{"id":107,"date":"2025-04-09T17:09:26","date_gmt":"2025-04-09T17:09:26","guid":{"rendered":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/chapter\/2-1-the-rectangular-coordinate-systems-and-graphs-college-algebra-2e-openstax\/"},"modified":"2026-03-19T15:56:28","modified_gmt":"2026-03-19T15:56:28","slug":"2-1-the-rectangular-coordinate-systems-and-graphs","status":"publish","type":"chapter","link":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/chapter\/2-1-the-rectangular-coordinate-systems-and-graphs\/","title":{"raw":"2.1 The Rectangular Coordinate Systems and Graphs","rendered":"2.1 The Rectangular Coordinate Systems and Graphs"},"content":{"raw":"<div id=\"main-content\" class=\"MainContent__ContentStyles-sc-6yy1if-0 NnXKu\" tabindex=\"-1\" data-dynamic-style=\"true\">\r\n<div id=\"page_b64cdbd9-26f9-4dfc-99d2-ed5a9f99bae8\" class=\"chapter-content-module\" data-type=\"page\" data-book-content=\"true\">\r\n<div class=\"ui-has-child-title\" data-type=\"abstract\"><section>\r\n<div class=\"textbox textbox--learning-objectives\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Learning Objectives<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nIn this section, you will:\r\n<ul>\r\n \t<li>Plot ordered pairs in a Cartesian coordinate system.<\/li>\r\n \t<li>Graph equations by plotting points.<\/li>\r\n \t<li>Graph equations with a graphing utility.<\/li>\r\n \t<li>Find x-intercepts and y-intercepts.<\/li>\r\n \t<li>Use the distance formula.<\/li>\r\n \t<li>Use the midpoint formula.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\n\r\n[caption id=\"attachment_334\" align=\"aligncenter\" width=\"300\"]<img class=\"wp-image-334 size-medium\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/2.1-Fig.-1-300x278.jpg\" alt=\"\" width=\"300\" height=\"278\" \/> Figure 1[\/caption]\r\n\r\n<\/section><\/div>\r\n<p id=\"fs-id2906377\">Tracie set out from Elmhurst, IL, to go to Franklin Park. On the way, she made a few stops to do errands. Each stop is indicated by a red dot in Figure 1. Laying a rectangular coordinate grid over the map, we can see that each stop aligns with an intersection of grid lines. In this section, we will learn how to use grid lines to describe locations and changes in locations.<\/p>\r\n\r\n<section id=\"fs-id1392675\" data-depth=\"1\">\r\n<h1 data-type=\"title\">Plotting Ordered Pairs in the Cartesian Coordinate System<\/h1>\r\n<p id=\"fs-id2500615\">An old story describes how seventeenth-century philosopher\/mathematician Ren\u00e9 Descartes, while sick in bed, invented the system that has become the foundation of algebra. According to the story, Descartes was staring at a fly crawling on the ceiling when he realized that he could describe the fly\u2019s location in relation to the perpendicular lines formed by the adjacent walls of his room. He viewed the perpendicular lines as horizontal and vertical axes. Further, by dividing each axis into equal unit lengths, Descartes saw that it was possible to locate any object in a two-dimensional plane using just two numbers\u2014the displacement from the horizontal axis and the displacement from the vertical axis.<\/p>\r\n<p id=\"fs-id1960277\">While there is evidence that ideas similar to Descartes\u2019 grid system existed centuries earlier, it was Descartes who introduced the components that comprise the <strong><span id=\"term-00001\" data-type=\"term\">Cartesian coordinate system<\/span><\/strong>, a grid system having perpendicular axes. Descartes named the horizontal axis the <strong><span id=\"term-00002\" data-type=\"term\"><em data-effect=\"italics\">x-<\/em>axis<\/span><\/strong> and the vertical axis the <strong><span id=\"term-00003\" data-type=\"term\"><em data-effect=\"italics\">y-<\/em>axis<\/span><\/strong>.<\/p>\r\n<p id=\"fs-id1167648\">The Cartesian coordinate system, also called the rectangular coordinate system, is based on a two-dimensional plane consisting of the <em data-effect=\"italics\">x<\/em>-axis and the <em data-effect=\"italics\">y<\/em>-axis. Perpendicular to each other, the axes divide the plane into four sections. Each section is called a <strong><span id=\"term-00004\" data-type=\"term\">quadrant<\/span><\/strong>; the quadrants are numbered counterclockwise as shown in Figure 2.<\/p>\r\n&nbsp;\r\n\r\n[caption id=\"attachment_35\" align=\"aligncenter\" width=\"300\"]<img class=\"wp-image-35 size-medium\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/03\/2.1-Fig.-2-300x266.jpg\" alt=\"\" width=\"300\" height=\"266\" \/> Figure 2[\/caption]\r\n<p id=\"fs-id1423341\">The center of the plane is the point at which the two axes cross. It is known as the <strong><span id=\"term-00005\" data-type=\"term\">origin<\/span><\/strong>, or point [latex] (0, 0). [\/latex] From the origin, each axis is further divided into equal units: increasing, positive numbers to the right on the <em data-effect=\"italics\">x-<\/em>axis and up the <em data-effect=\"italics\">y-<\/em>axis; decreasing, negative numbers to the left on the <em data-effect=\"italics\">x-<\/em>axis and down the <em data-effect=\"italics\">y-<\/em>axis. The axes extend to positive and negative infinity as shown by the arrowheads in Figure 3.<\/p>\r\n\r\n\r\n[caption id=\"attachment_36\" align=\"aligncenter\" width=\"288\"]<img class=\"wp-image-36 size-medium\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/03\/2.1-Fig.-3-288x300.jpg\" alt=\"\" width=\"288\" height=\"300\" \/> Figure 3[\/caption]\r\n<p id=\"fs-id1787344\">Each point in the plane is identified by its <strong><span id=\"term-00006\" data-type=\"term\"><em data-effect=\"italics\">x-<\/em>coordinate<\/span><\/strong>, or horizontal displacement from the origin, and its <strong><span id=\"term-00007\" data-type=\"term\"><em data-effect=\"italics\">y-<\/em>coordinate<\/span><\/strong>, or vertical displacement from the origin. Together, we write them as an <strong><span id=\"term-00008\" data-type=\"term\">ordered pair<\/span><\/strong> indicating the combined distance from the origin in the form [latex] (x, y). [\/latex] An ordered pair is also known as a coordinate pair because it consists of <em data-effect=\"italics\">x-<\/em> and <em data-effect=\"italics\">y<\/em>-coordinates. For example, we can represent the point [latex] (3, -1) [\/latex] in the plane by moving three units to the right of the origin in the horizontal direction, and one unit down in the vertical direction. See Figure 4.<\/p>\r\n\r\n\r\n[caption id=\"attachment_37\" align=\"aligncenter\" width=\"288\"]<img class=\"wp-image-37 size-medium\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/03\/2.1-Fig.-4-288x300.jpg\" alt=\"\" width=\"288\" height=\"300\" \/> Figure 4[\/caption]\r\n<p id=\"fs-id3155322\">When dividing the axes into equally spaced increments, note that the <em data-effect=\"italics\">x-<\/em>axis may be considered separately from the <em data-effect=\"italics\">y-<\/em>axis. In other words, while the <em data-effect=\"italics\">x-<\/em>axis may be divided and labeled according to consecutive integers, the <em data-effect=\"italics\">y-<\/em>axis may be divided and labeled by increments of 2, or 10, or 100. In fact, the axes may represent other units, such as years against the balance in a savings account, or quantity against cost, and so on. Consider the rectangular coordinate system primarily as a method for showing the relationship between two quantities.<\/p>\r\n\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Cartesian Coordinate System<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nA two-dimensional plane where the\r\n<ul>\r\n \t<li>x-axis is the horizontal axis<\/li>\r\n \t<li>y-axis is the vertical axis<\/li>\r\n<\/ul>\r\nA point in the plane is defined as an ordered pair,[latex] (x, y), [\/latex] such that <em>x<\/em> is determined by its horizontal distance from the origin and <em>y <\/em>is determined by its vertical distance from the origin.\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/div>\r\n<div id=\"main-content\" class=\"MainContent__ContentStyles-sc-6yy1if-0 NnXKu\" tabindex=\"-1\" data-dynamic-style=\"true\">\r\n<div id=\"page_b64cdbd9-26f9-4dfc-99d2-ed5a9f99bae8\" class=\"chapter-content-module\" data-type=\"page\" data-book-content=\"true\"><section id=\"fs-id1392675\" data-depth=\"1\">\r\n<div id=\"Example_02_01_01\" class=\"ui-has-child-title\" data-type=\"example\"><section>\r\n<div class=\"body\">\r\n<div id=\"fs-id2270902\" class=\"unnumbered\" data-type=\"exercise\"><section>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 1: Plotting Points in a Rectangular Coordinate System<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nPlot the points\u00a0[latex] (-2, 4), (3, 3) [\/latex] and [latex] (0, -3) [\/latex] in the plane.\r\n\r\n&nbsp;\r\n\r\n<details><summary>Solution (click to expand)<\/summary>To plot the point\u00a0[latex] (-2, 4), [\/latex] begin at the origin. The <em>x<\/em>-coordinate is -2, so move two units to the left. The <em>y<\/em>-coordinate is 4, so then move four units up in the positive <em>y<\/em> direction.\r\n\r\nTo plot the point [latex] (3, 3), [\/latex] begin again at the origin. The <em>x<\/em>-coordinate is 3, so more three units to the right. The <em>y<\/em>-coordinate is also 3, so move three units up in the positive <em>y<\/em> direction.\r\n\r\nTo plot the point [latex] (0, -3), [\/latex] begin again at the origin. The <em>x<\/em>-coordinate is 0. This tells us not to move in either direction along the\u00a0<em>x<\/em>-axis. The <em>y<\/em>-coordinate is -3, so move three units down in the negative\u00a0<em>y<\/em> direction. See the graph in Figure 5.\r\n\r\n&nbsp;\r\n\r\n[caption id=\"attachment_44\" align=\"aligncenter\" width=\"288\"]<img class=\"wp-image-44 size-medium\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/03\/2.1-Fig.-5-288x300.jpg\" alt=\"\" width=\"288\" height=\"300\" \/> Figure 5[\/caption]\r\n<h2>Analysis<\/h2>\r\nNote that when either coordinate is zero, the point must be on an axis. If the <em>x<\/em>-coordinate is zero, the point is on the\u00a0<em>y<\/em>-axis. If the <em>y<\/em>-coordinate is zero, the point is on the\u00a0<em>x<\/em>-axis.\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><\/div>\r\n<\/div>\r\n<section id=\"fs-id1940663\" data-depth=\"1\">\r\n<h1 data-type=\"title\">Graphing Equations by Plotting Points<\/h1>\r\n<p id=\"fs-id1951777\">We can plot a set of points to represent an equation. When such an equation contains both an <em data-effect=\"italics\">x <\/em>variable and a <em data-effect=\"italics\">y <\/em>variable, it is called an <span id=\"term-00009\" data-type=\"term\">equation in two variables<\/span>. Its graph is called a <span id=\"term-00010\" data-type=\"term\">graph in two variables<\/span>. Any graph on a two-dimensional plane is a graph in two variables.<\/p>\r\n<p id=\"fs-id1811199\">Suppose we want to graph the equation [latex] y=2x-1. [\/latex] We can begin by substituting a value for <em data-effect=\"italics\">x<\/em> into the equation and determining the resulting value of <em data-effect=\"italics\">y<\/em>. Each pair of <em data-effect=\"italics\">x<\/em>- and <em data-effect=\"italics\">y<\/em>-values is an ordered pair that can be plotted. Table 1 lists values of <em data-effect=\"italics\">x<\/em> from \u20133 to 3 and the resulting values for <em data-effect=\"italics\">y<\/em>.<\/p>\r\n\r\n<div id=\"Table_02_01_01\" class=\"os-table\">\r\n<table class=\"grid aligncenter\" style=\"height: 120px;\" data-id=\"Table_02_01_01\"><caption>Table 1<\/caption>\r\n<tbody>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"text-align: center; height: 15px; width: 170.117px;\" data-align=\"center\">[latex] x [\/latex]<\/td>\r\n<td style=\"text-align: center; height: 15px; width: 263px;\" data-align=\"center\">[latex] y=2x-1 [\/latex]<\/td>\r\n<td style=\"text-align: center; height: 15px; width: 213.483px;\" data-align=\"center\">[latex] (x, y) [\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"text-align: center; height: 15px; width: 170.117px;\" data-align=\"center\">[latex] -3 [\/latex]<\/td>\r\n<td style=\"text-align: center; height: 15px; width: 263px;\" data-align=\"center\">[latex] y=2(-3)-1=7 [\/latex]<\/td>\r\n<td style=\"text-align: center; height: 15px; width: 213.483px;\" data-align=\"center\">[latex] (-3, -7) [\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"text-align: center; height: 15px; width: 170.117px;\" data-align=\"center\">[latex] -2 [\/latex]<\/td>\r\n<td style=\"text-align: center; height: 15px; width: 263px;\" data-align=\"center\">[latex] y=2(-2)-1=-5 [\/latex]<\/td>\r\n<td style=\"text-align: center; height: 15px; width: 213.483px;\" data-align=\"center\">[latex] (-2, -5) [\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"text-align: center; height: 15px; width: 170.117px;\" data-align=\"center\">[latex] -1 [\/latex]<\/td>\r\n<td style=\"text-align: center; height: 15px; width: 263px;\" data-align=\"center\">[latex] y=2(-1)-1=-3 [\/latex]<\/td>\r\n<td style=\"text-align: center; height: 15px; width: 213.483px;\" data-align=\"center\">[latex] (-1, -3) [\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"text-align: center; height: 15px; width: 170.117px;\" data-align=\"center\">[latex] 0 [\/latex]<\/td>\r\n<td style=\"text-align: center; height: 15px; width: 263px;\" data-align=\"center\">[latex] y=2(0)-1=-1 [\/latex]<\/td>\r\n<td style=\"text-align: center; height: 15px; width: 213.483px;\" data-align=\"center\">[latex] (0, -1) [\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"text-align: center; height: 15px; width: 170.117px;\" data-align=\"center\">[latex] 1 [\/latex]<\/td>\r\n<td style=\"text-align: center; height: 15px; width: 263px;\" data-align=\"center\">[latex] y=2(1)-1=1 [\/latex]<\/td>\r\n<td style=\"text-align: center; height: 15px; width: 213.483px;\" data-align=\"center\">[latex] (1, 1) [\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"text-align: center; height: 15px; width: 170.117px;\" data-align=\"center\">[latex] 2 [\/latex]<\/td>\r\n<td style=\"text-align: center; height: 15px; width: 263px;\" data-align=\"center\">[latex] y=2(2)-1=3 [\/latex]<\/td>\r\n<td style=\"text-align: center; height: 15px; width: 213.483px;\" data-align=\"center\">[latex] (2, 3) [\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"text-align: center; height: 15px; width: 170.117px;\" data-align=\"center\">[latex] 3 [\/latex]<\/td>\r\n<td style=\"text-align: center; height: 15px; width: 263px;\" data-align=\"center\">[latex] y=2(3)-1=5 [\/latex]<\/td>\r\n<td style=\"text-align: center; height: 15px; width: 213.483px;\" data-align=\"center\">[latex] (3, 5) [\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<p id=\"fs-id1386116\">We can plot the points in the table. The points for this particular equation form a line, so we can connect them. See Figure 6<strong>. <\/strong>This is not true for all equations.<\/p>\r\n\r\n\r\n[caption id=\"attachment_38\" align=\"aligncenter\" width=\"255\"]<img class=\"wp-image-38 size-medium\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/03\/2.1-Fig.-6-255x300.jpg\" alt=\"\" width=\"255\" height=\"300\" \/> Figure 6[\/caption]\r\n<p id=\"fs-id2522154\">Note that the <em data-effect=\"italics\">x-<\/em>values chosen are arbitrary, regardless of the type of equation we are graphing. Of course, some situations may require particular values of <em data-effect=\"italics\">x<\/em> to be plotted in order to see a particular result. Otherwise, it is logical to choose values that can be calculated easily, and it is always a good idea to choose values that are both negative and positive. There is no rule dictating how many points to plot, although we need at least two to graph a line. Keep in mind, however, that the more points we plot, the more accurately we can sketch the graph.<\/p>\r\n\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">How To<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nGiven an equation, graph by plotting points\r\n<ol>\r\n \t<li>Make a table with one column labeled <em>x<\/em>, a second column labeled with the equation, and a third column listing the resulting ordered pairs.<\/li>\r\n \t<li>Enter <em>x<\/em>-values down the first column using positive and negative values. Selecting the\u00a0<em>x<\/em>-values in numerical order will make the graphing simpler.<\/li>\r\n \t<li>Select <em>x<\/em>-values that will yield\u00a0<em>y<\/em>-values with little effort, preferable ones that can be calculated mentally.<\/li>\r\n \t<li>Plot the ordered pairs.<\/li>\r\n \t<li>Connect the points if they form a line.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<\/section>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 2: Graphing an Equation in Two Variables by Plotting Points<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nGraph the equation [latex] y=-x+2 [\/latex] by plotting points.\r\n\r\n&nbsp;\r\n\r\n<details><summary>Solution (click to expand)<\/summary>First, we construct a table similar to Table 2. Choose <em>x<\/em> values and calculate\u00a0<em>y<\/em>.\r\n<table class=\"grid aligncenter\" style=\"border-collapse: collapse; width: 100%; height: 120px;\" border=\"0\"><caption>Table 2<\/caption>\r\n<tbody>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"width: 33.3333%; height: 15px; text-align: center;\">[latex] x [\/latex]<\/td>\r\n<td style=\"width: 33.3333%; height: 15px; text-align: center;\">[latex] y=-x+2 [\/latex]<\/td>\r\n<td style=\"width: 33.3333%; height: 15px; text-align: center;\">[latex] (x, y) [\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"width: 33.3333%; height: 15px; text-align: center;\">[latex] -5 [\/latex]<\/td>\r\n<td style=\"width: 33.3333%; height: 15px; text-align: center;\">[latex] y=-(-5)+2=7 [\/latex]<\/td>\r\n<td style=\"width: 33.3333%; height: 15px; text-align: center;\">[latex] (-5, 7) [\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"width: 33.3333%; height: 15px; text-align: center;\">[latex] -3 [\/latex]<\/td>\r\n<td style=\"width: 33.3333%; height: 15px; text-align: center;\">[latex] y=-(-3)+2=5 [\/latex]<\/td>\r\n<td style=\"width: 33.3333%; height: 15px; text-align: center;\">[latex] (-3, 5) [\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"width: 33.3333%; height: 15px; text-align: center;\">[latex] -1 [\/latex]<\/td>\r\n<td style=\"width: 33.3333%; height: 15px; text-align: center;\">[latex] y=-(-1)+2=3 [\/latex]<\/td>\r\n<td style=\"width: 33.3333%; height: 15px; text-align: center;\">[latex] (-1, 3) [\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"width: 33.3333%; height: 15px; text-align: center;\">[latex] 0 [\/latex]<\/td>\r\n<td style=\"width: 33.3333%; height: 15px; text-align: center;\">[latex] y=-(0)+2=2 [\/latex]<\/td>\r\n<td style=\"width: 33.3333%; height: 15px; text-align: center;\">[latex] (0, 2) [\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"width: 33.3333%; height: 15px; text-align: center;\">[latex] 1 [\/latex]<\/td>\r\n<td style=\"width: 33.3333%; height: 15px; text-align: center;\">[latex] y=-(1)+2=1 [\/latex]<\/td>\r\n<td style=\"width: 33.3333%; height: 15px; text-align: center;\">[latex] (1, 1) [\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"width: 33.3333%; height: 15px; text-align: center;\">[latex] 3 [\/latex]<\/td>\r\n<td style=\"width: 33.3333%; height: 15px; text-align: center;\">[latex] y=-(3)+2=-1 [\/latex]<\/td>\r\n<td style=\"width: 33.3333%; height: 15px; text-align: center;\">[latex] (3, -1) [\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"width: 33.3333%; height: 15px; text-align: center;\">[latex] 5 [\/latex]<\/td>\r\n<td style=\"width: 33.3333%; height: 15px; text-align: center;\">[latex] y=-(5)+2=-3 [\/latex]<\/td>\r\n<td style=\"width: 33.3333%; height: 15px; text-align: center;\">[latex] (5, -1) [\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nNow, plot the points. Connect them if they form a line. See Figure 7.