway.com\/Graph\r\n\r\nhttps:\/\/www.symbolab.com\/graphing-calculator\r\n\r\n[embed]https:\/\/youtu.be\/cEIOdi2R4fE[\/embed]\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try it now<\/h3>\r\nThese graphing utilities have features that allow you to turn a constant (number) into a variable. Follow these steps to learn how:\r\n<ol>\r\n \t<li>Graph the line [latex]y=-\\frac{2}{3}x-\\frac{4}{3}[\/latex].<\/li>\r\n \t<li>On the next line enter\u00a0[latex]y=-a x-\\frac{4}{3}[\/latex]. You will see a button pop up that says \"add slider: a\", click on the button. You will see the next line populated with the variable a and the interval on which a can take values.<\/li>\r\n \t<li>What part of a line does the variable a represent? The slope or the y-intercept?<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Plot linear equations in two variables on the coordinate plane.<\/li>\n<li>Use intercepts to plot lines.<\/li>\n<li>Use a graphing utility to graph a linear equation on a coordinate plane.<\/li>\n<\/ul>\n<\/div>\n<div>\n<p>We can plot a set of points to represent an equation. When such an equation contains both an <em>x <\/em>variable and a <em>y <\/em>variable, it is called an <strong>equation in two variables<\/strong>. Its graph is called a <strong>graph in two variables<\/strong>. Any graph on a two-dimensional plane is a graph in two variables.<\/p>\n<p>Suppose we want to graph the equation [latex]y=2x - 1[\/latex]. We can begin by substituting a value for <em>x<\/em> into the equation and determining the resulting value of <em>y<\/em>. Each pair of <em>x\u00a0<\/em>and <em>y-<\/em>values is an ordered pair that can be plotted. The table below\u00a0lists values of <em>x<\/em> from \u20133 to 3 and the resulting values for <em>y<\/em>.<\/p>\n<table summary=\"This is a table with 8 rows and 3 columns. The first row has columns labeled: x, y = 2x-1, (x, y). The entries in the second row are: negative 3; y = 2 times negative 3 minus 1 = negative 7; (-3, -7). The entries in the third row are: negative 2; y = 2 times negative 2 minus 1 = negative 5; (-2, -5). The entries in the fourth row are: negative1; y = 2 times negative 1 minus 1 = negative 3; (-1, -3). The entries in the fifth row are: 0; y = 2 times 0 minus 1 = negative 1; (0, -1). The entries in the sixth row are: 1; y = 2 times 1 minus 1 = 1; (1, 1). The entries in the seventh row are: 2; y = 2 times 2 minus 1 = 3; (2, 3). The entries in the eight row are: 3, y = 2 times 3 minus 1 = 5; (3,5)\">\n<tbody>\n<tr>\n<td>[latex]x[\/latex]<\/td>\n<td>[latex]y=2x - 1[\/latex]<\/td>\n<td>[latex]\\left(x,y\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]-3[\/latex]<\/td>\n<td>[latex]y=2\\left(-3\\right)-1=-7[\/latex]<\/td>\n<td>[latex]\\left(-3,-7\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]-2[\/latex]<\/td>\n<td>[latex]y=2\\left(-2\\right)-1=-5[\/latex]<\/td>\n<td>[latex]\\left(-2,-5\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]-1[\/latex]<\/td>\n<td>[latex]y=2\\left(-1\\right)-1=-3[\/latex]<\/td>\n<td>[latex]\\left(-1,-3\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]0[\/latex]<\/td>\n<td>[latex]y=2\\left(0\\right)-1=-1[\/latex]<\/td>\n<td>[latex]\\left(0,-1\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]1[\/latex]<\/td>\n<td>[latex]y=2\\left(1\\right)-1=1[\/latex]<\/td>\n<td>[latex]\\left(1,1\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]2[\/latex]<\/td>\n<td>[latex]y=2\\left(2\\right)-1=3[\/latex]<\/td>\n<td>[latex]\\left(2,3\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]3[\/latex]<\/td>\n<td>[latex]y=2\\left(3\\right)-1=5[\/latex]<\/td>\n<td>[latex]\\left(3,5\\right)[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>We can plot these points from the table. The points for this particular equation form a line, so we can connect them.<strong>\u00a0<\/strong>This is not true for all equations.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/12042408\/CNX_CAT_Figure_02_01_006.jpg\" alt=\"This is a graph of a line on an x, y coordinate plane. The x- and y-axis range from negative 8 to 8. A line passes through the points (-3, -7); (-2, -5); (-1, -3); (0, -1); (1, 1); (2, 3); and (3, 5).