{"id":724,"date":"2016-06-01T20:49:13","date_gmt":"2016-06-01T20:49:13","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/intermediatealgebra\/?post_type=chapter&#038;p=724"},"modified":"2019-07-24T21:03:18","modified_gmt":"2019-07-24T21:03:18","slug":"read-graphing-using-ordered-pairs","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/intermediatealgebra\/chapter\/read-graphing-using-ordered-pairs\/","title":{"raw":"Graph Linear Equations","rendered":"Graph Linear Equations"},"content":{"raw":"<div class=\"bcc-box bcc-highlight\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Create a table of ordered pairs from a two-variable linear equation<\/li>\r\n \t<li>Graph a two-variable linear equation using a table of ordered pairs<\/li>\r\n \t<li>Determine whether an ordered pair is a solution of an equation<\/li>\r\n<\/ul>\r\n<\/div>\r\nYou have likely used a coordinate plane before.\u00a0The coordinate plane consists of a horizontal axis and a vertical axis, number lines that intersect at right angles. (They are perpendicular to each other.)\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/06\/01182918\/image002.jpg\" alt=\"A graph with an x-axis running horizontally and a y-axis running vertically. The location where these axes cross is labeled the origin, and is the point zero, zero. The axes also divide the graph into four equal quadrants. The top right area is quadrant one. The top left area is quadrant two. The bottom left area is quadrant three. The bottom right area is quadrant four.\" width=\"417\" height=\"378\" \/>\r\n\r\nThe horizontal axis in the coordinate plane is called the <b><em>x<\/em>-axis<\/b>. The vertical axis is called the <b><em>y<\/em>-axis<\/b>. The point at which the two axes intersect is called the <b>origin<\/b>. The origin is at\u00a0[latex]0[\/latex] on the <i>x-<\/i>axis and\u00a0[latex]0[\/latex] on the <i>y-<\/i>axis.\r\n\r\nLocations on the coordinate plane are described as <b>ordered pairs<\/b>. An ordered pair tells you the location of a point by relating the point\u2019s location along the <i>x-<\/i>axis (the first value of the ordered pair) and along the <i>y<\/i>-axis (the second value of the ordered pair).\r\n\r\nIn an ordered pair, such as (<i>x<\/i>, <i>y<\/i>), the first value is called the <b><em>x<\/em>-coordinate<\/b> and the second value is the <b><em>y<\/em>-coordinate<\/b>. Note that the <i>x-<\/i>coordinate is listed before the <i>y-<\/i>coordinate. Since the origin has an <i>x-<\/i>coordinate of\u00a0[latex]0[\/latex] and a <i>y-<\/i>coordinate of\u00a0[latex]0[\/latex], its ordered pair is written as ([latex]0, 0[\/latex]).\r\n\r\nConsider the point below.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/06\/01182920\/image003-1.jpg\" alt=\"Grid with x-axis and y-axis. A blue dotted line extends from the origin, which is the point (0,0) along the horizontal x-axis to 4. A red dotted line goes up vertically from 4 on the x-axis to 3 on the y-axis. That point is labeled (4, 3).\" width=\"417\" height=\"378\" \/>\r\n\r\nTo identify the location of this point, start at the origin ([latex]0, 0[\/latex]) and move right along the <i>x-<\/i>axis until you are under the point. Look at the label on the <i>x-<\/i>axis. The\u00a0[latex]4[\/latex] indicates that, from the origin, you have traveled four units to the right along the <i>x<\/i>-axis. This is the <i>x-<\/i>coordinate, the first number in the ordered pair.\r\n\r\nFrom\u00a0[latex]4[\/latex] on the <i>x-<\/i>axis move up to the point and notice the number with which it aligns on the <i>y-<\/i>axis. The\u00a0[latex]3[\/latex] indicates that, after leaving the <i>x<\/i>-axis, you traveled\u00a0[latex]3[\/latex] units up in the vertical direction, the direction of the <i>y<\/i>-axis. This number is the <i>y-<\/i>coordinate, the second number in the ordered pair. With an <i>x-<\/i>coordinate of\u00a0[latex]4[\/latex] and a <i>y-<\/i>coordinate of\u00a0[latex]3[\/latex], you have the ordered pair ([latex]4, 3[\/latex]).\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nDescribe the point shown as an ordered pair.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/06\/01182922\/image004-1.jpg\" alt=\"A point that is 2 spaces above the x-axis and 5 spaces to the right of the y-axis.\" width=\"417\" height=\"378\" \/>\r\n\r\n[reveal-answer q=\"668288\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"668288\"]\r\n\r\nBegin at the origin and move along the <i>x-<\/i>axis. This is the <i>x-<\/i>coordinate and is written first in the ordered pair.\r\n<p style=\"text-align: center;\">([latex]5, y[\/latex])<\/p>\r\nMove from\u00a0[latex]5[\/latex] up to the ordered pair and read the number on the <i>y-<\/i>axis. This is the <i>y-<\/i>coordinate and is written second in the ordered pair.