\r\n\r\n&nbsp;\r\n\r\n[caption id=\"attachment_45\" align=\"aligncenter\" width=\"300\"]<img class=\"wp-image-45 size-medium\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/03\/2.1-Fig.-7-300x230.jpg\" alt=\"\" width=\"300\" height=\"230\" \/> Figure 7[\/caption]\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<section id=\"fs-id1901322\" data-depth=\"1\">\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Try it #1<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nConstruct a table and graph the equation by plotting points: [latex] y=\\frac{1}{2}x+2 [\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<h1 data-type=\"title\">Graphing Equations with a Graphing Utility<\/h1>\r\n<p id=\"fs-id1722871\">Most graphing calculators require similar techniques to graph an equation. The equations sometimes have to be manipulated so they are written in the style [latex] y=\\rule{2cm}{0.10mm}. [\/latex] The TI-84 Plus, and many other calculator makes and models, have a mode function, which allows the window (the screen for viewing the graph) to be altered so the pertinent parts of a graph can be seen.<\/p>\r\n<p id=\"fs-id1539399\">For example, the equation [latex] y=2x-20 [\/latex] has been entered in the TI-84 Plus shown in Figure 8<strong>a. <\/strong>In Figure 8<strong>b, <\/strong>the resulting graph is shown. Notice that we cannot see on the screen where the graph crosses the axes. The standard window screen on the TI-84 Plus shows [latex] 10\\le x\\le10, [\/latex] and [latex] -1\\le y\\le10. [\/latex] See Figure 8<strong>c<\/strong>.<\/p>\r\n&nbsp;\r\n<div id=\"Figure_02_01_09\" class=\"os-figure\">\r\n<div>\r\n\r\n[caption id=\"attachment_405\" align=\"aligncenter\" width=\"612\"]<img class=\"wp-image-405\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/2.2-Fig.-8a-300x78.jpg\" alt=\"\" width=\"612\" height=\"159\" \/> Figure 8. a. Enter the equation. b. This is the graph in the original window. c. These are the original settings.[\/caption]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><section id=\"fs-id1901322\" data-depth=\"1\">\r\n<p id=\"fs-id1416938\">By changing the window to show more of the positive <em data-effect=\"italics\">x-<\/em>axis and more of the negative <em data-effect=\"italics\">y-<\/em>axis, we have a much better view of the graph and the <em data-effect=\"italics\">x-<\/em> and <em data-effect=\"italics\">y-<\/em>intercepts. See Figure 9<strong>a<\/strong> and Figure 9<strong>b.<\/strong><\/p>\r\n&nbsp;\r\n\r\n[caption id=\"attachment_406\" align=\"aligncenter\" width=\"578\"]<img class=\"wp-image-406\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/2.9-Fig.-9-300x119.jpg\" alt=\"\" width=\"578\" height=\"229\" \/> Figure 9. a. This screen shows the new window settings. b. We can clearly view the intercepts in the new window.[\/caption]\r\n\r\n<\/section>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 3: Using a Graphing Utility to Graph an Equation<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nUse a graphing utility to graph the equation: [latex] y=-\\frac{2}{3}x+\\frac{4}{3}. [\/latex]\r\n\r\n&nbsp;\r\n\r\n<details><summary>Solution (click to expand)<\/summary>Enter the equation in the <em>y<\/em>- function of the calculator. Set the window settings so that both the\u00a0<em>x-<\/em> and <em>y-<\/em> intercepts are showing in the window. See Figure 10.\r\n\r\n&nbsp;\r\n\r\n[caption id=\"attachment_407\" align=\"aligncenter\" width=\"300\"]<img class=\"wp-image-407 size-medium\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/2.1-Fig.-10-300x166.jpg\" alt=\"\" width=\"300\" height=\"166\" \/> Figure 10[\/caption]\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<h1>Finding <em data-effect=\"italics\">x-<\/em>intercepts and <em data-effect=\"italics\">y-<\/em>intercepts<\/h1>\r\n<section id=\"fs-id1340475\" data-depth=\"1\">\r\n<p id=\"fs-id2503271\">The <span id=\"term-00011\" data-type=\"term\">intercepts<\/span> of a graph are points at which the graph crosses the axes. The <span id=\"term-00012\" data-type=\"term\"><em data-effect=\"italics\">x-<\/em>intercept<\/span> is the point at which the graph crosses the <em data-effect=\"italics\">x-<\/em>axis. At this point, the <em data-effect=\"italics\">y-<\/em>coordinate is zero. The <span id=\"term-00013\" data-type=\"term\"><em data-effect=\"italics\">y-<\/em>intercept<\/span> is the point at which the graph crosses the <em data-effect=\"italics\">y-<\/em>axis. At this point, the <em data-effect=\"italics\">x-<\/em>coordinate is zero.<\/p>\r\n<p id=\"fs-id1448434\">To determine the <em data-effect=\"italics\">x-<\/em>intercept, we set <em data-effect=\"italics\">y <\/em>equal to zero and solve for <em data-effect=\"italics\">x<\/em>. Similarly, to determine the <em data-effect=\"italics\">y-<\/em>intercept, we set <em data-effect=\"italics\">x <\/em>equal to zero and solve for <em data-effect=\"italics\">y<\/em>. For example, lets find the intercepts of the equation [latex] y=3x-1. [\/latex]<\/p>\r\n<p id=\"fs-id1493312\">To find the <em data-effect=\"italics\">x-<\/em>intercept, set [latex] y=0 [\/latex]<\/p>\r\n<p style=\"text-align: left;\">[latex] \\hspace{12em}y=3x-1 [\/latex]\r\n[latex] \\hspace{12em}0=3x-1 [\/latex]\r\n[latex] \\hspace{12em}1=3x [\/latex]\r\n[latex] \\hspace{12em}\\frac{1}{3}=x [\/latex]\r\n[latex] \\hspace{11.5em}(\\frac{1}{3}, 0) \\hspace{2em}x-\\text{intercept} [\/latex]<\/p>\r\n\r\n<div id=\"fs-id3064821\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><\/div>\r\n<p id=\"fs-id2905820\">To find the <em data-effect=\"italics\">y-<\/em>intercept, set [latex] x=0 [\/latex]<\/p>\r\n<p style=\"text-align: left;\">[latex] \\hspace{12em}y=3x-1 [\/latex]\r\n[latex] \\hspace{12em}y=3(0)-1 [\/latex]\r\n[latex] \\hspace{12em}y=-1 [\/latex]\r\n[latex] \\hspace{12em}(0, -1) \\hspace{2em}y-\\text{intercept} [\/latex]<\/p>\r\n\r\n<div id=\"fs-id1798574\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><\/div>\r\n<p id=\"fs-id1730534\">We can confirm that our results make sense by observing a graph of the equation as in Figure 11. Notice that the graph crosses the axes where we predicted it would.<\/p>\r\n&nbsp;\r\n\r\n[caption id=\"attachment_408\" align=\"aligncenter\" width=\"300\"]<img class=\"wp-image-408 size-medium\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/2.1-Fig.-11-300x234.jpg\" alt=\"\" width=\"300\" height=\"234\" \/> Figure 11[\/caption]\r\n\r\n<\/section>\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Given an Equation, Find the Intercepts<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<ul>\r\n \t<li>Find the <em>x<\/em>-intercept by setting\u00a0[latex] y=0 [\/latex] and solving for <em>x.<\/em><\/li>\r\n \t<li>Find the <em>y<\/em>-intercept by setting [latex] x=0 [\/latex] and solving for <em>y.<\/em><\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 4: Finding the Intercepts of the Given Equation<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nFind the intercepts of the equation\u00a0[latex] y=-3x-4. [\/latex] Then sketch the graph using only the intercepts.\r\n\r\n&nbsp;\r\n\r\n<details><summary>Solution (click to expand)<\/summary>Set [latex] y=0 [\/latex] to find the <em>x<\/em>-intercept.\r\n<p style=\"text-align: left;\">[latex] \\hspace{12em}y=-3x-4 [\/latex]\r\n[latex] \\hspace{12em}0=-3x-4 [\/latex]\r\n[latex] \\hspace{12em}4=-3x [\/latex]\r\n[latex] \\hspace{11em}-\\frac{4}{3}=x [\/latex]\r\n[latex] \\hspace{12em}(-\\frac{4}{3}, 0) \\hspace{2em}x-\\text{intercept} [\/latex]<\/p>\r\nSet [latex] x=0 [\/latex] to find the <em>y<\/em>-intercept.\r\n<p style=\"text-align: left;\">[latex] \\hspace{12em}y=-3x-4 [\/latex]\r\n[latex] \\hspace{12em}y=-3(0)-4 [\/latex]\r\n[latex] \\hspace{12em}y=-4 [\/latex]\r\n[latex] \\hspace{12em}(0, -4) \\hspace{2em}y-\\text{intercept} [\/latex]<\/p>\r\nPlot both points, and draw a line passing through them as in Figure 12.\r\n\r\n&nbsp;\r\n\r\n[caption id=\"attachment_409\" align=\"aligncenter\" width=\"300\"]<img class=\"wp-image-409 size-medium\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/2.1-Fig.-12-300x253.jpg\" alt=\"\" width=\"300\" height=\"253\" \/> Figure 12[\/caption]\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Try it #2<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nFind the intercepts of the equation and sketch the graph: [latex] y=-\\frac{3}{4}x+3. [\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1518804\" class=\"precalculus try ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"Try It\"><section>\r\n<div class=\"os-note-body\">\r\n<div id=\"ti_02_01_02\" class=\"unnumbered os-hasSolution\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id2437906\" data-type=\"problem\">\r\n<h1>Using the Distance Formula<\/h1>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<section id=\"fs-id1280821\" data-depth=\"1\">\r\n<p id=\"fs-id1277804\">Derived from the <span id=\"term-00014\" class=\"no-emphasis\" data-type=\"term\">Pythagorean Theorem<\/span>, the <strong><span id=\"term-00015\" data-type=\"term\">distance formula<\/span> <\/strong>is used to find the distance between two points in the plane. The Pythagorean Theorem,\u00a0[latex] a^2+b^2=c^2 [\/latex] is based on a right triangle where <em data-effect=\"italics\">a <\/em>and <em data-effect=\"italics\">b<\/em> are the lengths of the legs adjacent to the right angle, and <em data-effect=\"italics\">c<\/em> is the length of the hypotenuse. See Figure 13.<\/p>\r\n&nbsp;\r\n\r\n[caption id=\"attachment_410\" align=\"aligncenter\" width=\"300\"]<img class=\"wp-image-410 size-medium\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/2.1-Fig.-13-300x211.jpg\" alt=\"\" width=\"300\" height=\"211\" \/> Figure 13[\/caption]\r\n<p id=\"fs-id1151919\">The relationship of sides\u00a0[latex] |x_2-x_1| [\/latex] and [latex] |y_2-y_1| [\/latex] to side <em data-effect=\"italics\">d<\/em> is the same as that of sides <em data-effect=\"italics\">a <\/em>and <em data-effect=\"italics\">b <\/em>to side <em data-effect=\"italics\">c.<\/em> We use the absolute value symbol to indicate that the length is a positive number because the absolute value of any number is positive. (For example, [latex] |-3|=3. [\/latex]) The symbols [latex] |x_2-x_1| [\/latex] and [latex] |y_2-y_1| [\/latex] indicate that the lengths of the sides of the triangle are positive. To find the length <em data-effect=\"italics\">c<\/em>, take the square root of both sides of the Pythagorean Theorem.<\/p>\r\n<p style=\"text-align: center;\">[latex] c^2=a^2+b^2\\rightarrow c=\\sqrt{a^2+b^2} [\/latex]<\/p>\r\n<p id=\"fs-id2666328\">It follows that the distance formula is given as<\/p>\r\n<p style=\"text-align: center;\">[latex] d^2=(x_2-x_1)^2+(y_2-y_1)^2\\rightarrow =\\sqrt{(x_2-x_1)^2+(y_2-y_1)^2} [\/latex]<\/p>\r\n<p id=\"fs-id1832560\">We do not have to use the absolute value symbols in this definition because any number squared is positive.<\/p>\r\n\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">The Distance Formula<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nGiven endpoints\u00a0[latex] (x_1, y_1) [\/latex] and\u00a0[latex] (x_2, y_2), [\/latex] the distance between two points is given by\r\n<p style=\"text-align: center;\">[latex] d=\\sqrt{(x_2-x_1)^2+(y_2-y_1)^2} [\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 5: Finding the Distance between Two Points<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nFind the distance between the two points\u00a0[latex](-3, -1) [\/latex] and [latex] (2, 3). [\/latex]\r\n\r\n&nbsp;\r\n\r\n<details><summary>Solution (click to expand)<\/summary>Let us first look at the graph of the two points. Connect the points to form a right triangle as in Figure 14.\r\n\r\n&nbsp;\r\n\r\n[caption id=\"attachment_411\" align=\"aligncenter\" width=\"300\"]<img class=\"wp-image-411 size-medium\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/2.1-Fig.-14-300x234.jpg\" alt=\"\" width=\"300\" height=\"234\" \/> Figure 14[\/caption]\r\n\r\nThen, calculate the length of <em>d<\/em> using the distance formula.\r\n<p style=\"text-align: left;\">[latex] \\hspace{10em}d=\\sqrt{(x_2-x_1)^2+(y_2-y_1)^2} [\/latex]\r\n[latex] \\hspace{10em}d=\\sqrt{(2-(-3))^2+(3-(-1))^2} [\/latex]\r\n[latex] \\hspace{10.7em}=\\sqrt{(5)^2+(4)^2} [\/latex]\r\n[latex] \\hspace{10.7em}=\\sqrt{25+16} [\/latex]\r\n[latex] \\hspace{10.7em}=\\sqrt{41} [\/latex]<\/p>\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">Try it #3<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nFind the distance between two points:\u00a0[latex] (1, 4) [\/latex] and [latex] (11, 9). [\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 6: Finding the Distance between Two Locations<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nLet's return to the situation introduced at the beginning of this section.\r\n\r\nTracie set out from Elmhurt, IL, to go to Franklin Park. On the way, she made a few stops to do errands. Each stop is indicated by a red dot in Figure 1. Find the total distance that Tracie traveled. Compare this with the distance between her starting and final positions.\r\n\r\n&nbsp;\r\n\r\n<details><summary>Solution (click to expand)<\/summary>The first thing we should do is identify ordered pairs to describe each position. If we set the starting position as the origin, we can identify each of the other points by counting units east (right) and north (up) on the grid. For example, the first stop is 1 block east and 1 block north, so it is at\u00a0[latex] (1, 1). [\/latex] The next stop is 5 blocks to the east, so it is at\u00a0[latex] (5, 1). [\/latex] After that, she traveled 3 blocks east and 2 blocks north to [latex] (8, 3). [\/latex] Lastly, she traveled 4 blocks north to [latex] (8, 7). [\/latex] We can label these points on the grid as in Figure 15.\r\n\r\n&nbsp;\r\n\r\n[caption id=\"attachment_436\" align=\"aligncenter\" width=\"368\"]<img class=\"wp-image-436 \" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/2.1-Fig.-15-300x278.jpg\" alt=\"\" width=\"368\" height=\"341\" \/> Figure 15[\/caption]\r\n\r\nNext, we can calculate the distance. Note that each grid unit represents 1,000 feet.\r\n<ul>\r\n \t<li>From her starting location to her first stop at\u00a0[latex] (1, 1) [\/latex] Tracie might have driven north 1,000 feet and then east 1,000 feet, or vice versa. Either way, she drove 2,000 feet to her first stop.<\/li>\r\n \t<li>Her second stop is at\u00a0[latex] (5, 1). [\/latex] So from [latex] (1, 1) [\/latex] to\u00a0[latex] (5, 1), [\/latex] Tracie drove east 4,000 feet.<\/li>\r\n \t<li>Her third stop is at\u00a0[latex] (8, 3). [\/latex] There are a number of routes from [latex] (5, 1) [\/latex] to [latex] (8, 3). [\/latex] Whatever route Tracie decided to use, the distance is the same, as there are no angular streets between the two points. Let's say she drove east 3,000 feet and then north 2,000 feet for a total of 5,000 feet.<\/li>\r\n \t<li>Tracie's final stop is at\u00a0[latex] (8, 7). [\/latex] This is a straight drive north from\u00a0[latex] (8, 3) [\/latex] for a total of 4,000 feet.<\/li>\r\n<\/ul>\r\nNext, we will add the distances listed in Table 3.\r\n<table class=\"grid\" style=\"border-collapse: collapse; width: 100%;\" border=\"0\"><caption>Table 3<\/caption>\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 50%;\"><strong>From\/To<\/strong><\/td>\r\n<td style=\"width: 50%;\"><strong>Number of Feet Driven<\/strong><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%; text-align: center;\">[latex] (0, 0) [\/latex]\u00a0to [latex] (1, 1) [\/latex]<\/td>\r\n<td style=\"width: 50%; text-align: center;\">2,000<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%; text-align: center;\">[latex] (1, 1) [\/latex]\u00a0to [latex] (5, 1) [\/latex]<\/td>\r\n<td style=\"width: 50%; text-align: center;\">4,000<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%; text-align: center;\">[latex] (5, 1) [\/latex] to [latex] (8, 3) [\/latex]<\/td>\r\n<td style=\"width: 50%; text-align: center;\">5,000<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%; text-align: center;\">[latex] (8, 3) [\/latex] to [latex] (8, 7) [\/latex]<\/td>\r\n<td style=\"width: 50%; text-align: center;\">4,000<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%; text-align: center;\">Total<\/td>\r\n<td style=\"width: 50%; text-align: center;\">15,000<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nThe total distance Tracie drove is 15,000 feet, or 2.84 miles. This is not, however the actual distance between her starting and ending positions. To find this distance, we can use the distance formula between the points\u00a0[latex] (0, 0) [\/latex] and [latex] (8, 7). [\/latex]\r\n<p style=\"text-align: left;\">[latex] \\hspace{10em}d=\\sqrt{(8-0)^2+(7-0)^2} [\/latex]\r\n[latex] \\hspace{10.8em}=\\sqrt{64+49}[\/latex]\r\n[latex] \\hspace{10.8em}=\\sqrt{113}[\/latex]\r\n[latex] \\hspace{10.8em}\\approx 10.63 \\hspace{1em}\\text{units}[\/latex]<\/p>\r\nAt 1,000 feet per grid unit, the distance between Elmhurst, IL, to Franklin Park is 10,630.14 feet or 2.01 miles. The distance formula results in a shorter calculation because it is based on the hypotenuse of a right triangle, a straight diagonal from the origin to the point\u00a0[latex] (8, 7). [\/latex] Perhaps you have heard the saying \"as the crow flies,\" which means the shortest distance between two points because a crow can fly in a straight line even though a person on the ground has to travel a longer distance on existing roadways.