\" width=\"731\" height=\"669\" \/><\/p>\n<p>Note that the <em>x-<\/em>values chosen are arbitrary regardless of the type of equation we are graphing. Of course, some situations may require particular values of <em>x<\/em> to be plotted in order to see a particular result. Otherwise, it is logical to choose values that can be calculated easily, and it is always a good idea to choose values that are both negative and positive. There is no rule dictating how many points to plot, although we need at least two to graph a line. Keep in mind, however, that the more points we plot, the more accurately we can sketch the graph.<\/p>\n<div class=\"textbox\">\n<h3>How To: Given an equation, graph by plotting points<\/h3>\n<ol>\n<li>Make a table with one column labeled <em>x<\/em>, a second column labeled with the equation, and a third column listing the resulting ordered pairs.<\/li>\n<li>Enter <em>x-<\/em>values down the first column using positive and negative values. Selecting the <em>x-<\/em>values in numerical order will make graphing easier.<\/li>\n<li>Select <em>x-<\/em>values that will yield <em>y-<\/em>values with little effort, preferably ones that can be calculated mentally.<\/li>\n<li>Plot the ordered pairs.<\/li>\n<li>Connect the points if they form a line.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Graphing an Equation in Two Variables by Plotting Points<\/h3>\n<p>Graph the equation [latex]y=-x+2[\/latex] by plotting points.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q792137\">Show Solution<\/span><\/p>\n<div id=\"q792137\" class=\"hidden-answer\" style=\"display: none\">\n<p>First, we construct a table similar to the one below. Choose <em>x<\/em> values and calculate <em>y.<\/em><\/p>\n<table summary=\"The table shows 8 rows and 3 columns. The entries in the first row are: x; y = negative x plus 2; and (x, y). The entries in the second row are: negative 5; y = the opposite of negative 5 plus 2 = 7; (-5, 7). The entries in the third row are: negative 3; y = the opposite of negative 3 plus 2 = 5; (-3, 5). The entries in the fourth row are: -1; y = the opposite of negative 1 plus 2 = 3; (-1, 3). The entries in the fifth row are: 0; y = opposite of zero plus 2 = 2; (0, 2). The entries in the sixth row are: 1; y = the opposite of 1 plus 2 = 1; (1, 1). The entries in the seventh row are: 3; y = the opposite of 3 plus 2 = negative 1; (3, -1). The entries in the eighth row are: 5; y = the opposite of 5 plus 2 = negative 3; (5, -3).\">\n<tbody>\n<tr>\n<td>[latex]x[\/latex]<\/td>\n<td>[latex]y=-x+2[\/latex]<\/td>\n<td>[latex]\\left(x,y\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]-5[\/latex]<\/td>\n<td>[latex]y=-\\left(-5\\right)+2=7[\/latex]<\/td>\n<td>[latex]\\left(-5,7\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]-3[\/latex]<\/td>\n<td>[latex]y=-\\left(-3\\right)+2=5[\/latex]<\/td>\n<td>[latex]\\left(-3,5\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]-1[\/latex]<\/td>\n<td>[latex]y=-\\left(-1\\right)+2=3[\/latex]<\/td>\n<td>[latex]\\left(-1,3\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]0[\/latex]<\/td>\n<td>[latex]y=-\\left(0\\right)+2=2[\/latex]<\/td>\n<td>[latex]\\left(0,2\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]1[\/latex]<\/td>\n<td>[latex]y=-\\left(1\\right)+2=1[\/latex]<\/td>\n<td>[latex]\\left(1,1\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]3[\/latex]<\/td>\n<td>[latex]y=-\\left(3\\right)+2=-1[\/latex]<\/td>\n<td>[latex]\\left(3,-1\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]5[\/latex]<\/td>\n<td>[latex]y=-\\left(5\\right)+2=-3[\/latex]<\/td>\n<td>[latex]\\left(5,-3\\right)[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Now, plot the points. Connect them if they form a line.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/12042411\/CNX_CAT_Figure_02_01_007.jpg\" alt=\"This image is a graph of a line on an x, y coordinate plane. The x-axis includes numbers that range from negative 7 to 7. The y-axis includes numbers that range from negative 5 to 8. A line passes through the points: (-5, 7); (-3, 5); (-1, 3); (0, 2); (1, 1); (3, -1); and (5, -3).\" width=\"731\" height=\"556\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Construct a table and graph the equation by plotting points: [latex]y=\\frac{1}{2}x+2[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q811886\">Show Solution<\/span><\/p>\n<div id=\"q811886\" class=\"hidden-answer\" style=\"display: none\">\n<table summary=\"The table shows 6 rows and 3 columns. The entries in the first row are: x; y = x divided by 2 plus 2, (x,y). The entries in the second row are: negative 2; y = (negative 2) divided by 2 plus 2 = 1; (-2, 1). The entries in the third row are: negative 1; y = (negative 1) divided by 2 plus 2 = 3\/2; (-1,3\/2). The entries in the fourth row are: 0; y = (0)\/2 + 2 = 2; (0,2). The entries in the fifth row are: 1; y = (1)\/2 + 2 = 5\/2; (1,5\/2). The entries in the sixth row are: 2; y = (2)\/2 + 2 = 3; (2,3).