\r\n<p style=\"text-align: center;\">([latex]5, 2[\/latex])<\/p>\r\nThe ordered pair of the point is ([latex]5, 2[\/latex]).[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3 style=\"text-align: center;\">Example<\/h3>\r\nPlot the point [latex](\u22124,\u22122)[\/latex].\r\n\r\n[reveal-answer q=\"118522\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"118522\"]\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/06\/01182926\/image006.jpg\" alt=\"Graph with blue arrow pointing from origin to four units to the left. A red arrow points down 2 spaces to the point negative 4, negative 2.\" width=\"417\" height=\"378\" \/>\r\n\r\nThe <i>x-<\/i>coordinate is [latex]\u22124[\/latex] because it comes first in the ordered pair. Start at the origin and move\u00a0[latex]4[\/latex] units in a negative direction (left) along the <i>x-<\/i>axis.\r\n\r\nThe <i>y-<\/i>coordinate is [latex]\u22122[\/latex] because it comes second in the ordered pair. Now move\u00a0[latex]2[\/latex] units in a negative direction (down). If you look over to the <i>y-<\/i>axis, you should be lined up with [latex]\u22122[\/latex] on the axis.\r\n\r\nDraw a point at this location and label the point [latex](\u22124,\u22122)[\/latex].[\/hidden-answer]\r\n\r\n<\/div>\r\nhttps:\/\/youtu.be\/p_MESleS3mw\r\n\r\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.\r\n<div>\r\n\r\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 us look at what a linear relationship is.\r\n<h3>Plotting Points to Graph Linear Relationships<\/h3>\r\n<\/div>\r\nA linear relationship is a relationship between variables such that when plotted on a coordinate plane, the points lie on a line. Start by looking at a series of points in Quadrant I on the coordinate plane.\r\n\r\nThese series of points can also be represented in a table. In the table below, the <em>x-<\/em>\u00a0and <em>y<\/em>-coordinates of each ordered pair on the graph is recorded.\r\n<div style=\"margin: auto;\">\r\n<table>\r\n<tbody>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"height: 15px; text-align: center;\"><b><strong><em>x<\/em><\/strong><\/b><b><strong>-coordinate<\/strong><\/b><\/td>\r\n<td style=\"height: 15px; text-align: center;\"><b><strong><em>y<\/em><\/strong><\/b><b><strong>-coordinate<\/strong><\/b><\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"height: 15px; text-align: center;\">[latex]0[\/latex]<\/td>\r\n<td style=\"height: 15px; text-align: center;\">[latex]0[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"height: 15px; text-align: center;\">[latex]1[\/latex]<\/td>\r\n<td style=\"height: 15px; text-align: center;\">[latex]2[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"height: 15px; text-align: center;\">[latex]2[\/latex]<\/td>\r\n<td style=\"height: 15px; text-align: center;\">[latex]4[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15.4688px;\">\r\n<td style=\"height: 15.4688px; text-align: center;\">[latex]3[\/latex]<\/td>\r\n<td style=\"height: 15.4688px; text-align: center;\">[latex]6[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"height: 15px; text-align: center;\">[latex]4[\/latex]<\/td>\r\n<td style=\"height: 15px; text-align: center;\">[latex]8[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\nNotice that each <em>y<\/em>-coordinate is twice the corresponding <em>x<\/em>-value. All of these <em>x-<\/em>\u00a0and <em>y<\/em>-values follow the same pattern, and, when placed on a coordinate plane, they all line up as shown below.\r\n\r\n<img class=\"aligncenter wp-image-1458\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/06\/01183156\/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\" \/>\r\n\r\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 ([latex]1.5, 3[\/latex]), ([latex]2.5, 5[\/latex]), and ([latex]3.5, 7[\/latex]) should all appear on the line too, correct? Look to see what happens below.\r\n\r\n<img class=\"aligncenter wp-image-1459\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/06\/01183200\/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\" \/>\r\n\r\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 the graph seen below.\r\n\r\n<img class=\"aligncenter wp-image-1460\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/06\/01183205\/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\" \/>\r\n\r\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.\r\n\r\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 a straight line.