\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<h1>Using the Midpoint Formula<\/h1>\r\n<section id=\"fs-id2507035\" data-depth=\"1\">\r\n<p id=\"fs-id1151538\">When the endpoints of a line segment are known, we can find the point midway between them. This point is known as the midpoint and the formula is known as the <strong><span id=\"term-00016\" data-type=\"term\">midpoint formula<\/span><\/strong>. Given the endpoints of a line segment,\u00a0[latex] (x_1, y_1) [\/latex] and [latex] (x_2, y_2) [\/latex] the midpoint formula states how to find the coordinates of the midpoint [latex] M. [\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex] M=(\\frac{x_1+x_2}{2}, \\frac{y_1+y_2}{2}) [\/latex]<\/p>\r\n<p id=\"fs-id2837053\">A graphical view of a midpoint is shown in <a class=\"autogenerated-content\" href=\"2-1-the-rectangular-coordinate-systems-and-graphs#Figure_02_01_018\">Figure 16<\/a>. Notice that the line segments on either side of the midpoint are congruent.<\/p>\r\n&nbsp;\r\n<div id=\"Figure_02_01_018\" class=\"os-figure\">\r\n<div class=\"os-caption-container\">\r\n\r\n[caption id=\"attachment_435\" align=\"aligncenter\" width=\"285\"]<img class=\"wp-image-435 size-medium\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/2.1.16-285x300.jpg\" alt=\"\" width=\"285\" height=\"300\" \/> Figure 16[\/caption]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 7: Finding the Midpoint of the Line Segment<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nFind the midpoint of the line segment with the endpoints\u00a0[latex] (7, -2) [\/latex] and [latex] (9, 5). [\/latex]\r\n\r\n&nbsp;\r\n\r\n<details><summary>Solution (click to expand)<\/summary>Use the formula to find the midpoint of the line segment.\r\n<p style=\"text-align: center;\">[latex] (\\frac{x_1+x_2}{2}, \\frac{y_1+y_2}{2})=(\\frac{7+9}{2}, \\frac{-2+5}{2}) [\/latex]\r\n[latex] \\hspace{3.2em}=(8, \\frac{3}{2}) [\/latex]<\/p>\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<section id=\"fs-id2507035\" data-depth=\"1\">\r\n<div id=\"Example_02_01_07\" class=\"ui-has-child-title\" data-type=\"example\"><section>\r\n<div class=\"body\">\r\n<div id=\"fs-id3008576\" class=\"unnumbered\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id3008579\" data-type=\"problem\">\r\n<div class=\"os-problem-container\">\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Try it #4<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nFind the midpoint of the line segment with endpoints\u00a0[latex] (-2, -1) [\/latex] and [latex] (-8, 6). [\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 8: Finding the Center of a Circle<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nThe diameter of a circle has endpoints\u00a0[latex] (-1, -4) [\/latex] and [latex] (5, -4). [\/latex] Find the center of the circle.\r\n\r\n&nbsp;\r\n\r\n<details><summary>Solution (click to expand)<\/summary>The center of a circle is the center, or midpoint, of its diameter. Thus, the midpoint formula will yield the center point.\r\n<p style=\"text-align: center;\">[latex] (\\frac{x_1+x_2}{2}, \\frac{y_1+y_2}{2}) [\/latex]\r\n[latex] (\\frac{-1+5}{2}, \\frac{-4-4}{2}) = (\\frac{4}{2}, -\\frac{8}{2})=(2, -4) [\/latex]<\/p>\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--key-takeaways\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Media<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nAccess these online resources for additional instruction and practice with the Cartesian coordinate system.\r\n<ul>\r\n \t<li><a href=\"http:\/\/openstax.org\/l\/coordplotpnts\">Plotting points on the coordinate plane<\/a><\/li>\r\n \t<li><a href=\"http:\/\/openstax.org\/l\/xyintsgraph\">Find x and y intercepts based on the graph of a line<\/a><\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\n<div id=\"Example_02_01_08\" class=\"ui-has-child-title\" data-type=\"example\"><\/div>\r\n<div class=\"os-eos os-section-exercises-container\" data-uuid-key=\".section-exercises\">\r\n<h1 data-type=\"document-title\" data-rex-keep=\"true\"><span class=\"os-text\">2.1 Section Exercises<\/span><\/h1>\r\n<section id=\"fs-id1553580\" class=\"section-exercises\" data-depth=\"1\"><section id=\"fs-id2496131\" data-depth=\"2\">\r\n<h2 data-type=\"title\">Verbal<\/h2>\r\n<div id=\"fs-id1355431\" class=\"os-hasSolution\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1355432\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1355431-solution\">1<\/a><span class=\"os-divider\">. <\/span>Is it possible for a point plotted in the Cartesian coordinate system to not lie in one of the four quadrants? Explain.\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2435397\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id2435398\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">2<\/span><span class=\"os-divider\">. <\/span>Describe the process for finding the <em data-effect=\"italics\">x-<\/em>intercept and the <em data-effect=\"italics\">y<\/em>-intercept of a graph algebraically.\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1351774\" class=\"os-hasSolution\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id2682309\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1351774-solution\">3<\/a><span class=\"os-divider\">. <\/span>Describe in your own words what the <em data-effect=\"italics\">y<\/em>-intercept of a graph is.\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1823207\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1823208\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">4<\/span><span class=\"os-divider\">. <\/span>When using the distance formula\u00a0[latex] d=\\sqrt{(x_2-x_1)^2+(y_2-y_1)^2)}, [\/latex] explain the correct order of operations that are to be performed to obtain the correct answer.\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><section id=\"fs-id3263952\" data-depth=\"2\">\r\n<h2 data-type=\"title\">Algebraic<\/h2>\r\n<p id=\"fs-id1789811\">For each of the following exercises, find the <em data-effect=\"italics\">x<\/em>-intercept and the <em data-effect=\"italics\">y<\/em>-intercept without graphing. Write the coordinates of each intercept.<\/p>\r\n\r\n<div id=\"fs-id3039754\" class=\"os-hasSolution\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id3039755\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id3039754-solution\">5<\/a><span class=\"os-divider\">.<\/span> [latex] y=-3x+6 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1499169\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1499170\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">6<\/span><span class=\"os-divider\">.<\/span> [latex] 4y=2x-1 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1007678\" class=\"os-hasSolution\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1591152\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1007678-solution\">7<\/a><span class=\"os-divider\">.<\/span> [latex] 3x-2y=6 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2948006\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id2948007\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">8<\/span><span class=\"os-divider\">.<\/span> [latex] 4x-3=2y [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1723142\" class=\"os-hasSolution\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1723143\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1723142-solution\">9<\/a><span class=\"os-divider\">.<\/span> [latex] 3x+8y=9 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1197890\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1197891\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">10<\/span><span class=\"os-divider\">.<\/span> [latex] 2x-\\frac{2}{3}=\\frac{3}{4}y+3 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<p id=\"fs-id3207567\">For each of the following exercises, solve the equation for <em data-effect=\"italics\">y<\/em> in terms of <em data-effect=\"italics\">x<\/em>.<\/p>\r\n\r\n<div id=\"fs-id3042175\" class=\"os-hasSolution\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id3042176\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id3042175-solution\">11<\/a><span class=\"os-divider\">.<\/span> [latex] 4x+2y=8 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1238095\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1238096\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">12<\/span><span class=\"os-divider\">.<\/span> [latex] 3x-2y=6 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2512516\" class=\"os-hasSolution\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1441357\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id2512516-solution\">13<\/a><span class=\"os-divider\">.<\/span> [latex] 2x=5-3y [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1467950\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id2521131\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">14<\/span><span class=\"os-divider\">.<\/span> [latex] x-2y=7 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1278706\" class=\"os-hasSolution\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1402803\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1278706-solution\">15<\/a><span class=\"os-divider\">.<\/span> [latex] 5y+4=10x [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2388766\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id2388767\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">16<\/span><span class=\"os-divider\">.<\/span> [latex] 5x+2y=0 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<p id=\"fs-id1702384\">For each of the following exercises, find the distance between the two points. Simplify your answers, and write the exact answer in simplest radical form for irrational answers.<\/p>\r\n\r\n<div id=\"fs-id1798648\" class=\"os-hasSolution\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1798650\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1798648-solution\">17<\/a><span class=\"os-divider\">. <\/span> [latex] (-4, 1) [\/latex] and [latex] (3, -4) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1762329\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id2947043\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">18<\/span><span class=\"os-divider\">. <\/span> [latex] (2, -5) [\/latex] and [latex] (7, 4) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1932412\" class=\"os-hasSolution\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id2266178\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1932412-solution\">19 <\/a> [latex] (5, 0) [\/latex] and [latex] (5, 6) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1538085\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1538086\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">20<\/span><span class=\"os-divider\">. <\/span> [latex] (-4, 3) [\/latex] and [latex] (10, 3) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2628482\" class=\"os-hasSolution\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id2628483\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id2628482-solution\">21<\/a><span class=\"os-divider\">. <\/span>Find the distance between the two points given using your calculator, and round your answer to the nearest hundredth.\r\n<div class=\"os-problem-container\">\r\n\r\n[latex] (19, 12) [\/latex] and [latex] (41, 71) [\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<p id=\"fs-id2753833\">For each of the following exercises, find the coordinates of the midpoint of the line segment that joins the two given points.<\/p>\r\n\r\n<div id=\"fs-id2389564\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id2389565\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">22. <\/span> [latex] (-5, -6) [\/latex] and [latex] (4, 2) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1386855\" class=\"os-hasSolution\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1386856\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1386855-solution\">23<\/a><span class=\"os-divider\">. <\/span> [latex] (-1, 1) [\/latex] and [latex] (7, -4) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2425333\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id2425334\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">24<\/span><span class=\"os-divider\">. <\/span> [latex] (-5, -3) [\/latex] and [latex] (-2, -8) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2736528\" class=\"os-hasSolution\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id2736529\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id2736528-solution\">25<\/a><span class=\"os-divider\">. <\/span> [latex] (0, 7) [\/latex] and [latex] (4, -9) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1960033\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1960034\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">26<\/span><span class=\"os-divider\">. <\/span> [latex] (-43, 17) [\/latex] and [latex] (23, -34) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><section id=\"fs-id2434980\" data-depth=\"2\">\r\n<h2 data-type=\"title\">Graphical<\/h2>\r\n<p id=\"fs-id1919734\">For each of the following exercises, identify the information requested.<\/p>\r\n\r\n<div id=\"fs-id1940524\" class=\"os-hasSolution\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1940525\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1940524-solution\">27<\/a><span class=\"os-divider\">. <\/span>What are the coordinates of the origin?\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1965313\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1965314\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">28<\/span><span class=\"os-divider\">. <\/span>If a point is located on the <em data-effect=\"italics\">y<\/em>-axis, what is the <em data-effect=\"italics\">x<\/em>-coordinate?\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1575040\" class=\"os-hasSolution\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1575041\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1575040-solution\">29<\/a><span class=\"os-divider\">. <\/span>If a point is located on the <em data-effect=\"italics\">x<\/em>-axis, what is the <em data-effect=\"italics\">y<\/em>-coordinate?\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<p id=\"fs-id1475798\">For each of the following exercises, plot the three points on the given coordinate plane. State whether the three points you plotted appear to be collinear (on the same line).<\/p>\r\n\r\n<div id=\"fs-id1182006\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1182007\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">30<\/span><span class=\"os-divider\">.<\/span> [latex] (4, 1), (-2, -3), (5, 0) [\/latex]\r\n<div class=\"os-problem-container\">\r\n\r\n<span id=\"fs-id2387110\" data-type=\"media\" data-alt=\"This is an image of a blank x, y coordinate plane with the x and y axes ranging from negative 5 to 5.\"><img class=\"alignnone size-medium wp-image-1136\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/2.1.30-1-288x300.jpg\" alt=\"\" width=\"288\" height=\"300\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1341660\" class=\"os-hasSolution\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1341661\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1341660-solution\">31<\/a><span class=\"os-divider\">. <\/span> [latex] (-1, 2)), (0, 4), (2, 1) [\/latex]\r\n<div class=\"os-problem-container\">\r\n\r\n<img class=\"alignnone size-medium wp-image-430\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/2.1.30-288x300.jpg\" alt=\"\" width=\"288\" height=\"300\" \/>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2440944\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id2440946\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">32<\/span><span class=\"os-divider\">. <\/span> [latex] (-3, 0), (-3, 4), (-3, -3) [\/latex]\r\n<div class=\"os-problem-container\">\r\n\r\n<img class=\"alignnone size-medium wp-image-430\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/2.1.30-288x300.jpg\" alt=\"\" width=\"288\" height=\"300\" \/>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2453414\" class=\"os-hasSolution\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id2453415\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id2453414-solution\">33<\/a><span class=\"os-divider\">. <\/span>Name the coordinates of the points graphed.\r\n<div class=\"os-problem-container\">\r\n\r\n<span id=\"fs-id1333042\" data-type=\"media\" data-alt=\"This is an image of an x, y coordinate plane where the x and y-axis range from negative 5 to 5. Three points are plotted: A, B, and C.\"><img class=\"alignnone size-medium wp-image-430\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/2.1.30-288x300.jpg\" alt=\"\" width=\"288\" height=\"300\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id3155305\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id3155306\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">34<\/span><span class=\"os-divider\">. <\/span>Name the quadrant in which the following points would be located. If the point is on an axis, name the axis.\r\n\r\n(a) [latex] (-3, -4) [\/latex]\r\n\r\n(b) [latex] (-5, 0) [\/latex]\r\n\r\n(c) [latex] (1, -4) [\/latex]\r\n\r\n(d) [latex] (-2, 7) [\/latex]\r\n\r\n(e) [latex] (0, -3) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<p id=\"fs-id1569563\">For each of the following exercises, construct a table and graph the equation by plotting at least three points.<\/p>\r\n\r\n<div id=\"fs-id1569567\" class=\"os-hasSolution\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id3182657\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1569567-solution\">35<\/a><span class=\"os-divider\">.<\/span> [latex] y=\\frac{1}{3}x+2 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id3070032\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id3070034\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">36<\/span><span class=\"os-divider\">.<\/span> [latex] y=-3x+1 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1445099\" class=\"os-hasSolution\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1477477\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1445099-solution\">37<\/a><span class=\"os-divider\">.<\/span> [latex] 2y=x+3 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><section id=\"fs-id1387762\" data-depth=\"2\">\r\n<h2 data-type=\"title\">Numeric<\/h2>\r\n<p id=\"fs-id2528941\">For each of the following exercises, find and plot the <em data-effect=\"italics\">x-<\/em> and <em data-effect=\"italics\">y<\/em>-intercepts, and graph the straight line based on those two points.<\/p>\r\n\r\n<div id=\"fs-id766182\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id766183\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">38<\/span><span class=\"os-divider\">.<\/span> [latex] 4x-3y=12 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id3176745\" class=\"os-hasSolution\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id3176746\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id3176745-solution\">39<\/a><span class=\"os-divider\">.