{"id":277,"date":"2023-02-01T00:03:51","date_gmt":"2023-02-01T00:03:51","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/ct-state-quantitative-reasoning\/chapter\/graphing-linear-equations\/"},"modified":"2023-02-01T00:03:51","modified_gmt":"2023-02-01T00:03:51","slug":"graphing-linear-equations","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/ct-state-quantitative-reasoning\/chapter\/graphing-linear-equations\/","title":{"raw":"Graphing Linear Equations","rendered":"Graphing Linear Equations"},"content":{"raw":"\n<div class=\"bcc-box bcc-highlight\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n \t<li>Graph an Equation Using Ordered Pairs\n<ul>\n \t<li>Create a table of ordered pairs from a two-variable linear equation<\/li>\n \t<li>Graph a two-variable linear equation using a table of ordered pairs<\/li>\n \t<li>Determine whether an ordered pair is a solution of an equation<\/li>\n<\/ul>\n<\/li>\n \t<li>Graph Linear Equations in Different Forms\n<ul>\n \t<li>Solve for <em>y<\/em>, then graph a two-variable linear equation<\/li>\n \t<li>Graph horizontal and vertical lines<\/li>\n<\/ul>\n<\/li>\n \t<li>Graph an Equation Using Intercepts\n<ul>\n \t<li>Recognize when an ordered pair is a <em>y<\/em>-intercept or an <em>x<\/em>-intercept<\/li>\n \t<li>Graph a linear equation using <em>x<\/em>- and <em>y<\/em>-intercepts<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<h2><span style=\"line-height: 1.5\">Graphing Using Ordered Pairs<\/span><\/h2>\nGraphing ordered pairs is only the beginning of the story. Once you know how to place points on a grid, you can use them to make sense of all kinds of mathematical relationships.\n<div>\n\nYou can use a <b><strong>coordinate plane<\/strong><\/b> to plot points and to map various relationships, such as the relationship between an object\u2019s distance and the elapsed time. Many mathematical relationships are <b><strong>linear relationships<\/strong><\/b>. Let\u2019s look at what a linear relationship is.\n<h3 id=\"title1\">Plotting points to graph linear relationships<\/h3>\n<\/div>\nA linear relationship is a relationship between variables such that when plotted on a coordinate plane, the points lie on a line. Let\u2019s start by looking at a series of points in Quadrant I on the coordinate plane.\n\nLook at the five <b><strong>ordered pairs<\/strong><\/b> (and their <em>x<\/em>- and <em>y<\/em>-coordinates) below. Do you see any pattern to the location of the points? If this pattern continued, what other points could be on the line?\n\nYou probably identified that if this pattern continued the next ordered pair would be at (5, 10). This makes sense because the point (5, 10) \u201clines up\u201d with the other points in the series\u2014it is literally on the same line as the others. Applying the same logic, you may identify that the ordered pairs (6, 12) and (7, 14) would also belong if this coordinate plane were larger; they, too, will line up with the other points.\n\nThese series of points can also be represented in a table. In the table below, the <em>x-<\/em>&nbsp;and <em>y<\/em>-coordinates of each ordered pair on the graph is recorded.\n<div align=\"center\">\n<table>\n<tbody>\n<tr>\n<td><b><strong><em>x<\/em><\/strong><\/b><b><strong>-coordinate<\/strong><\/b><\/td>\n<td><b><strong><em>y<\/em><\/strong><\/b><b><strong>-coordinate<\/strong><\/b><\/td>\n<\/tr>\n<tr>\n<td>0<\/td>\n<td>0<\/td>\n<\/tr>\n<tr>\n<td>1<\/td>\n<td>2<\/td>\n<\/tr>\n<tr>\n<td>2<\/td>\n<td>4<\/td>\n<\/tr>\n<tr>\n<td>3<\/td>\n<td>6<\/td>\n<\/tr>\n<tr>\n<td>4<\/td>\n<td>8<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\nNotice that each <em>y<\/em>-coordinate is twice the corresponding <em>x<\/em>-value. All of these <em>x-<\/em>&nbsp;and <em>y<\/em>-values follow the same pattern, and, when placed on a coordinate plane, they all line up.\n\n<img class=\"aligncenter wp-image-1458\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/03\/08205615\/Graphing-Linear-Equations-1.png\" alt=\"Graph with the point (0,0); the point (1,2); the point (2,4); the point (3,6); and the point (4,8).\" width=\"358\" height=\"276\">\n\nOnce you know the pattern that relates the <em>x-<\/em> and <em>y-<\/em>values, you can find a <em>y<\/em>-value for any <em>x<\/em>-value that lies on the line. So if the rule of this pattern is that each <em>y<\/em>-coordinate is <em>twice<\/em> the corresponding <em>x<\/em>-value, then the ordered pairs (1.5, 3), (2.5, 5), and (3.5, 7) should all appear on the line too, correct? Look to see what happens.\n\n<img class=\"aligncenter wp-image-1459\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/03\/08210025\/Graphing-Linear-Equations-2.png\" alt=\"Graph with the point (0,0); the point (1,2); the point (1.5, 3); the point (2,4); the point (2.5, 5); the point (3,6); the point (3.5, 7); and the point (4,8).\" width=\"386\" height=\"297\">\n\nIf you were to keep adding ordered pairs (<em>x<\/em>, <em>y<\/em>) where the <em>y<\/em>-value was twice the <em>x<\/em>-value, you would end up with a graph like this.\n\n<img class=\"aligncenter wp-image-1460\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/03\/08210231\/Graphing-Linear-Equations-3.png\" alt=\"A line drawn through the point (0,0); the point (1,2); the point (2,4); the point (3,6); and the point (4,8).\" width=\"386\" height=\"297\">\n\nLook at how all of the points blend together to create a line. You can think of a line, then, as a collection of an infinite number of individual points that share the same mathematical relationship. In this case, the relationship is that the <em>y<\/em>-value is twice the <em>x<\/em>-value.\n\nThere are multiple ways to represent a linear relationship\u2014a table, a linear graph, and there is also a <b><strong>linear equation<\/strong><\/b>. A linear equation is an equation with two variables whose ordered pairs graph as a straight line.\n\nThere are several ways to create a graph from a linear equation. One way is to create a table of values for <em>x<\/em> and <em>y<\/em>, and then plot these ordered pairs on the coordinate plane. Two points are enough to determine a line. However, it\u2019s always a good idea to plot more than two points to avoid possible errors.