{"id":1747,"date":"2023-10-12T00:32:05","date_gmt":"2023-10-12T00:32:05","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/tulsacc-precalculus\/chapter\/introduction-points-and-lines-in-the-plane\/"},"modified":"2023-11-01T00:56:11","modified_gmt":"2023-11-01T00:56:11","slug":"introduction-points-and-lines-in-the-plane","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/tulsacc-precalculus\/chapter\/introduction-points-and-lines-in-the-plane\/","title":{"raw":"Points and Lines in the Plane","rendered":"Points and Lines in the Plane"},"content":{"raw":"<div class=\"bcc-box bcc-highlight\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Define the components of the Cartesian coordinate system.<\/li>\r\n \t<li>Plot points on the Cartesian coordinate plane.<\/li>\r\n \t<li>Determine the slope of a line given two points.<\/li>\r\n \t<li>Use the distance formula to find the distance between two points in the plane.<\/li>\r\n \t<li>Use the midpoint formula to find the midpoint between two points.<\/li>\r\n \t<li>Plot linear equations in two variables on the coordinate plane.<\/li>\r\n \t<li>Use intercepts to plot lines.<\/li>\r\n \t<li>Use a graphing utility to graph a linear equation on a coordinate plane.<\/li>\r\n<\/ul>\r\n<\/div>\r\nTracie 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 the figure below. 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.\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/09\/25200240\/CNX_CAT_Figure_02_01_001.jpg\" alt=\"Road map of a city with street names on an x, y coordinate grid. Various points are marked in red on the grid lines indicating different locations on the map.\" width=\"731\" height=\"480\" \/>\r\n<h2>Plotting Points on the Coordinate Plane<\/h2>\r\nAn old story describes how seventeenth-century philosopher\/mathematician Ren\u00e9 Descartes invented the system that has become the foundation of algebra while sick in bed. 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.\r\n\r\nWhile 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>Cartesian coordinate system<\/strong>, a grid system having perpendicular axes. Descartes named the horizontal axis the <strong><em>x-<\/em>axis<\/strong> and the vertical axis the <strong><em>y-<\/em>axis<\/strong>.\r\n\r\nThe Cartesian coordinate system, also called the rectangular coordinate system, is based on a two-dimensional plane consisting of the <em>x<\/em>-axis and the <em>y<\/em>-axis. Perpendicular to each other, the axes divide the plane into four sections. Each section is called a <strong>quadrant<\/strong>; the quadrants are numbered counterclockwise as shown in the figure below.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/12042358\/CNX_CAT_Figure_02_01_002.jpg\" alt=\"This is an image of an x, y plane with the axes labeled. The upper right section is labeled: Quadrant I. The upper left section is labeled: Quadrant II. The lower left section is labeled: Quadrant III. The lower right section is labeled: Quadrant IV.\" width=\"487\" height=\"442\" \/> <b>The Cartesian coordinate system with all four quadrants labeled.<\/b>[\/caption]\r\n\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n<iframe id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=92752&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"650\"><\/iframe>\r\n\r\n<\/div>\r\nThe center of the plane is the point at which the two axes cross. It is known as the <strong>origin\u00a0<\/strong>or point [latex]\\left(0,0\\right)[\/latex]. From the origin, each axis is further divided into equal units: increasing, positive numbers to the right on the <em>x-<\/em>axis and up the <em>y-<\/em>axis; decreasing, negative numbers to the left on the <em>x-<\/em>axis and down the <em>y-<\/em>axis. The axes extend to positive and negative infinity as shown by the arrowheads in the figure below.\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/12042401\/CNX_CAT_Figure_02_01_003.jpg\" alt=\"This is an image of an x, y coordinate plane. The x and y axis range from negative 5 to 5.\" width=\"487\" height=\"442\" \/>\r\n\r\nEach point in the plane is identified by its <strong><em>x-<\/em>coordinate<\/strong>,\u00a0or horizontal displacement from the origin, and its <strong><em>y-<\/em>coordinate<\/strong>, or vertical displacement from the origin. Together we write them as an <strong>ordered pair<\/strong> indicating the combined distance from the origin in the form [latex]\\left(x,y\\right)[\/latex]. An ordered pair is also known as a coordinate pair because it consists of <em>x\u00a0<\/em>and <em>y<\/em>-coordinates. For example, we can represent the point [latex]\\left(3,-1\\right)[\/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.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/12042403\/CNX_CAT_Figure_02_01_004.jpg\" alt=\"This is an image of an x, y coordinate plane. The x and y axis range from negative 5 to 5. The point (3, -1) is labeled. An arrow extends rightward from the origin 3 units and another arrow extends downward one unit from the end of that arrow to the point.\" width=\"487\" height=\"442\" \/> <b>An illustration of how to plot the point (3,-1).<\/b>[\/caption]\r\n\r\nWhen dividing the axes into equally spaced increments, note that the <em>x-<\/em>axis may be considered separately from the <em>y-<\/em>axis. In other words, while the <em>x-<\/em>axis may be divided and labeled according to consecutive integers, the <em>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. Consider the rectangular coordinate system primarily as a method for showing the relationship between two quantities.\r\n<div class=\"textbox\">\r\n<h3>A General Note: Cartesian Coordinate System<\/h3>\r\nA two-dimensional plane where the\r\n<ul>\r\n \t<li><em>x<\/em>-axis is the horizontal axis<\/li>\r\n \t<li><em>y<\/em>-axis is the vertical axis<\/li>\r\n<\/ul>\r\nA point in the plane is defined as an ordered pair, [latex]\\left(x,y\\right)[\/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 class=\"textbox exercises\">\r\n<h3>Example: Plotting Points in a Rectangular Coordinate System<\/h3>\r\nPlot the points [latex]\\left(-2,4\\right)[\/latex], [latex]\\left(3,3\\right)[\/latex], and [latex]\\left(0,-3\\right)[\/latex] in the coordinate plane.\r\n[reveal-answer q=\"380739\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"380739\"]\r\n\r\nTo plot the point [latex]\\left(-2,4\\right)[\/latex], begin at the origin. The <em>x<\/em>-coordinate is \u20132, 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]\\left(3,3\\right)[\/latex], begin again at the origin. The <em>x<\/em>-coordinate is 3, so move 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]\\left(0,-3\\right)[\/latex], begin again at the origin. The <em>x<\/em>-coordinate is 0. This tells us not to move in either direction along the <em>x<\/em>-axis. The <em>y<\/em>-coordinate is \u20133, so move three units down in the negative <em>y<\/em> direction.\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/12042406\/CNX_CAT_Figure_02_01_005.jpg\" alt=\"This is an image of a graph on an x, y coordinate plane. The x and y axes range from negative 5 to 5. The points (-2, 4); (3, 3); and (0, -3) are labeled. Arrows extend from the origin to the points.\" width=\"487\" height=\"442\" \/>\r\n<h4>Analysis of the Solution<\/h4>\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 <em>y<\/em>-axis. If the <em>y<\/em>-coordinate is zero, the point is on the <em>x<\/em>-axis.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n<iframe id=\"mom2\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=92753&amp;theme=oea&amp;iframe_resize_id=mom2\" width=\"100%\" height=\"650\"><\/iframe>\r\n\r\n<\/div>\r\n<h2>The Slope of a Line<\/h2>\r\nThe <strong>slope<\/strong> of a line refers to the ratio of the vertical change in <em>y<\/em> over the horizontal change in <em>x<\/em> between any two points on a line. It indicates the direction in which a line slants as well as its steepness. Slope is sometimes described as rise over run.\r\n<div style=\"text-align: center;\">[latex]m=\\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}[\/latex]<\/div>\r\nIf the slope is positive, the line slants upward to the right. If the slope is negative, the line slants downward to the right. As the slope increases, the line becomes steeper. Some examples are shown below. The lines indicate the following slopes: [latex]m=-3[\/latex], [latex]m=2[\/latex], and [latex]m=\\frac{1}{3}[\/latex].\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/11185922\/CNX_CAT_Figure_02_02_002.jpg\" alt=\"Coordinate plane with the x and y axes ranging from negative 10 to 10. Three linear functions are plotted: y = negative 3 times x minus 2; y = 2 times x plus 1; and y = x over 3 plus 2.\" width=\"487\" height=\"442\" \/>\r\n<div class=\"textbox\">\r\n<h3>A General Note: The Slope of a Line<\/h3>\r\nThe slope of a line, <em>m<\/em>, represents the change in <em>y<\/em> over the change in <em>x.<\/em> Given two points, [latex]\\left({x}_{1},{y}_{1}\\right)[\/latex] and [latex]\\left({x}_{2},{y}_{2}\\right)[\/latex], the following formula determines the slope of a line containing these points:\r\n<div style=\"text-align: center;\">[latex]m=\\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}[\/latex]<\/div>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Finding the Slope of a Line Given Two Points<\/h3>\r\nFind the slope of a line that passes through the points [latex]\\left(2,-1\\right)[\/latex] and [latex]\\left(-5,3\\right)[\/latex].\r\n[reveal-answer q=\"688301\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"688301\"]\r\n\r\nWe substitute the <em>y-<\/em>values and the <em>x-<\/em>values into the formula.\r\n<div style=\"text-align: center;\">[latex]\\begin{array}{l}m\\hfill&amp;=\\frac{3-\\left(-1\\right)}{-5 - 2}\\hfill \\\\ \\hfill&amp;=\\frac{4}{-7}\\hfill \\\\ \\hfill&amp;=-\\frac{4}{7}\\hfill \\end{array}[\/latex]<\/div>\r\nThe slope is [latex]-\\frac{4}{7}[\/latex].\r\n<h4>Analysis of the Solution<\/h4>\r\nIt does not matter which point is called [latex]\\left({x}_{1},{y}_{1}\\right)[\/latex] or [latex]\\left({x}_{2},{y}_{2}\\right)[\/latex]. As long as we are consistent with the order of the <em>y<\/em> terms and the order of the <em>x<\/em> terms in the numerator and denominator, the calculation will yield the same result.\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nFind the slope of the line that passes through the points [latex]\\left(-2,6\\right)[\/latex] and [latex]\\left(1,4\\right)[\/latex].\r\n\r\n[reveal-answer q=\"196055\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"196055\"]\r\n\r\nslope[latex]=m=\\dfrac{-2}{3}=-\\dfrac{2}{3}[\/latex][\/hidden-answer]\r\n<iframe id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=1719&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"350\" data-mce-fragment=\"1\"><\/iframe>\r\n\r\n<\/div>\r\n<h2>Distance in the Plane<\/h2>\r\nDerived from the <strong>Pythagorean Theorem<\/strong>, the <strong>distance formula<\/strong> is used to find the distance between two points in the plane. The Pythagorean Theorem, [latex]{a}^{2}+{b}^{2}={c}^{2}[\/latex], is based on a right triangle where <em>a <\/em>and <em>b<\/em> are the lengths of the legs adjacent to the right angle, and <em>c<\/em> is the length of the hypotenuse.