<\/span> [latex] x-2y=8 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1272882\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1272883\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">40<\/span><span class=\"os-divider\">.<\/span> [latex] y-5=5x [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1951937\" class=\"os-hasSolution\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1951938\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1951937-solution\">41<\/a><span class=\"os-divider\">.<\/span> [latex] 3y=-2x+6 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1517685\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1929175\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">42<\/span><span class=\"os-divider\">.<\/span> [latex] y=\\frac{x-3}{2} [\/latex]\r\n<div class=\"os-problem-container\">\r\n\r\nFor each of the following exercises, use the graph in the figure below.\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<span id=\"fs-id1832449\" data-type=\"media\" data-alt=\"This is an image of an x, y coordinate plane with the x and y axes ranging from negative 5 to 5. The points (-3, 4) and (5, 2) are plotted. A line connects these two points.\"><img class=\"alignnone size-medium wp-image-431\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/2.1.43-288x300.jpg\" alt=\"\" width=\"288\" height=\"300\" \/><\/span>\r\n<div id=\"fs-id1267899\" class=\"os-hasSolution\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id2432278\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1267899-solution\">43<\/a><span class=\"os-divider\">. <\/span>Find the distance between the two endpoints using the distance formula. Round to three decimal places.\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1895437\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1815372\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">44<\/span><span class=\"os-divider\">. <\/span>Find the coordinates of the midpoint of the line segment connecting the two points.\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1815376\" class=\"os-hasSolution\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1418769\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1815376-solution\">45<\/a><span class=\"os-divider\">. <\/span>Find the distance that [latex] (-3, 4) [\/latex] is from the origin.\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1892569\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1892570\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">46<\/span><span class=\"os-divider\">. <\/span>Find the distance that [latex] (5, 2) [\/latex] is from the origin. Round to three decimal places.\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2437516\" class=\"os-hasSolution\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id2437517\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id2437516-solution\">47<\/a><span class=\"os-divider\">. <\/span>Which point is closer to the origin?\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><section id=\"fs-id2437604\" data-depth=\"2\">\r\n<h2 data-type=\"title\">Technology<\/h2>\r\n<p id=\"fs-id1333818\">For the following exercises, use your graphing calculator to input the linear graphs in the Y= graph menu.<\/p>\r\n<p id=\"fs-id1333822\">After graphing it, use the 2<sup>nd<\/sup> CALC button and 1:value button, hit enter. At the lower part of the screen you will see \u201cx=\u201d and a blinking cursor. You may enter any number for <em data-effect=\"italics\">x<\/em> and it will display the <em data-effect=\"italics\">y<\/em> value for any <em data-effect=\"italics\">x<\/em> value you input. Use this and plug in <em data-effect=\"italics\">x<\/em> = 0, thus finding the <em data-effect=\"italics\">y<\/em>-intercept, for each of the following graphs.<\/p>\r\n\r\n<div id=\"fs-id1686604\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1686605\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">48<\/span><span class=\"os-divider\">.<\/span> [latex] Y_1=-2x+5 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1181943\" class=\"os-hasSolution\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1181944\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1181943-solution\">49<\/a> [latex] Y_1=\\frac{3x-8}{4} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2800083\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1421792\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">50.<\/span> [latex] Y_1=\\frac{x+5}{2} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<p id=\"fs-id2785067\">For the following exercises, use your graphing calculator to input the linear graphs in the Y= graph menu.<\/p>\r\n<p id=\"fs-id2384791\">After graphing it, use the 2<sup>nd<\/sup> <strong>CALC<\/strong> button and 2:zero button, hit <strong>ENTER<\/strong>. At the lower part of the screen you will see \u201cleft bound?\u201d and a blinking cursor on the graph of the line. Move this cursor to the left of the <em data-effect=\"italics\">x<\/em>-intercept, hit <strong>ENTER<\/strong>. Now it says \u201cright bound?\u201d Move the cursor to the right of the <em data-effect=\"italics\">x<\/em>-intercept, hit <strong>ENTER<\/strong>. Now it says \u201cguess?\u201d Move your cursor to the left somewhere in between the left and right bound near the <em data-effect=\"italics\">x<\/em>-intercept. Hit <strong>ENTER<\/strong>. At the bottom of your screen it will display the coordinates of the <em data-effect=\"italics\">x-<\/em>intercept or the \u201czero\u201d to the <em data-effect=\"italics\">y<\/em>-value. Use this to find the <em data-effect=\"italics\">x<\/em>-intercept.<\/p>\r\n<p id=\"fs-id2016100\">Note: With linear\/straight line functions the zero is not really a \u201cguess,\u201d but it is necessary to enter a \u201cguess\u201d so it will search and find the exact <em data-effect=\"italics\">x<\/em>-intercept between your right and left boundaries. With other types of functions (more than one <em data-effect=\"italics\">x<\/em>-intercept), they may be irrational numbers so \u201cguess\u201d is more appropriate to give it the correct limits to find a very close approximation between the left and right boundaries.<\/p>\r\n\r\n<div id=\"fs-id1425403\" class=\"os-hasSolution\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1425404\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1425403-solution\">51<\/a><span class=\"os-divider\">.<\/span> [latex] Y_1=-8x+6 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1845247\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1845248\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">52<\/span><span class=\"os-divider\">.<\/span> [latex] Y_1=4x-7 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2387375\" class=\"os-hasSolution\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id2387376\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id2387375-solution\">53<\/a><span class=\"os-divider\">.<\/span> [latex] Y_1=\\frac{3x+5}{4} [\/latex]\r\n<div class=\"os-problem-container\">\r\n\r\nRound your answer to the nearest thousandth.\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><section id=\"fs-id1932603\" data-depth=\"2\">\r\n<h2 data-type=\"title\">Extensions<\/h2>\r\n<div id=\"fs-id1932608\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1333979\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">54<\/span><span class=\"os-divider\">. <\/span>Someone drove 10 mi directly east from their home, made a left turn at an intersection, and then traveled 5 mi north to their place of work. If a road was made directly from the home to the place of work, what would its distance be to the nearest tenth of a mile?\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2513497\" class=\"os-hasSolution\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id2513498\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id2513497-solution\">55<\/a><span class=\"os-divider\">. <\/span>If the road was made in the previous exercise, how much shorter would the person\u2019s one-way trip be every day?\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2508338\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id2508339\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">56<\/span><span class=\"os-divider\">. <\/span>Given these four points: [latex] A(1, 3), B(-3, 5), C(4, 7) [\/latex] and [latex] D(5, -4) [\/latex] find the coordinates of the midpoint of line segments [latex] \\overline{AB} [\/latex] and [latex] \\overline{CD}. [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1343510\" class=\"os-hasSolution\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1343511\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1343510-solution\">57<\/a><span class=\"os-divider\">. <\/span>After finding the two midpoints in the previous exercise, find the distance between the two midpoints to the nearest thousandth.\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1312332\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1312333\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">58<\/span><span class=\"os-divider\">. <\/span>Given the graph of the rectangle shown and the coordinates of its vertices, prove that the diagonals of the rectangle are of equal length.\r\n<div class=\"os-problem-container\">\r\n\r\n<span id=\"fs-id3207847\" data-type=\"media\" data-alt=\"This is an image of an x, y coordinate plane with the x and y axes ranging from negative 12 to 12. The points (-6, 5); (10, 5); (-6, -1) and (10, -1) are plotted and labeled. These points are connected to form a rectangle. Dotted lines extend from each corner point to their opposite point.\"><img class=\"alignnone size-medium wp-image-432\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/2.1.58-300x175.jpg\" alt=\"\" width=\"300\" height=\"175\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1228174\" class=\"os-hasSolution\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1228176\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1228174-solution\">59<\/a><span class=\"os-divider\">. <\/span>In the previous exercise, find the coordinates of the midpoint for each diagonal.\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><section id=\"fs-id2454474\" data-depth=\"2\">\r\n<h2 data-type=\"title\">Real-World Applications<\/h2>\r\n<div id=\"fs-id2722618\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id2722619\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">60<\/span><span class=\"os-divider\">. <\/span>The coordinates on a map for San Francisco are [latex] (53, 17) [\/latex] and those for Sacramento are [latex] (128, 78) [\/latex] Note that coordinates represent miles. Find the distance between the cities to the nearest mile.\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1511688\" class=\"os-hasSolution\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1511690\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1511688-solution\">61<\/a><span class=\"os-divider\">. <\/span>If San Jose\u2019s coordinates are [latex] (76, -12) [\/latex] where the coordinates represent miles, find the distance between San Jose and San Francisco to the nearest mile.\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2925632\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id2925633\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">62<\/span><span class=\"os-divider\">. <\/span>A small craft in Lake Ontario sends out a distress signal. The coordinates of the boat in trouble were [latex] (49, 64) [\/latex] One rescue boat is at the coordinates [latex] (60, 82) [\/latex] and a second Coast Guard craft is at coordinates [latex] (58, 47). [\/latex] Assuming both rescue craft travel at the same rate, which one would get to the distressed boat the fastest?\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1551836\" class=\"os-hasSolution\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1551837\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1551836-solution\">63<\/a><span class=\"os-divider\">. <\/span>A person on the top of a building wants to have a guy wire extend to a point on the ground 20 ft from the building. To the nearest foot, how long will the wire have to be if the building is 50 ft tall?\r\n<div class=\"os-problem-container\">\r\n\r\n<span id=\"fs-id1477367\" data-type=\"media\" data-alt=\"A right triangle with its bottom left point sitting on the point (0,0). The upper right hand corner is labeled (20,50). The base has a length of 20 units and the triangle has a height of 50 units.\"><img class=\"alignnone size-full wp-image-433\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/2.1.63.jpg\" alt=\"\" width=\"242\" height=\"203\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1717974\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1717976\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">64<\/span><span class=\"os-divider\">. <\/span>If we rent a truck and pay a $75\/day fee plus $.20 for every mile we travel, write a linear equation that would express the total cost per day [latex] y, [\/latex] using [latex] x [\/latex] to represent the number of miles we travel. Graph this function on your graphing calculator and find the total cost for one day if we travel 70 mi.\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><\/section><\/div>\r\n&nbsp;","rendered":"<div id=\"main-content\" class=\"MainContent__ContentStyles-sc-6yy1if-0 NnXKu\" tabindex=\"-1\" data-dynamic-style=\"true\">\n<div id=\"page_b64cdbd9-26f9-4dfc-99d2-ed5a9f99bae8\" class=\"chapter-content-module\" data-type=\"page\" data-book-content=\"true\">\n<div class=\"ui-has-child-title\" data-type=\"abstract\">\n<section>\n<div class=\"textbox textbox--learning-objectives\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Learning Objectives<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>In this section, you will:<\/p>\n<ul>\n<li>Plot ordered pairs in a Cartesian coordinate system.<\/li>\n<li>Graph equations by plotting points.<\/li>\n<li>Graph equations with a graphing utility.<\/li>\n<li>Find x-intercepts and y-intercepts.<\/li>\n<li>Use the distance formula.<\/li>\n<li>Use the midpoint formula.<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<figure id=\"attachment_334\" aria-describedby=\"caption-attachment-334\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-334 size-medium\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/2.1-Fig.-1-300x278.jpg\" alt=\"\" width=\"300\" height=\"278\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/2.1-Fig.-1-300x278.jpg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/2.1-Fig.-1-65x60.jpg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/2.1-Fig.-1-225x208.jpg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/2.1-Fig.-1-350x324.jpg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/2.1-Fig.-1.jpg 649w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-334\" class=\"wp-caption-text\">Figure 1<\/figcaption><\/figure>\n<\/section>\n<\/div>\n<p id=\"fs-id2906377\">Tracie set out from Elmhurst, IL, to go to Franklin Park. On the way, she made a few stops to do errands. Each stop is indicated by a red dot in Figure 1. Laying a rectangular coordinate grid over the map, we can see that each stop aligns with an intersection of grid lines. In this section, we will learn how to use grid lines to describe locations and changes in locations.<\/p>\n<section id=\"fs-id1392675\" data-depth=\"1\">\n<h1 data-type=\"title\">Plotting Ordered Pairs in the Cartesian Coordinate System<\/h1>\n<p id=\"fs-id2500615\">An old story describes how seventeenth-century philosopher\/mathematician Ren\u00e9 Descartes, while sick in bed, invented the system that has become the foundation of algebra. According to the story, Descartes was staring at a fly crawling on the ceiling when he realized that he could describe the fly\u2019s location in relation to the perpendicular lines formed by the adjacent walls of his room. He viewed the perpendicular lines as horizontal and vertical axes. Further, by dividing each axis into equal unit lengths, Descartes saw that it was possible to locate any object in a two-dimensional plane using just two numbers\u2014the displacement from the horizontal axis and the displacement from the vertical axis.<\/p>\n<p id=\"fs-id1960277\">While there is evidence that ideas similar to Descartes\u2019 grid system existed centuries earlier, it was Descartes who introduced the components that comprise the <strong><span id=\"term-00001\" data-type=\"term\">Cartesian coordinate system<\/span><\/strong>, a grid system having perpendicular axes. Descartes named the horizontal axis the <strong><span id=\"term-00002\" data-type=\"term\"><em data-effect=\"italics\">x-<\/em>axis<\/span><\/strong> and the vertical axis the <strong><span id=\"term-00003\" data-type=\"term\"><em data-effect=\"italics\">y-<\/em>axis<\/span><\/strong>.<\/p>\n<p id=\"fs-id1167648\">The Cartesian coordinate system, also called the rectangular coordinate system, is based on a two-dimensional plane consisting of the <em data-effect=\"italics\">x<\/em>-axis and the <em data-effect=\"italics\">y<\/em>-axis. Perpendicular to each other, the axes divide the plane into four sections. Each section is called a <strong><span id=\"term-00004\" data-type=\"term\">quadrant<\/span><\/strong>; the quadrants are numbered counterclockwise as shown in Figure 2.<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_35\" aria-describedby=\"caption-attachment-35\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-35 size-medium\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/03\/2.1-Fig.-2-300x266.jpg\" alt=\"\" width=\"300\" height=\"266\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/03\/2.1-Fig.-2-300x266.jpg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/03\/2.1-Fig.-2-65x58.jpg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/03\/2.1-Fig.-2-225x199.jpg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/03\/2.1-Fig.-2-350x310.jpg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/03\/2.1-Fig.-2.jpg 418w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-35\" class=\"wp-caption-text\">Figure 2<\/figcaption><\/figure>\n<p id=\"fs-id1423341\">The center of the plane is the point at which the two axes cross. It is known as the <strong><span id=\"term-00005\" data-type=\"term\">origin<\/span><\/strong>, or point [latex](0, 0).[\/latex] From the origin, each axis is further divided into equal units: increasing, positive numbers to the right on the <em data-effect=\"italics\">x-<\/em>axis and up the <em data-effect=\"italics\">y-<\/em>axis; decreasing, negative numbers to the left on the <em data-effect=\"italics\">x-<\/em>axis and down the <em data-effect=\"italics\">y-<\/em>axis. The axes extend to positive and negative infinity as shown by the arrowheads in Figure 3.