\n\nThen you draw a line through the points to show all of the points that are on the line. The arrows at each end of the graph indicate that the line continues endlessly in both directions. Every point on this line is a solution to the linear equation.\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\nGraph the linear equation [latex]y=\u22121.5x[\/latex].\n\n[reveal-answer q=\"983342\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"983342\"]Evaluate [latex]y=\u22121.5x[\/latex]&nbsp;for different values of <em>x<\/em>, and create a table of corresponding <em>x<\/em> and <em>y<\/em> values.\n<table>\n<tbody>\n<tr>\n<td><em>x<\/em> values<\/td>\n<td>[latex]\u22121.5x[\/latex]<\/td>\n<td><em>y<\/em> values<\/td>\n<\/tr>\n<tr>\n<td>[latex]0[\/latex]<\/td>\n<td>[latex]\u22121.5(0)[\/latex]<\/td>\n<td>[latex]0[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]2[\/latex]<\/td>\n<td>[latex]\u22121.5(2)[\/latex]<\/td>\n<td>[latex]\u22123[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]4[\/latex]<\/td>\n<td>[latex]\u22121.5(4)[\/latex]<\/td>\n<td>[latex]\u22126[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]6[\/latex]<\/td>\n<td>[latex]\u22121.5(6)[\/latex]<\/td>\n<td>[latex]\u22129[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\nSince the coefficient of <em>x<\/em> is [latex]\u22121.5[\/latex], it is convenient to choose multiples of 2 for <em>x<\/em>. This ensures that <em>y<\/em> is an integer, and makes the line easier to graph.\n\nConvert the table to ordered pairs. Then plot the ordered pairs.\n<p style=\"text-align: center\">[latex](0,0)[\/latex]<\/p>\n<p style=\"text-align: center\">[latex](2,\u22123)[\/latex]<\/p>\n<p style=\"text-align: center\">[latex](4,\u22126)[\/latex]<\/p>\n<p style=\"text-align: center\">[latex](6,\u22129)[\/latex]<\/p>\nDraw a line through the points to indicate all of the points on the line.\n<h4>Answer\n<img class=\"aligncenter wp-image-1467\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/03\/08213504\/Graphing-Linear-Equations-52.png\" alt=\"Graphing Linear Equations (5)\" width=\"353\" height=\"271\">[\/hidden-answer]<\/h4>\n<\/div>\n<h3>Graph the linear equation<\/h3>\nhttps:\/\/youtu.be\/f5yvGPEWpvE\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\nGraph the linear equation [latex]y=2x+3[\/latex].\n\n[reveal-answer q=\"834421\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"834421\"]Evaluate [latex]y=2x+3[\/latex]&nbsp;for different values of <em>x<\/em>, and create a table of corresponding <em>x<\/em> and <em>y<\/em> values.\n<table>\n<tbody>\n<tr>\n<td><em>x<\/em> values<\/td>\n<td>[latex]2x+3[\/latex]<\/td>\n<td><em>y<\/em> values<\/td>\n<\/tr>\n<tr>\n<td>0<\/td>\n<td>2(0) + 3<\/td>\n<td>3<\/td>\n<\/tr>\n<tr>\n<td>1<\/td>\n<td>2(1) + 3<\/td>\n<td>5<\/td>\n<\/tr>\n<tr>\n<td>2<\/td>\n<td>2(2) + 3<\/td>\n<td>7<\/td>\n<\/tr>\n<tr>\n<td>3<\/td>\n<td>2(3) + 3<\/td>\n<td>9<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: center\">(0, 3)<\/p>\n<p style=\"text-align: center\">(1, 5)<\/p>\n<p style=\"text-align: center\">(2, 7)<\/p>\n<p style=\"text-align: center\">(3, 9)<\/p>\nConvert the table to ordered pairs.&nbsp;Plot the ordered pairs.\n\n<img class=\"aligncenter wp-image-1468\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/03\/08213653\/Graphing-Linear-Equations-6.png\" alt=\"Graph showing the point (0,3); the point (1,5); the point (2,7); and the point (3,9).\" width=\"317\" height=\"244\">\n\nDraw a line through the points to indicate all of the points on the line.\n<h4>Answer<\/h4>\n<img class=\"aligncenter wp-image-1469\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/03\/08213931\/Graphing-Linear-Equations-7.png\" alt=\"Line drawn through the point (0,3); the point (1,5); the point (2,7); and the point (3,9). The line is labeled y=2x+3.\" width=\"275\" height=\"211\">[\/hidden-answer]\n\n<\/div>\n<h3 id=\"title2\">Ordered Pairs as Solutions<\/h3>\nSo far, you have considered the following ideas about lines: a line is a visual representation of a linear equation, and the line itself is made up of an infinite number of points (or ordered pairs). The picture below shows the line of the linear equation [latex]y=2x\u20135[\/latex]&nbsp;with some of the specific points on the line.\n\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064253\/image021-1.jpg\" alt=\"Line drawn through the points 0, negative 5; the point 1, negative 3; the point 2, negative 1; the point (4,3); and the point 5,5). The line is labeled y=2x-5.\" width=\"428\" height=\"423\">\n\nEvery point on the line is a solution to the equation [latex]y=2x\u20135[\/latex]. You can try any of the points that are labeled like the ordered pair, [latex](1,\u22123)[\/latex].\n<p style=\"text-align: center\">[latex]\\begin{array}{l}\\,\\,\\,\\,y=2x-5\\\\-3=2\\left(1\\right)-5\\\\-3=2-5\\\\-3=-3\\\\\\text{This is true.}\\end{array}[\/latex]<\/p>\nYou can also try ANY of the other points on the line. Every point on the line is a solution to the equation [latex]y=2x\u20135[\/latex]. All this means is that determining whether an ordered pair is a solution of an equation is pretty straightforward. If the ordered pair is on the line created by the linear equation, then it is a solution to the equation. But if the ordered pair is not on the line\u2014no matter how close it may look\u2014then it is not a solution to the equation.\n<div class=\"textbox shaded\">\n<h3 id=\"Identifying Solutions\">Identifying Solutions<\/h3>\nTo find out whether an ordered pair is a solution of a linear equation, you can do the following:\n<ul>\n \t<li>Graph the linear equation, and graph the ordered pair. If the ordered pair appears to be on the graph of a line, then it is a possible solution of the linear equation. If the ordered pair does not lie on the graph of a line, then it is not a solution.