{"id":379,"date":"2018-04-17T02:43:24","date_gmt":"2018-04-17T02:43:24","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/wm-accountingformanagers\/?post_type=chapter&#038;p=379"},"modified":"2024-04-26T22:09:30","modified_gmt":"2024-04-26T22:09:30","slug":"calculate-the-rate-of-change","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/wm-accountingformanagers\/chapter\/calculate-the-rate-of-change\/","title":{"raw":"Rate of Change","rendered":"Rate of Change"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning outcome<\/h3>\r\n<ul>\r\n \t<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Calculate the rate of change using data points and graphical representations&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:13185,&quot;3&quot;:[null,0],&quot;10&quot;:2,&quot;11&quot;:4,&quot;12&quot;:0,&quot;15&quot;:&quot;Calibri&quot;,&quot;16&quot;:10}\">Calculate the rate of change using data points and graphical representations<\/span><\/li>\r\n<\/ul>\r\n<\/div>\r\nFrequently we want to compare two different types of measurements, such as miles to gallons. To make this comparison, we use a rate. Examples of rates are [latex]120[\/latex] miles in [latex]2[\/latex] hours, [latex]160[\/latex] words in [latex]4[\/latex] minutes, and [latex]\\text{\\$5}[\/latex] dollars per [latex]64[\/latex] ounces.\r\n<div class=\"textbox shaded\">\r\n<h3>Rate<\/h3>\r\nA rate compares two quantities of different units. A rate is usually written as a fraction.\r\n\r\n<\/div>\r\nWhen writing a fraction as a rate, we put the first given amount with its units in the numerator and the second amount with its units in the denominator. When rates are simplified, the units remain in the numerator and denominator.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nBob drove his car [latex]525[\/latex] miles in [latex]9[\/latex] hours. Write this rate as a fraction.\r\n\r\nSolution\r\n<table id=\"eip-id1168466124872\" class=\"unnumbered unstyled\" summary=\".\" data-label=\"\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]\\text{525 miles in 9 hours}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Write as a fraction, with [latex]525[\/latex] miles in the numerator and [latex]9[\/latex] hours in the denominator.<\/td>\r\n<td>[latex]{\\Large\\frac{\\text{525 miles}}{\\text{9 hours}}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]{\\Large\\frac{\\text{175 miles}}{\\text{3 hours}}}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nSo [latex]525[\/latex] miles in [latex]9[\/latex] hours is equivalent to [latex]{\\Large\\frac{\\text{175 miles}}{\\text{3 hours}}}[\/latex]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146615[\/ohm_question]\r\n\r\n<\/div>\r\nhttps:\/\/youtu.be\/jlEJU-l5DWw\r\n<p data-type=\"title\">Let's examine how a rate is represented on a graph and determine how to identify it.<\/p>\r\n<p data-type=\"title\">Using rubber bands on a geoboard gives a concrete way to model lines on a coordinate grid. By stretching a rubber band between two pegs on a geoboard, we can discover how to find the slope of a line. And when you ride a bicycle, you <u data-effect=\"underline\">feel<\/u> the slope as you pump uphill or coast downhill.<\/p>\r\nWe\u2019ll start by stretching a rubber band between two pegs to make a line as shown in the image below.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/25224529\/CNX_BMath_Figure_11_04_001.png\" alt=\"The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style loop connecting the point in column 1 row 4 and the point in column 4 row 2.\" data-media-type=\"image\/png\" \/>\r\nDoes it look like a line?\r\n\r\nNow we stretch one part of the rubber band straight up from the left peg and around a third peg to make the sides of a right triangle as shown in the image below. We carefully make a [latex]90^ \\circ [\/latex] angle around the third peg, so that one side is vertical and the other is horizontal.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/25224530\/CNX_BMath_Figure_11_04_002.png\" alt=\"The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style triangle connecting three of the three points at column 1 row 2, column 1 row 4,and column 4 row 2.