<\/p>\n<figure id=\"attachment_36\" aria-describedby=\"caption-attachment-36\" style=\"width: 288px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-36 size-medium\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/03\/2.1-Fig.-3-288x300.jpg\" alt=\"\" width=\"288\" height=\"300\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/03\/2.1-Fig.-3-288x300.jpg 288w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/03\/2.1-Fig.-3-65x68.jpg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/03\/2.1-Fig.-3-225x234.jpg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/03\/2.1-Fig.-3-350x365.jpg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/03\/2.1-Fig.-3.jpg 357w\" sizes=\"auto, (max-width: 288px) 100vw, 288px\" \/><figcaption id=\"caption-attachment-36\" class=\"wp-caption-text\">Figure 3<\/figcaption><\/figure>\n<p id=\"fs-id1787344\">Each point in the plane is identified by its <strong><span id=\"term-00006\" data-type=\"term\"><em data-effect=\"italics\">x-<\/em>coordinate<\/span><\/strong>, or horizontal displacement from the origin, and its <strong><span id=\"term-00007\" data-type=\"term\"><em data-effect=\"italics\">y-<\/em>coordinate<\/span><\/strong>, or vertical displacement from the origin. Together, we write them as an <strong><span id=\"term-00008\" data-type=\"term\">ordered pair<\/span><\/strong> indicating the combined distance from the origin in the form [latex](x, y).[\/latex] An ordered pair is also known as a coordinate pair because it consists of <em data-effect=\"italics\">x-<\/em> and <em data-effect=\"italics\">y<\/em>-coordinates. For example, we can represent the point [latex](3, -1)[\/latex] in the plane by moving three units to the right of the origin in the horizontal direction, and one unit down in the vertical direction. See Figure 4.<\/p>\n<figure id=\"attachment_37\" aria-describedby=\"caption-attachment-37\" style=\"width: 288px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-37 size-medium\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/03\/2.1-Fig.-4-288x300.jpg\" alt=\"\" width=\"288\" height=\"300\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/03\/2.1-Fig.-4-288x300.jpg 288w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/03\/2.1-Fig.-4-65x68.jpg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/03\/2.1-Fig.-4-225x234.jpg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/03\/2.1-Fig.-4-350x365.jpg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/03\/2.1-Fig.-4.jpg 357w\" sizes=\"auto, (max-width: 288px) 100vw, 288px\" \/><figcaption id=\"caption-attachment-37\" class=\"wp-caption-text\">Figure 4<\/figcaption><\/figure>\n<p id=\"fs-id3155322\">When dividing the axes into equally spaced increments, note that the <em data-effect=\"italics\">x-<\/em>axis may be considered separately from the <em data-effect=\"italics\">y-<\/em>axis. In other words, while the <em data-effect=\"italics\">x-<\/em>axis may be divided and labeled according to consecutive integers, the <em data-effect=\"italics\">y-<\/em>axis may be divided and labeled by increments of 2, or 10, or 100. In fact, the axes may represent other units, such as years against the balance in a savings account, or quantity against cost, and so on. Consider the rectangular coordinate system primarily as a method for showing the relationship between two quantities.<\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Cartesian Coordinate System<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>A two-dimensional plane where the<\/p>\n<ul>\n<li>x-axis is the horizontal axis<\/li>\n<li>y-axis is the vertical axis<\/li>\n<\/ul>\n<p>A point in the plane is defined as an ordered pair,[latex](x, y),[\/latex] such that <em>x<\/em> is determined by its horizontal distance from the origin and <em>y <\/em>is determined by its vertical distance from the origin.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/div>\n<div class=\"MainContent__ContentStyles-sc-6yy1if-0 NnXKu\" tabindex=\"-1\" data-dynamic-style=\"true\">\n<div class=\"chapter-content-module\" data-type=\"page\" data-book-content=\"true\">\n<section data-depth=\"1\">\n<div id=\"Example_02_01_01\" class=\"ui-has-child-title\" data-type=\"example\">\n<section>\n<div class=\"body\">\n<div id=\"fs-id2270902\" class=\"unnumbered\" data-type=\"exercise\">\n<section>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 1: Plotting Points in a Rectangular Coordinate System<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Plot the points\u00a0[latex](-2, 4), (3, 3)[\/latex] and [latex](0, -3)[\/latex] in the plane.<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary>Solution (click to expand)<\/summary>\n<p>To plot the point\u00a0[latex](-2, 4),[\/latex] begin at the origin. The <em>x<\/em>-coordinate is -2, so move two units to the left. The <em>y<\/em>-coordinate is 4, so then move four units up in the positive <em>y<\/em> direction.<\/p>\n<p>To plot the point [latex](3, 3),[\/latex] begin again at the origin. The <em>x<\/em>-coordinate is 3, so more three units to the right. The <em>y<\/em>-coordinate is also 3, so move three units up in the positive <em>y<\/em> direction.<\/p>\n<p>To plot the point [latex](0, -3),[\/latex] begin again at the origin. The <em>x<\/em>-coordinate is 0. This tells us not to move in either direction along the\u00a0<em>x<\/em>-axis. The <em>y<\/em>-coordinate is -3, so move three units down in the negative\u00a0<em>y<\/em> direction. See the graph in Figure 5.<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_44\" aria-describedby=\"caption-attachment-44\" style=\"width: 288px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-44 size-medium\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/03\/2.1-Fig.-5-288x300.jpg\" alt=\"\" width=\"288\" height=\"300\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/03\/2.1-Fig.-5-288x300.jpg 288w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/03\/2.1-Fig.-5-65x68.jpg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/03\/2.1-Fig.-5-225x234.jpg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/03\/2.1-Fig.-5-350x365.jpg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/03\/2.1-Fig.-5.jpg 357w\" sizes=\"auto, (max-width: 288px) 100vw, 288px\" \/><figcaption id=\"caption-attachment-44\" class=\"wp-caption-text\">Figure 5<\/figcaption><\/figure>\n<h2>Analysis<\/h2>\n<p>Note that when either coordinate is zero, the point must be on an axis. If the <em>x<\/em>-coordinate is zero, the point is on the\u00a0<em>y<\/em>-axis. If the <em>y<\/em>-coordinate is zero, the point is on the\u00a0<em>x<\/em>-axis.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<\/div>\n<\/div>\n<section id=\"fs-id1940663\" data-depth=\"1\">\n<h1 data-type=\"title\">Graphing Equations by Plotting Points<\/h1>\n<p id=\"fs-id1951777\">We can plot a set of points to represent an equation. When such an equation contains both an <em data-effect=\"italics\">x <\/em>variable and a <em data-effect=\"italics\">y <\/em>variable, it is called an <span id=\"term-00009\" data-type=\"term\">equation in two variables<\/span>. Its graph is called a <span id=\"term-00010\" data-type=\"term\">graph in two variables<\/span>. Any graph on a two-dimensional plane is a graph in two variables.<\/p>\n<p id=\"fs-id1811199\">Suppose we want to graph the equation [latex]y=2x-1.[\/latex] We can begin by substituting a value for <em data-effect=\"italics\">x<\/em> into the equation and determining the resulting value of <em data-effect=\"italics\">y<\/em>. Each pair of <em data-effect=\"italics\">x<\/em>&#8211; and <em data-effect=\"italics\">y<\/em>-values is an ordered pair that can be plotted. Table 1 lists values of <em data-effect=\"italics\">x<\/em> from \u20133 to 3 and the resulting values for <em data-effect=\"italics\">y<\/em>.<\/p>\n<div id=\"Table_02_01_01\" class=\"os-table\">\n<table class=\"grid aligncenter\" style=\"height: 120px;\" data-id=\"Table_02_01_01\">\n<caption>Table 1<\/caption>\n<tbody>\n<tr style=\"height: 15px;\">\n<td style=\"text-align: center; height: 15px; width: 170.117px;\" data-align=\"center\">[latex]x[\/latex]<\/td>\n<td style=\"text-align: center; height: 15px; width: 263px;\" data-align=\"center\">[latex]y=2x-1[\/latex]<\/td>\n<td style=\"text-align: center; height: 15px; width: 213.483px;\" data-align=\"center\">[latex](x, y)[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"text-align: center; height: 15px; width: 170.117px;\" data-align=\"center\">[latex]-3[\/latex]<\/td>\n<td style=\"text-align: center; height: 15px; width: 263px;\" data-align=\"center\">[latex]y=2(-3)-1=7[\/latex]<\/td>\n<td style=\"text-align: center; height: 15px; width: 213.483px;\" data-align=\"center\">[latex](-3, -7)[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"text-align: center; height: 15px; width: 170.117px;\" data-align=\"center\">[latex]-2[\/latex]<\/td>\n<td style=\"text-align: center; height: 15px; width: 263px;\" data-align=\"center\">[latex]y=2(-2)-1=-5[\/latex]<\/td>\n<td style=\"text-align: center; height: 15px; width: 213.483px;\" data-align=\"center\">[latex](-2, -5)[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"text-align: center; height: 15px; width: 170.117px;\" data-align=\"center\">[latex]-1[\/latex]<\/td>\n<td style=\"text-align: center; height: 15px; width: 263px;\" data-align=\"center\">[latex]y=2(-1)-1=-3[\/latex]<\/td>\n<td style=\"text-align: center; height: 15px; width: 213.483px;\" data-align=\"center\">[latex](-1, -3)[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"text-align: center; height: 15px; width: 170.117px;\" data-align=\"center\">[latex]0[\/latex]<\/td>\n<td style=\"text-align: center; height: 15px; width: 263px;\" data-align=\"center\">[latex]y=2(0)-1=-1[\/latex]<\/td>\n<td style=\"text-align: center; height: 15px; width: 213.483px;\" data-align=\"center\">[latex](0, -1)[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"text-align: center; height: 15px; width: 170.117px;\" data-align=\"center\">[latex]1[\/latex]<\/td>\n<td style=\"text-align: center; height: 15px; width: 263px;\" data-align=\"center\">[latex]y=2(1)-1=1[\/latex]<\/td>\n<td style=\"text-align: center; height: 15px; width: 213.483px;\" data-align=\"center\">[latex](1, 1)[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"text-align: center; height: 15px; width: 170.117px;\" data-align=\"center\">[latex]2[\/latex]<\/td>\n<td style=\"text-align: center; height: 15px; width: 263px;\" data-align=\"center\">[latex]y=2(2)-1=3[\/latex]<\/td>\n<td style=\"text-align: center; height: 15px; width: 213.483px;\" data-align=\"center\">[latex](2, 3)[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"text-align: center; height: 15px; width: 170.117px;\" data-align=\"center\">[latex]3[\/latex]<\/td>\n<td style=\"text-align: center; height: 15px; width: 263px;\" data-align=\"center\">[latex]y=2(3)-1=5[\/latex]<\/td>\n<td style=\"text-align: center; height: 15px; width: 213.483px;\" data-align=\"center\">[latex](3, 5)[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p id=\"fs-id1386116\">We can plot the points in the table. The points for this particular equation form a line, so we can connect them. See Figure 6<strong>. <\/strong>This is not true for all equations.<\/p>\n<figure id=\"attachment_38\" aria-describedby=\"caption-attachment-38\" style=\"width: 255px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-38 size-medium\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/03\/2.1-Fig.-6-255x300.jpg\" alt=\"\" width=\"255\" height=\"300\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/03\/2.1-Fig.-6-255x300.jpg 255w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/03\/2.1-Fig.-6-65x76.jpg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/03\/2.1-Fig.-6-225x265.jpg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/03\/2.1-Fig.-6-350x412.jpg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/03\/2.1-Fig.-6.jpg 357w\" sizes=\"auto, (max-width: 255px) 100vw, 255px\" \/><figcaption id=\"caption-attachment-38\" class=\"wp-caption-text\">Figure 6<\/figcaption><\/figure>\n<p id=\"fs-id2522154\">Note that the <em data-effect=\"italics\">x-<\/em>values chosen are arbitrary, regardless of the type of equation we are graphing. Of course, some situations may require particular values of <em data-effect=\"italics\">x<\/em> to be plotted in order to see a particular result. Otherwise, it is logical to choose values that can be calculated easily, and it is always a good idea to choose values that are both negative and positive. There is no rule dictating how many points to plot, although we need at least two to graph a line. Keep in mind, however, that the more points we plot, the more accurately we can sketch the graph.<\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">How To<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Given an equation, graph by plotting points<\/p>\n<ol>\n<li>Make a table with one column labeled <em>x<\/em>, a second column labeled with the equation, and a third column listing the resulting ordered pairs.<\/li>\n<li>Enter <em>x<\/em>-values down the first column using positive and negative values. Selecting the\u00a0<em>x<\/em>-values in numerical order will make the graphing simpler.<\/li>\n<li>Select <em>x<\/em>-values that will yield\u00a0<em>y<\/em>-values with little effort, preferable ones that can be calculated mentally.<\/li>\n<li>Plot the ordered pairs.<\/li>\n<li>Connect the points if they form a line.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/section>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 2: Graphing an Equation in Two Variables by Plotting Points<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Graph the equation [latex]y=-x+2[\/latex] by plotting points.<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary>Solution (click to expand)<\/summary>\n<p>First, we construct a table similar to Table 2. Choose <em>x<\/em> values and calculate\u00a0<em>y<\/em>.<\/p>\n<table class=\"grid aligncenter\" style=\"border-collapse: collapse; width: 100%; height: 120px;\">\n<caption>Table 2<\/caption>\n<tbody>\n<tr style=\"height: 15px;\">\n<td style=\"width: 33.3333%; height: 15px; text-align: center;\">[latex]x[\/latex]<\/td>\n<td style=\"width: 33.3333%; height: 15px; text-align: center;\">[latex]y=-x+2[\/latex]<\/td>\n<td style=\"width: 33.3333%; height: 15px; text-align: center;\">[latex](x, y)[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"width: 33.3333%; height: 15px; text-align: center;\">[latex]-5[\/latex]<\/td>\n<td style=\"width: 33.3333%; height: 15px; text-align: center;\">[latex]y=-(-5)+2=7[\/latex]<\/td>\n<td style=\"width: 33.3333%; height: 15px; text-align: center;\">[latex](-5, 7)[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"width: 33.3333%; height: 15px; text-align: center;\">[latex]-3[\/latex]<\/td>\n<td style=\"width: 33.3333%; height: 15px; text-align: center;\">[latex]y=-(-3)+2=5[\/latex]<\/td>\n<td style=\"width: 33.3333%; height: 15px; text-align: center;\">[latex](-3, 5)[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"width: 33.3333%; height: 15px; text-align: center;\">[latex]-1[\/latex]<\/td>\n<td style=\"width: 33.3333%; height: 15px; text-align: center;\">[latex]y=-(-1)+2=3[\/latex]<\/td>\n<td style=\"width: 33.3333%; height: 15px; text-align: center;\">[latex](-1, 3)[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"width: 33.3333%; height: 15px; text-align: center;\">[latex]0[\/latex]<\/td>\n<td style=\"width: 33.3333%; height: 15px; text-align: center;\">[latex]y=-(0)+2=2[\/latex]<\/td>\n<td style=\"width: 33.3333%; height: 15px; text-align: center;\">[latex](0, 2)[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"width: 33.3333%; height: 15px; text-align: center;\">[latex]1[\/latex]<\/td>\n<td style=\"width: 33.3333%; height: 15px; text-align: center;\">[latex]y=-(1)+2=1[\/latex]<\/td>\n<td style=\"width: 33.3333%; height: 15px; text-align: center;\">[latex](1, 1)[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"width: 33.3333%; height: 15px; text-align: center;\">[latex]3[\/latex]<\/td>\n<td style=\"width: 33.3333%; height: 15px; text-align: center;\">[latex]y=-(3)+2=-1[\/latex]<\/td>\n<td style=\"width: 33.3333%; height: 15px; text-align: center;\">[latex](3, -1)[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"width: 33.3333%; height: 15px; text-align: center;\">[latex]5[\/latex]<\/td>\n<td style=\"width: 33.3333%; height: 15px; text-align: center;\">[latex]y=-(5)+2=-3[\/latex]<\/td>\n<td style=\"width: 33.3333%; height: 15px; text-align: center;\">[latex](5, -1)[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Now, plot the points. Connect them if they form a line. See Figure 7.<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_45\" aria-describedby=\"caption-attachment-45\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-45 size-medium\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/03\/2.1-Fig.-7-300x230.jpg\" alt=\"\" width=\"300\" height=\"230\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/03\/2.1-Fig.-7-300x230.jpg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/03\/2.1-Fig.-7-65x50.jpg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/03\/2.1-Fig.-7-225x173.jpg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/03\/2.1-Fig.-7-350x269.jpg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/03\/2.1-Fig.-7.jpg 414w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-45\" class=\"wp-caption-text\">Figure 7<\/figcaption><\/figure>\n<\/details>\n<\/div>\n<\/div>\n<section id=\"fs-id1901322\" data-depth=\"1\">\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Try it #1<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Construct a table and graph the equation by plotting points: [latex]y=\\frac{1}{2}x+2[\/latex]<\/p>\n<\/div>\n<\/div>\n<h1 data-type=\"title\">Graphing Equations with a Graphing Utility<\/h1>\n<p id=\"fs-id1722871\">Most graphing calculators require similar techniques to graph an equation. The equations sometimes have to be manipulated so they are written in the style [latex]y=\\rule{2cm}{0.10mm}.[\/latex] The TI-84 Plus, and many other calculator makes and models, have a mode function, which allows the window (the screen for viewing the graph) to be altered so the pertinent parts of a graph can be seen.<\/p>\n<p id=\"fs-id1539399\">For example, the equation [latex]y=2x-20[\/latex] has been entered in the TI-84 Plus shown in Figure 8<strong>a. <\/strong>In Figure 8<strong>b, <\/strong>the resulting graph is shown. Notice that we cannot see on the screen where the graph crosses the axes. The standard window screen on the TI-84 Plus shows [latex]10\\le x\\le10,[\/latex] and [latex]-1\\le y\\le10.[\/latex] See Figure 8<strong>c<\/strong>.