<\/li>\n \t<li>Substitute the (<i>x<\/i>, <i>y<\/i>) values into the equation. If the equation yields a true statement, then the ordered pair is a solution of the linear equation. If the ordered pair does not yield a true statement then it is not a solution.<\/li>\n<\/ul>\n<\/div>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\nDetermine whether [latex](\u22122,4)[\/latex] is a solution to the equation [latex]4y+5x=3[\/latex].\n\n[reveal-answer q=\"980260\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"980260\"]For this problem, you will use the substitution method. Substitute [latex]x=\u22122[\/latex]&nbsp;and [latex]y=4[\/latex]&nbsp;into the equation.\n<p style=\"text-align: center\">[latex]\\begin{array}{r}4y+5x=3\\\\4\\left(4\\right)+5\\left(\u22122\\right)=3\\end{array}[\/latex]<\/p>\nEvaluate.\n<p style=\"text-align: center\">[latex]\\begin{array}{r}16+\\left(\u221210\\right)=3\\\\6=3\\end{array}[\/latex]<\/p>\nThe statement is not true, so [latex](\u22122,4)[\/latex] is not a solution to the equation [latex]4y+5x=3[\/latex].\n<h4>Answer<\/h4>\n[latex](\u22122,4)[\/latex] is not a solution to the equation [latex]4y+5x=3[\/latex].[\/hidden-answer]\n\n<\/div>\n<h3 id=\"video2\">Determine If an Ordered Pair is a Solution to a Linear Equation<\/h3>\nhttps:\/\/youtu.be\/9aWGxt7OnB8\n<h2 id=\"title2\">Solve for <em>y<\/em>, then graph a linear equation<\/h2>\nThe linear equations we have graphed so far are in the form [latex]y=mx+b[\/latex] where <em>m<\/em> and <em>b<\/em> are real numbers. In this section we will graph linear equations that appear in different forms than we have seen.\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\nGraph the linear equation [latex]y+3x=5[\/latex].\n\n[reveal-answer q=\"61530\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"61530\"]First, solve [latex]y+3x=5[\/latex] for <em>y<\/em>, then the equation will look familiar and you can create a table of ordered pairs.\n<p style=\"text-align: center\">[latex]\\begin{array}{r}y+3x-3x=5-3x\\\\y=5-3x\\end{array}[\/latex]<\/p>\nEvaluate [latex]y=5\u20133x[\/latex]&nbsp;for different values of <em>x<\/em>, and create a table of corresponding <em>x<\/em> and <em>y<\/em> values.\n<table>\n<tbody>\n<tr>\n<td><em>x <\/em>values<\/td>\n<td>[latex]5\u20133x[\/latex]<\/td>\n<td><em>y<\/em> values<\/td>\n<\/tr>\n<tr>\n<td>[latex]0[\/latex]<\/td>\n<td>[latex]5\u20133(0)[\/latex]<\/td>\n<td>[latex]5[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]1[\/latex]<\/td>\n<td>[latex]5\u20133(1)[\/latex]<\/td>\n<td>[latex]2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]2[\/latex]<\/td>\n<td>[latex]5\u20133(2)[\/latex]<\/td>\n<td>[latex]\u22121[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]3[\/latex]<\/td>\n<td>[latex]5\u20133(3)[\/latex]<\/td>\n<td>[latex]\u22124[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\nPlot the ordered pairs (shown below).\n<p style=\"text-align: center\">[latex](0,5)[\/latex]<\/p>\n<p style=\"text-align: center\">[latex](1,2)[\/latex]<\/p>\n<p style=\"text-align: center\">[latex](2,\u22121)[\/latex]<\/p>\n<p style=\"text-align: center\">[latex](3,\u22124)[\/latex]<\/p>\n<img class=\"aligncenter wp-image-1470\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/03\/08215139\/Graphing-Linear-Equations-8.png\" alt=\"Graph showing the point (0,5), the point (1,2), the point (2,-1), and the point (3,-4).\" width=\"393\" height=\"303\">\n\nDraw a line through the points to indicate all of the points on the line.\n<h4>Answer<\/h4>\n<img class=\"aligncenter wp-image-1471\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/03\/08215447\/Graphing-Linear-Equations-9.png\" alt=\"Graphing Linear Equations (9)\" width=\"389\" height=\"300\">[\/hidden-answer]\n\n<\/div>\n<h2 id=\"video2\">Video: Solve for y, then graph a linear equation<\/h2>\nhttps:\/\/youtu.be\/6yL3gfPbOt8\n<h2 id=\"title3\">Horizontal and Vertical Lines<\/h2>\nThe linear equations [latex]x=2[\/latex]&nbsp;and [latex]y=\u22123[\/latex]&nbsp;only have one variable in each of them. However, because these are linear equations, then they will graph on a coordinate plane just as the linear equations above do. Just think of the equation [latex]x=2[\/latex]&nbsp;as [latex]x=0y+2[\/latex]&nbsp;and think of [latex]y=\u22123[\/latex]&nbsp;as [latex]y=0x\u20133[\/latex].\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\nGraph [latex]y=\u22123[\/latex].\n\n[reveal-answer q=\"140758\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"140758\"]\n<table>\n<tbody>\n<tr>\n<td><em>x<\/em> values<\/td>\n<td>[latex]0x\u20133[\/latex]<\/td>\n<td><em>y<\/em> values<\/td>\n<\/tr>\n<tr>\n<td>[latex]0[\/latex]<\/td>\n<td>[latex]0(0)\u20133[\/latex]<\/td>\n<td>[latex]\u22123[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]1[\/latex]<\/td>\n<td>[latex]0(1)\u20133[\/latex]<\/td>\n<td>[latex]\u22123[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]2[\/latex]<\/td>\n<td>[latex]0(2)\u20133[\/latex]<\/td>\n<td>[latex]\u22123[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]3[\/latex]<\/td>\n<td>[latex]0(3)\u20133[\/latex]<\/td>\n<td>[latex]\u22123[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\nWrite [latex]y=\u22123[\/latex]&nbsp;as [latex]y=0x\u20133[\/latex], and evaluate <em>y<\/em> when <em>x<\/em> has several values. Or just realize that [latex]y=\u22123[\/latex]&nbsp;means every <em>y<\/em> value will be [latex]\u22123[\/latex], no matter what <em>x<\/em> is.\n<p style=\"text-align: center\">[latex](0,\u22123)[\/latex]<\/p>\n<p style=\"text-align: center\">[latex](1,\u22123)[\/latex]<\/p>\n<p style=\"text-align: center\">[latex](2,\u22123)[\/latex]<\/p>\n<p style=\"text-align: center\">[latex](3,\u22123)[\/latex]<\/p>\nPlot the ordered pairs (shown below).\n\n<img class=\"aligncenter wp-image-1473\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/03\/08220042\/Graphing-Linear-Equations-10.png\" alt=\"Graph with the point (1,-3), the point (2,-3), and the point (3,-3).\" width=\"356\" height=\"274\">\n\nDraw a line through the points to indicate all of the points on the line.