\">\n<tbody>\n<tr>\n<td>[latex]x[\/latex]<\/td>\n<td>[latex]y=\\frac{1}{2}x+2[\/latex]<\/td>\n<td>[latex]\\left(x,y\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]-2[\/latex]<\/td>\n<td>[latex]y=\\frac{1}{2}\\left(-2\\right)+2=1[\/latex]<\/td>\n<td>[latex]\\left(-2,1\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]-1[\/latex]<\/td>\n<td>[latex]y=\\frac{1}{2}\\left(-1\\right)+2=\\frac{3}{2}[\/latex]<\/td>\n<td>[latex]\\left(-1,\\frac{3}{2}\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]0[\/latex]<\/td>\n<td>[latex]y=\\frac{1}{2}\\left(0\\right)+2=2[\/latex]<\/td>\n<td>[latex]\\left(0,2\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]1[\/latex]<\/td>\n<td>[latex]y=\\frac{1}{2}\\left(1\\right)+2=\\frac{5}{2}[\/latex]<\/td>\n<td>[latex]\\left(1,\\frac{5}{2}\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]2[\/latex]<\/td>\n<td>[latex]y=\\frac{1}{2}\\left(2\\right)+2=3[\/latex]<\/td>\n<td>[latex]\\left(2,3\\right)[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/12042413\/CNX_CAT_Figure_02_01_008.jpg\" alt=\"This is an image of a graph on an x, y coordinate plane. The x and y-axis range from negative 5 to 5. A line passes through the points (-2, 1); (-1, 3\/2); (0, 2); (1, 5\/2); and (2, 3).\" width=\"487\" height=\"442\" \/><\/p>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom3\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=110939&amp;theme=oea&amp;iframe_resize_id=mom3\" width=\"100%\" height=\"350\"><\/iframe><\/p>\n<\/div>\n<h2>Using Intercepts to Plot Lines in the Coordinate Plane<\/h2>\n<p>The <strong>intercepts<\/strong> of a graph are points where the graph crosses the axes. The <strong><em>x-<\/em>intercept<\/strong> is the point where the graph crosses the <em>x-<\/em>axis. At this point, the <em>y-<\/em>coordinate is zero. The <strong><em>y-<\/em>intercept<\/strong> is the point where the graph crosses the <em>y-<\/em>axis. At this point, the <em>x-<\/em>coordinate is zero.<\/p>\n<p>To determine the <em>x-<\/em>intercept, we set <em>y <\/em>equal to zero and solve for <em>x<\/em>. Similarly, to determine the <em>y-<\/em>intercept, we set <em>x <\/em>equal to zero and solve for <em>y<\/em>. For example, lets find the intercepts of the equation [latex]y=3x - 1[\/latex].<\/p>\n<p>To find the <em>x-<\/em>intercept, set [latex]y=0[\/latex].<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{llllll}y=3x - 1\\hfill & \\hfill \\\\ 0=3x - 1\\hfill & \\hfill \\\\ 1=3x\\hfill & \\hfill \\\\ \\frac{1}{3}=x\\hfill & \\hfill \\\\ \\left(\\frac{1}{3},0\\right)\\hfill & x\\text{-intercept}\\hfill \\end{array}[\/latex]<\/div>\n<p>To find the <em>y-<\/em>intercept, set [latex]x=0[\/latex].<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{lllll}y=3x - 1\\hfill & \\hfill \\\\ y=3\\left(0\\right)-1\\hfill & \\hfill \\\\ y=-1\\hfill & \\hfill \\\\ \\left(0,-1\\right)\\hfill & y\\text{-intercept}\\hfill \\end{array}[\/latex]<\/div>\n<p>We can confirm that our results make sense by observing a graph of the equation. Notice that the graph crosses the axes where we predicted it would.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/12042423\/CNX_CAT_Figure_02_01_012.jpg\" alt=\"This is an image of a line graph on an x, y coordinate plane. The x and y-axis range from negative 4 to 4. The function y = 3x \u2013 1 is plotted on the coordinate plane\" width=\"487\" height=\"366\" \/><\/p>\n<div class=\"textbox\">\n<h3>How To: Given an equation, find the intercepts<\/h3>\n<ol>\n<li>Find the <em>x<\/em>-intercept by setting [latex]y=0[\/latex] and solving for [latex]x[\/latex].<\/li>\n<li>Find the <em>y-<\/em>intercept by setting [latex]x=0[\/latex] and solving for [latex]y[\/latex].<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Finding the Intercepts of the Given Equation<\/h3>\n<p>Find the intercepts of the equation [latex]y=-3x - 4[\/latex]. Then sketch the graph using only the intercepts.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q814560\">Show Solution<\/span><\/p>\n<div id=\"q814560\" class=\"hidden-answer\" style=\"display: none\">\nSet [latex]y=0[\/latex] to find the <em>x-<\/em>intercept.<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{l}y=-3x - 4\\hfill \\\\ 0=-3x - 4\\hfill \\\\ 4=-3x\\hfill \\\\ -\\frac{4}{3}=x\\hfill \\\\ \\left(-\\frac{4}{3},0\\right)x\\text{-intercept}\\hfill \\end{array}[\/latex]<\/div>\n<p>Set [latex]x=0[\/latex] to find the <em>y-<\/em>intercept.