<\/p>\n<p>&nbsp;<\/p>\n<div id=\"Figure_02_01_09\" class=\"os-figure\">\n<div>\n<figure id=\"attachment_405\" aria-describedby=\"caption-attachment-405\" style=\"width: 612px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-405\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/2.2-Fig.-8a-300x78.jpg\" alt=\"\" width=\"612\" height=\"159\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/2.2-Fig.-8a-300x78.jpg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/2.2-Fig.-8a-768x200.jpg 768w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/2.2-Fig.-8a-65x17.jpg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/2.2-Fig.-8a-225x59.jpg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/2.2-Fig.-8a-350x91.jpg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/2.2-Fig.-8a.jpg 807w\" sizes=\"auto, (max-width: 612px) 100vw, 612px\" \/><figcaption id=\"caption-attachment-405\" class=\"wp-caption-text\">Figure 8. a. Enter the equation. b. This is the graph in the original window. c. These are the original settings.<\/figcaption><\/figure>\n<\/div>\n<\/div>\n<\/section>\n<section data-depth=\"1\">\n<p id=\"fs-id1416938\">By changing the window to show more of the positive <em data-effect=\"italics\">x-<\/em>axis and more of the negative <em data-effect=\"italics\">y-<\/em>axis, we have a much better view of the graph and the <em data-effect=\"italics\">x-<\/em> and <em data-effect=\"italics\">y-<\/em>intercepts. See Figure 9<strong>a<\/strong> and Figure 9<strong>b.<\/strong><\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_406\" aria-describedby=\"caption-attachment-406\" style=\"width: 578px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-406\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/2.9-Fig.-9-300x119.jpg\" alt=\"\" width=\"578\" height=\"229\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/2.9-Fig.-9-300x119.jpg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/2.9-Fig.-9-65x26.jpg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/2.9-Fig.-9-225x89.jpg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/2.9-Fig.-9-350x139.jpg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/2.9-Fig.-9.jpg 529w\" sizes=\"auto, (max-width: 578px) 100vw, 578px\" \/><figcaption id=\"caption-attachment-406\" class=\"wp-caption-text\">Figure 9. a. This screen shows the new window settings. b. We can clearly view the intercepts in the new window.<\/figcaption><\/figure>\n<\/section>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 3: Using a Graphing Utility to Graph an Equation<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Use a graphing utility to graph the equation: [latex]y=-\\frac{2}{3}x+\\frac{4}{3}.[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary>Solution (click to expand)<\/summary>\n<p>Enter the equation in the <em>y<\/em>&#8211; function of the calculator. Set the window settings so that both the\u00a0<em>x-<\/em> and <em>y-<\/em> intercepts are showing in the window. See Figure 10.<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_407\" aria-describedby=\"caption-attachment-407\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-407 size-medium\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/2.1-Fig.-10-300x166.jpg\" alt=\"\" width=\"300\" height=\"166\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/2.1-Fig.-10-300x166.jpg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/2.1-Fig.-10-65x36.jpg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/2.1-Fig.-10-225x125.jpg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/2.1-Fig.-10-350x194.jpg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/2.1-Fig.-10.jpg 357w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-407\" class=\"wp-caption-text\">Figure 10<\/figcaption><\/figure>\n<\/details>\n<\/div>\n<\/div>\n<h1>Finding <em data-effect=\"italics\">x-<\/em>intercepts and <em data-effect=\"italics\">y-<\/em>intercepts<\/h1>\n<section id=\"fs-id1340475\" data-depth=\"1\">\n<p id=\"fs-id2503271\">The <span id=\"term-00011\" data-type=\"term\">intercepts<\/span> of a graph are points at which the graph crosses the axes. The <span id=\"term-00012\" data-type=\"term\"><em data-effect=\"italics\">x-<\/em>intercept<\/span> is the point at which the graph crosses the <em data-effect=\"italics\">x-<\/em>axis. At this point, the <em data-effect=\"italics\">y-<\/em>coordinate is zero. The <span id=\"term-00013\" data-type=\"term\"><em data-effect=\"italics\">y-<\/em>intercept<\/span> is the point at which the graph crosses the <em data-effect=\"italics\">y-<\/em>axis. At this point, the <em data-effect=\"italics\">x-<\/em>coordinate is zero.<\/p>\n<p id=\"fs-id1448434\">To determine the <em data-effect=\"italics\">x-<\/em>intercept, we set <em data-effect=\"italics\">y <\/em>equal to zero and solve for <em data-effect=\"italics\">x<\/em>. Similarly, to determine the <em data-effect=\"italics\">y-<\/em>intercept, we set <em data-effect=\"italics\">x <\/em>equal to zero and solve for <em data-effect=\"italics\">y<\/em>. For example, lets find the intercepts of the equation [latex]y=3x-1.[\/latex]<\/p>\n<p id=\"fs-id1493312\">To find the <em data-effect=\"italics\">x-<\/em>intercept, set [latex]y=0[\/latex]<\/p>\n<p style=\"text-align: left;\">[latex]\\hspace{12em}y=3x-1[\/latex]<br \/>\n[latex]\\hspace{12em}0=3x-1[\/latex]<br \/>\n[latex]\\hspace{12em}1=3x[\/latex]<br \/>\n[latex]\\hspace{12em}\\frac{1}{3}=x[\/latex]<br \/>\n[latex]\\hspace{11.5em}(\\frac{1}{3}, 0) \\hspace{2em}x-\\text{intercept}[\/latex]<\/p>\n<div id=\"fs-id3064821\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><\/div>\n<p id=\"fs-id2905820\">To find the <em data-effect=\"italics\">y-<\/em>intercept, set [latex]x=0[\/latex]<\/p>\n<p style=\"text-align: left;\">[latex]\\hspace{12em}y=3x-1[\/latex]<br \/>\n[latex]\\hspace{12em}y=3(0)-1[\/latex]<br \/>\n[latex]\\hspace{12em}y=-1[\/latex]<br \/>\n[latex]\\hspace{12em}(0, -1) \\hspace{2em}y-\\text{intercept}[\/latex]<\/p>\n<div id=\"fs-id1798574\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><\/div>\n<p id=\"fs-id1730534\">We can confirm that our results make sense by observing a graph of the equation as in Figure 11. Notice that the graph crosses the axes where we predicted it would.<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_408\" aria-describedby=\"caption-attachment-408\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-408 size-medium\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/2.1-Fig.-11-300x234.jpg\" alt=\"\" width=\"300\" height=\"234\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/2.1-Fig.-11-300x234.jpg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/2.1-Fig.-11-65x51.jpg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/2.1-Fig.-11-225x176.jpg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/2.1-Fig.-11-350x274.jpg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/2.1-Fig.-11.jpg 357w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-408\" class=\"wp-caption-text\">Figure 11<\/figcaption><\/figure>\n<\/section>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Given an Equation, Find the Intercepts<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<ul>\n<li>Find the <em>x<\/em>-intercept by setting\u00a0[latex]y=0[\/latex] and solving for <em>x.<\/em><\/li>\n<li>Find the <em>y<\/em>-intercept by setting [latex]x=0[\/latex] and solving for <em>y.<\/em><\/li>\n<\/ul>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 4: Finding the Intercepts of the Given Equation<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Find the intercepts of the equation\u00a0[latex]y=-3x-4.[\/latex] Then sketch the graph using only the intercepts.<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary>Solution (click to expand)<\/summary>\n<p>Set [latex]y=0[\/latex] to find the <em>x<\/em>-intercept.<\/p>\n<p style=\"text-align: left;\">[latex]\\hspace{12em}y=-3x-4[\/latex]<br \/>\n[latex]\\hspace{12em}0=-3x-4[\/latex]<br \/>\n[latex]\\hspace{12em}4=-3x[\/latex]<br \/>\n[latex]\\hspace{11em}-\\frac{4}{3}=x[\/latex]<br \/>\n[latex]\\hspace{12em}(-\\frac{4}{3}, 0) \\hspace{2em}x-\\text{intercept}[\/latex]<\/p>\n<p>Set [latex]x=0[\/latex] to find the <em>y<\/em>-intercept.<\/p>\n<p style=\"text-align: left;\">[latex]\\hspace{12em}y=-3x-4[\/latex]<br \/>\n[latex]\\hspace{12em}y=-3(0)-4[\/latex]<br \/>\n[latex]\\hspace{12em}y=-4[\/latex]<br \/>\n[latex]\\hspace{12em}(0, -4) \\hspace{2em}y-\\text{intercept}[\/latex]<\/p>\n<p>Plot both points, and draw a line passing through them as in Figure 12.<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_409\" aria-describedby=\"caption-attachment-409\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-409 size-medium\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/2.1-Fig.-12-300x253.jpg\" alt=\"\" width=\"300\" height=\"253\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/2.1-Fig.-12-300x253.jpg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/2.1-Fig.-12-65x55.jpg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/2.1-Fig.-12-225x190.jpg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/2.1-Fig.-12-350x295.jpg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/2.1-Fig.-12.jpg 357w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-409\" class=\"wp-caption-text\">Figure 12<\/figcaption><\/figure>\n<\/details>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Try it #2<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Find the intercepts of the equation and sketch the graph: [latex]y=-\\frac{3}{4}x+3.[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1518804\" class=\"precalculus try ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"Try It\">\n<section>\n<div class=\"os-note-body\">\n<div id=\"ti_02_01_02\" class=\"unnumbered os-hasSolution\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id2437906\" data-type=\"problem\">\n<h1>Using the Distance Formula<\/h1>\n<\/div>\n<\/section>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<section id=\"fs-id1280821\" data-depth=\"1\">\n<p id=\"fs-id1277804\">Derived from the <span id=\"term-00014\" class=\"no-emphasis\" data-type=\"term\">Pythagorean Theorem<\/span>, the <strong><span id=\"term-00015\" data-type=\"term\">distance formula<\/span> <\/strong>is used to find the distance between two points in the plane. The Pythagorean Theorem,\u00a0[latex]a^2+b^2=c^2[\/latex] is based on a right triangle where <em data-effect=\"italics\">a <\/em>and <em data-effect=\"italics\">b<\/em> are the lengths of the legs adjacent to the right angle, and <em data-effect=\"italics\">c<\/em> is the length of the hypotenuse. See Figure 13.<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_410\" aria-describedby=\"caption-attachment-410\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-410 size-medium\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/2.1-Fig.-13-300x211.jpg\" alt=\"\" width=\"300\" height=\"211\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/2.1-Fig.-13-300x211.jpg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/2.1-Fig.-13-65x46.jpg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/2.1-Fig.-13-225x158.jpg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/2.1-Fig.-13-350x246.jpg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/2.1-Fig.-13.jpg 447w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-410\" class=\"wp-caption-text\">Figure 13<\/figcaption><\/figure>\n<p id=\"fs-id1151919\">The relationship of sides\u00a0[latex]|x_2-x_1|[\/latex] and [latex]|y_2-y_1|[\/latex] to side <em data-effect=\"italics\">d<\/em> is the same as that of sides <em data-effect=\"italics\">a <\/em>and <em data-effect=\"italics\">b <\/em>to side <em data-effect=\"italics\">c.<\/em> We use the absolute value symbol to indicate that the length is a positive number because the absolute value of any number is positive. (For example, [latex]|-3|=3.[\/latex]) The symbols [latex]|x_2-x_1|[\/latex] and [latex]|y_2-y_1|[\/latex] indicate that the lengths of the sides of the triangle are positive. To find the length <em data-effect=\"italics\">c<\/em>, take the square root of both sides of the Pythagorean Theorem.<\/p>\n<p style=\"text-align: center;\">[latex]c^2=a^2+b^2\\rightarrow c=\\sqrt{a^2+b^2}[\/latex]<\/p>\n<p id=\"fs-id2666328\">It follows that the distance formula is given as<\/p>\n<p style=\"text-align: center;\">[latex]d^2=(x_2-x_1)^2+(y_2-y_1)^2\\rightarrow =\\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}[\/latex]<\/p>\n<p id=\"fs-id1832560\">We do not have to use the absolute value symbols in this definition because any number squared is positive.<\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">The Distance Formula<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Given endpoints\u00a0[latex](x_1, y_1)[\/latex] and\u00a0[latex](x_2, y_2),[\/latex] the distance between two points is given by<\/p>\n<p style=\"text-align: center;\">[latex]d=\\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 5: Finding the Distance between Two Points<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Find the distance between the two points\u00a0[latex](-3, -1)[\/latex] and [latex](2, 3).[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary>Solution (click to expand)<\/summary>\n<p>Let us first look at the graph of the two points. Connect the points to form a right triangle as in Figure 14.<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_411\" aria-describedby=\"caption-attachment-411\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-411 size-medium\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/2.1-Fig.-14-300x234.jpg\" alt=\"\" width=\"300\" height=\"234\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/2.1-Fig.-14-300x234.jpg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/2.1-Fig.-14-65x51.jpg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/2.1-Fig.-14-225x176.jpg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/2.1-Fig.-14-350x274.jpg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/2.1-Fig.-14.jpg 357w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-411\" class=\"wp-caption-text\">Figure 14<\/figcaption><\/figure>\n<p>Then, calculate the length of <em>d<\/em> using the distance formula.<\/p>\n<p style=\"text-align: left;\">[latex]\\hspace{10em}d=\\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}[\/latex]<br \/>\n[latex]\\hspace{10em}d=\\sqrt{(2-(-3))^2+(3-(-1))^2}[\/latex]<br \/>\n[latex]\\hspace{10.7em}=\\sqrt{(5)^2+(4)^2}[\/latex]<br \/>\n[latex]\\hspace{10.7em}=\\sqrt{25+16}[\/latex]<br \/>\n[latex]\\hspace{10.7em}=\\sqrt{41}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">Try it #3<\/header>\n<div class=\"textbox__content\">\n<p>Find the distance between two points:\u00a0[latex](1, 4)[\/latex] and [latex](11, 9).[\/latex]<\/p>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 6: Finding the Distance between Two Locations<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Let&#8217;s return to the situation introduced at the beginning of this section.<\/p>\n<p>Tracie set out from Elmhurt, IL, to go to Franklin Park. On the way, she made a few stops to do errands. Each stop is indicated by a red dot in Figure 1. Find the total distance that Tracie traveled. Compare this with the distance between her starting and final positions.<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary>Solution (click to expand)<\/summary>\n<p>The first thing we should do is identify ordered pairs to describe each position. If we set the starting position as the origin, we can identify each of the other points by counting units east (right) and north (up) on the grid. For example, the first stop is 1 block east and 1 block north, so it is at\u00a0[latex](1, 1).[\/latex] The next stop is 5 blocks to the east, so it is at\u00a0[latex](5, 1).[\/latex] After that, she traveled 3 blocks east and 2 blocks north to [latex](8, 3).[\/latex] Lastly, she traveled 4 blocks north to [latex](8, 7).[\/latex] We can label these points on the grid as in Figure 15.<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_436\" aria-describedby=\"caption-attachment-436\" style=\"width: 368px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-436\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/2.1-Fig.-15-300x278.jpg\" alt=\"\" width=\"368\" height=\"341\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/2.1-Fig.-15-300x278.jpg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/2.1-Fig.-15-65x60.jpg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/2.1-Fig.-15-225x208.jpg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/2.1-Fig.-15-350x324.jpg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/2.1-Fig.-15.jpg 649w\" sizes=\"auto, (max-width: 368px) 100vw, 368px\" \/><figcaption id=\"caption-attachment-436\" class=\"wp-caption-text\">Figure 15<\/figcaption><\/figure>\n<p>Next, we can calculate the distance. Note that each grid unit represents 1,000 feet.<\/p>\n<ul>\n<li>From her starting location to her first stop at\u00a0[latex](1, 1)[\/latex] Tracie might have driven north 1,000 feet and then east 1,000 feet, or vice versa. Either way, she drove 2,000 feet to her first stop.<\/li>\n<li>Her second stop is at\u00a0[latex](5, 1).[\/latex] So from [latex](1, 1)[\/latex] to\u00a0[latex](5, 1),[\/latex] Tracie drove east 4,000 feet.<\/li>\n<li>Her third stop is at\u00a0[latex](8, 3).[\/latex] There are a number of routes from [latex](5, 1)[\/latex] to [latex](8, 3).[\/latex] Whatever route Tracie decided to use, the distance is the same, as there are no angular streets between the two points. Let&#8217;s say she drove east 3,000 feet and then north 2,000 feet for a total of 5,000 feet.<\/li>\n<li>Tracie&#8217;s final stop is at\u00a0[latex](8, 7).[\/latex] This is a straight drive north from\u00a0[latex](8, 3)[\/latex] for a total of 4,000 feet.<\/li>\n<\/ul>\n<p>Next, we will add the distances listed in Table 3.<\/p>\n<table class=\"grid\" style=\"border-collapse: collapse; width: 100%;\">\n<caption>Table 3<\/caption>\n<tbody>\n<tr>\n<td style=\"width: 50%;\"><strong>From\/To<\/strong><\/td>\n<td style=\"width: 50%;\"><strong>Number of Feet Driven<\/strong><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%; text-align: center;\">[latex](0, 0)[\/latex]\u00a0to [latex](1, 1)[\/latex]<\/td>\n<td style=\"width: 50%; text-align: center;\">2,000<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%; text-align: center;\">[latex](1, 1)[\/latex]\u00a0to [latex](5, 1)[\/latex]<\/td>\n<td style=\"width: 50%; text-align: center;\">4,000<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%; text-align: center;\">[latex](5, 1)[\/latex] to [latex](8, 3)[\/latex]<\/td>\n<td style=\"width: 50%; text-align: center;\">5,000<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%; text-align: center;\">[latex](8, 3)[\/latex] to [latex](8, 7)[\/latex]<\/td>\n<td style=\"width: 50%; text-align: center;\">4,000<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%; text-align: center;\">Total<\/td>\n<td style=\"width: 50%; text-align: center;\">15,000<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>The total distance Tracie drove is 15,000 feet, or 2.84 miles. This is not, however the actual distance between her starting and ending positions. To find this distance, we can use the distance formula between the points\u00a0[latex](0, 0)[\/latex] and [latex](8, 7).