\n<h4>Answer<\/h4>\n<img class=\"aligncenter wp-image-1474\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/03\/08220257\/Graphing-Linear-Equations-11.png\" alt=\"Graphing Linear Equations (11)\" width=\"351\" height=\"270\">Notice that [latex]y=\u22123[\/latex]&nbsp;graphs as a horizontal line.[\/hidden-answer]\n\n<\/div>\n<p id=\"video3\">In the following video you will see more examples of graphing horizontal and vertical lines.<\/p>\nhttps:\/\/youtu.be\/2A2fhImjOBc\n<h2 id=\"Intercepts\">Intercepts<\/h2>\nThe intercepts of a line are the points where the line intercepts, or crosses, the horizontal and vertical axes. To help you remember what \u201cintercept\u201d means, think about the word \u201cintersect.\u201d The two words sound alike and in this case mean the same thing.\n\nThe straight line on the graph below intercepts the two coordinate axes. The point where the line crosses the <i>x<\/i>-axis is called the <b><i>x<\/i>-intercept<\/b>. The <b><i>y<\/i>-intercept<\/b> is the point where the line crosses the <i>y<\/i>-axis.\n\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064249\/image018-1.jpg\" alt=\"A line going through two points. One point is on the x-axis and is labeled the x-intercept. The other point is on the y-axis and is labeled y-intercept.\" width=\"329\" height=\"320\">\n\nThe <i>x<\/i>-intercept above is the point [latex](\u22122,0)[\/latex]. The <i>y<\/i>-intercept above is the point (0, 2).\n\nNotice that the <i>y<\/i>-intercept always occurs where [latex]x=0[\/latex], and the <i>x<\/i>-intercept always occurs where [latex]y=0[\/latex].\n\nTo find the <em>x<\/em>- and <em>y<\/em>-intercepts of a linear equation, you can substitute 0 for <i>y<\/i> and for <i>x<\/i> respectively.\n\nFor example, the linear equation [latex]3y+2x=6[\/latex]&nbsp;has an <i>x<\/i> intercept when [latex]y=0[\/latex], so [latex]3\\left(0\\right)+2x=6\\\\[\/latex].\n<p style=\"text-align: center\">[latex]\\begin{array}{r}2x=6\\\\x=3\\end{array}[\/latex]<\/p>\nThe <em>x<\/em>-intercept is [latex](3,0)[\/latex].\n\nLikewise the <i>y<\/i>-intercept occurs when [latex]x=0[\/latex].\n<p style=\"text-align: center\">[latex]\\begin{array}{r}3y+2\\left(0\\right)=6\\\\3y=6\\\\y=2\\end{array}[\/latex]<\/p>\nThe <i>y<\/i>-intercept is [latex](0,2)[\/latex].\n<h3 id=\"Using Intercepts to Graph Lines\">Using Intercepts to Graph Lines<\/h3>\nYou can use intercepts to graph linear equations. Once you have found the two intercepts, draw a line through them.\n\nLet\u2019s do it with the equation [latex]3y+2x=6[\/latex]. You figured out that the intercepts of the line this equation represents are [latex](0,2)[\/latex] and [latex](3,0)[\/latex]. That\u2019s all you need to know.\n\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064250\/image019-1.jpg\" alt=\"A line drawn through the points (0,2) and (3,0). The point (0,2) is labeled y-intercept and the point (3,0) is labeled x-intercept. The line is labeled 3y+2x=6.\" width=\"340\" height=\"344\">\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\nGraph [latex]5y+3x=30[\/latex]&nbsp;using the <em>x<\/em> and <em>y<\/em>-intercepts.\n\n[reveal-answer q=\"153435\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"153435\"]When an equation is in [latex]Ax+By=C[\/latex]&nbsp;form, you can easily find the <i>x<\/i>- and <i>y<\/i>-intercepts and then graph.\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>\nTo find the <i>y<\/i>-intercept, set [latex]x=0[\/latex]&nbsp;and solve for <i>y<\/i>.\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>\nTo find the <i>x<\/i>-intercept, set [latex]y=0[\/latex] and solve for <i>x<\/i>.\n<h4>Answer<\/h4>\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=\"\" width=\"425\" height=\"430\">[\/hidden-answer]\n\n<\/div>\nhttps:\/\/youtu.be\/k8r-q_T6UFk\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\nGraph [latex]y=2x-4[\/latex] using the <em>x<\/em> and <em>y<\/em>-intercepts.\n\n[reveal-answer q=\"476848\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"476848\"]First, find the <em>y<\/em>-intercept. Set <em>x<\/em> equal to zero and solve for <em>y<\/em>.\n<p style=\"text-align: center\">[latex]\\begin{array}{l}y=2x-4\\\\y=2\\left(0\\right)-4\\\\y=0-4\\\\y=-4\\\\y\\text{-intercept}\\left(0,-4\\right)\\end{array}[\/latex]<\/p>\nTo find the <i>x<\/i>-intercept, set [latex]y=0[\/latex]&nbsp;and solve for <i>x<\/i>.\n<p style=\"text-align: center\">[latex]\\begin{array}{l}y=2x-4\\\\0=2x-4\\\\4=2x\\\\x=2\\\\x\\text{-intercept}\\left(2,0\\right)\\end{array}[\/latex]<\/p>\n\n<h4>Answer<\/h4>\nmake graph for this example[\/hidden-answer]\n<\/div>\n<h2>Summary<\/h2>\nThe coordinate plane is a system for graphing and describing points and lines. The coordinate plane is comprised of a horizontal (<i>x<\/i>-) axis and a vertical (<i>y-<\/i>) axis. The intersection of these lines creates the origin, which is the point [latex](0,0)[\/latex]. The coordinate plane is split into four quadrants. Together, these features of the coordinate system allow for the graphical representation and communication about points, lines, and other algebraic concepts.\n\n\n","rendered":"<div class=\"bcc-box bcc-highlight\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Graph an Equation Using Ordered Pairs\n<ul>\n<li>Create a table of ordered pairs from a two-variable linear equation<\/li>\n<li>Graph a two-variable linear equation using a table of ordered pairs<\/li>\n<li>Determine whether an ordered pair is a solution of an equation<\/li>\n<\/ul>\n<\/li>\n<li>Graph Linear Equations in Different Forms\n<ul>\n<li>Solve for <em>y<\/em>, then graph a two-variable linear equation<\/li>\n<li>Graph horizontal and vertical lines<\/li>\n<\/ul>\n<\/li>\n<li>Graph an Equation Using Intercepts\n<ul>\n<li>Recognize when an ordered pair is a <em>y<\/em>-intercept or an <em>x<\/em>-intercept<\/li>\n<li>Graph a linear equation using <em>x<\/em>&#8211; and <em>y<\/em>-intercepts<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<h2><span style=\"line-height: 1.5\">Graphing Using Ordered Pairs<\/span><\/h2>\n<p>Graphing ordered pairs is only the beginning of the story. Once you know how to place points on a grid, you can use them to make sense of all kinds of mathematical relationships.