\" data-media-type=\"image\/png\" \/>\r\nTo find the slope of the line, we measure the distance along the vertical and horizontal legs of the triangle. The vertical distance is called the <em data-effect=\"italics\">rise<\/em> and the horizontal distance is called the <em data-effect=\"italics\">run<\/em>, as shown below.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/25224531\/CNX_BMath_Figure_11_04_003.png\" alt=\"This figure shows two arrows. The first arrow is vertical and is labeled rise. The second arrow is horizontal and labeled run. The second arrow points from the end of the first arrow.\" width=\"154\" height=\"111\" data-media-type=\"image\/png\" \/>\r\nTo help remember the terms, it may help to think of the images shown below.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/25224533\/CNX_BMath_Figure_11_04_004.png\" alt=\"A hot air balloon goes straight up, as if along the y-axis. Rise. A jogger runs straight across, as if along the x-axis. Run.\" width=\"364\" height=\"279\" data-media-type=\"image\/png\" \/>\r\nOn our geoboard, the rise is [latex]2[\/latex] units because the rubber band goes up [latex]2[\/latex] spaces on the vertical leg. See the image below.\r\n\r\nWhat is the run? Be sure to count the spaces between the pegs rather than the pegs themselves! The rubber band goes across [latex]3[\/latex] spaces on the horizontal leg, so the run is [latex]3[\/latex] units.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/25224536\/CNX_BMath_Figure_11_04_005.png\" alt=\"The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style triangle connecting three of the three points at column 1 row 2, column 1 row 4, and column 4 row 2. The triangle has a rise of 2 units and a run of 3 units.\" data-media-type=\"image\/png\" \/>\r\nThe slope of a line is the ratio of the rise to the run. So the slope of our line is [latex]{\\Large\\frac{2}{3}}[\/latex]. In mathematics, the slope is always represented by the letter [latex]m[\/latex].\r\n<div class=\"textbox shaded\">\r\n<h3>Slope of a line or rate of change<\/h3>\r\nThe slope of a line is [latex]m={\\Large\\frac{\\text{rise}}{\\text{run}}}[\/latex].\r\n\r\nThe rise measures the vertical change and the run measures the horizontal change.\r\n\r\n<\/div>\r\nWhat is the slope of the line on the geoboard in the image above?\r\n<p style=\"text-align: center;\">[latex]m={\\Large\\frac{\\text{rise}}{\\text{run}}}[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]m={\\Large\\frac{2}{3}}[\/latex]<\/p>\r\n<p style=\"text-align: center;\">The line has slope [latex]{\\Large\\frac{2}{3}}[\/latex]<\/p>\r\nIf we start by going up the rise is positive, and if we stretch it down the rise is negative. We will count the run from left to right, just like you read this paragraph, so the run will be positive.\r\n\r\nSince the slope formula has rise over run, it may be easier to always count out the rise first and then the run.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nWhat is the slope of the line on the geoboard shown?\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/25224537\/CNX_BMath_Figure_11_04_006.png\" alt=\"The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style loop connecting the point in column 1 row 5 and the point in column 5 row 2.\" data-media-type=\"image\/png\" \/>\r\n\r\nSolution\r\nUse the definition of slope.\r\n[latex]m={\\Large\\frac{\\text{rise}}{\\text{run}}}[\/latex]\r\n\r\nStart at the left peg and make a right triangle by stretching the rubber band up and to the right to reach the second peg.\r\nCount the rise and the run as shown.\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/25224538\/CNX_BMath_Figure_11_04_007.png\" alt=\"The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style triangle connecting three of the three points at column 1 row 2, column 1 row 5,and column 5 row 2.\" data-media-type=\"image\/png\" \/>\r\n[latex]\\begin{array}{cccc}\\text{The rise is }3\\text{ units}.\\hfill &amp; &amp; &amp; m={\\Large\\frac{3}{\\text{run}}}\\hfill \\\\ \\text{The run is}4\\text{ units}.