[\/latex]<\/p>\n<p style=\"text-align: left;\">[latex]\\hspace{10em}d=\\sqrt{(8-0)^2+(7-0)^2}[\/latex]<br \/>\n[latex]\\hspace{10.8em}=\\sqrt{64+49}[\/latex]<br \/>\n[latex]\\hspace{10.8em}=\\sqrt{113}[\/latex]<br \/>\n[latex]\\hspace{10.8em}\\approx 10.63 \\hspace{1em}\\text{units}[\/latex]<\/p>\n<p>At 1,000 feet per grid unit, the distance between Elmhurst, IL, to Franklin Park is 10,630.14 feet or 2.01 miles. The distance formula results in a shorter calculation because it is based on the hypotenuse of a right triangle, a straight diagonal from the origin to the point\u00a0[latex](8, 7).[\/latex] Perhaps you have heard the saying &#8220;as the crow flies,&#8221; which means the shortest distance between two points because a crow can fly in a straight line even though a person on the ground has to travel a longer distance on existing roadways.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<h1>Using the Midpoint Formula<\/h1>\n<section id=\"fs-id2507035\" data-depth=\"1\">\n<p id=\"fs-id1151538\">When the endpoints of a line segment are known, we can find the point midway between them. This point is known as the midpoint and the formula is known as the <strong><span id=\"term-00016\" data-type=\"term\">midpoint formula<\/span><\/strong>. Given the endpoints of a line segment,\u00a0[latex](x_1, y_1)[\/latex] and [latex](x_2, y_2)[\/latex] the midpoint formula states how to find the coordinates of the midpoint [latex]M.[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]M=(\\frac{x_1+x_2}{2}, \\frac{y_1+y_2}{2})[\/latex]<\/p>\n<p id=\"fs-id2837053\">A graphical view of a midpoint is shown in <a class=\"autogenerated-content\" href=\"2-1-the-rectangular-coordinate-systems-and-graphs#Figure_02_01_018\">Figure 16<\/a>. Notice that the line segments on either side of the midpoint are congruent.<\/p>\n<p>&nbsp;<\/p>\n<div id=\"Figure_02_01_018\" class=\"os-figure\">\n<div class=\"os-caption-container\">\n<figure id=\"attachment_435\" aria-describedby=\"caption-attachment-435\" style=\"width: 285px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-435 size-medium\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/2.1.16-285x300.jpg\" alt=\"\" width=\"285\" height=\"300\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/2.1.16-285x300.jpg 285w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/2.1.16-65x68.jpg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/2.1.16-225x237.jpg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/2.1.16.jpg 291w\" sizes=\"auto, (max-width: 285px) 100vw, 285px\" \/><figcaption id=\"caption-attachment-435\" class=\"wp-caption-text\">Figure 16<\/figcaption><\/figure>\n<\/div>\n<\/div>\n<\/section>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 7: Finding the Midpoint of the Line Segment<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Find the midpoint of the line segment with the endpoints\u00a0[latex](7, -2)[\/latex] and [latex](9, 5).[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary>Solution (click to expand)<\/summary>\n<p>Use the formula to find the midpoint of the line segment.<\/p>\n<p style=\"text-align: center;\">[latex](\\frac{x_1+x_2}{2}, \\frac{y_1+y_2}{2})=(\\frac{7+9}{2}, \\frac{-2+5}{2})[\/latex]<br \/>\n[latex]\\hspace{3.2em}=(8, \\frac{3}{2})[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<section data-depth=\"1\">\n<div id=\"Example_02_01_07\" class=\"ui-has-child-title\" data-type=\"example\">\n<section>\n<div class=\"body\">\n<div id=\"fs-id3008576\" class=\"unnumbered\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id3008579\" data-type=\"problem\">\n<div class=\"os-problem-container\">\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Try it #4<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Find the midpoint of the line segment with endpoints\u00a0[latex](-2, -1)[\/latex] and [latex](-8, 6).[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 8: Finding the Center of a Circle<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>The diameter of a circle has endpoints\u00a0[latex](-1, -4)[\/latex] and [latex](5, -4).[\/latex] Find the center of the circle.<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary>Solution (click to expand)<\/summary>\n<p>The center of a circle is the center, or midpoint, of its diameter. Thus, the midpoint formula will yield the center point.<\/p>\n<p style=\"text-align: center;\">[latex](\\frac{x_1+x_2}{2}, \\frac{y_1+y_2}{2})[\/latex]<br \/>\n[latex](\\frac{-1+5}{2}, \\frac{-4-4}{2}) = (\\frac{4}{2}, -\\frac{8}{2})=(2, -4)[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--key-takeaways\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Media<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Access these online resources for additional instruction and practice with the Cartesian coordinate system.<\/p>\n<ul>\n<li><a href=\"http:\/\/openstax.org\/l\/coordplotpnts\">Plotting points on the coordinate plane<\/a><\/li>\n<li><a href=\"http:\/\/openstax.org\/l\/xyintsgraph\">Find x and y intercepts based on the graph of a line<\/a><\/li>\n<\/ul>\n<\/div>\n<\/div>\n<div id=\"Example_02_01_08\" class=\"ui-has-child-title\" data-type=\"example\"><\/div>\n<div class=\"os-eos os-section-exercises-container\" data-uuid-key=\".section-exercises\">\n<h1 data-type=\"document-title\" data-rex-keep=\"true\"><span class=\"os-text\">2.1 Section Exercises<\/span><\/h1>\n<section id=\"fs-id1553580\" class=\"section-exercises\" data-depth=\"1\">\n<section id=\"fs-id2496131\" data-depth=\"2\">\n<h2 data-type=\"title\">Verbal<\/h2>\n<div id=\"fs-id1355431\" class=\"os-hasSolution\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1355432\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1355431-solution\">1<\/a><span class=\"os-divider\">. <\/span>Is it possible for a point plotted in the Cartesian coordinate system to not lie in one of the four quadrants? Explain.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2435397\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id2435398\" data-type=\"problem\">\n<p><span class=\"os-number\">2<\/span><span class=\"os-divider\">. <\/span>Describe the process for finding the <em data-effect=\"italics\">x-<\/em>intercept and the <em data-effect=\"italics\">y<\/em>-intercept of a graph algebraically.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1351774\" class=\"os-hasSolution\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id2682309\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1351774-solution\">3<\/a><span class=\"os-divider\">. <\/span>Describe in your own words what the <em data-effect=\"italics\">y<\/em>-intercept of a graph is.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1823207\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1823208\" data-type=\"problem\">\n<p><span class=\"os-number\">4<\/span><span class=\"os-divider\">. <\/span>When using the distance formula\u00a0[latex]d=\\sqrt{(x_2-x_1)^2+(y_2-y_1)^2)},[\/latex] explain the correct order of operations that are to be performed to obtain the correct answer.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<section id=\"fs-id3263952\" data-depth=\"2\">\n<h2 data-type=\"title\">Algebraic<\/h2>\n<p id=\"fs-id1789811\">For each of the following exercises, find the <em data-effect=\"italics\">x<\/em>-intercept and the <em data-effect=\"italics\">y<\/em>-intercept without graphing. Write the coordinates of each intercept.<\/p>\n<div id=\"fs-id3039754\" class=\"os-hasSolution\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id3039755\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id3039754-solution\">5<\/a><span class=\"os-divider\">.<\/span> [latex]y=-3x+6[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1499169\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1499170\" data-type=\"problem\">\n<p><span class=\"os-number\">6<\/span><span class=\"os-divider\">.<\/span> [latex]4y=2x-1[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1007678\" class=\"os-hasSolution\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1591152\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1007678-solution\">7<\/a><span class=\"os-divider\">.<\/span> [latex]3x-2y=6[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2948006\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id2948007\" data-type=\"problem\">\n<p><span class=\"os-number\">8<\/span><span class=\"os-divider\">.<\/span> [latex]4x-3=2y[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1723142\" class=\"os-hasSolution\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1723143\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1723142-solution\">9<\/a><span class=\"os-divider\">.<\/span> [latex]3x+8y=9[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1197890\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1197891\" data-type=\"problem\">\n<p><span class=\"os-number\">10<\/span><span class=\"os-divider\">.<\/span> [latex]2x-\\frac{2}{3}=\\frac{3}{4}y+3[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<p id=\"fs-id3207567\">For each of the following exercises, solve the equation for <em data-effect=\"italics\">y<\/em> in terms of <em data-effect=\"italics\">x<\/em>.<\/p>\n<div id=\"fs-id3042175\" class=\"os-hasSolution\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id3042176\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id3042175-solution\">11<\/a><span class=\"os-divider\">.<\/span> [latex]4x+2y=8[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1238095\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1238096\" data-type=\"problem\">\n<p><span class=\"os-number\">12<\/span><span class=\"os-divider\">.<\/span> [latex]3x-2y=6[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2512516\" class=\"os-hasSolution\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1441357\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id2512516-solution\">13<\/a><span class=\"os-divider\">.<\/span> [latex]2x=5-3y[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1467950\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id2521131\" data-type=\"problem\">\n<p><span class=\"os-number\">14<\/span><span class=\"os-divider\">.<\/span> [latex]x-2y=7[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1278706\" class=\"os-hasSolution\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1402803\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1278706-solution\">15<\/a><span class=\"os-divider\">.<\/span> [latex]5y+4=10x[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2388766\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id2388767\" data-type=\"problem\">\n<p><span class=\"os-number\">16<\/span><span class=\"os-divider\">.<\/span> [latex]5x+2y=0[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<p id=\"fs-id1702384\">For each of the following exercises, find the distance between the two points. Simplify your answers, and write the exact answer in simplest radical form for irrational answers.<\/p>\n<div id=\"fs-id1798648\" class=\"os-hasSolution\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1798650\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1798648-solution\">17<\/a><span class=\"os-divider\">. <\/span> [latex](-4, 1)[\/latex] and [latex](3, -4)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1762329\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id2947043\" data-type=\"problem\">\n<p><span class=\"os-number\">18<\/span><span class=\"os-divider\">. <\/span> [latex](2, -5)[\/latex] and [latex](7, 4)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1932412\" class=\"os-hasSolution\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id2266178\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1932412-solution\">19 <\/a> [latex](5, 0)[\/latex] and [latex](5, 6)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1538085\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1538086\" data-type=\"problem\">\n<p><span class=\"os-number\">20<\/span><span class=\"os-divider\">. <\/span> [latex](-4, 3)[\/latex] and [latex](10, 3)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2628482\" class=\"os-hasSolution\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id2628483\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id2628482-solution\">21<\/a><span class=\"os-divider\">. <\/span>Find the distance between the two points given using your calculator, and round your answer to the nearest hundredth.<\/p>\n<div class=\"os-problem-container\">\n<p>[latex](19, 12)[\/latex] and [latex](41, 71)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<p id=\"fs-id2753833\">For each of the following exercises, find the coordinates of the midpoint of the line segment that joins the two given points.<\/p>\n<div id=\"fs-id2389564\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id2389565\" data-type=\"problem\">\n<p><span class=\"os-number\">22. <\/span> [latex](-5, -6)[\/latex] and [latex](4, 2)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1386855\" class=\"os-hasSolution\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1386856\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1386855-solution\">23<\/a><span class=\"os-divider\">. <\/span> [latex](-1, 1)[\/latex] and [latex](7, -4)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2425333\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id2425334\" data-type=\"problem\">\n<p><span class=\"os-number\">24<\/span><span class=\"os-divider\">. <\/span> [latex](-5, -3)[\/latex] and [latex](-2, -8)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2736528\" class=\"os-hasSolution\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id2736529\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id2736528-solution\">25<\/a><span class=\"os-divider\">. <\/span> [latex](0, 7)[\/latex] and [latex](4, -9)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1960033\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1960034\" data-type=\"problem\">\n<p><span class=\"os-number\">26<\/span><span class=\"os-divider\">. <\/span> [latex](-43, 17)[\/latex] and [latex](23, -34)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<section id=\"fs-id2434980\" data-depth=\"2\">\n<h2 data-type=\"title\">Graphical<\/h2>\n<p id=\"fs-id1919734\">For each of the following exercises, identify the information requested.<\/p>\n<div id=\"fs-id1940524\" class=\"os-hasSolution\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1940525\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1940524-solution\">27<\/a><span class=\"os-divider\">. <\/span>What are the coordinates of the origin?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1965313\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1965314\" data-type=\"problem\">\n<p><span class=\"os-number\">28<\/span><span class=\"os-divider\">. <\/span>If a point is located on the <em data-effect=\"italics\">y<\/em>-axis, what is the <em data-effect=\"italics\">x<\/em>-coordinate?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1575040\" class=\"os-hasSolution\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1575041\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1575040-solution\">29<\/a><span class=\"os-divider\">. <\/span>If a point is located on the <em data-effect=\"italics\">x<\/em>-axis, what is the <em data-effect=\"italics\">y<\/em>-coordinate?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<p id=\"fs-id1475798\">For each of the following exercises, plot the three points on the given coordinate plane. State whether the three points you plotted appear to be collinear (on the same line).<\/p>\n<div id=\"fs-id1182006\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1182007\" data-type=\"problem\">\n<p><span class=\"os-number\">30<\/span><span class=\"os-divider\">.<\/span> [latex](4, 1), (-2, -3), (5, 0)[\/latex]<\/p>\n<div class=\"os-problem-container\">\n<p><span id=\"fs-id2387110\" data-type=\"media\" data-alt=\"This is an image of a blank x, y coordinate plane with the x and y axes ranging from negative 5 to 5.\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-1136\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/2.1.30-1-288x300.jpg\" alt=\"\" width=\"288\" height=\"300\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/2.1.30-1-288x300.jpg 288w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/2.1.30-1-65x68.jpg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/2.1.30-1-225x234.jpg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/2.1.30-1-350x365.jpg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/2.1.30-1.jpg 357w\" sizes=\"auto, (max-width: 288px) 100vw, 288px\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1341660\" class=\"os-hasSolution\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1341661\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1341660-solution\">31<\/a><span class=\"os-divider\">. <\/span> [latex](-1, 2)), (0, 4), (2, 1)[\/latex]<\/p>\n<div class=\"os-problem-container\">\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-430\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/2.1.30-288x300.jpg\" alt=\"\" width=\"288\" height=\"300\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/2.1.30-288x300.jpg 288w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/2.1.30-65x68.jpg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/2.1.30-225x234.jpg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/2.1.30-350x365.jpg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/2.1.30.jpg 357w\" sizes=\"auto, (max-width: 288px) 100vw, 288px\" \/><\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2440944\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id2440946\" data-type=\"problem\">\n<p><span class=\"os-number\">32<\/span><span class=\"os-divider\">. <\/span> [latex](-3, 0), (-3, 4), (-3, -3)[\/latex]<\/p>\n<div class=\"os-problem-container\">\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-430\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/2.1.30-288x300.jpg\" alt=\"\" width=\"288\" height=\"300\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/2.1.30-288x300.jpg 288w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/2.1.30-65x68.jpg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/2.1.30-225x234.jpg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/2.1.30-350x365.jpg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/2.1.30.jpg 357w\" sizes=\"auto, (max-width: 288px) 100vw, 288px\" \/><\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2453414\" class=\"os-hasSolution\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id2453415\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id2453414-solution\">33<\/a><span class=\"os-divider\">. <\/span>Name the coordinates of the points graphed.