<\/p>\n<div>\n<p>You can use a <b><strong>coordinate plane<\/strong><\/b> to plot points and to map various relationships, such as the relationship between an object\u2019s distance and the elapsed time. Many mathematical relationships are <b><strong>linear relationships<\/strong><\/b>. Let\u2019s look at what a linear relationship is.<\/p>\n<h3 id=\"title1\">Plotting points to graph linear relationships<\/h3>\n<\/div>\n<p>A linear relationship is a relationship between variables such that when plotted on a coordinate plane, the points lie on a line. Let\u2019s start by looking at a series of points in Quadrant I on the coordinate plane.<\/p>\n<p>Look at the five <b><strong>ordered pairs<\/strong><\/b> (and their <em>x<\/em>&#8211; and <em>y<\/em>-coordinates) below. Do you see any pattern to the location of the points? If this pattern continued, what other points could be on the line?<\/p>\n<p>You probably identified that if this pattern continued the next ordered pair would be at (5, 10). This makes sense because the point (5, 10) \u201clines up\u201d with the other points in the series\u2014it is literally on the same line as the others. Applying the same logic, you may identify that the ordered pairs (6, 12) and (7, 14) would also belong if this coordinate plane were larger; they, too, will line up with the other points.<\/p>\n<p>These series of points can also be represented in a table. In the table below, the <em>x-<\/em>&nbsp;and <em>y<\/em>-coordinates of each ordered pair on the graph is recorded.<\/p>\n<div style=\"margin: auto;\">\n<table>\n<tbody>\n<tr>\n<td><b><strong><em>x<\/em><\/strong><\/b><b><strong>-coordinate<\/strong><\/b><\/td>\n<td><b><strong><em>y<\/em><\/strong><\/b><b><strong>-coordinate<\/strong><\/b><\/td>\n<\/tr>\n<tr>\n<td>0<\/td>\n<td>0<\/td>\n<\/tr>\n<tr>\n<td>1<\/td>\n<td>2<\/td>\n<\/tr>\n<tr>\n<td>2<\/td>\n<td>4<\/td>\n<\/tr>\n<tr>\n<td>3<\/td>\n<td>6<\/td>\n<\/tr>\n<tr>\n<td>4<\/td>\n<td>8<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p>Notice that each <em>y<\/em>-coordinate is twice the corresponding <em>x<\/em>-value. All of these <em>x-<\/em>&nbsp;and <em>y<\/em>-values follow the same pattern, and, when placed on a coordinate plane, they all line up.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-1458\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/03\/08205615\/Graphing-Linear-Equations-1.png\" alt=\"Graph with the point (0,0); the point (1,2); the point (2,4); the point (3,6); and the point (4,8).\" width=\"358\" height=\"276\" \/><\/p>\n<p>Once you know the pattern that relates the <em>x-<\/em> and <em>y-<\/em>values, you can find a <em>y<\/em>-value for any <em>x<\/em>-value that lies on the line. So if the rule of this pattern is that each <em>y<\/em>-coordinate is <em>twice<\/em> the corresponding <em>x<\/em>-value, then the ordered pairs (1.5, 3), (2.5, 5), and (3.5, 7) should all appear on the line too, correct? Look to see what happens.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-1459\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/03\/08210025\/Graphing-Linear-Equations-2.png\" alt=\"Graph with the point (0,0); the point (1,2); the point (1.5, 3); the point (2,4); the point (2.5, 5); the point (3,6); the point (3.5, 7); and the point (4,8).\" width=\"386\" height=\"297\" \/><\/p>\n<p>If you were to keep adding ordered pairs (<em>x<\/em>, <em>y<\/em>) where the <em>y<\/em>-value was twice the <em>x<\/em>-value, you would end up with a graph like this.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-1460\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/03\/08210231\/Graphing-Linear-Equations-3.png\" alt=\"A line drawn through the point (0,0); the point (1,2); the point (2,4); the point (3,6); and the point (4,8).\" width=\"386\" height=\"297\" \/><\/p>\n<p>Look at how all of the points blend together to create a line. You can think of a line, then, as a collection of an infinite number of individual points that share the same mathematical relationship. In this case, the relationship is that the <em>y<\/em>-value is twice the <em>x<\/em>-value.<\/p>\n<p>There are multiple ways to represent a linear relationship\u2014a table, a linear graph, and there is also a <b><strong>linear equation<\/strong><\/b>. A linear equation is an equation with two variables whose ordered pairs graph as a straight line.<\/p>\n<p>There are several ways to create a graph from a linear equation. One way is to create a table of values for <em>x<\/em> and <em>y<\/em>, and then plot these ordered pairs on the coordinate plane. Two points are enough to determine a line. However, it\u2019s always a good idea to plot more than two points to avoid possible errors.<\/p>\n<p>Then you draw a line through the points to show all of the points that are on the line. The arrows at each end of the graph indicate that the line continues endlessly in both directions. Every point on this line is a solution to the linear equation.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Graph the linear equation [latex]y=\u22121.5x[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q983342\">Show Solution<\/span><\/p>\n<div id=\"q983342\" class=\"hidden-answer\" style=\"display: none\">Evaluate [latex]y=\u22121.5x[\/latex]&nbsp;for different values of <em>x<\/em>, and create a table of corresponding <em>x<\/em> and <em>y<\/em> values.<\/p>\n<table>\n<tbody>\n<tr>\n<td><em>x<\/em> values<\/td>\n<td>[latex]\u22121.5x[\/latex]<\/td>\n<td><em>y<\/em> values<\/td>\n<\/tr>\n<tr>\n<td>[latex]0[\/latex]<\/td>\n<td>[latex]\u22121.5(0)[\/latex]<\/td>\n<td>[latex]0[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]2[\/latex]<\/td>\n<td>[latex]\u22121.5(2)[\/latex]<\/td>\n<td>[latex]\u22123[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]4[\/latex]<\/td>\n<td>[latex]\u22121.5(4)[\/latex]<\/td>\n<td>[latex]\u22126[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]6[\/latex]<\/td>\n<td>[latex]\u22121.5(6)[\/latex]<\/td>\n<td>[latex]\u22129[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Since the coefficient of <em>x<\/em> is [latex]\u22121.5[\/latex], it is convenient to choose multiples of 2 for <em>x<\/em>. This ensures that <em>y<\/em> is an integer, and makes the line easier to graph.