<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{l}y=-3x - 4\\hfill \\\\ y=-3\\left(0\\right)-4\\hfill \\\\ y=-4\\hfill \\\\ \\left(0,-4\\right)y\\text{-intercept}\\hfill \\end{array}[\/latex]<\/div>\n<p>Plot both points and draw a line passing through them.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/12042425\/CNX_CAT_Figure_02_01_013.jpg\" alt=\"This is an image of a line graph on an x, y coordinate plane. The x-axis ranges from negative 5 to 5. The y-axis ranges from negative 6 to 3. The line passes through the points (-4\/3, 0) and (0, -4).\" width=\"487\" height=\"406\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Find the intercepts of the equation and sketch the graph: [latex]y=-\\frac{3}{4}x+3[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q80464\">Show Solution<\/span><\/p>\n<div id=\"q80464\" class=\"hidden-answer\" style=\"display: none\">\n<p><em>x<\/em>-intercept is [latex]\\left(4,0\\right)[\/latex]; <em>y-<\/em>intercept is [latex]\\left(0,3\\right)[\/latex].<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/09\/25200257\/CNX_CAT_Figure_02_01_014.jpg\" alt=\"This is an image of a line graph on an x, y coordinate plane. The x and y axes range from negative 4 to 6. The function y = -3x\/4 + 3 is plotted.\" width=\"487\" height=\"447\" \/><\/p>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom4\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=92757&amp;theme=oea&amp;iframe_resize_id=mom4\" width=\"100%\" height=\"450\"><\/iframe><\/p>\n<\/div>\n<h2>Using a Graphing Utility to Plot Lines<\/h2>\n<p><iframe loading=\"lazy\" class=\"resizable\" src=\"https:\/\/www.geogebra.org\/graphing\" frameborder=\"0\" width=\"500\" height=\"750\"><\/iframe><\/p>\n<p>https:\/\/www.desmos.com\/calculator<\/p>\n<p>https:\/\/www.mathway.com\/Graph<\/p>\n<p>https:\/\/www.symbolab.com\/graphing-calculator<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Learn Desmos: Lines\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/cEIOdi2R4fE?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Try it now<\/h3>\n<p>These graphing utilities have features that allow you to turn a constant (number) into a variable. Follow these steps to learn how:<\/p>\n<ol>\n<li>Graph the line [latex]y=-\\frac{2}{3}x-\\frac{4}{3}[\/latex].<\/li>\n<li>On the next line enter\u00a0[latex]y=-a x-\\frac{4}{3}[\/latex]. You will see a button pop up that says &#8220;add slider: a&#8221;, click on the button. You will see the next line populated with the variable a and the interval on which a can take values.<\/li>\n<li>What part of a line does the variable a represent? The slope or the y-intercept?<\/li>\n<\/ol>\n<\/div>\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-84\">\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>Revision and Adaptation. <strong>Provided 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, Shared previously<\/div><ul class=\"citation-list\"><li>College Algebra. <strong>Authored by<\/strong>: Abramson, Jay et al.. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\">http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/a>. <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\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/li><li>Question ID 110939. <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>. <strong>License Terms<\/strong>: IMathAS Community License CC- BY + GPL<\/li><li>Question ID 92757. <strong>Authored by<\/strong>: Michael Jenck. <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>: IMathAS Community License CC- BY + GPL<\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">All rights reserved content<\/div><ul class=\"citation-list\"><li>Learn Desmos: Lines. <strong>Authored by<\/strong>: Desmos. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/cEIOdi2R4fE\">https:\/\/youtu.be\/cEIOdi2R4fE<\/a>. <strong>License<\/strong>: <em>All Rights Reserved<\/em>. <strong>License Terms<\/strong>: Standard YouTube License<\/li><li>Learn Desmos: Sliders. <strong>Authored by<\/strong>: Desmos. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/9MChp2P0vMA\">https:\/\/youtu.be\/9MChp2P0vMA<\/a>. <strong>License<\/strong>: <em>All Rights Reserved<\/em>. <strong>License Terms<\/strong>: Standard YouTube License<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t 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