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/12042428\/CNX_CAT_Figure_02_01_015.jpg\" alt=\"This is an image of a triangle on an x, y coordinate plane. The x and y axes range from 0 to 7. The points (x sub 1, y sub 1); (x sub 2, y sub 1); and (x sub 2, y sub 2) are labeled and connected to form a triangle. Along the base of the triangle, the following equation is displayed: the absolute value of x sub 2 minus x sub 1 equals a. The hypotenuse of the triangle is labeled: d = c. The remaining side is labeled: the absolute value of y sub 2 minus y sub 1 equals b.\" width=\"487\" height=\"331\" \/>\r\n\r\nThe relationship of sides [latex]|{x}_{2}-{x}_{1}|[\/latex] and [latex]|{y}_{2}-{y}_{1}|[\/latex] to side <em>d<\/em> is the same as that of sides <em>a <\/em>and <em>b <\/em>to side <em>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>c<\/em>, take the square root of both sides of the Pythagorean Theorem.\r\n<div style=\"text-align: center;\">[latex]{c}^{2}={a}^{2}+{b}^{2}\\rightarrow c=\\sqrt{{a}^{2}+{b}^{2}}[\/latex]<\/div>\r\nIt follows that the distance formula is given as\r\n<div style=\"text-align: center;\">[latex]{d}^{2}={\\left({x}_{2}-{x}_{1}\\right)}^{2}+{\\left({y}_{2}-{y}_{1}\\right)}^{2}\\to d=\\sqrt{{\\left({x}_{2}-{x}_{1}\\right)}^{2}+{\\left({y}_{2}-{y}_{1}\\right)}^{2}}[\/latex]<\/div>\r\nWe do not have to use the absolute value symbols in this definition because any number squared is positive.\r\n<div class=\"textbox\">\r\n<h3>A General Note: The Distance Formula<\/h3>\r\nGiven endpoints [latex]\\left({x}_{1},{y}_{1}\\right)[\/latex] and [latex]\\left({x}_{2},{y}_{2}\\right)[\/latex], the distance between two points is given by\r\n<div style=\"text-align: center;\">[latex]d=\\sqrt{{\\left({x}_{2}-{x}_{1}\\right)}^{2}+{\\left({y}_{2}-{y}_{1}\\right)}^{2}}[\/latex]<\/div>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Finding the Distance between Two Points<\/h3>\r\nFind the distance between the points [latex]\\left(-3,-1\\right)[\/latex] and [latex]\\left(2,3\\right)[\/latex].\r\n\r\n[reveal-answer q=\"737169\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"737169\"]\r\n\r\nLet us first look at the graph of the two points. Connect the points to form a right triangle.\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/12042430\/CNX_CAT_Figure_02_01_016.jpg\" alt=\"This is an image of a triangle on an x, y coordinate plane. The x-axis ranges from negative 4 to 4. The y-axis ranges from negative 2 to 4. The points (-3, -1); (2, -1); and (2, 3) are plotted and labeled on the graph. The points are connected to form a triangle\" width=\"487\" height=\"289\" \/>\r\n\r\nThen, calculate the length of <em>d <\/em>using the distance formula.\r\n<div style=\"text-align: center;\">[latex]\\begin{array}{l}d=\\sqrt{{\\left({x}_{2}-{x}_{1}\\right)}^{2}+{\\left({y}_{2}-{y}_{1}\\right)}^{2}}\\hfill \\\\ d=\\sqrt{{\\left(2-\\left(-3\\right)\\right)}^{2}+{\\left(3-\\left(-1\\right)\\right)}^{2}}\\hfill \\\\ =\\sqrt{{\\left(5\\right)}^{2}+{\\left(4\\right)}^{2}}\\hfill \\\\ =\\sqrt{25+16}\\hfill \\\\ =\\sqrt{41}\\hfill \\end{array}[\/latex]<\/div>\r\n<div>[\/hidden-answer]<\/div>\r\n<\/div>\r\n<div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n<div>\r\n\r\nFind the distance between two points: [latex]\\left(1,4\\right)[\/latex] and [latex]\\left(11,9\\right)[\/latex].\r\n[reveal-answer q=\"934526\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"934526\"]\r\n\r\n[latex]\\sqrt{125}=5\\sqrt{5}[\/latex][\/hidden-answer]\r\n\r\n<iframe id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=19140&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"350\"><\/iframe>\r\n\r\n<\/div>\r\n<div>\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Finding the Distance between Two Locations<\/h3>\r\nLet\u2019s return to the situation introduced at the beginning of this section.\r\n\r\nTracie 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. Find the total distance that Tracie traveled. Compare this with the distance between her starting and final positions.\r\n\r\n[reveal-answer q=\"411250\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"411250\"]\r\n\r\nThe first thing we should do is identify ordered pairs to describe each position. If we set the starting position at 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 [latex]\\left(1,1\\right)[\/latex]. The next stop is 5 blocks to the east so it is at [latex]\\left(5,1\\right)[\/latex]. After that, she traveled 3 blocks east and 2 blocks north to [latex]\\left(8,3\\right)[\/latex]. Lastly, she traveled 4 blocks north to [latex]\\left(8,7\\right)[\/latex]. We can label these points on the grid.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/12042433\/CNX_CAT_Figure_02_01_017.jpg\" alt=\"This is an image of a road map of a city. The point (1, 1) is on North Avenue and Bertau Avenue. The point (5, 1) is on North Avenue and Wolf Road. The point (8, 3) is on Mannheim Road and McLean Street. The point (8, 7) is on Mannheim Road and Schiller Avenue.\" width=\"731\" height=\"480\" \/>\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 [latex]\\left(1,1\\right)[\/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 [latex]\\left(5,1\\right)[\/latex]. So from [latex]\\left(1,1\\right)[\/latex] to [latex]\\left(5,1\\right)[\/latex], Tracie drove east 4,000 feet.<\/li>\r\n \t<li>Her third stop is at [latex]\\left(8,3\\right)[\/latex]. There are a number of routes from [latex]\\left(5,1\\right)[\/latex] to [latex]\\left(8,3\\right)[\/latex]. Whatever route Tracie decided to use, the distance is the same, as there are no angular streets between the two points. Let\u2019s 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\u2019s final stop is at [latex]\\left(8,7\\right)[\/latex]. This is a straight drive north from [latex]\\left(8,3\\right)[\/latex] for a total of 4,000 feet.<\/li>\r\n<\/ul>\r\nNext, we will add the distances listed in the table.\r\n<table summary=\"A table with 6 rows and 2 columns. The entries in the first row are: From\/To and Number of Feet Driven. The entries in the second row are: (0, 0) to (1, 1) and 2,000. The entries in the third row are: (1, 1) to (5, 1) and 4,000. The entries in the fourth row are: (5, 1) to (8, 3) and 5,000. The entries in the fourth row are: (8, 3) to (8, 7) and 4,000. The entries in the sixth row are: Total and 15,000.\">\r\n<thead>\r\n<tr>\r\n<th>From\/To<\/th>\r\n<th>Number of Feet Driven<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>[latex]\\left(0,0\\right)[\/latex] to [latex]\\left(1,1\\right)[\/latex]<\/td>\r\n<td>2,000<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]\\left(1,1\\right)[\/latex] to [latex]\\left(5,1\\right)[\/latex]<\/td>\r\n<td>4,000<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]\\left(5,1\\right)[\/latex] to [latex]\\left(8,3\\right)[\/latex]<\/td>\r\n<td>5,000<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]\\left(8,3\\right)[\/latex] to [latex]\\left(8,7\\right)[\/latex]<\/td>\r\n<td>4,000<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Total<\/td>\r\n<td>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 [latex]\\left(0,0\\right)[\/latex] and [latex]\\left(8,7\\right)[\/latex].\r\n<div style=\"text-align: center;\">[latex]\\begin{array}{l}d=\\sqrt{{\\left(8 - 0\\right)}^{2}+{\\left(7 - 0\\right)}^{2}}\\hfill \\\\ =\\sqrt{64+49}\\hfill \\\\ =\\sqrt{113}\\hfill \\\\ =10.63\\text{ units}\\hfill \\end{array}[\/latex]<\/div>\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 [latex]\\left(8,7\\right)[\/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[\/hidden-answer]\r\n\r\n<\/div>\r\n<h3>Using the Midpoint Formula<\/h3>\r\nWhen 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>midpoint formula<\/strong>. Given the endpoints of a line segment, [latex]\\left({x}_{1},{y}_{1}\\right)[\/latex] and [latex]\\left({x}_{2},{y}_{2}\\right)[\/latex], the midpoint formula states how to find the coordinates of the midpoint [latex]M[\/latex].\r\n<div style=\"text-align: center;\">[latex]M=\\left(\\frac{{x}_{1}+{x}_{2}}{2},\\frac{{y}_{1}+{y}_{2}}{2}\\right)[\/latex]<\/div>\r\nA graphical view of a midpoint is shown below. Notice that the line segments on either side of the midpoint are congruent.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/12042436\/CNX_CAT_Figure_02_01_018.jpg\" alt=\"This is a line graph on an x, y coordinate plane with the x and y axes ranging from 0 to 6. The points (x sub 1, y sub 1), (x sub 2, y sub 2), and (x sub 1 plus x sub 2 all over 2, y sub 1 plus y sub 2 all over 2) are plotted. A straight line runs through these three points. Pairs of short parallel lines bisect the two sections of the line to note that they are equivalent.\" width=\"487\" height=\"290\" \/>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Finding the Midpoint of a Line Segment<\/h3>\r\nFind the midpoint of the line segment with the endpoints [latex]\\left(7,-2\\right)[\/latex] and [latex]\\left(9,5\\right)[\/latex].\r\n\r\n[reveal-answer q=\"788934\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"788934\"]\r\n\r\nUse the formula to find the midpoint of the line segment.\r\n<div style=\"text-align: center;\">[latex]\\begin{array}{l}\\left(\\frac{{x}_{1}+{x}_{2}}{2},\\frac{{y}_{1}+{y}_{2}}{2}\\right)\\hfill&amp;=\\left(\\frac{7+9}{2},\\frac{-2+5}{2}\\right)\\hfill \\\\ \\hfill&amp;=\\left(8,\\frac{3}{2}\\right)\\hfill \\end{array}[\/latex]<\/div>\r\n<div>[\/hidden-answer]<\/div>\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nFind the midpoint of the line segment with endpoints [latex]\\left(-2,-1\\right)[\/latex] and [latex]\\left(-8,6\\right)[\/latex].\r\n\r\n[reveal-answer q=\"964077\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"964077\"]\r\n\r\n[latex]\\left(-5,\\frac{5}{2}\\right)[\/latex][\/hidden-answer]\r\n<iframe id=\"mom2\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=2308&amp;theme=oea&amp;iframe_resize_id=mom2\" width=\"100%\" height=\"250\"><\/iframe>\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Finding the Center of a Circle<\/h3>\r\nThe diameter of a circle has endpoints [latex]\\left(-1,-4\\right)[\/latex] and [latex]\\left(5,-4\\right)[\/latex]. Find the center of the circle.\r\n\r\n[reveal-answer q=\"418175\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"418175\"]\r\n\r\nThe center of a circle is the center or midpoint of its diameter. Thus, the midpoint formula will yield the center point.\r\n<div style=\"text-align: center;\">[latex]\\begin{array}{l}\\left(\\frac{{x}_{1}+{x}_{2}}{2},\\frac{{y}_{1}+{y}_{2}}{2}\\right)\\\\ \\left(\\frac{-1+5}{2},\\frac{-4 - 4}{2}\\right)=\\left(\\frac{4}{2},-\\frac{8}{2}\\right)=\\left(2,-4\\right)\\end{array}[\/latex]<\/div>\r\n<div>[\/hidden-answer]<\/div>\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n<iframe id=\"mom3\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=110938&amp;theme=oea&amp;iframe_resize_id=mom3\" width=\"100%\" height=\"250\"><\/iframe>\r\n\r\n<\/div>\r\n<h2>Graphing Linear Equations<\/h2>\r\nWe can plot a set of points to represent an equation. When such an equation contains both an <em>x <\/em>variable and a <em>y <\/em>variable, it is called an <strong>equation in two variables<\/strong>. Its graph is called a <strong>graph in two variables<\/strong>. Any graph on a two-dimensional plane is a graph in two variables.