<\/p>\n<div class=\"os-problem-container\">\n<p><span id=\"fs-id1333042\" data-type=\"media\" data-alt=\"This is an image of an x, y coordinate plane where the x and y-axis range from negative 5 to 5. Three points are plotted: A, B, and C.\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-430\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/2.1.30-288x300.jpg\" alt=\"\" width=\"288\" height=\"300\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/2.1.30-288x300.jpg 288w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/2.1.30-65x68.jpg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/2.1.30-225x234.jpg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/2.1.30-350x365.jpg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/2.1.30.jpg 357w\" sizes=\"auto, (max-width: 288px) 100vw, 288px\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id3155305\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id3155306\" data-type=\"problem\">\n<p><span class=\"os-number\">34<\/span><span class=\"os-divider\">. <\/span>Name the quadrant in which the following points would be located. If the point is on an axis, name the axis.<\/p>\n<p>(a) [latex](-3, -4)[\/latex]<\/p>\n<p>(b) [latex](-5, 0)[\/latex]<\/p>\n<p>(c) [latex](1, -4)[\/latex]<\/p>\n<p>(d) [latex](-2, 7)[\/latex]<\/p>\n<p>(e) [latex](0, -3)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<p id=\"fs-id1569563\">For each of the following exercises, construct a table and graph the equation by plotting at least three points.<\/p>\n<div id=\"fs-id1569567\" class=\"os-hasSolution\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id3182657\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1569567-solution\">35<\/a><span class=\"os-divider\">.<\/span> [latex]y=\\frac{1}{3}x+2[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id3070032\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id3070034\" data-type=\"problem\">\n<p><span class=\"os-number\">36<\/span><span class=\"os-divider\">.<\/span> [latex]y=-3x+1[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1445099\" class=\"os-hasSolution\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1477477\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1445099-solution\">37<\/a><span class=\"os-divider\">.<\/span> [latex]2y=x+3[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<section id=\"fs-id1387762\" data-depth=\"2\">\n<h2 data-type=\"title\">Numeric<\/h2>\n<p id=\"fs-id2528941\">For each of the following exercises, find and plot the <em data-effect=\"italics\">x-<\/em> and <em data-effect=\"italics\">y<\/em>-intercepts, and graph the straight line based on those two points.<\/p>\n<div id=\"fs-id766182\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id766183\" data-type=\"problem\">\n<p><span class=\"os-number\">38<\/span><span class=\"os-divider\">.<\/span> [latex]4x-3y=12[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id3176745\" class=\"os-hasSolution\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id3176746\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id3176745-solution\">39<\/a><span class=\"os-divider\">.<\/span> [latex]x-2y=8[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1272882\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1272883\" data-type=\"problem\">\n<p><span class=\"os-number\">40<\/span><span class=\"os-divider\">.<\/span> [latex]y-5=5x[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1951937\" class=\"os-hasSolution\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1951938\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1951937-solution\">41<\/a><span class=\"os-divider\">.<\/span> [latex]3y=-2x+6[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1517685\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1929175\" data-type=\"problem\">\n<p><span class=\"os-number\">42<\/span><span class=\"os-divider\">.<\/span> [latex]y=\\frac{x-3}{2}[\/latex]<\/p>\n<div class=\"os-problem-container\">\n<p>For each of the following exercises, use the graph in the figure below.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<p><span id=\"fs-id1832449\" data-type=\"media\" data-alt=\"This is an image of an x, y coordinate plane with the x and y axes ranging from negative 5 to 5. The points (-3, 4) and (5, 2) are plotted. A line connects these two points.\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-431\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/2.1.43-288x300.jpg\" alt=\"\" width=\"288\" height=\"300\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/2.1.43-288x300.jpg 288w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/2.1.43-65x68.jpg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/2.1.43-225x234.jpg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/2.1.43-350x365.jpg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/2.1.43.jpg 357w\" sizes=\"auto, (max-width: 288px) 100vw, 288px\" \/><\/span><\/p>\n<div id=\"fs-id1267899\" class=\"os-hasSolution\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id2432278\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1267899-solution\">43<\/a><span class=\"os-divider\">. <\/span>Find the distance between the two endpoints using the distance formula. Round to three decimal places.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1895437\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1815372\" data-type=\"problem\">\n<p><span class=\"os-number\">44<\/span><span class=\"os-divider\">. <\/span>Find the coordinates of the midpoint of the line segment connecting the two points.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1815376\" class=\"os-hasSolution\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1418769\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1815376-solution\">45<\/a><span class=\"os-divider\">. <\/span>Find the distance that [latex](-3, 4)[\/latex] is from the origin.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1892569\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1892570\" data-type=\"problem\">\n<p><span class=\"os-number\">46<\/span><span class=\"os-divider\">. <\/span>Find the distance that [latex](5, 2)[\/latex] is from the origin. Round to three decimal places.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2437516\" class=\"os-hasSolution\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id2437517\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id2437516-solution\">47<\/a><span class=\"os-divider\">. <\/span>Which point is closer to the origin?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<section id=\"fs-id2437604\" data-depth=\"2\">\n<h2 data-type=\"title\">Technology<\/h2>\n<p id=\"fs-id1333818\">For the following exercises, use your graphing calculator to input the linear graphs in the Y= graph menu.<\/p>\n<p id=\"fs-id1333822\">After graphing it, use the 2<sup>nd<\/sup> CALC button and 1:value button, hit enter. At the lower part of the screen you will see \u201cx=\u201d and a blinking cursor. You may enter any number for <em data-effect=\"italics\">x<\/em> and it will display the <em data-effect=\"italics\">y<\/em> value for any <em data-effect=\"italics\">x<\/em> value you input. Use this and plug in <em data-effect=\"italics\">x<\/em> = 0, thus finding the <em data-effect=\"italics\">y<\/em>-intercept, for each of the following graphs.<\/p>\n<div id=\"fs-id1686604\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1686605\" data-type=\"problem\">\n<p><span class=\"os-number\">48<\/span><span class=\"os-divider\">.<\/span> [latex]Y_1=-2x+5[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1181943\" class=\"os-hasSolution\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1181944\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1181943-solution\">49<\/a> [latex]Y_1=\\frac{3x-8}{4}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2800083\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1421792\" data-type=\"problem\">\n<p><span class=\"os-number\">50.<\/span> [latex]Y_1=\\frac{x+5}{2}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<p id=\"fs-id2785067\">For the following exercises, use your graphing calculator to input the linear graphs in the Y= graph menu.<\/p>\n<p id=\"fs-id2384791\">After graphing it, use the 2<sup>nd<\/sup> <strong>CALC<\/strong> button and 2:zero button, hit <strong>ENTER<\/strong>. At the lower part of the screen you will see \u201cleft bound?\u201d and a blinking cursor on the graph of the line. Move this cursor to the left of the <em data-effect=\"italics\">x<\/em>-intercept, hit <strong>ENTER<\/strong>. Now it says \u201cright bound?\u201d Move the cursor to the right of the <em data-effect=\"italics\">x<\/em>-intercept, hit <strong>ENTER<\/strong>. Now it says \u201cguess?\u201d Move your cursor to the left somewhere in between the left and right bound near the <em data-effect=\"italics\">x<\/em>-intercept. Hit <strong>ENTER<\/strong>. At the bottom of your screen it will display the coordinates of the <em data-effect=\"italics\">x-<\/em>intercept or the \u201czero\u201d to the <em data-effect=\"italics\">y<\/em>-value. Use this to find the <em data-effect=\"italics\">x<\/em>-intercept.<\/p>\n<p id=\"fs-id2016100\">Note: With linear\/straight line functions the zero is not really a \u201cguess,\u201d but it is necessary to enter a \u201cguess\u201d so it will search and find the exact <em data-effect=\"italics\">x<\/em>-intercept between your right and left boundaries. With other types of functions (more than one <em data-effect=\"italics\">x<\/em>-intercept), they may be irrational numbers so \u201cguess\u201d is more appropriate to give it the correct limits to find a very close approximation between the left and right boundaries.<\/p>\n<div id=\"fs-id1425403\" class=\"os-hasSolution\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1425404\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1425403-solution\">51<\/a><span class=\"os-divider\">.<\/span> [latex]Y_1=-8x+6[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1845247\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1845248\" data-type=\"problem\">\n<p><span class=\"os-number\">52<\/span><span class=\"os-divider\">.<\/span> [latex]Y_1=4x-7[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2387375\" class=\"os-hasSolution\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id2387376\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id2387375-solution\">53<\/a><span class=\"os-divider\">.<\/span> [latex]Y_1=\\frac{3x+5}{4}[\/latex]<\/p>\n<div class=\"os-problem-container\">\n<p>Round your answer to the nearest thousandth.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<section id=\"fs-id1932603\" data-depth=\"2\">\n<h2 data-type=\"title\">Extensions<\/h2>\n<div id=\"fs-id1932608\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1333979\" data-type=\"problem\">\n<p><span class=\"os-number\">54<\/span><span class=\"os-divider\">. <\/span>Someone drove 10 mi directly east from their home, made a left turn at an intersection, and then traveled 5 mi north to their place of work. If a road was made directly from the home to the place of work, what would its distance be to the nearest tenth of a mile?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2513497\" class=\"os-hasSolution\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id2513498\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id2513497-solution\">55<\/a><span class=\"os-divider\">. <\/span>If the road was made in the previous exercise, how much shorter would the person\u2019s one-way trip be every day?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2508338\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id2508339\" data-type=\"problem\">\n<p><span class=\"os-number\">56<\/span><span class=\"os-divider\">. <\/span>Given these four points: [latex]A(1, 3), B(-3, 5), C(4, 7)[\/latex] and [latex]D(5, -4)[\/latex] find the coordinates of the midpoint of line segments [latex]\\overline{AB}[\/latex] and [latex]\\overline{CD}.[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1343510\" class=\"os-hasSolution\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1343511\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1343510-solution\">57<\/a><span class=\"os-divider\">. <\/span>After finding the two midpoints in the previous exercise, find the distance between the two midpoints to the nearest thousandth.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1312332\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1312333\" data-type=\"problem\">\n<p><span class=\"os-number\">58<\/span><span class=\"os-divider\">. <\/span>Given the graph of the rectangle shown and the coordinates of its vertices, prove that the diagonals of the rectangle are of equal length.<\/p>\n<div class=\"os-problem-container\">\n<p><span id=\"fs-id3207847\" data-type=\"media\" data-alt=\"This is an image of an x, y coordinate plane with the x and y axes ranging from negative 12 to 12. The points (-6, 5); (10, 5); (-6, -1) and (10, -1) are plotted and labeled. These points are connected to form a rectangle. Dotted lines extend from each corner point to their opposite point.\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-432\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/2.1.58-300x175.jpg\" alt=\"\" width=\"300\" height=\"175\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/2.1.58-300x175.jpg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/2.1.58-65x38.jpg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/2.1.58-225x131.jpg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/2.1.58-350x204.jpg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/2.1.58.jpg 505w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1228174\" class=\"os-hasSolution\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1228176\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1228174-solution\">59<\/a><span class=\"os-divider\">. <\/span>In the previous exercise, find the coordinates of the midpoint for each diagonal.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<section id=\"fs-id2454474\" data-depth=\"2\">\n<h2 data-type=\"title\">Real-World Applications<\/h2>\n<div id=\"fs-id2722618\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id2722619\" data-type=\"problem\">\n<p><span class=\"os-number\">60<\/span><span class=\"os-divider\">. <\/span>The coordinates on a map for San Francisco are [latex](53, 17)[\/latex] and those for Sacramento are [latex](128, 78)[\/latex] Note that coordinates represent miles. Find the distance between the cities to the nearest mile.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1511688\" class=\"os-hasSolution\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1511690\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1511688-solution\">61<\/a><span class=\"os-divider\">. <\/span>If San Jose\u2019s coordinates are [latex](76, -12)[\/latex] where the coordinates represent miles, find the distance between San Jose and San Francisco to the nearest mile.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2925632\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id2925633\" data-type=\"problem\">\n<p><span class=\"os-number\">62<\/span><span class=\"os-divider\">. <\/span>A small craft in Lake Ontario sends out a distress signal. The coordinates of the boat in trouble were [latex](49, 64)[\/latex] One rescue boat is at the coordinates [latex](60, 82)[\/latex] and a second Coast Guard craft is at coordinates [latex](58, 47).[\/latex] Assuming both rescue craft travel at the same rate, which one would get to the distressed boat the fastest?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1551836\" class=\"os-hasSolution\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1551837\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1551836-solution\">63<\/a><span class=\"os-divider\">. <\/span>A person on the top of a building wants to have a guy wire extend to a point on the ground 20 ft from the building. To the nearest foot, how long will the wire have to be if the building is 50 ft tall?<\/p>\n<div class=\"os-problem-container\">\n<p><span id=\"fs-id1477367\" data-type=\"media\" data-alt=\"A right triangle with its bottom left point sitting on the point (0,0). The upper right hand corner is labeled (20,50). The base has a length of 20 units and the triangle has a height of 50 units.\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-433\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/2.1.63.jpg\" alt=\"\" width=\"242\" height=\"203\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/2.1.63.jpg 242w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/2.1.63-65x55.jpg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/2.1.63-225x189.jpg 225w\" sizes=\"auto, (max-width: 242px) 100vw, 242px\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1717974\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1717976\" data-type=\"problem\">\n<p><span class=\"os-number\">64<\/span><span class=\"os-divider\">. <\/span>If we rent a truck and pay a $75\/day fee plus $.20 for every mile we travel, write a linear equation that would express the total cost per day [latex]y,[\/latex] using [latex]x[\/latex] to represent the number of miles we travel. Graph this function on your graphing calculator and find the total cost for one day if we travel 70 mi.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<\/section>\n<\/div>\n<p>&nbsp;<\/p>\n","protected":false},"author":158,"menu_order":1,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-107","chapter","type-chapter","status-publish","hentry"],"part":51,"_links":{"self":[{"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/chapters\/107","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/wp\/v2\/users\/158"}],"version-history":[{"count":84,"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/chapters\/107\/revisions"}],"predecessor-version":[{"id":1147,"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/chapters\/107\/revisions\/1147"}],"part":[{"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/parts\/51"}],"metadata":[{"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/chapters\/107\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/wp\/v2\/media?parent=107"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=107"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/wp\/v2\/contributor?post=107"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/wp\/v2\/license?post=107"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}