\r\n\r\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\u00a0<\/em>and then plot these ordered pairs on the coordinate plane. Two points are enough to determine a line. However, it is always a good idea to plot more than two points to avoid possible errors.\r\n\r\nAfter graphing the points, draw a line through the points to show that all of the points 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.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nGraph the linear equation [latex]y=2x+3[\/latex].\r\n\r\n[reveal-answer q=\"834421\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"834421\"]\r\n\r\nEvaluate [latex]y=2x+3[\/latex]\u00a0for different values of <em>x<\/em>, and create a table of corresponding <em>x<\/em> and <em>y<\/em> values.\r\n<table style=\"width: 456px;\">\r\n<tbody>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"height: 15px; text-align: center; width: 128px;\"><strong><em>x<\/em> values<\/strong><\/td>\r\n<td style=\"height: 15px; text-align: center; width: 180px;\"><strong>[latex]2x+3[\/latex]<\/strong><\/td>\r\n<td style=\"height: 15px; text-align: center; width: 128px;\"><strong><em>y<\/em> values<\/strong><\/td>\r\n<td style=\"text-align: center; width: 20px;\"><strong><em>Ordered Pairs<\/em><\/strong><\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"height: 15px; text-align: center; width: 128px;\">[latex]0[\/latex]<\/td>\r\n<td style=\"height: 15px; text-align: center; width: 180px;\">[latex]2(0) + 3[\/latex]<\/td>\r\n<td style=\"height: 15px; text-align: center; width: 128px;\">[latex]3[\/latex]<\/td>\r\n<td style=\"text-align: center; width: 20px;\">([latex]0, 3[\/latex])<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15.3125px;\">\r\n<td style=\"height: 15.3125px; text-align: center; width: 128px;\">[latex]1[\/latex]<\/td>\r\n<td style=\"height: 15.3125px; text-align: center; width: 180px;\">[latex]2(1) + 3[\/latex]<\/td>\r\n<td style=\"height: 15.3125px; text-align: center; width: 128px;\">[latex]5[\/latex]<\/td>\r\n<td style=\"text-align: center; width: 20px;\">([latex]1, 5[\/latex])<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"height: 15px; text-align: center; width: 128px;\">[latex]2[\/latex]<\/td>\r\n<td style=\"height: 15px; text-align: center; width: 180px;\">[latex]2(2) + 3[\/latex]<\/td>\r\n<td style=\"height: 15px; text-align: center; width: 128px;\">[latex]7[\/latex]<\/td>\r\n<td style=\"text-align: center; width: 20px;\">([latex]2, 7[\/latex])<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"height: 15px; text-align: center; width: 128px;\">[latex]3[\/latex]<\/td>\r\n<td style=\"height: 15px; text-align: center; width: 180px;\">[latex]2(3) + 3[\/latex]<\/td>\r\n<td style=\"height: 15px; text-align: center; width: 128px;\">[latex]9[\/latex]<\/td>\r\n<td style=\"text-align: center; width: 20px;\">([latex]3, 9[\/latex])<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nPlot the ordered pairs.\r\n\r\n<img class=\"aligncenter wp-image-1468\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/06\/01183213\/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\" \/>\r\n\r\nDraw a line through the points to indicate all of the points on the line to obtain the graph seen below.\r\n\r\n<img class=\"aligncenter wp-image-1469\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/06\/01183217\/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]\r\n\r\n<\/div>\r\n<h3>Ordered Pairs as Solutions<\/h3>\r\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]\u00a0with some of the specific points on the line.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/06\/01183219\/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\" \/>\r\n\r\nEvery point on the line is a solution to the equation [latex]y=2x\u20135[\/latex]. You can plug the ordered pair [latex](1,\u22123)[\/latex] into the equation and the equation will be satisfied.\r\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>\r\nYou can also plug ANY of the other points on the line into the equation. 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.\r\n\r\nTo find out whether an ordered pair is a solution of a linear equation, you can do the following:\r\n<ul>\r\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>\r\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>\r\n<\/ul>\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nDetermine whether [latex](\u22122,4)[\/latex] is a solution to the equation [latex]4y+5x=3[\/latex].\r\n\r\n[reveal-answer q=\"980260\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"980260\"]\r\n\r\nFor this problem, you will use the substitution method. Substitute [latex]x=\u22122[\/latex]\u00a0and [latex]y=4[\/latex]\u00a0into the equation.\r\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>\r\nEvaluate.