\r\n\r\nSuppose we want to graph the equation [latex]y=2x - 1[\/latex]. We can begin by substituting a value for <em>x<\/em> into the equation and determining the resulting value of <em>y<\/em>. Each pair of <em>x\u00a0<\/em>and <em>y-<\/em>values is an ordered pair that can be plotted. The table below\u00a0lists values of <em>x<\/em> from \u20133 to 3 and the resulting values for <em>y<\/em>.\r\n<table summary=\"This is a table with 8 rows and 3 columns. The first row has columns labeled: x, y = 2x-1, (x, y). The entries in the second row are: negative 3; y = 2 times negative 3 minus 1 = negative 7; (-3, -7). The entries in the third row are: negative 2; y = 2 times negative 2 minus 1 = negative 5; (-2, -5). The entries in the fourth row are: negative1; y = 2 times negative 1 minus 1 = negative 3; (-1, -3). The entries in the fifth row are: 0; y = 2 times 0 minus 1 = negative 1; (0, -1). The entries in the sixth row are: 1; y = 2 times 1 minus 1 = 1; (1, 1). The entries in the seventh row are: 2; y = 2 times 2 minus 1 = 3; (2, 3). The entries in the eight row are: 3, y = 2 times 3 minus 1 = 5; (3,5)\">\r\n<tbody>\r\n<tr>\r\n<td>[latex]x[\/latex]<\/td>\r\n<td>[latex]y=2x - 1[\/latex]<\/td>\r\n<td>[latex]\\left(x,y\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]-3[\/latex]<\/td>\r\n<td>[latex]y=2\\left(-3\\right)-1=-7[\/latex]<\/td>\r\n<td>[latex]\\left(-3,-7\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]-2[\/latex]<\/td>\r\n<td>[latex]y=2\\left(-2\\right)-1=-5[\/latex]<\/td>\r\n<td>[latex]\\left(-2,-5\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]-1[\/latex]<\/td>\r\n<td>[latex]y=2\\left(-1\\right)-1=-3[\/latex]<\/td>\r\n<td>[latex]\\left(-1,-3\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]0[\/latex]<\/td>\r\n<td>[latex]y=2\\left(0\\right)-1=-1[\/latex]<\/td>\r\n<td>[latex]\\left(0,-1\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]1[\/latex]<\/td>\r\n<td>[latex]y=2\\left(1\\right)-1=1[\/latex]<\/td>\r\n<td>[latex]\\left(1,1\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]2[\/latex]<\/td>\r\n<td>[latex]y=2\\left(2\\right)-1=3[\/latex]<\/td>\r\n<td>[latex]\\left(2,3\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]3[\/latex]<\/td>\r\n<td>[latex]y=2\\left(3\\right)-1=5[\/latex]<\/td>\r\n<td>[latex]\\left(3,5\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nWe can plot these points from the table. The points for this particular equation form a line, so we can connect them.<strong>\u00a0<\/strong>This is not true for all equations.\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/12042408\/CNX_CAT_Figure_02_01_006.jpg\" alt=\"This is a graph of a line on an x, y coordinate plane. The x- and y-axis range from negative 8 to 8. A line passes through the points (-3, -7); (-2, -5); (-1, -3); (0, -1); (1, 1); (2, 3); and (3, 5).\" width=\"731\" height=\"669\" \/>\r\n\r\nNote that the <em>x-<\/em>values chosen are arbitrary regardless of the type of equation we are graphing. Of course, some situations may require particular values of <em>x<\/em> to be plotted in order to see a particular result. Otherwise, it is logical to choose values that can be calculated easily, and it is always a good idea to choose values that are both negative and positive. There is no rule dictating how many points to plot, although we need at least two to graph a line. Keep in mind, however, that the more points we plot, the more accurately we can sketch the graph.\r\n<div class=\"textbox\">\r\n<h3>How To: Given an equation, graph by plotting points<\/h3>\r\n<ol>\r\n \t<li>Make a table with one column labeled <em>x<\/em>, a second column labeled with the equation, and a third column listing the resulting ordered pairs.<\/li>\r\n \t<li>Enter <em>x-<\/em>values down the first column using positive and negative values. Selecting the <em>x-<\/em>values in numerical order will make graphing easier.<\/li>\r\n \t<li>Select <em>x-<\/em>values that will yield <em>y-<\/em>values with little effort, preferably ones that can be calculated mentally.<\/li>\r\n \t<li>Plot the ordered pairs.<\/li>\r\n \t<li>Connect the points if they form a line.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Graphing an Equation in Two Variables by Plotting Points<\/h3>\r\nGraph the equation [latex]y=-x+2[\/latex] by plotting points.\r\n[reveal-answer q=\"792137\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"792137\"]\r\n\r\nFirst, we construct a table similar to the one below. Choose <em>x<\/em> values and calculate <em>y.<\/em>\r\n<table summary=\"The table shows 8 rows and 3 columns. The entries in the first row are: x; y = negative x plus 2; and (x, y). The entries in the second row are: negative 5; y = the opposite of negative 5 plus 2 = 7; (-5, 7). The entries in the third row are: negative 3; y = the opposite of negative 3 plus 2 = 5; (-3, 5). The entries in the fourth row are: -1; y = the opposite of negative 1 plus 2 = 3; (-1, 3). The entries in the fifth row are: 0; y = opposite of zero plus 2 = 2; (0, 2). The entries in the sixth row are: 1; y = the opposite of 1 plus 2 = 1; (1, 1). The entries in the seventh row are: 3; y = the opposite of 3 plus 2 = negative 1; (3, -1). The entries in the eighth row are: 5; y = the opposite of 5 plus 2 = negative 3; (5, -3).\">\r\n<tbody>\r\n<tr>\r\n<td>[latex]x[\/latex]<\/td>\r\n<td>[latex]y=-x+2[\/latex]<\/td>\r\n<td>[latex]\\left(x,y\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]-5[\/latex]<\/td>\r\n<td>[latex]y=-\\left(-5\\right)+2=7[\/latex]<\/td>\r\n<td>[latex]\\left(-5,7\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]-3[\/latex]<\/td>\r\n<td>[latex]y=-\\left(-3\\right)+2=5[\/latex]<\/td>\r\n<td>[latex]\\left(-3,5\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]-1[\/latex]<\/td>\r\n<td>[latex]y=-\\left(-1\\right)+2=3[\/latex]<\/td>\r\n<td>[latex]\\left(-1,3\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]0[\/latex]<\/td>\r\n<td>[latex]y=-\\left(0\\right)+2=2[\/latex]<\/td>\r\n<td>[latex]\\left(0,2\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]1[\/latex]<\/td>\r\n<td>[latex]y=-\\left(1\\right)+2=1[\/latex]<\/td>\r\n<td>[latex]\\left(1,1\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]3[\/latex]<\/td>\r\n<td>[latex]y=-\\left(3\\right)+2=-1[\/latex]<\/td>\r\n<td>[latex]\\left(3,-1\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]5[\/latex]<\/td>\r\n<td>[latex]y=-\\left(5\\right)+2=-3[\/latex]<\/td>\r\n<td>[latex]\\left(5,-3\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nNow, plot the points. Connect them if they form a line.\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/12042411\/CNX_CAT_Figure_02_01_007.jpg\" alt=\"This image is a graph of a line on an x, y coordinate plane. The x-axis includes numbers that range from negative 7 to 7. The y-axis includes numbers that range from negative 5 to 8. A line passes through the points: (-5, 7); (-3, 5); (-1, 3); (0, 2); (1, 1); (3, -1); and (5, -3).\" width=\"731\" height=\"556\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nConstruct a table and graph the equation by plotting points: [latex]y=\\frac{1}{2}x+2[\/latex].\r\n[reveal-answer q=\"811886\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"811886\"]\r\n<table summary=\"The table shows 6 rows and 3 columns. The entries in the first row are: x; y = x divided by 2 plus 2, (x,y). The entries in the second row are: negative 2; y = (negative 2) divided by 2 plus 2 = 1; (-2, 1). The entries in the third row are: negative 1; y = (negative 1) divided by 2 plus 2 = 3\/2; (-1,3\/2). The entries in the fourth row are: 0; y = (0)\/2 + 2 = 2; (0,2). The entries in the fifth row are: 1; y = (1)\/2 + 2 = 5\/2; (1,5\/2). The entries in the sixth row are: 2; y = (2)\/2 + 2 = 3; (2,3).\">\r\n<tbody>\r\n<tr>\r\n<td>[latex]x[\/latex]<\/td>\r\n<td>[latex]y=\\frac{1}{2}x+2[\/latex]<\/td>\r\n<td>[latex]\\left(x,y\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]-2[\/latex]<\/td>\r\n<td>[latex]y=\\frac{1}{2}\\left(-2\\right)+2=1[\/latex]<\/td>\r\n<td>[latex]\\left(-2,1\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]-1[\/latex]<\/td>\r\n<td>[latex]y=\\frac{1}{2}\\left(-1\\right)+2=\\frac{3}{2}[\/latex]<\/td>\r\n<td>[latex]\\left(-1,\\frac{3}{2}\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]0[\/latex]<\/td>\r\n<td>[latex]y=\\frac{1}{2}\\left(0\\right)+2=2[\/latex]<\/td>\r\n<td>[latex]\\left(0,2\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]1[\/latex]<\/td>\r\n<td>[latex]y=\\frac{1}{2}\\left(1\\right)+2=\\frac{5}{2}[\/latex]<\/td>\r\n<td>[latex]\\left(1,\\frac{5}{2}\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]2[\/latex]<\/td>\r\n<td>[latex]y=\\frac{1}{2}\\left(2\\right)+2=3[\/latex]<\/td>\r\n<td>[latex]\\left(2,3\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/12042413\/CNX_CAT_Figure_02_01_008.jpg\" alt=\"This is an image of a graph on an x, y coordinate plane. The x and y-axis range from negative 5 to 5. A line passes through the points (-2, 1); (-1, 3\/2); (0, 2); (1, 5\/2); and (2, 3).\" width=\"487\" height=\"442\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<iframe id=\"mom3\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=110939&amp;theme=oea&amp;iframe_resize_id=mom3\" width=\"100%\" height=\"350\"><\/iframe>\r\n\r\n<\/div>\r\n<h3>Using Intercepts to Plot Lines in the Coordinate Plane<\/h3>\r\nThe <strong>intercepts<\/strong> of a graph are points where the graph crosses the axes. The <strong><em>x-<\/em>intercept<\/strong> is the point where the graph crosses the <em>x-<\/em>axis. At this point, the <em>y-<\/em>coordinate is zero. The <strong><em>y-<\/em>intercept<\/strong> is the point where the graph crosses the <em>y-<\/em>axis. At this point, the <em>x-<\/em>coordinate is zero.\r\n\r\nTo determine the <em>x-<\/em>intercept, we set <em>y <\/em>equal to zero and solve for <em>x<\/em>. Similarly, to determine the <em>y-<\/em>intercept, we set <em>x <\/em>equal to zero and solve for <em>y<\/em>. For example, lets find the intercepts of the equation [latex]y=3x - 1[\/latex].\r\n\r\nTo find the <em>x-<\/em>intercept, set [latex]y=0[\/latex].\r\n<div style=\"text-align: center;\">[latex]\\begin{array}{llllll}y=3x - 1\\hfill &amp; \\hfill \\\\ 0=3x - 1\\hfill &amp; \\hfill \\\\ 1=3x\\hfill &amp; \\hfill \\\\ \\frac{1}{3}=x\\hfill &amp; \\hfill \\\\ \\left(\\frac{1}{3},0\\right)\\hfill &amp; x\\text{-intercept}\\hfill \\end{array}[\/latex]<\/div>\r\nTo find the <em>y-<\/em>intercept, set [latex]x=0[\/latex].\r\n<div style=\"text-align: center;\">[latex]\\begin{array}{lllll}y=3x - 1\\hfill &amp; \\hfill \\\\ y=3\\left(0\\right)-1\\hfill &amp; \\hfill \\\\ y=-1\\hfill &amp; \\hfill \\\\ \\left(0,-1\\right)\\hfill &amp; y\\text{-intercept}\\hfill \\end{array}[\/latex]<\/div>\r\nWe can confirm that our results make sense by observing a graph of the equation. Notice that the graph crosses the axes where we predicted it would.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/12042423\/CNX_CAT_Figure_02_01_012.jpg\" alt=\"This is an image of a line graph on an x, y coordinate plane. The x and y-axis range from negative 4 to 4. The function y = 3x \u2013 1 is plotted on the coordinate plane\" width=\"487\" height=\"366\" \/>\r\n<div class=\"textbox\">\r\n<h3>How To: Given an equation, find the intercepts<\/h3>\r\n<ol>\r\n \t<li>Find the <em>x<\/em>-intercept by setting [latex]y=0[\/latex] and solving for [latex]x[\/latex].