\\hfill &amp; &amp; &amp; m={\\Large\\frac{3}{4}}\\hfill \\\\ &amp; &amp; &amp; \\text{The slope is }{\\Large\\frac{3}{4}}\\hfill \\end{array}[\/latex]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nWhat is the slope of the line on the geoboard shown?\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/25224542\/CNX_BMath_Figure_11_04_010.png\" alt=\"The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style loop connecting the point in column 1 row 3 and the point in column 4 row 4.\" data-media-type=\"image\/png\" \/>\r\n[reveal-answer q=\"698295\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"698295\"]\r\n\r\nSolution\r\nUse the definition of slope.\r\n[latex]m={\\Large\\frac{\\text{rise}}{\\text{run}}}[\/latex]\r\n\r\nStart at the left peg and make a right triangle by stretching the rubber band to the peg on the right. This time we need to stretch the rubber band down to make the vertical leg, so the rise is negative.\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/25224543\/CNX_BMath_Figure_11_04_011.png\" alt=\"The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style triangle connecting three of the three points at column 1 row 3, column 1 row 4,and column 4 row 4.\" data-media-type=\"image\/png\" \/>\r\n[latex]\\begin{array}{cccc}\\text{The rise is }-1.\\hfill &amp; &amp; &amp; m={\\Large\\frac{-1}{\\text{run}}}\\hfill \\\\ \\text{The run is}3.\\hfill &amp; &amp; &amp; m={\\Large\\frac{-1}{3}}\\hfill \\\\ &amp; &amp; &amp; m=-{\\Large\\frac{1}{3}}\\hfill \\\\ &amp; &amp; &amp; \\text{The slope is }-{\\Large\\frac{1}{3}}\\hfill \\end{array}[\/latex]\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[ohm_question]147013[\/ohm_question]\r\n\r\n<\/div>\r\nNotice that in the first example, the slope is positive and in the second example the slope is negative. Do you notice any difference in the two lines shown in the images below.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/25224547\/CNX_BMath_Figure_11_04_059.png\" alt=\"Two grids of 5 by 5 pegs. The first grid is labeled A. Below a, there is the equation m equals 3 fourths. Within the grid, a line is drawn from the bottom-right corner peg to the peg in the second row and fifth column. Below b, there is the equation m equals negative one third. Within the grid, a line is drawn from the first column of the third row to the fourth column of the fourth row.\" width=\"529\" height=\"328\" data-media-type=\"image\/png\" \/>\r\nAs you read from left to right, the line in Figure A, is going up; it has positive slope. The line Figure B is going down; it has negative slope.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/25224549\/CNX_BMath_Figure_11_04_060_img.png\" alt=\"A diagnol arrow points up and to the right, and it is labeled positive slope. Another diagnol arrow points down and to the right, and it is labeled negative slope.\" width=\"427\" height=\"122\" data-media-type=\"image\/png\" \/>\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nUse a geoboard to model a line with slope [latex]{\\Large\\frac{1}{2}}[\/latex].\r\n[reveal-answer q=\"225850\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"225850\"]\r\n\r\nSolution\r\nTo model a line with a specific slope on a geoboard, we need to know the rise and the run.\r\n<table id=\"eip-id1172468198215\" class=\"unnumbered unstyled\" summary=\".\" data-label=\"\">\r\n<tbody>\r\n<tr>\r\n<td>Use the slope formula.<\/td>\r\n<td>[latex]m=\\Large\\frac{\\text{rise}}{\\text{run}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Replace [latex]m[\/latex] with [latex]{\\Large\\frac{1}{2}}[\/latex] .<\/td>\r\n<td>[latex]\\Large\\frac{1}{2}\\normalsize =\\Large\\frac{\\text{rise}}{\\text{run}}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nSo, the rise is [latex]1[\/latex] unit and the run is [latex]2[\/latex] units.\r\nStart at a peg in the lower left of the geoboard. Stretch the rubber band up [latex]1[\/latex] unit, and then right [latex]2[\/latex] units.\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/25224550\/CNX_BMath_Figure_11_04_014.png\" alt=\"The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style triangle connecting three of the three points at column 1 row 3, column 1 row 4,and column 3 row 3.