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}16+\\left(\u221210\\right)=3\\\\6=3\\end{array}[\/latex]<\/p>\r\nThe statement is not true, so [latex](\u22122,4)[\/latex] is not a solution to the equation [latex]4y+5x=3[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nBelow is a video further explaining how to determine if an ordered pair is a solution to a linear equation.\r\n\r\nhttps:\/\/youtu.be\/9aWGxt7OnB8","rendered":"<div class=\"bcc-box bcc-highlight\">\n<h3>Learning Outcomes<\/h3>\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<\/div>\n<p>You have likely used a coordinate plane before.\u00a0The coordinate plane consists of a horizontal axis and a vertical axis, number lines that intersect at right angles. (They are perpendicular to each other.)<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/06\/01182918\/image002.jpg\" alt=\"A graph with an x-axis running horizontally and a y-axis running vertically. The location where these axes cross is labeled the origin, and is the point zero, zero. The axes also divide the graph into four equal quadrants. The top right area is quadrant one. The top left area is quadrant two. The bottom left area is quadrant three. The bottom right area is quadrant four.\" width=\"417\" height=\"378\" \/><\/p>\n<p>The horizontal axis in the coordinate plane is called the <b><em>x<\/em>-axis<\/b>. The vertical axis is called the <b><em>y<\/em>-axis<\/b>. The point at which the two axes intersect is called the <b>origin<\/b>. The origin is at\u00a0[latex]0[\/latex] on the <i>x-<\/i>axis and\u00a0[latex]0[\/latex] on the <i>y-<\/i>axis.<\/p>\n<p>Locations on the coordinate plane are described as <b>ordered pairs<\/b>. An ordered pair tells you the location of a point by relating the point\u2019s location along the <i>x-<\/i>axis (the first value of the ordered pair) and along the <i>y<\/i>-axis (the second value of the ordered pair).<\/p>\n<p>In an ordered pair, such as (<i>x<\/i>, <i>y<\/i>), the first value is called the <b><em>x<\/em>-coordinate<\/b> and the second value is the <b><em>y<\/em>-coordinate<\/b>. Note that the <i>x-<\/i>coordinate is listed before the <i>y-<\/i>coordinate. Since the origin has an <i>x-<\/i>coordinate of\u00a0[latex]0[\/latex] and a <i>y-<\/i>coordinate of\u00a0[latex]0[\/latex], its ordered pair is written as ([latex]0, 0[\/latex]).<\/p>\n<p>Consider the point below.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/06\/01182920\/image003-1.jpg\" alt=\"Grid with x-axis and y-axis. A blue dotted line extends from the origin, which is the point (0,0) along the horizontal x-axis to 4. A red dotted line goes up vertically from 4 on the x-axis to 3 on the y-axis. That point is labeled (4, 3).\" width=\"417\" height=\"378\" \/><\/p>\n<p>To identify the location of this point, start at the origin ([latex]0, 0[\/latex]) and move right along the <i>x-<\/i>axis until you are under the point. Look at the label on the <i>x-<\/i>axis. The\u00a0[latex]4[\/latex] indicates that, from the origin, you have traveled four units to the right along the <i>x<\/i>-axis. This is the <i>x-<\/i>coordinate, the first number in the ordered pair.<\/p>\n<p>From\u00a0[latex]4[\/latex] on the <i>x-<\/i>axis move up to the point and notice the number with which it aligns on the <i>y-<\/i>axis. The\u00a0[latex]3[\/latex] indicates that, after leaving the <i>x<\/i>-axis, you traveled\u00a0[latex]3[\/latex] units up in the vertical direction, the direction of the <i>y<\/i>-axis. This number is the <i>y-<\/i>coordinate, the second number in the ordered pair. With an <i>x-<\/i>coordinate of\u00a0[latex]4[\/latex] and a <i>y-<\/i>coordinate of\u00a0[latex]3[\/latex], you have the ordered pair ([latex]4, 3[\/latex]).<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Describe the point shown as an ordered pair.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/06\/01182922\/image004-1.jpg\" alt=\"A point that is 2 spaces above the x-axis and 5 spaces to the right of the y-axis.\" width=\"417\" height=\"378\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q668288\">Show Solution<\/span><\/p>\n<div id=\"q668288\" class=\"hidden-answer\" style=\"display: none\">\n<p>Begin at the origin and move along the <i>x-<\/i>axis. This is the <i>x-<\/i>coordinate and is written first in the ordered pair.<\/p>\n<p style=\"text-align: center;\">([latex]5, y[\/latex])<\/p>\n<p>Move from\u00a0[latex]5[\/latex] up to the ordered pair and read the number on the <i>y-<\/i>axis. This is the <i>y-<\/i>coordinate and is written second in the ordered pair.<\/p>\n<p style=\"text-align: center;\">([latex]5, 2[\/latex])<\/p>\n<p>The ordered pair of the point is ([latex]5, 2[\/latex]).