<\/li>\r\n \t<li>Find the <em>y-<\/em>intercept by setting [latex]x=0[\/latex] and solving for [latex]y[\/latex].<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Finding the Intercepts of the Given Equation<\/h3>\r\nFind the intercepts of the equation [latex]y=-3x - 4[\/latex]. Then sketch the graph using only the intercepts.\r\n\r\n[reveal-answer q=\"814560\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"814560\"]\r\nSet [latex]y=0[\/latex] to find the <em>x-<\/em>intercept.\r\n<div style=\"text-align: center;\">[latex]\\begin{array}{l}y=-3x - 4\\hfill \\\\ 0=-3x - 4\\hfill \\\\ 4=-3x\\hfill \\\\ -\\frac{4}{3}=x\\hfill \\\\ \\left(-\\frac{4}{3},0\\right)x\\text{-intercept}\\hfill \\end{array}[\/latex]<\/div>\r\nSet [latex]x=0[\/latex] to find the <em>y-<\/em>intercept.\r\n<div style=\"text-align: center;\">[latex]\\begin{array}{l}y=-3x - 4\\hfill \\\\ y=-3\\left(0\\right)-4\\hfill \\\\ y=-4\\hfill \\\\ \\left(0,-4\\right)y\\text{-intercept}\\hfill \\end{array}[\/latex]<\/div>\r\nPlot both points and draw a line passing through them.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/12042425\/CNX_CAT_Figure_02_01_013.jpg\" alt=\"This is an image of a line graph on an x, y coordinate plane. The x-axis ranges from negative 5 to 5. The y-axis ranges from negative 6 to 3. The line passes through the points (-4\/3, 0) and (0, -4).\" width=\"487\" height=\"406\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nFind the intercepts of the equation and sketch the graph: [latex]y=-\\frac{3}{4}x+3[\/latex].\r\n\r\n[reveal-answer q=\"80464\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"80464\"]\r\n\r\n<em>x<\/em>-intercept is [latex]\\left(4,0\\right)[\/latex]; <em>y-<\/em>intercept is [latex]\\left(0,3\\right)[\/latex].\r\n\r\n<img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/09\/25200257\/CNX_CAT_Figure_02_01_014.jpg\" alt=\"This is an image of a line graph on an x, y coordinate plane. The x and y axes range from negative 4 to 6. The function y = -3x\/4 + 3 is plotted.\" width=\"487\" height=\"447\" \/>\r\n\r\n[\/hidden-answer]\r\n<iframe id=\"mom4\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=92757&amp;theme=oea&amp;iframe_resize_id=mom4\" width=\"100%\" height=\"350\"><\/iframe>\r\n\r\n<\/div>\r\n<h3>Using a Graphing Utility to Plot Lines<\/h3>\r\nMost graphing calculators require similar techniques to graph an equation. The equations sometimes have to be manipulated so they are written in the style y=_____. 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.\r\n\r\nFor example, the equation [latex]y=2x - 20[\/latex] has been entered in the TI-84 Plus as shown below. 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\\le 10[\/latex], and [latex]-10\\le y\\le 10[\/latex].\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"731\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/12042416\/CNX_CAT_Figure_02_01_009abcN.jpg\" alt=\"This is an image of three side-by-side calculator screen captures. The first screen is the plot screen with the function y sub 1 equals two times x minus twenty. The second screen shows the plotted line on the coordinate plane. The third screen shows the window edit screen with the following settings: Xmin = -10; Xmax = 10; Xscl = 1; Ymin = -10; Ymax = 10; Yscl = 1; Xres = 1.\" width=\"731\" height=\"215\" \/> (a) Enter the equation. (b) This is the graph in the original window. (c) These are the original settings.[\/caption]\r\n\r\nBy changing the window to show more of the positive <em>x-<\/em>axis and more of the negative <em>y-<\/em>axis, we have a much better view of the graph and the <em>x<\/em>\u00a0and <em>y-<\/em>intercepts. See (a) and (b) below.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/12042418\/CNX_CAT_Figure_02_01_010ab.jpg\" alt=\"This is an image of two side-by-side calculator screen captures. The first screen is the window edit screen with the following settings: Xmin = negative 5; Xmax = 15; Xscl = 1; Ymin = -30; Ymax = 10; Yscl = 1; Xres =1. The second screen shows the plot of the previous graph, but is more centered on the line.\" width=\"487\" height=\"213\" \/> (a) This screen shows the new window settings. (b) We can clearly view the intercepts in the new window.[\/caption]\r\n\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Using a Graphing Utility to Graph an Equation<\/h3>\r\nUse a graphing utility to graph the equation: [latex]y=-\\frac{2}{3}x-\\frac{4}{3}[\/latex].\r\n[reveal-answer q=\"294810\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"294810\"]\r\n\r\nEnter the equation in the <em>y=<\/em> function of the calculator. Set the window settings so that both the <em>x<\/em>\u00a0and <em>y-<\/em> intercepts are showing in the window.\r\n<div class=\"mceTemp\"><\/div>\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/12042421\/CNX_CAT_Figure_02_01_011.jpg\" alt=\"This image is of a line graph on an x, y coordinate plane. The x-axis has numbers that range from negative 3 to 4. The y-axis has numbers that range from negative 3 to 3. The function y = -2x\/3 + 4\/3 is plotted.\" width=\"487\" height=\"343\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div>\r\n<h2>Key Concepts<\/h2>\r\n<\/div>\r\n<ul>\r\n \t<li>We can locate or plot points in the Cartesian coordinate system using ordered pairs which are defined as displacement from the <em>x-<\/em>axis and displacement from the <em>y-<\/em>axis.<\/li>\r\n \t<li>An equation can be graphed in the plane by creating a table of values and plotting points.<\/li>\r\n \t<li>Using a graphing calculator or a computer program makes graphing equations faster and more accurate. Equations usually have to be entered in the form <em>y=<\/em>_____.<\/li>\r\n \t<li>Finding the <em>x- <\/em>and <em>y-<\/em>intercepts can define the graph of a line. These are the points where the graph crosses the axes.<\/li>\r\n \t<li>The distance formula is derived from the Pythagorean Theorem and is used to find the length of a line segment.<\/li>\r\n \t<li>The midpoint formula provides a method of finding the coordinates of the midpoint by dividing the sum of the <em>x<\/em>-coordinates and the sum of the <em>y<\/em>-coordinates of the endpoints by 2.<\/li>\r\n<\/ul>\r\n<h2>Glossary<\/h2>\r\n<dl id=\"fs-id1165131990658\" class=\"definition\">\r\n \t<dt><strong>Cartesian coordinate system<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165131990661\">a grid system designed with perpendicular axes invented by Ren\u00e9 Descartes<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165132943522\" class=\"definition\">\r\n \t<dt><strong>equation in two variables<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165132943525\">a mathematical statement, typically written in <em>x <\/em>and <em>y<\/em>, in which two expressions are equal<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165132943528\" class=\"definition\">\r\n \t<dt><strong>distance formula<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165134297639\">a formula that can be used to find the length of a line segment if the endpoints are known<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165134297646\" class=\"definition\">\r\n \t<dt><strong>graph in two variables<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165135486042\">the graph of an equation in two variables, which is always shown in two variables in the two-dimensional plane<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165137644987\" class=\"definition\">\r\n \t<dt><strong>intercepts<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165137644990\">the points at which the graph of an equation crosses the <em>x<\/em>-axis and the <em>y<\/em>-axis<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165132943528\" class=\"definition\">\r\n \t<dt><strong>midpoint formula<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165134297639\">\u00a0a formula to find the point that divides a line segment into two parts of equal length<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165132943528\" class=\"definition\">\r\n \t<dt><strong>ordered pair<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165134297639\">a pair of numbers indicating horizontal displacement and vertical displacement from the origin; also known as a coordinate pair, [latex]\\left(x,y\\right)[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165134297646\" class=\"definition\">\r\n \t<dt><strong>origin<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165135486042\">the point where the two axes cross in the center of the plane, described by the ordered pair [latex]\\left(0,0\\right)[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165137644987\" class=\"definition\">\r\n \t<dt><strong>quadrant<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165137644990\">one quarter of the coordinate plane, created when the axes divide the plane into four sections<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165134297646\" class=\"definition\">\r\n \t<dt><strong><em>x<\/em>-axis<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165135486042\">the common name of the horizontal axis on a coordinate plane; a number line increasing from left to right<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165137644987\" class=\"definition\">\r\n \t<dt><strong><em>x-<\/em>coordinate<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165137644990\">the first coordinate of an ordered pair, representing the horizontal displacement and direction from the origin<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165134297646\" class=\"definition\">\r\n \t<dt><strong><em>x-<\/em>intercept<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165135486042\">the point where a graph intersects the <em>x-<\/em>axis; an ordered pair with a <em>y<\/em>-coordinate of zero<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165137644987\" class=\"definition\">\r\n \t<dt><strong><em>y<\/em>-axis<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165137644990\">the common name of the vertical axis on a coordinate plane; a number line increasing from bottom to top<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165134297646\" class=\"definition\">\r\n \t<dt><strong><em>y-<\/em>coordinate<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165135486042\">\u00a0the second coordinate of an ordered pair, representing the vertical displacement and direction from the origin<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165137644987\" class=\"definition\">\r\n \t<dt><strong><em>y<\/em>-intercept<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165137644990\">a point where a graph intercepts the <em>y-<\/em>axis; an ordered pair with an <em>x<\/em>-coordinate of zero<\/dd>\r\n<\/dl>","rendered":"<div class=\"bcc-box bcc-highlight\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Define the components of the Cartesian coordinate system.<\/li>\n<li>Plot points on the Cartesian coordinate plane.<\/li>\n<li>Determine the slope of a line given two points.