\" data-media-type=\"image\/png\" \/>\r\nThe hypotenuse of the right triangle formed by the rubber band represents a line with a slope of [latex]{\\Large\\frac{1}{2}}[\/latex].\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\nUse a geoboard to model a line with the given slope: [latex]m=\\Large\\frac{1}{3}[\/latex].\r\n[reveal-answer q=\"152695\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"152695\"]\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/25224551\/CNX_BMath_Figure_11_04_015_img.png\" alt=\"The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style triangle connecting three of the three points at column 2 row 3, column 2 row 4,and column 5 row 3.\" data-media-type=\"image\/png\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n&nbsp;\r\n\r\nUse a geoboard to model a line with the given slope: [latex]m=\\Large\\frac{3}{2}[\/latex].\r\n[reveal-answer q=\"64394\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"64394\"]\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/25224552\/CNX_BMath_Figure_11_04_016_img.png\" alt=\"The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style triangle connecting three of the three points at column 1 row 1, column 1 row 4,and column 3 row 1.\" data-media-type=\"image\/png\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nUse a geoboard to model a line with slope [latex]{\\Large\\frac{-1}{4}}[\/latex]\r\n[reveal-answer q=\"953715\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"953715\"]\r\n\r\nSolution\r\n<table id=\"eip-id1172466948346\" class=\"unnumbered unstyled\" summary=\".\" data-label=\"\">\r\n<tbody>\r\n<tr>\r\n<td>Use the slope formula.<\/td>\r\n<td>[latex]m=\\Large\\frac{\\text{rise}}{\\text{run}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Replace [latex]m[\/latex] with [latex]-{\\Large\\frac{1}{4}}[\/latex] .<\/td>\r\n<td>[latex]-\\Large\\frac{1}{4}\\normalsize =\\Large\\frac{\\text{rise}}{\\text{run}}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nSo, the rise is [latex]-1[\/latex] and the run is [latex]4[\/latex].\r\nSince the rise is negative, we choose a starting peg on the upper left that will give us room to count down. We stretch the rubber band down [latex]1[\/latex] unit, then to the right [latex]4[\/latex] units.\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/25224553\/CNX_BMath_Figure_11_04_017_img.png\" alt=\"The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style triangle connecting three of the three points at column 1 row 2, column 1 row 3,and column 5 row 3.\" data-media-type=\"image\/png\" \/>\r\nThe hypotenuse of the right triangle formed by the rubber band represents a line whose slope is [latex]-{\\Large\\frac{1}{4}}[\/latex].\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\nUse a geoboard to model a line with the given slope: [latex]m={\\Large\\frac{-3}{2}}[\/latex].\r\n[reveal-answer q=\"99402\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"99402\"]\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/25224555\/CNX_BMath_Figure_11_04_018_img.png\" alt=\"The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style triangle connecting three of the three points at column 2 row 3, column 2 row 5,and column 3 row 5.\" data-media-type=\"image\/png\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n&nbsp;\r\n\r\nUse a geoboard to model a line with the given slope: [latex]m={\\Large\\frac{-1}{3}}[\/latex].\r\n[reveal-answer q=\"714001\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"714001\"]\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/25224556\/CNX_BMath_Figure_11_04_019_img.png\" alt=\"The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style triangle connecting three of the three points at column 1 row 1, column 1 row 2,and column 4 row 2.\" data-media-type=\"image\/png\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<p data-type=\"title\">Now we\u2019ll look at some graphs on a coordinate grid to find their slopes. The method will be very similar to what we just modeled on our geoboards.<\/p>\r\nTo find the slope, we must count out the rise and the run. But where do we start?