<\/p>\n<p>Convert the table to ordered pairs. Then plot the ordered pairs.<\/p>\n<p style=\"text-align: center\">[latex](0,0)[\/latex]<\/p>\n<p style=\"text-align: center\">[latex](2,\u22123)[\/latex]<\/p>\n<p style=\"text-align: center\">[latex](4,\u22126)[\/latex]<\/p>\n<p style=\"text-align: center\">[latex](6,\u22129)[\/latex]<\/p>\n<p>Draw a line through the points to indicate all of the points on the line.<\/p>\n<h4>Answer<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-1467\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/03\/08213504\/Graphing-Linear-Equations-52.png\" alt=\"Graphing Linear Equations (5)\" width=\"353\" height=\"271\" \/><\/div>\n<\/div>\n<\/h4>\n<\/div>\n<h3>Graph the linear equation<\/h3>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Graph Basic Linear Equations by Completing a Table of Values\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/f5yvGPEWpvE?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 the linear equation [latex]y=2x+3[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q834421\">Show Solution<\/span><\/p>\n<div id=\"q834421\" class=\"hidden-answer\" style=\"display: none\">Evaluate [latex]y=2x+3[\/latex]&nbsp;for different values of <em>x<\/em>, and create a table of corresponding <em>x<\/em> and <em>y<\/em> values.<\/p>\n<table>\n<tbody>\n<tr>\n<td><em>x<\/em> values<\/td>\n<td>[latex]2x+3[\/latex]<\/td>\n<td><em>y<\/em> values<\/td>\n<\/tr>\n<tr>\n<td>0<\/td>\n<td>2(0) + 3<\/td>\n<td>3<\/td>\n<\/tr>\n<tr>\n<td>1<\/td>\n<td>2(1) + 3<\/td>\n<td>5<\/td>\n<\/tr>\n<tr>\n<td>2<\/td>\n<td>2(2) + 3<\/td>\n<td>7<\/td>\n<\/tr>\n<tr>\n<td>3<\/td>\n<td>2(3) + 3<\/td>\n<td>9<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: center\">(0, 3)<\/p>\n<p style=\"text-align: center\">(1, 5)<\/p>\n<p style=\"text-align: center\">(2, 7)<\/p>\n<p style=\"text-align: center\">(3, 9)<\/p>\n<p>Convert the table to ordered pairs.&nbsp;Plot the ordered pairs.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-1468\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/03\/08213653\/Graphing-Linear-Equations-6.png\" alt=\"Graph showing the point (0,3); the point (1,5); the point (2,7); and the point (3,9).\" width=\"317\" height=\"244\" \/><\/p>\n<p>Draw a line through the points to indicate all of the points on the line.<\/p>\n<h4>Answer<\/h4>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-1469\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/03\/08213931\/Graphing-Linear-Equations-7.png\" alt=\"Line drawn through the point (0,3); the point (1,5); the point (2,7); and the point (3,9). The line is labeled y=2x+3.\" width=\"275\" height=\"211\" \/><\/div>\n<\/div>\n<\/div>\n<h3 id=\"title2\">Ordered Pairs as Solutions<\/h3>\n<p>So far, you have considered the following ideas about lines: a line is a visual representation of a linear equation, and the line itself is made up of an infinite number of points (or ordered pairs). The picture below shows the line of the linear equation [latex]y=2x\u20135[\/latex]&nbsp;with some of the specific points on the line.<\/p>\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\/04064253\/image021-1.jpg\" alt=\"Line drawn through the points 0, negative 5; the point 1, negative 3; the point 2, negative 1; the point (4,3); and the point 5,5). The line is labeled y=2x-5.\" width=\"428\" height=\"423\" \/><\/p>\n<p>Every point on the line is a solution to the equation [latex]y=2x\u20135[\/latex]. You can try any of the points that are labeled like the ordered pair, [latex](1,\u22123)[\/latex].<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{l}\\,\\,\\,\\,y=2x-5\\\\-3=2\\left(1\\right)-5\\\\-3=2-5\\\\-3=-3\\\\\\text{This is true.}\\end{array}[\/latex]<\/p>\n<p>You can also try ANY of the other points on the line. Every point on the line is a solution to the equation [latex]y=2x\u20135[\/latex]. All this means is that determining whether an ordered pair is a solution of an equation is pretty straightforward. If the ordered pair is on the line created by the linear equation, then it is a solution to the equation. But if the ordered pair is not on the line\u2014no matter how close it may look\u2014then it is not a solution to the equation.<\/p>\n<div class=\"textbox shaded\">\n<h3 id=\"Identifying Solutions\">Identifying Solutions<\/h3>\n<p>To find out whether an ordered pair is a solution of a linear equation, you can do the following:<\/p>\n<ul>\n<li>Graph the linear equation, and graph the ordered pair. If the ordered pair appears to be on the graph of a line, then it is a possible solution of the linear equation. If the ordered pair does not lie on the graph of a line, then it is not a solution.<\/li>\n<li>Substitute the (<i>x<\/i>, <i>y<\/i>) values into the equation. If the equation yields a true statement, then the ordered pair is a solution of the linear equation. If the ordered pair does not yield a true statement then it is not a solution.<\/li>\n<\/ul>\n<\/div>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Determine whether [latex](\u22122,4)[\/latex] is a solution to the equation [latex]4y+5x=3[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q980260\">Show Solution<\/span><\/p>\n<div id=\"q980260\" class=\"hidden-answer\" style=\"display: none\">For this problem, you will use the substitution method. Substitute [latex]x=\u22122[\/latex]&nbsp;and [latex]y=4[\/latex]&nbsp;into the equation.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{r}4y+5x=3\\\\4\\left(4\\right)+5\\left(\u22122\\right)=3\\end{array}[\/latex]<\/p>\n<p>Evaluate.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{r}16+\\left(\u221210\\right)=3\\\\6=3\\end{array}[\/latex]<\/p>\n<p>The statement is not true, so [latex](\u22122,4)[\/latex] is not a solution to the equation [latex]4y+5x=3[\/latex].<\/p>\n<h4>Answer<\/h4>\n<p>[latex](\u22122,4)[\/latex] is not a solution to the equation [latex]4y+5x=3[\/latex].