<\/li>\n<li>Use the distance formula to find the distance between two points in the plane.<\/li>\n<li>Use the midpoint formula to find the midpoint between two points.<\/li>\n<li>Plot linear equations in two variables on the coordinate plane.<\/li>\n<li>Use intercepts to plot lines.<\/li>\n<li>Use a graphing utility to graph a linear equation on a coordinate plane.<\/li>\n<\/ul>\n<\/div>\n<p>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 the figure below. 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<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/09\/25200240\/CNX_CAT_Figure_02_01_001.jpg\" alt=\"Road map of a city with street names on an x, y coordinate grid. Various points are marked in red on the grid lines indicating different locations on the map.\" width=\"731\" height=\"480\" \/><\/p>\n<h2>Plotting Points on the Coordinate Plane<\/h2>\n<p>An old story describes how seventeenth-century philosopher\/mathematician Ren\u00e9 Descartes invented the system that has become the foundation of algebra while sick in bed. 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>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>Cartesian coordinate system<\/strong>, a grid system having perpendicular axes. Descartes named the horizontal axis the <strong><em>x-<\/em>axis<\/strong> and the vertical axis the <strong><em>y-<\/em>axis<\/strong>.<\/p>\n<p>The Cartesian coordinate system, also called the rectangular coordinate system, is based on a two-dimensional plane consisting of the <em>x<\/em>-axis and the <em>y<\/em>-axis. Perpendicular to each other, the axes divide the plane into four sections. Each section is called a <strong>quadrant<\/strong>; the quadrants are numbered counterclockwise as shown in the figure below.<\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/12042358\/CNX_CAT_Figure_02_01_002.jpg\" alt=\"This is an image of an x, y plane with the axes labeled. The upper right section is labeled: Quadrant I. The upper left section is labeled: Quadrant II. The lower left section is labeled: Quadrant III. The lower right section is labeled: Quadrant IV.\" width=\"487\" height=\"442\" \/><\/p>\n<p class=\"wp-caption-text\"><b>The Cartesian coordinate system with all four quadrants labeled.<\/b><\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=92752&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"650\"><\/iframe><\/p>\n<\/div>\n<p>The center of the plane is the point at which the two axes cross. It is known as the <strong>origin\u00a0<\/strong>or point [latex]\\left(0,0\\right)[\/latex]. From the origin, each axis is further divided into equal units: increasing, positive numbers to the right on the <em>x-<\/em>axis and up the <em>y-<\/em>axis; decreasing, negative numbers to the left on the <em>x-<\/em>axis and down the <em>y-<\/em>axis. The axes extend to positive and negative infinity as shown by the arrowheads in the figure below.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/12042401\/CNX_CAT_Figure_02_01_003.jpg\" alt=\"This is an image of an x, y coordinate plane. The x and y axis range from negative 5 to 5.\" width=\"487\" height=\"442\" \/><\/p>\n<p>Each point in the plane is identified by its <strong><em>x-<\/em>coordinate<\/strong>,\u00a0or horizontal displacement from the origin, and its <strong><em>y-<\/em>coordinate<\/strong>, or vertical displacement from the origin. Together we write them as an <strong>ordered pair<\/strong> indicating the combined distance from the origin in the form [latex]\\left(x,y\\right)[\/latex]. An ordered pair is also known as a coordinate pair because it consists of <em>x\u00a0<\/em>and <em>y<\/em>-coordinates. For example, we can represent the point [latex]\\left(3,-1\\right)[\/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.<\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/12042403\/CNX_CAT_Figure_02_01_004.jpg\" alt=\"This is an image of an x, y coordinate plane. The x and y axis range from negative 5 to 5. The point (3, -1) is labeled. An arrow extends rightward from the origin 3 units and another arrow extends downward one unit from the end of that arrow to the point.\" width=\"487\" height=\"442\" \/><\/p>\n<p class=\"wp-caption-text\"><b>An illustration of how to plot the point (3,-1).<\/b><\/p>\n<\/div>\n<p>When dividing the axes into equally spaced increments, note that the <em>x-<\/em>axis may be considered separately from the <em>y-<\/em>axis. In other words, while the <em>x-<\/em>axis may be divided and labeled according to consecutive integers, the <em>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. Consider the rectangular coordinate system primarily as a method for showing the relationship between two quantities.<\/p>\n<div class=\"textbox\">\n<h3>A General Note: Cartesian Coordinate System<\/h3>\n<p>A two-dimensional plane where the<\/p>\n<ul>\n<li><em>x<\/em>-axis is the horizontal axis<\/li>\n<li><em>y<\/em>-axis is the vertical axis<\/li>\n<\/ul>\n<p>A point in the plane is defined as an ordered pair, [latex]\\left(x,y\\right)[\/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 class=\"textbox exercises\">\n<h3>Example: Plotting Points in a Rectangular Coordinate System<\/h3>\n<p>Plot the points [latex]\\left(-2,4\\right)[\/latex], [latex]\\left(3,3\\right)[\/latex], and [latex]\\left(0,-3\\right)[\/latex] in the coordinate plane.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q380739\">Show Solution<\/span><\/p>\n<div id=\"q380739\" class=\"hidden-answer\" style=\"display: none\">\n<p>To plot the point [latex]\\left(-2,4\\right)[\/latex], begin at the origin. The <em>x<\/em>-coordinate is \u20132, 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]\\left(3,3\\right)[\/latex], begin again at the origin. The <em>x<\/em>-coordinate is 3, so move 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]\\left(0,-3\\right)[\/latex], begin again at the origin. The <em>x<\/em>-coordinate is 0. This tells us not to move in either direction along the <em>x<\/em>-axis. The <em>y<\/em>-coordinate is \u20133, so move three units down in the negative <em>y<\/em> direction.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/12042406\/CNX_CAT_Figure_02_01_005.jpg\" alt=\"This is an image of a graph on an x, y coordinate plane. The x and y axes range from negative 5 to 5. The points (-2, 4); (3, 3); and (0, -3) are labeled. Arrows extend from the origin to the points.\" width=\"487\" height=\"442\" \/><\/p>\n<h4>Analysis of the Solution<\/h4>\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 <em>y<\/em>-axis. If the <em>y<\/em>-coordinate is zero, the point is on the <em>x<\/em>-axis.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"mom2\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=92753&amp;theme=oea&amp;iframe_resize_id=mom2\" width=\"100%\" height=\"650\"><\/iframe><\/p>\n<\/div>\n<h2>The Slope of a Line<\/h2>\n<p>The <strong>slope<\/strong> of a line refers to the ratio of the vertical change in <em>y<\/em> over the horizontal change in <em>x<\/em> between any two points on a line. It indicates the direction in which a line slants as well as its steepness. Slope is sometimes described as rise over run.<\/p>\n<div style=\"text-align: center;\">[latex]m=\\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}[\/latex]<\/div>\n<p>If the slope is positive, the line slants upward to the right. If the slope is negative, the line slants downward to the right. As the slope increases, the line becomes steeper. Some examples are shown below. The lines indicate the following slopes: [latex]m=-3[\/latex], [latex]m=2[\/latex], and [latex]m=\\frac{1}{3}[\/latex].<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/11185922\/CNX_CAT_Figure_02_02_002.jpg\" alt=\"Coordinate plane with the x and y axes ranging from negative 10 to 10. Three linear functions are plotted: y = negative 3 times x minus 2; y = 2 times x plus 1; and y = x over 3 plus 2.\" width=\"487\" height=\"442\" \/><\/p>\n<div class=\"textbox\">\n<h3>A General Note: The Slope of a Line<\/h3>\n<p>The slope of a line, <em>m<\/em>, represents the change in <em>y<\/em> over the change in <em>x.<\/em> Given two points, [latex]\\left({x}_{1},{y}_{1}\\right)[\/latex] and [latex]\\left({x}_{2},{y}_{2}\\right)[\/latex], the following formula determines the slope of a line containing these points:<\/p>\n<div style=\"text-align: center;\">[latex]m=\\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}[\/latex]<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Finding the Slope of a Line Given Two Points<\/h3>\n<p>Find the slope of a line that passes through the points [latex]\\left(2,-1\\right)[\/latex] and [latex]\\left(-5,3\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q688301\">Show Solution<\/span><\/p>\n<div id=\"q688301\" class=\"hidden-answer\" style=\"display: none\">\n<p>We substitute the <em>y-<\/em>values and the <em>x-<\/em>values into the formula.<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{l}m\\hfill&=\\frac{3-\\left(-1\\right)}{-5 - 2}\\hfill \\\\ \\hfill&=\\frac{4}{-7}\\hfill \\\\ \\hfill&=-\\frac{4}{7}\\hfill \\end{array}[\/latex]<\/div>\n<p>The slope is [latex]-\\frac{4}{7}[\/latex].<\/p>\n<h4>Analysis of the Solution<\/h4>\n<p>It does not matter which point is called [latex]\\left({x}_{1},{y}_{1}\\right)[\/latex] or [latex]\\left({x}_{2},{y}_{2}\\right)[\/latex]. As long as we are consistent with the order of the <em>y<\/em> terms and the order of the <em>x<\/em> terms in the numerator and denominator, the calculation will yield the same result.\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Find the slope of the line that passes through the points [latex]\\left(-2,6\\right)[\/latex] and [latex]\\left(1,4\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q196055\">Show Solution<\/span><\/p>\n<div id=\"q196055\" class=\"hidden-answer\" style=\"display: none\">\n<p>slope[latex]=m=\\dfrac{-2}{3}=-\\dfrac{2}{3}[\/latex]<\/p><\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=1719&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"350\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<\/div>\n<h2>Distance in the Plane<\/h2>\n<p>Derived from the <strong>Pythagorean Theorem<\/strong>, the <strong>distance formula<\/strong> is used to find the distance between two points in the plane. The Pythagorean Theorem, [latex]{a}^{2}+{b}^{2}={c}^{2}[\/latex], is based on a right triangle where <em>a <\/em>and <em>b<\/em> are the lengths of the legs adjacent to the right angle, and <em>c<\/em> is the length of the hypotenuse.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/12042428\/CNX_CAT_Figure_02_01_015.jpg\" alt=\"This is an image of a triangle on an x, y coordinate plane. The x and y axes range from 0 to 7. The points (x sub 1, y sub 1); (x sub 2, y sub 1); and (x sub 2, y sub 2) are labeled and connected to form a triangle. Along the base of the triangle, the following equation is displayed: the absolute value of x sub 2 minus x sub 1 equals a. The hypotenuse of the triangle is labeled: d = c. The remaining side is labeled: the absolute value of y sub 2 minus y sub 1 equals b.\" width=\"487\" height=\"331\" \/><\/p>\n<p>The relationship of sides [latex]|{x}_{2}-{x}_{1}|[\/latex] and [latex]|{y}_{2}-{y}_{1}|[\/latex] to side <em>d<\/em> is the same as that of sides <em>a <\/em>and <em>b <\/em>to side <em>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>c<\/em>, take the square root of both sides of the Pythagorean Theorem.