<\/p><\/div>\n<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3 style=\"text-align: center;\">Example<\/h3>\n<p>Plot the point [latex](\u22124,\u22122)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q118522\">Show Solution<\/span><\/p>\n<div id=\"q118522\" class=\"hidden-answer\" style=\"display: none\">\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/06\/01182926\/image006.jpg\" alt=\"Graph with blue arrow pointing from origin to four units to the left. A red arrow points down 2 spaces to the point negative 4, negative 2.\" width=\"417\" height=\"378\" \/><\/p>\n<p>The <i>x-<\/i>coordinate is [latex]\u22124[\/latex] because it comes first in the ordered pair. Start at the origin and move\u00a0[latex]4[\/latex] units in a negative direction (left) along the <i>x-<\/i>axis.<\/p>\n<p>The <i>y-<\/i>coordinate is [latex]\u22122[\/latex] because it comes second in the ordered pair. Now move\u00a0[latex]2[\/latex] units in a negative direction (down). If you look over to the <i>y-<\/i>axis, you should be lined up with [latex]\u22122[\/latex] on the axis.<\/p>\n<p>Draw a point at this location and label the point [latex](\u22124,\u22122)[\/latex].<\/p><\/div>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Plot Points Given as Ordered Pairs on the Coordinate Plane\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/p_MESleS3mw?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\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 us look at what a linear relationship is.<\/p>\n<h3>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. Start by looking at a series of points in Quadrant I on the coordinate plane.<\/p>\n<p>These series of points can also be represented in a table. In the table below, the <em>x-<\/em>\u00a0and <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 style=\"height: 15px;\">\n<td style=\"height: 15px; text-align: center;\"><b><strong><em>x<\/em><\/strong><\/b><b><strong>-coordinate<\/strong><\/b><\/td>\n<td style=\"height: 15px; text-align: center;\"><b><strong><em>y<\/em><\/strong><\/b><b><strong>-coordinate<\/strong><\/b><\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"height: 15px; text-align: center;\">[latex]0[\/latex]<\/td>\n<td style=\"height: 15px; text-align: center;\">[latex]0[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"height: 15px; text-align: center;\">[latex]1[\/latex]<\/td>\n<td style=\"height: 15px; text-align: center;\">[latex]2[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"height: 15px; text-align: center;\">[latex]2[\/latex]<\/td>\n<td style=\"height: 15px; text-align: center;\">[latex]4[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15.4688px;\">\n<td style=\"height: 15.4688px; text-align: center;\">[latex]3[\/latex]<\/td>\n<td style=\"height: 15.4688px; text-align: center;\">[latex]6[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"height: 15px; text-align: center;\">[latex]4[\/latex]<\/td>\n<td style=\"height: 15px; text-align: center;\">[latex]8[\/latex]<\/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>\u00a0and <em>y<\/em>-values follow the same pattern, and, when placed on a coordinate plane, they all line up as shown below.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-1458\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/06\/01183156\/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 ([latex]1.5, 3[\/latex]), ([latex]2.5, 5[\/latex]), and ([latex]3.5, 7[\/latex]) should all appear on the line too, correct? Look to see what happens below.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-1459\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/06\/01183200\/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 the graph seen below.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-1460\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/06\/01183205\/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 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\u00a0<\/em>and then plot these ordered pairs on the coordinate plane. Two points are enough to determine a line. However, it is always a good idea to plot more than two points to avoid possible errors.<\/p>\n<p>After graphing the points, draw a line through the points to show that all of the points 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=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\">\n<p>Evaluate [latex]y=2x+3[\/latex]\u00a0for different values of <em>x<\/em>, and create a table of corresponding <em>x<\/em> and <em>y<\/em> values.