<\/p><\/div>\n<\/div>\n<\/div>\n<h3 id=\"video2\">Determine If an Ordered Pair is a Solution to a Linear Equation<\/h3>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Determine If an Ordered Pair is a Solution to a Linear Equation\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/9aWGxt7OnB8?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2 id=\"title2\">Solve for <em>y<\/em>, then graph a linear equation<\/h2>\n<p>The linear equations we have graphed so far are in the form [latex]y=mx+b[\/latex] where <em>m<\/em> and <em>b<\/em> are real numbers. In this section we will graph linear equations that appear in different forms than we have seen.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Graph the linear equation [latex]y+3x=5[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q61530\">Show Solution<\/span><\/p>\n<div id=\"q61530\" class=\"hidden-answer\" style=\"display: none\">First, solve [latex]y+3x=5[\/latex] for <em>y<\/em>, then the equation will look familiar and you can create a table of ordered pairs.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{r}y+3x-3x=5-3x\\\\y=5-3x\\end{array}[\/latex]<\/p>\n<p>Evaluate [latex]y=5\u20133x[\/latex]&nbsp;for different values of <em>x<\/em>, and create a table of corresponding <em>x<\/em> and <em>y<\/em> values.<\/p>\n<table>\n<tbody>\n<tr>\n<td><em>x <\/em>values<\/td>\n<td>[latex]5\u20133x[\/latex]<\/td>\n<td><em>y<\/em> values<\/td>\n<\/tr>\n<tr>\n<td>[latex]0[\/latex]<\/td>\n<td>[latex]5\u20133(0)[\/latex]<\/td>\n<td>[latex]5[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]1[\/latex]<\/td>\n<td>[latex]5\u20133(1)[\/latex]<\/td>\n<td>[latex]2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]2[\/latex]<\/td>\n<td>[latex]5\u20133(2)[\/latex]<\/td>\n<td>[latex]\u22121[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]3[\/latex]<\/td>\n<td>[latex]5\u20133(3)[\/latex]<\/td>\n<td>[latex]\u22124[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Plot the ordered pairs (shown below).<\/p>\n<p style=\"text-align: center\">[latex](0,5)[\/latex]<\/p>\n<p style=\"text-align: center\">[latex](1,2)[\/latex]<\/p>\n<p style=\"text-align: center\">[latex](2,\u22121)[\/latex]<\/p>\n<p style=\"text-align: center\">[latex](3,\u22124)[\/latex]<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-1470\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/03\/08215139\/Graphing-Linear-Equations-8.png\" alt=\"Graph showing the point (0,5), the point (1,2), the point (2,-1), and the point (3,-4).\" width=\"393\" height=\"303\" \/><\/p>\n<p>Draw a line through the points to indicate all of the points on the line.<\/p>\n<h4>Answer<\/h4>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-1471\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/03\/08215447\/Graphing-Linear-Equations-9.png\" alt=\"Graphing Linear Equations (9)\" width=\"389\" height=\"300\" \/><\/div>\n<\/div>\n<\/div>\n<h2 id=\"video2\">Video: Solve for y, then graph a linear equation<\/h2>\n<p><iframe loading=\"lazy\" id=\"oembed-3\" title=\"Ex 3:  Graph a Linear Equation in Standard Form Using a Table of Values\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/6yL3gfPbOt8?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2 id=\"title3\">Horizontal and Vertical Lines<\/h2>\n<p>The linear equations [latex]x=2[\/latex]&nbsp;and [latex]y=\u22123[\/latex]&nbsp;only have one variable in each of them. However, because these are linear equations, then they will graph on a coordinate plane just as the linear equations above do. Just think of the equation [latex]x=2[\/latex]&nbsp;as [latex]x=0y+2[\/latex]&nbsp;and think of [latex]y=\u22123[\/latex]&nbsp;as [latex]y=0x\u20133[\/latex].<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Graph [latex]y=\u22123[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q140758\">Show Solution<\/span><\/p>\n<div id=\"q140758\" class=\"hidden-answer\" style=\"display: none\">\n<table>\n<tbody>\n<tr>\n<td><em>x<\/em> values<\/td>\n<td>[latex]0x\u20133[\/latex]<\/td>\n<td><em>y<\/em> values<\/td>\n<\/tr>\n<tr>\n<td>[latex]0[\/latex]<\/td>\n<td>[latex]0(0)\u20133[\/latex]<\/td>\n<td>[latex]\u22123[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]1[\/latex]<\/td>\n<td>[latex]0(1)\u20133[\/latex]<\/td>\n<td>[latex]\u22123[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]2[\/latex]<\/td>\n<td>[latex]0(2)\u20133[\/latex]<\/td>\n<td>[latex]\u22123[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]3[\/latex]<\/td>\n<td>[latex]0(3)\u20133[\/latex]<\/td>\n<td>[latex]\u22123[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Write [latex]y=\u22123[\/latex]&nbsp;as [latex]y=0x\u20133[\/latex], and evaluate <em>y<\/em> when <em>x<\/em> has several values. Or just realize that [latex]y=\u22123[\/latex]&nbsp;means every <em>y<\/em> value will be [latex]\u22123[\/latex], no matter what <em>x<\/em> is.<\/p>\n<p style=\"text-align: center\">[latex](0,\u22123)[\/latex]<\/p>\n<p style=\"text-align: center\">[latex](1,\u22123)[\/latex]<\/p>\n<p style=\"text-align: center\">[latex](2,\u22123)[\/latex]<\/p>\n<p style=\"text-align: center\">[latex](3,\u22123)[\/latex]<\/p>\n<p>Plot the ordered pairs (shown below).<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-1473\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/03\/08220042\/Graphing-Linear-Equations-10.png\" alt=\"Graph with the point (1,-3), the point (2,-3), and the point (3,-3).\" width=\"356\" height=\"274\" \/><\/p>\n<p>Draw a line through the points to indicate all of the points on the line.<\/p>\n<h4>Answer<\/h4>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-1474\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/03\/08220257\/Graphing-Linear-Equations-11.png\" alt=\"Graphing Linear Equations (11)\" width=\"351\" height=\"270\" \/>Notice that [latex]y=\u22123[\/latex]&nbsp;graphs as a horizontal line.<\/div>\n<\/div>\n<\/div>\n<p id=\"video3\">In the following video you will see more examples of graphing horizontal and vertical lines.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-4\" title=\"Graphing Horzontal and Vertical Lines (L8.6)\" width=\"500\" height=\"375\" src=\"https:\/\/www.youtube.com\/embed\/2A2fhImjOBc?