<\/p>\n<div style=\"text-align: center;\">[latex]{c}^{2}={a}^{2}+{b}^{2}\\rightarrow c=\\sqrt{{a}^{2}+{b}^{2}}[\/latex]<\/div>\n<p>It follows that the distance formula is given as<\/p>\n<div style=\"text-align: center;\">[latex]{d}^{2}={\\left({x}_{2}-{x}_{1}\\right)}^{2}+{\\left({y}_{2}-{y}_{1}\\right)}^{2}\\to d=\\sqrt{{\\left({x}_{2}-{x}_{1}\\right)}^{2}+{\\left({y}_{2}-{y}_{1}\\right)}^{2}}[\/latex]<\/div>\n<p>We do not have to use the absolute value symbols in this definition because any number squared is positive.<\/p>\n<div class=\"textbox\">\n<h3>A General Note: The Distance Formula<\/h3>\n<p>Given endpoints [latex]\\left({x}_{1},{y}_{1}\\right)[\/latex] and [latex]\\left({x}_{2},{y}_{2}\\right)[\/latex], the distance between two points is given by<\/p>\n<div style=\"text-align: center;\">[latex]d=\\sqrt{{\\left({x}_{2}-{x}_{1}\\right)}^{2}+{\\left({y}_{2}-{y}_{1}\\right)}^{2}}[\/latex]<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Finding the Distance between Two Points<\/h3>\n<p>Find the distance between the points [latex]\\left(-3,-1\\right)[\/latex] and [latex]\\left(2,3\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q737169\">Show Solution<\/span><\/p>\n<div id=\"q737169\" class=\"hidden-answer\" style=\"display: none\">\n<p>Let us first look at the graph of the two points. Connect the points to form a right triangle.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/12042430\/CNX_CAT_Figure_02_01_016.jpg\" alt=\"This is an image of a triangle on an x, y coordinate plane. The x-axis ranges from negative 4 to 4. The y-axis ranges from negative 2 to 4. The points (-3, -1); (2, -1); and (2, 3) are plotted and labeled on the graph. The points are connected to form a triangle\" width=\"487\" height=\"289\" \/><\/p>\n<p>Then, calculate the length of <em>d <\/em>using the distance formula.<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{l}d=\\sqrt{{\\left({x}_{2}-{x}_{1}\\right)}^{2}+{\\left({y}_{2}-{y}_{1}\\right)}^{2}}\\hfill \\\\ d=\\sqrt{{\\left(2-\\left(-3\\right)\\right)}^{2}+{\\left(3-\\left(-1\\right)\\right)}^{2}}\\hfill \\\\ =\\sqrt{{\\left(5\\right)}^{2}+{\\left(4\\right)}^{2}}\\hfill \\\\ =\\sqrt{25+16}\\hfill \\\\ =\\sqrt{41}\\hfill \\end{array}[\/latex]<\/div>\n<div><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<div>\n<p>Find the distance between two points: [latex]\\left(1,4\\right)[\/latex] and [latex]\\left(11,9\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q934526\">Show Solution<\/span><\/p>\n<div id=\"q934526\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\sqrt{125}=5\\sqrt{5}[\/latex]<\/p><\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=19140&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"350\"><\/iframe><\/p>\n<\/div>\n<div>\n<p>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Finding the Distance between Two Locations<\/h3>\n<p>Let\u2019s return to the situation introduced at the beginning of this section.<\/p>\n<p>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. Find the total distance that Tracie traveled. Compare this with the distance between her starting and final positions.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q411250\">Show Solution<\/span><\/p>\n<div id=\"q411250\" class=\"hidden-answer\" style=\"display: none\">\n<p>The first thing we should do is identify ordered pairs to describe each position. If we set the starting position at 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 [latex]\\left(1,1\\right)[\/latex]. The next stop is 5 blocks to the east so it is at [latex]\\left(5,1\\right)[\/latex]. After that, she traveled 3 blocks east and 2 blocks north to [latex]\\left(8,3\\right)[\/latex]. Lastly, she traveled 4 blocks north to [latex]\\left(8,7\\right)[\/latex]. We can label these points on the grid.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/12042433\/CNX_CAT_Figure_02_01_017.jpg\" alt=\"This is an image of a road map of a city. The point (1, 1) is on North Avenue and Bertau Avenue. The point (5, 1) is on North Avenue and Wolf Road. The point (8, 3) is on Mannheim Road and McLean Street. The point (8, 7) is on Mannheim Road and Schiller Avenue.\" width=\"731\" height=\"480\" \/><\/p>\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 [latex]\\left(1,1\\right)[\/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 [latex]\\left(5,1\\right)[\/latex]. So from [latex]\\left(1,1\\right)[\/latex] to [latex]\\left(5,1\\right)[\/latex], Tracie drove east 4,000 feet.<\/li>\n<li>Her third stop is at [latex]\\left(8,3\\right)[\/latex]. There are a number of routes from [latex]\\left(5,1\\right)[\/latex] to [latex]\\left(8,3\\right)[\/latex]. Whatever route Tracie decided to use, the distance is the same, as there are no angular streets between the two points. Let\u2019s say she drove east 3,000 feet and then north 2,000 feet for a total of 5,000 feet.<\/li>\n<li>Tracie\u2019s final stop is at [latex]\\left(8,7\\right)[\/latex]. This is a straight drive north from [latex]\\left(8,3\\right)[\/latex] for a total of 4,000 feet.<\/li>\n<\/ul>\n<p>Next, we will add the distances listed in the table.<\/p>\n<table summary=\"A table with 6 rows and 2 columns. The entries in the first row are: From\/To and Number of Feet Driven. The entries in the second row are: (0, 0) to (1, 1) and 2,000. The entries in the third row are: (1, 1) to (5, 1) and 4,000. The entries in the fourth row are: (5, 1) to (8, 3) and 5,000. The entries in the fourth row are: (8, 3) to (8, 7) and 4,000. The entries in the sixth row are: Total and 15,000.\">\n<thead>\n<tr>\n<th>From\/To<\/th>\n<th>Number of Feet Driven<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>[latex]\\left(0,0\\right)[\/latex] to [latex]\\left(1,1\\right)[\/latex]<\/td>\n<td>2,000<\/td>\n<\/tr>\n<tr>\n<td>[latex]\\left(1,1\\right)[\/latex] to [latex]\\left(5,1\\right)[\/latex]<\/td>\n<td>4,000<\/td>\n<\/tr>\n<tr>\n<td>[latex]\\left(5,1\\right)[\/latex] to [latex]\\left(8,3\\right)[\/latex]<\/td>\n<td>5,000<\/td>\n<\/tr>\n<tr>\n<td>[latex]\\left(8,3\\right)[\/latex] to [latex]\\left(8,7\\right)[\/latex]<\/td>\n<td>4,000<\/td>\n<\/tr>\n<tr>\n<td>Total<\/td>\n<td>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 [latex]\\left(0,0\\right)[\/latex] and [latex]\\left(8,7\\right)[\/latex].<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{l}d=\\sqrt{{\\left(8 - 0\\right)}^{2}+{\\left(7 - 0\\right)}^{2}}\\hfill \\\\ =\\sqrt{64+49}\\hfill \\\\ =\\sqrt{113}\\hfill \\\\ =10.63\\text{ units}\\hfill \\end{array}[\/latex]<\/div>\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 [latex]\\left(8,7\\right)[\/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<\/div>\n<\/div>\n<\/div>\n<h3>Using the Midpoint Formula<\/h3>\n<p>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>midpoint formula<\/strong>. Given the endpoints of a line segment, [latex]\\left({x}_{1},{y}_{1}\\right)[\/latex] and [latex]\\left({x}_{2},{y}_{2}\\right)[\/latex], the midpoint formula states how to find the coordinates of the midpoint [latex]M[\/latex].<\/p>\n<div style=\"text-align: center;\">[latex]M=\\left(\\frac{{x}_{1}+{x}_{2}}{2},\\frac{{y}_{1}+{y}_{2}}{2}\\right)[\/latex]<\/div>\n<p>A graphical view of a midpoint is shown below. Notice that the line segments on either side of the midpoint are congruent.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/12042436\/CNX_CAT_Figure_02_01_018.jpg\" alt=\"This is a line graph on an x, y coordinate plane with the x and y axes ranging from 0 to 6. The points (x sub 1, y sub 1), (x sub 2, y sub 2), and (x sub 1 plus x sub 2 all over 2, y sub 1 plus y sub 2 all over 2) are plotted. A straight line runs through these three points. Pairs of short parallel lines bisect the two sections of the line to note that they are equivalent.\" width=\"487\" height=\"290\" \/><\/p>\n<div class=\"textbox exercises\">\n<h3>Example: Finding the Midpoint of a Line Segment<\/h3>\n<p>Find the midpoint of the line segment with the endpoints [latex]\\left(7,-2\\right)[\/latex] and [latex]\\left(9,5\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q788934\">Show Solution<\/span><\/p>\n<div id=\"q788934\" class=\"hidden-answer\" style=\"display: none\">\n<p>Use the formula to find the midpoint of the line segment.<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{l}\\left(\\frac{{x}_{1}+{x}_{2}}{2},\\frac{{y}_{1}+{y}_{2}}{2}\\right)\\hfill&=\\left(\\frac{7+9}{2},\\frac{-2+5}{2}\\right)\\hfill \\\\ \\hfill&=\\left(8,\\frac{3}{2}\\right)\\hfill \\end{array}[\/latex]<\/div>\n<div><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Find the midpoint of the line segment with endpoints [latex]\\left(-2,-1\\right)[\/latex] and [latex]\\left(-8,6\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q964077\">Show Solution<\/span><\/p>\n<div id=\"q964077\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\left(-5,\\frac{5}{2}\\right)[\/latex]<\/p><\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom2\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=2308&amp;theme=oea&amp;iframe_resize_id=mom2\" width=\"100%\" height=\"250\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Finding the Center of a Circle<\/h3>\n<p>The diameter of a circle has endpoints [latex]\\left(-1,-4\\right)[\/latex] and [latex]\\left(5,-4\\right)[\/latex]. Find the center of the circle.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q418175\">Show Solution<\/span><\/p>\n<div id=\"q418175\" class=\"hidden-answer\" style=\"display: none\">\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<div style=\"text-align: center;\">[latex]\\begin{array}{l}\\left(\\frac{{x}_{1}+{x}_{2}}{2},\\frac{{y}_{1}+{y}_{2}}{2}\\right)\\\\ \\left(\\frac{-1+5}{2},\\frac{-4 - 4}{2}\\right)=\\left(\\frac{4}{2},-\\frac{8}{2}\\right)=\\left(2,-4\\right)\\end{array}[\/latex]<\/div>\n<div><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"mom3\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=110938&amp;theme=oea&amp;iframe_resize_id=mom3\" width=\"100%\" height=\"250\"><\/iframe><\/p>\n<\/div>\n<h2>Graphing Linear Equations<\/h2>\n<p>We can plot a set of points to represent an equation. When such an equation contains both an <em>x <\/em>variable and a <em>y <\/em>variable, it is called an <strong>equation in two variables<\/strong>. Its graph is called a <strong>graph in two variables<\/strong>. Any graph on a two-dimensional plane is a graph in two variables.<\/p>\n<p>Suppose we want to graph the equation [latex]y=2x - 1[\/latex]. We can begin by substituting a value for <em>x<\/em> into the equation and determining the resulting value of <em>y<\/em>. Each pair of <em>x\u00a0<\/em>and <em>y-<\/em>values is an ordered pair that can be plotted. The table below\u00a0lists values of <em>x<\/em> from \u20133 to 3 and the resulting values for <em>y<\/em>.