<\/p>\n<table style=\"width: 456px;\">\n<tbody>\n<tr style=\"height: 15px;\">\n<td style=\"height: 15px; text-align: center; width: 128px;\"><strong><em>x<\/em> values<\/strong><\/td>\n<td style=\"height: 15px; text-align: center; width: 180px;\"><strong>[latex]2x+3[\/latex]<\/strong><\/td>\n<td style=\"height: 15px; text-align: center; width: 128px;\"><strong><em>y<\/em> values<\/strong><\/td>\n<td style=\"text-align: center; width: 20px;\"><strong><em>Ordered Pairs<\/em><\/strong><\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"height: 15px; text-align: center; width: 128px;\">[latex]0[\/latex]<\/td>\n<td style=\"height: 15px; text-align: center; width: 180px;\">[latex]2(0) + 3[\/latex]<\/td>\n<td style=\"height: 15px; text-align: center; width: 128px;\">[latex]3[\/latex]<\/td>\n<td style=\"text-align: center; width: 20px;\">([latex]0, 3[\/latex])<\/td>\n<\/tr>\n<tr style=\"height: 15.3125px;\">\n<td style=\"height: 15.3125px; text-align: center; width: 128px;\">[latex]1[\/latex]<\/td>\n<td style=\"height: 15.3125px; text-align: center; width: 180px;\">[latex]2(1) + 3[\/latex]<\/td>\n<td style=\"height: 15.3125px; text-align: center; width: 128px;\">[latex]5[\/latex]<\/td>\n<td style=\"text-align: center; width: 20px;\">([latex]1, 5[\/latex])<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"height: 15px; text-align: center; width: 128px;\">[latex]2[\/latex]<\/td>\n<td style=\"height: 15px; text-align: center; width: 180px;\">[latex]2(2) + 3[\/latex]<\/td>\n<td style=\"height: 15px; text-align: center; width: 128px;\">[latex]7[\/latex]<\/td>\n<td style=\"text-align: center; width: 20px;\">([latex]2, 7[\/latex])<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"height: 15px; text-align: center; width: 128px;\">[latex]3[\/latex]<\/td>\n<td style=\"height: 15px; text-align: center; width: 180px;\">[latex]2(3) + 3[\/latex]<\/td>\n<td style=\"height: 15px; text-align: center; width: 128px;\">[latex]9[\/latex]<\/td>\n<td style=\"text-align: center; width: 20px;\">([latex]3, 9[\/latex])<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>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\/wp-content\/uploads\/sites\/121\/2016\/06\/01183213\/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 to obtain the graph seen below.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-1469\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/06\/01183217\/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>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]\u00a0with 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\/wp-content\/uploads\/sites\/121\/2016\/06\/01183219\/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 plug the ordered pair [latex](1,\u22123)[\/latex] into the equation and the equation will be satisfied.<\/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 plug ANY of the other points on the line into the equation. 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<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 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\">\n<p>For this problem, you will use the substitution method. Substitute [latex]x=\u22122[\/latex]\u00a0and [latex]y=4[\/latex]\u00a0into 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<\/div>\n<\/div>\n<\/div>\n<p>Below is a video further explaining how to determine if an ordered pair is a solution to a linear equation.<\/p>\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\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-724\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Plot Points Given as Ordered Pairs on the Coordinate Plane. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <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><\/ul><div class=\"license-attribution-dropdown-subheading\">All rights reserved content<\/div><ul class=\"citation-list\"><li>Graph Basic Linear Equations by Completing a Table of Values. <strong>Authored by<\/strong>: mathispower4u. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/Graph%20the%20linear%20equation\">http:\/\/Graph%20the%20linear%20equation<\/a>. <strong>License<\/strong>: <em>All Rights Reserved<\/em>. <strong>License Terms<\/strong>: Standard YouTube License<\/li><li>Determine If an Ordered Pair is a Solution to a Linear Equation. <strong>Authored by<\/strong>: mathispower4u. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/9aWGxt7OnB8\">https:\/\/youtu.be\/9aWGxt7OnB8<\/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 <\/section>","protected":false},"author":21,"menu_order":3,"template":"","meta":{"_candela_citation":"[{\"type\":\"copyrighted_video\",\"description\":\"Graph Basic Linear Equations by Completing a Table of Values\",\"author\":\"mathispower4u\",\"organization\":\"\",\"url\":\"Graph the linear 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