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2 id=\"Intercepts\">Intercepts<\/h2>\n<p>The intercepts of a line are the points where the line intercepts, or crosses, the horizontal and vertical axes. To help you remember what \u201cintercept\u201d means, think about the word \u201cintersect.\u201d The two words sound alike and in this case mean the same thing.<\/p>\n<p>The straight line on the graph below intercepts the two coordinate axes. The point where the line crosses the <i>x<\/i>-axis is called the <b><i>x<\/i>-intercept<\/b>. The <b><i>y<\/i>-intercept<\/b> is the point where the line crosses the <i>y<\/i>-axis.<\/p>\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\/04064249\/image018-1.jpg\" alt=\"A line going through two points. One point is on the x-axis and is labeled the x-intercept. The other point is on the y-axis and is labeled y-intercept.\" width=\"329\" height=\"320\" \/><\/p>\n<p>The <i>x<\/i>-intercept above is the point [latex](\u22122,0)[\/latex]. The <i>y<\/i>-intercept above is the point (0, 2).<\/p>\n<p>Notice that the <i>y<\/i>-intercept always occurs where [latex]x=0[\/latex], and the <i>x<\/i>-intercept always occurs where [latex]y=0[\/latex].<\/p>\n<p>To find the <em>x<\/em>&#8211; and <em>y<\/em>-intercepts of a linear equation, you can substitute 0 for <i>y<\/i> and for <i>x<\/i> respectively.<\/p>\n<p>For example, the linear equation [latex]3y+2x=6[\/latex]&nbsp;has an <i>x<\/i> intercept when [latex]y=0[\/latex], so [latex]3\\left(0\\right)+2x=6\\\\[\/latex].<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{r}2x=6\\\\x=3\\end{array}[\/latex]<\/p>\n<p>The <em>x<\/em>-intercept is [latex](3,0)[\/latex].<\/p>\n<p>Likewise the <i>y<\/i>-intercept occurs when [latex]x=0[\/latex].<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{r}3y+2\\left(0\\right)=6\\\\3y=6\\\\y=2\\end{array}[\/latex]<\/p>\n<p>The <i>y<\/i>-intercept is [latex](0,2)[\/latex].<\/p>\n<h3 id=\"Using Intercepts to Graph Lines\">Using Intercepts to Graph Lines<\/h3>\n<p>You can use intercepts to graph linear equations. Once you have found the two intercepts, draw a line through them.<\/p>\n<p>Let\u2019s do it with the equation [latex]3y+2x=6[\/latex]. You figured out that the intercepts of the line this equation represents are [latex](0,2)[\/latex] and [latex](3,0)[\/latex]. That\u2019s all you need to know.<\/p>\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\/04064250\/image019-1.jpg\" alt=\"A line drawn through the points (0,2) and (3,0). The point (0,2) is labeled y-intercept and the point (3,0) is labeled x-intercept. The line is labeled 3y+2x=6.\" width=\"340\" height=\"344\" \/><\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Graph [latex]5y+3x=30[\/latex]&nbsp;using 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]&nbsp;form, 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]&nbsp;and 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=\"\" width=\"425\" height=\"430\" \/><\/div>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"oembed-5\" 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]y=2x-4[\/latex] using 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=\"q476848\">Show Solution<\/span><\/p>\n<div id=\"q476848\" class=\"hidden-answer\" style=\"display: none\">First, find the <em>y<\/em>-intercept. Set <em>x<\/em> equal to zero and solve for <em>y<\/em>.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{l}y=2x-4\\\\y=2\\left(0\\right)-4\\\\y=0-4\\\\y=-4\\\\y\\text{-intercept}\\left(0,-4\\right)\\end{array}[\/latex]<\/p>\n<p>To find the <i>x<\/i>-intercept, set [latex]y=0[\/latex]&nbsp;and solve for <i>x<\/i>.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{l}y=2x-4\\\\0=2x-4\\\\4=2x\\\\x=2\\\\x\\text{-intercept}\\left(2,0\\right)\\end{array}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>make graph for this example<\/p><\/div>\n<\/div>\n<\/div>\n<h2>Summary<\/h2>\n<p>The coordinate plane is a system for graphing and describing points and lines. The coordinate plane is comprised of a horizontal (<i>x<\/i>-) axis and a vertical (<i>y-<\/i>) axis. The intersection of these lines creates the origin, which is the point [latex](0,0)[\/latex]. The coordinate plane is split into four quadrants. Together, these features of the coordinate system allow for the graphical representation and communication about points, lines, and other algebraic concepts.<\/p>\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-277\">\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>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, Shared previously<\/div><ul class=\"citation-list\"><li>Determine the Ordered Pairs for Points Plotted on the Coordinate Plane. <strong>Authored by<\/strong>: mathispower4u. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/c9WVU34MY5Q\">https:\/\/youtu.be\/c9WVU34MY5Q<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Plot Points Given as Ordered Pairs on the Coordinate Plane. <strong>Authored by<\/strong>: mathispower4u. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/p_MESleS3mw\">https:\/\/youtu.be\/p_MESleS3mw<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Graph Basic Linear Equations by Completing a Table of Values. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) . <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/f5yvGPEWpvE\">https:\/\/youtu.be\/f5yvGPEWpvE<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Determine If an Ordered Pair is a Solution to a Linear Equation. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) . <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/9aWGxt7OnB8\">https:\/\/youtu.be\/9aWGxt7OnB8<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Graph Linear Equations Using Intercepts. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) . <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/k8r-q_T6UFk\">https:\/\/youtu.be\/k8r-q_T6UFk<\/a>. <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>\n\t\t\t\t\t\t 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