<\/p>\n<table summary=\"This is a table with 8 rows and 3 columns. The first row has columns labeled: x, y = 2x-1, (x, y). The entries in the second row are: negative 3; y = 2 times negative 3 minus 1 = negative 7; (-3, -7). The entries in the third row are: negative 2; y = 2 times negative 2 minus 1 = negative 5; (-2, -5). The entries in the fourth row are: negative1; y = 2 times negative 1 minus 1 = negative 3; (-1, -3). The entries in the fifth row are: 0; y = 2 times 0 minus 1 = negative 1; (0, -1). The entries in the sixth row are: 1; y = 2 times 1 minus 1 = 1; (1, 1). The entries in the seventh row are: 2; y = 2 times 2 minus 1 = 3; (2, 3). The entries in the eight row are: 3, y = 2 times 3 minus 1 = 5; (3,5)\">\n<tbody>\n<tr>\n<td>[latex]x[\/latex]<\/td>\n<td>[latex]y=2x - 1[\/latex]<\/td>\n<td>[latex]\\left(x,y\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]-3[\/latex]<\/td>\n<td>[latex]y=2\\left(-3\\right)-1=-7[\/latex]<\/td>\n<td>[latex]\\left(-3,-7\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]-2[\/latex]<\/td>\n<td>[latex]y=2\\left(-2\\right)-1=-5[\/latex]<\/td>\n<td>[latex]\\left(-2,-5\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]-1[\/latex]<\/td>\n<td>[latex]y=2\\left(-1\\right)-1=-3[\/latex]<\/td>\n<td>[latex]\\left(-1,-3\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]0[\/latex]<\/td>\n<td>[latex]y=2\\left(0\\right)-1=-1[\/latex]<\/td>\n<td>[latex]\\left(0,-1\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]1[\/latex]<\/td>\n<td>[latex]y=2\\left(1\\right)-1=1[\/latex]<\/td>\n<td>[latex]\\left(1,1\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]2[\/latex]<\/td>\n<td>[latex]y=2\\left(2\\right)-1=3[\/latex]<\/td>\n<td>[latex]\\left(2,3\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]3[\/latex]<\/td>\n<td>[latex]y=2\\left(3\\right)-1=5[\/latex]<\/td>\n<td>[latex]\\left(3,5\\right)[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>We can plot these points from the table. The points for this particular equation form a line, so we can connect them.<strong>\u00a0<\/strong>This is not true for all equations.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/12042408\/CNX_CAT_Figure_02_01_006.jpg\" alt=\"This is a graph of a line on an x, y coordinate plane. The x- and y-axis range from negative 8 to 8. A line passes through the points (-3, -7); (-2, -5); (-1, -3); (0, -1); (1, 1); (2, 3); and (3, 5).\" width=\"731\" height=\"669\" \/><\/p>\n<p>Note that the <em>x-<\/em>values chosen are arbitrary regardless of the type of equation we are graphing. Of course, some situations may require particular values of <em>x<\/em> to be plotted in order to see a particular result. Otherwise, it is logical to choose values that can be calculated easily, and it is always a good idea to choose values that are both negative and positive. There is no rule dictating how many points to plot, although we need at least two to graph a line. Keep in mind, however, that the more points we plot, the more accurately we can sketch the graph.<\/p>\n<div class=\"textbox\">\n<h3>How To: Given an equation, graph by plotting points<\/h3>\n<ol>\n<li>Make a table with one column labeled <em>x<\/em>, a second column labeled with the equation, and a third column listing the resulting ordered pairs.<\/li>\n<li>Enter <em>x-<\/em>values down the first column using positive and negative values. Selecting the <em>x-<\/em>values in numerical order will make graphing easier.<\/li>\n<li>Select <em>x-<\/em>values that will yield <em>y-<\/em>values with little effort, preferably ones that can be calculated mentally.<\/li>\n<li>Plot the ordered pairs.<\/li>\n<li>Connect the points if they form a line.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Graphing an Equation in Two Variables by Plotting Points<\/h3>\n<p>Graph the equation [latex]y=-x+2[\/latex] by plotting points.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q792137\">Show Solution<\/span><\/p>\n<div id=\"q792137\" class=\"hidden-answer\" style=\"display: none\">\n<p>First, we construct a table similar to the one below. Choose <em>x<\/em> values and calculate <em>y.<\/em><\/p>\n<table summary=\"The table shows 8 rows and 3 columns. The entries in the first row are: x; y = negative x plus 2; and (x, y). The entries in the second row are: negative 5; y = the opposite of negative 5 plus 2 = 7; (-5, 7). The entries in the third row are: negative 3; y = the opposite of negative 3 plus 2 = 5; (-3, 5). The entries in the fourth row are: -1; y = the opposite of negative 1 plus 2 = 3; (-1, 3). The entries in the fifth row are: 0; y = opposite of zero plus 2 = 2; (0, 2). The entries in the sixth row are: 1; y = the opposite of 1 plus 2 = 1; (1, 1). The entries in the seventh row are: 3; y = the opposite of 3 plus 2 = negative 1; (3, -1). The entries in the eighth row are: 5; y = the opposite of 5 plus 2 = negative 3; (5, -3).\">\n<tbody>\n<tr>\n<td>[latex]x[\/latex]<\/td>\n<td>[latex]y=-x+2[\/latex]<\/td>\n<td>[latex]\\left(x,y\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]-5[\/latex]<\/td>\n<td>[latex]y=-\\left(-5\\right)+2=7[\/latex]<\/td>\n<td>[latex]\\left(-5,7\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]-3[\/latex]<\/td>\n<td>[latex]y=-\\left(-3\\right)+2=5[\/latex]<\/td>\n<td>[latex]\\left(-3,5\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]-1[\/latex]<\/td>\n<td>[latex]y=-\\left(-1\\right)+2=3[\/latex]<\/td>\n<td>[latex]\\left(-1,3\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]0[\/latex]<\/td>\n<td>[latex]y=-\\left(0\\right)+2=2[\/latex]<\/td>\n<td>[latex]\\left(0,2\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]1[\/latex]<\/td>\n<td>[latex]y=-\\left(1\\right)+2=1[\/latex]<\/td>\n<td>[latex]\\left(1,1\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]3[\/latex]<\/td>\n<td>[latex]y=-\\left(3\\right)+2=-1[\/latex]<\/td>\n<td>[latex]\\left(3,-1\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]5[\/latex]<\/td>\n<td>[latex]y=-\\left(5\\right)+2=-3[\/latex]<\/td>\n<td>[latex]\\left(5,-3\\right)[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Now, plot the points. Connect them if they form a line.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/12042411\/CNX_CAT_Figure_02_01_007.jpg\" alt=\"This image is a graph of a line on an x, y coordinate plane. The x-axis includes numbers that range from negative 7 to 7. The y-axis includes numbers that range from negative 5 to 8. A line passes through the points: (-5, 7); (-3, 5); (-1, 3); (0, 2); (1, 1); (3, -1); and (5, -3).\" width=\"731\" height=\"556\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Construct a table and graph the equation by plotting points: [latex]y=\\frac{1}{2}x+2[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q811886\">Show Solution<\/span><\/p>\n<div id=\"q811886\" class=\"hidden-answer\" style=\"display: none\">\n<table summary=\"The table shows 6 rows and 3 columns. The entries in the first row are: x; y = x divided by 2 plus 2, (x,y). The entries in the second row are: negative 2; y = (negative 2) divided by 2 plus 2 = 1; (-2, 1). The entries in the third row are: negative 1; y = (negative 1) divided by 2 plus 2 = 3\/2; (-1,3\/2). The entries in the fourth row are: 0; y = (0)\/2 + 2 = 2; (0,2). The entries in the fifth row are: 1; y = (1)\/2 + 2 = 5\/2; (1,5\/2). The entries in the sixth row are: 2; y = (2)\/2 + 2 = 3; (2,3).\">\n<tbody>\n<tr>\n<td>[latex]x[\/latex]<\/td>\n<td>[latex]y=\\frac{1}{2}x+2[\/latex]<\/td>\n<td>[latex]\\left(x,y\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]-2[\/latex]<\/td>\n<td>[latex]y=\\frac{1}{2}\\left(-2\\right)+2=1[\/latex]<\/td>\n<td>[latex]\\left(-2,1\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]-1[\/latex]<\/td>\n<td>[latex]y=\\frac{1}{2}\\left(-1\\right)+2=\\frac{3}{2}[\/latex]<\/td>\n<td>[latex]\\left(-1,\\frac{3}{2}\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]0[\/latex]<\/td>\n<td>[latex]y=\\frac{1}{2}\\left(0\\right)+2=2[\/latex]<\/td>\n<td>[latex]\\left(0,2\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]1[\/latex]<\/td>\n<td>[latex]y=\\frac{1}{2}\\left(1\\right)+2=\\frac{5}{2}[\/latex]<\/td>\n<td>[latex]\\left(1,\\frac{5}{2}\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]2[\/latex]<\/td>\n<td>[latex]y=\\frac{1}{2}\\left(2\\right)+2=3[\/latex]<\/td>\n<td>[latex]\\left(2,3\\right)[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/12042413\/CNX_CAT_Figure_02_01_008.jpg\" alt=\"This is an image of a graph on an x, y coordinate plane. The x and y-axis range from negative 5 to 5. A line passes through the points (-2, 1); (-1, 3\/2); (0, 2); (1, 5\/2); and (2, 3).\" width=\"487\" height=\"442\" \/><\/p>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom3\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=110939&amp;theme=oea&amp;iframe_resize_id=mom3\" width=\"100%\" height=\"350\"><\/iframe><\/p>\n<\/div>\n<h3>Using Intercepts to Plot Lines in the Coordinate Plane<\/h3>\n<p>The <strong>intercepts<\/strong> of a graph are points where the graph crosses the axes. The <strong><em>x-<\/em>intercept<\/strong> is the point where the graph crosses the <em>x-<\/em>axis. At this point, the <em>y-<\/em>coordinate is zero. The <strong><em>y-<\/em>intercept<\/strong> is the point where the graph crosses the <em>y-<\/em>axis. At this point, the <em>x-<\/em>coordinate is zero.<\/p>\n<p>To determine the <em>x-<\/em>intercept, we set <em>y <\/em>equal to zero and solve for <em>x<\/em>. Similarly, to determine the <em>y-<\/em>intercept, we set <em>x <\/em>equal to zero and solve for <em>y<\/em>. For example, lets find the intercepts of the equation [latex]y=3x - 1[\/latex].<\/p>\n<p>To find the <em>x-<\/em>intercept, set [latex]y=0[\/latex].<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{llllll}y=3x - 1\\hfill & \\hfill \\\\ 0=3x - 1\\hfill & \\hfill \\\\ 1=3x\\hfill & \\hfill \\\\ \\frac{1}{3}=x\\hfill & \\hfill \\\\ \\left(\\frac{1}{3},0\\right)\\hfill & x\\text{-intercept}\\hfill \\end{array}[\/latex]<\/div>\n<p>To find the <em>y-<\/em>intercept, set [latex]x=0[\/latex].<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{lllll}y=3x - 1\\hfill & \\hfill \\\\ y=3\\left(0\\right)-1\\hfill & \\hfill \\\\ y=-1\\hfill & \\hfill \\\\ \\left(0,-1\\right)\\hfill & y\\text{-intercept}\\hfill \\end{array}[\/latex]<\/div>\n<p>We can confirm that our results make sense by observing a graph of the equation. Notice that the graph crosses the axes where we predicted it would.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/12042423\/CNX_CAT_Figure_02_01_012.jpg\" alt=\"This is an image of a line graph on an x, y coordinate plane. The x and y-axis range from negative 4 to 4. The function y = 3x \u2013 1 is plotted on the coordinate plane\" width=\"487\" height=\"366\" \/><\/p>\n<div class=\"textbox\">\n<h3>How To: Given an equation, find the intercepts<\/h3>\n<ol>\n<li>Find the <em>x<\/em>-intercept by setting [latex]y=0[\/latex] and solving for [latex]x[\/latex].<\/li>\n<li>Find the <em>y-<\/em>intercept by setting [latex]x=0[\/latex] and solving for [latex]y[\/latex].<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Finding the Intercepts of the Given Equation<\/h3>\n<p>Find the intercepts of the equation [latex]y=-3x - 4[\/latex]. Then sketch the graph using only the intercepts.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q814560\">Show Solution<\/span><\/p>\n<div id=\"q814560\" class=\"hidden-answer\" style=\"display: none\">\nSet [latex]y=0[\/latex] to find the <em>x-<\/em>intercept.<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{l}y=-3x - 4\\hfill \\\\ 0=-3x - 4\\hfill \\\\ 4=-3x\\hfill \\\\ -\\frac{4}{3}=x\\hfill \\\\ \\left(-\\frac{4}{3},0\\right)x\\text{-intercept}\\hfill \\end{array}[\/latex]<\/div>\n<p>Set [latex]x=0[\/latex] to find the <em>y-<\/em>intercept.<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{l}y=-3x - 4\\hfill \\\\ y=-3\\left(0\\right)-4\\hfill \\\\ y=-4\\hfill \\\\ \\left(0,-4\\right)y\\text{-intercept}\\hfill \\end{array}[\/latex]<\/div>\n<p>Plot both points and draw a line passing through them.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/12042425\/CNX_CAT_Figure_02_01_013.jpg\" alt=\"This is an image of a line graph on an x, y coordinate plane. The x-axis ranges from negative 5 to 5. The y-axis ranges from negative 6 to 3. The line passes through the points (-4\/3, 0) and (0, -4).\" width=\"487\" height=\"406\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Find the intercepts of the equation and sketch the graph: [latex]y=-\\frac{3}{4}x+3[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q80464\">Show Solution<\/span><\/p>\n<div id=\"q80464\" class=\"hidden-answer\" style=\"display: none\">\n<p><em>x<\/em>-intercept is [latex]\\left(4,0\\right)[\/latex]; <em>y-<\/em>intercept is [latex]\\left(0,3\\right)[\/latex].<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/09\/25200257\/CNX_CAT_Figure_02_01_014.jpg\" alt=\"This is an image of a line graph on an x, y coordinate plane. The x and y axes range from negative 4 to 6. The function y = -3x\/4 + 3 is plotted.\" width=\"487\" height=\"447\" \/><\/p>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom4\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=92757&amp;theme=oea&amp;iframe_resize_id=mom4\" width=\"100%\" height=\"350\"><\/iframe><\/p>\n<\/div>\n<h3>Using a Graphing Utility to Plot Lines<\/h3>\n<p>Most graphing calculators require similar techniques to graph an equation. The equations sometimes have to be manipulated so they are written in the style y=_____. 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>For example, the equation [latex]y=2x - 20[\/latex] has been entered in the TI-84 Plus as shown below. 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\\le 10[\/latex], and [latex]-10\\le y\\le 10[\/latex].<\/p>\n<div style=\"width: 741px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/12042416\/CNX_CAT_Figure_02_01_009abcN.jpg\" alt=\"This is an image of three side-by-side calculator screen captures. The first screen is the plot screen with the function y sub 1 equals two times x minus twenty. The second screen shows the plotted line on the coordinate plane. The third screen shows the window edit screen with the following settings: Xmin = -10; Xmax = 10; Xscl = 1; Ymin = -10; Ymax = 10; Yscl = 1; Xres = 1.\" width=\"731\" height=\"215\" \/><\/p>\n<p class=\"wp-caption-text\">(a) Enter the equation. (b) This is the graph in the original window. (c) These are the original settings.<\/p>\n<\/div>\n<p>By changing the window to show more of the positive <em>x-<\/em>axis and more of the negative <em>y-<\/em>axis, we have a much better view of the graph and the <em>x<\/em>\u00a0and <em>y-<\/em>intercepts. See (a) and (b) below.<\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/12042418\/CNX_CAT_Figure_02_01_010ab.jpg\" alt=\"This is an image of two side-by-side calculator screen captures. The first screen is the window edit screen with the following settings: Xmin = negative 5; Xmax = 15; Xscl = 1; Ymin = -30; Ymax = 10; Yscl = 1; Xres =1. The second screen shows the plot of the previous graph, but is more centered on the line.\" width=\"487\" height=\"213\" \/><\/p>\n<p class=\"wp-caption-text\">(a) This screen shows the new window settings. (b) We can clearly view the intercepts in the new window.<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Using a Graphing Utility to Graph an Equation<\/h3>\n<p>Use a graphing utility to graph the equation: [latex]y=-\\frac{2}{3}x-\\frac{4}{3}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q294810\">Show Solution<\/span><\/p>\n<div id=\"q294810\" class=\"hidden-answer\" style=\"display: none\">\n<p>Enter the equation in the <em>y=<\/em> function of the calculator. Set the window settings so that both the <em>x<\/em>\u00a0and <em>y-<\/em> intercepts are showing in the window.<\/p>\n<div class=\"mceTemp\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/12042421\/CNX_CAT_Figure_02_01_011.jpg\" alt=\"This image is of a line graph on an x, y coordinate plane. The x-axis has numbers that range from negative 3 to 4. The y-axis has numbers that range from negative 3 to 3. The function y = -2x\/3 + 4\/3 is plotted.\" width=\"487\" height=\"343\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div>\n<h2>Key Concepts<\/h2>\n<\/div>\n<ul>\n<li>We can locate or plot points in the Cartesian coordinate system using ordered pairs which are defined as displacement from the <em>x-<\/em>axis and displacement from the <em>y-<\/em>axis.<\/li>\n<li>An equation can be graphed in the plane by creating a table of values and plotting points.<\/li>\n<li>Using a graphing calculator or a computer program makes graphing equations faster and more accurate. Equations usually have to be entered in the form <em>y=<\/em>_____.<\/li>\n<li>Finding the <em>x- <\/em>and <em>y-<\/em>intercepts can define the graph of a line. These are the points where the graph crosses the axes.<\/li>\n<li>The distance formula is derived from the Pythagorean Theorem and is used to find the length of a line segment.<\/li>\n<li>The midpoint formula provides a method of finding the coordinates of the midpoint by dividing the sum of the <em>x<\/em>-coordinates and the sum of the <em>y<\/em>-coordinates of the endpoints by 2.<\/li>\n<\/ul>\n<h2>Glossary<\/h2>\n<dl id=\"fs-id1165131990658\" class=\"definition\">\n<dt><strong>Cartesian coordinate system<\/strong><\/dt>\n<dd id=\"fs-id1165131990661\">a grid system designed with perpendicular axes invented by Ren\u00e9 Descartes<\/dd>\n<\/dl>\n<dl id=\"fs-id1165132943522\" class=\"definition\">\n<dt><strong>equation in two variables<\/strong><\/dt>\n<dd id=\"fs-id1165132943525\">a mathematical statement, typically written in <em>x <\/em>and <em>y<\/em>, in which two expressions are equal<\/dd>\n<\/dl>\n<dl id=\"fs-id1165132943528\" class=\"definition\">\n<dt><strong>distance formula<\/strong><\/dt>\n<dd id=\"fs-id1165134297639\">a formula that can be used to find the length of a line segment if the endpoints are known<\/dd>\n<\/dl>\n<dl id=\"fs-id1165134297646\" class=\"definition\">\n<dt><strong>graph in two variables<\/strong><\/dt>\n<dd id=\"fs-id1165135486042\">the graph of an equation in two variables, which is always shown in two variables in the two-dimensional plane<\/dd>\n<\/dl>\n<dl id=\"fs-id1165137644987\" class=\"definition\">\n<dt><strong>intercepts<\/strong><\/dt>\n<dd id=\"fs-id1165137644990\">the points at which the graph of an equation crosses the <em>x<\/em>-axis and the <em>y<\/em>-axis<\/dd>\n<\/dl>\n<dl id=\"fs-id1165132943528\" class=\"definition\">\n<dt><strong>midpoint formula<\/strong><\/dt>\n<dd id=\"fs-id1165134297639\">\u00a0a formula to find the point that divides a line segment into two parts of equal length<\/dd>\n<\/dl>\n<dl id=\"fs-id1165132943528\" class=\"definition\">\n<dt><strong>ordered pair<\/strong><\/dt>\n<dd id=\"fs-id1165134297639\">a pair of numbers indicating horizontal displacement and vertical displacement from the origin; also known as a coordinate pair, [latex]\\left(x,y\\right)[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1165134297646\" class=\"definition\">\n<dt><strong>origin<\/strong><\/dt>\n<dd id=\"fs-id1165135486042\">the point where the two axes cross in the center of the plane, described by the ordered pair [latex]\\left(0,0\\right)[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1165137644987\" class=\"definition\">\n<dt><strong>quadrant<\/strong><\/dt>\n<dd id=\"fs-id1165137644990\">one quarter of the coordinate plane, created when the axes divide the plane into four sections<\/dd>\n<\/dl>\n<dl id=\"fs-id1165134297646\" class=\"definition\">\n<dt><strong><em>x<\/em>-axis<\/strong><\/dt>\n<dd id=\"fs-id1165135486042\">the common name of the horizontal axis on a coordinate plane; a number line increasing from left to right<\/dd>\n<\/dl>\n<dl id=\"fs-id1165137644987\" class=\"definition\">\n<dt><strong><em>x-<\/em>coordinate<\/strong><\/dt>\n<dd id=\"fs-id1165137644990\">the first coordinate of an ordered pair, representing the horizontal displacement and direction from the origin<\/dd>\n<\/dl>\n<dl id=\"fs-id1165134297646\" class=\"definition\">\n<dt><strong><em>x-<\/em>intercept<\/strong><\/dt>\n<dd id=\"fs-id1165135486042\">the point where a graph intersects the <em>x-<\/em>axis; an ordered pair with a <em>y<\/em>-coordinate of zero<\/dd>\n<\/dl>\n<dl id=\"fs-id1165137644987\" class=\"definition\">\n<dt><strong><em>y<\/em>-axis<\/strong><\/dt>\n<dd id=\"fs-id1165137644990\">the common name of the vertical axis on a coordinate plane; a number line increasing from bottom to top<\/dd>\n<\/dl>\n<dl id=\"fs-id1165134297646\" class=\"definition\">\n<dt><strong><em>y-<\/em>coordinate<\/strong><\/dt>\n<dd id=\"fs-id1165135486042\">\u00a0the second coordinate of an ordered pair, representing the vertical displacement and direction from the origin<\/dd>\n<\/dl>\n<dl id=\"fs-id1165137644987\" class=\"definition\">\n<dt><strong><em>y<\/em>-intercept<\/strong><\/dt>\n<dd id=\"fs-id1165137644990\">a point where a graph intercepts the <em>y-<\/em>axis; an ordered pair with an <em>x<\/em>-coordinate of zero<\/dd>\n<\/dl>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-1747\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Revision and Adaptation. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>College Algebra. <strong>Authored by<\/strong>: Abramson, Jay et al.. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\">http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download for free at http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/li><li>Question ID 92752, 92753, 92757. <strong>Authored by<\/strong>: Michael Jenck. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License, CC-BY + GPL<\/li><li>Question ID 19140. <strong>Authored by<\/strong>: Amy Lambert. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community LicenseCC-BY + GPL<\/li><li>Question ID 2308. <strong>Authored by<\/strong>: David Lippman. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License, CC-BY + GPL<\/li><li>Question ID 110938, 110939. <strong>Authored by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC- BY + GPL<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":708740,"menu_order":9,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Revision and Adaptation\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"College Algebra\",\"author\":\"Abramson, Jay et al.\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download for free at http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\"},{\"type\":\"cc\",\"description\":\"Question ID 92752, 92753, 92757\",\"author\":\"Michael Jenck\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"IMathAS Community License, CC-BY + GPL\"},{\"type\":\"cc\",\"description\":\"Question ID 19140\",\"author\":\"Amy 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