\r\n\r\nWe locate any two points on the line. We try to choose points with coordinates that are integers to make our calculations easier. We then start with the point on the left and sketch a right triangle, so we can count the rise and run.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nFind the slope of the line shown:\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/25224557\/CNX_BMath_Figure_11_04_020.png\" alt=\"The graph shows the x y-coordinate plane. The x-axis runs from -1 to 6. The y-axis runs from -4 to 2. A line passes through the points (0, negative 3) and (5, 1).\" width=\"192\" height=\"180\" data-media-type=\"image\/png\" \/>\r\n\r\nSolution\r\nLocate two points on the graph, choosing points whose coordinates are integers. We will use [latex]\\left(0,-3\\right)[\/latex] and [latex]\\left(5,1\\right)[\/latex].\r\n\r\nStarting with the point on the left, [latex]\\left(0,-3\\right)[\/latex], sketch a right triangle, going from the first point to the second point, [latex]\\left(5,1\\right)[\/latex].\r\n<table id=\"eip-id1168466130951\" class=\"unnumbered unstyled\" summary=\".\" data-label=\"\">\r\n<tbody>\r\n<tr>\r\n<td><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/25224558\/CNX_BMath_Figure_11_04_021.png\" alt=\"The graph shows the x y-coordinate plane. The x-axis runs from -1 to 6. The y-axis runs from -4 to 2. A line passes through the points (0, negative 3) and (5, 1). A line is drawn from each of the points to the point (0, 1). \" width=\"197\" height=\"181\" data-media-type=\"image\/png\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Count the rise on the vertical leg of the triangle.<\/td>\r\n<td>The rise is [latex]4[\/latex] units.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Count the run on the horizontal leg.<\/td>\r\n<td>The run is [latex]5[\/latex] units.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Use the slope formula.<\/td>\r\n<td>[latex]m={\\Large\\frac{\\text{rise}}{\\text{run}}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Substitute the values of the rise and run.<\/td>\r\n<td>[latex]m={\\Large\\frac{4}{5}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>The slope of the line is [latex]{\\Large\\frac{4}{5}}[\/latex] .<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\nNotice that the slope is positive since the line slants upward from left to right.\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]147014[\/ohm_question]\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">\r\n<h3>Find the slope from a graph<\/h3>\r\n<ol id=\"eip-id1168469837806\" class=\"stepwise\" data-number-style=\"arabic\">\r\n \t<li>Locate two points on the line whose coordinates are integers.<\/li>\r\n \t<li>Starting with the point on the left, sketch a right triangle, going from the first point to the second point.<\/li>\r\n \t<li>Count the rise and the run on the legs of the triangle.<\/li>\r\n \t<li>Take the ratio of rise to run to find the slope. [latex]m={\\Large\\frac{\\text{rise}}{\\text{run}}}[\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nFind the slope of the line shown:\r\n\r\n<img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/25224602\/CNX_BMath_Figure_11_04_024.png\" alt=\"The graph shows the x y-coordinate plane. The x-axis runs from -1 to 9. The y-axis runs from -1 to 7. A line passes through the points (0, 5) and (6, 1).\" width=\"259\" height=\"221\" data-media-type=\"image\/png\" \/>\r\n[reveal-answer q=\"966343\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"966343\"]\r\n\r\nSolution\r\nLocate two points on the graph. Look for points with coordinates that are integers. We can choose any points, but we will use [latex](0, 5)[\/latex] and [latex](3, 3)[\/latex]. Starting with the point on the left, sketch a right triangle, going from the first point to the second point.\r\n<table id=\"eip-id1168465988432\" class=\"unnumbered unstyled\" summary=\".\" data-label=\"\">\r\n<tbody>\r\n<tr>\r\n<td><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/25224603\/CNX_BMath_Figure_11_04_025.png\" alt=\"The graph shows the x y-coordinate plane. The x-axis runs from -1 to 9. The y-axis runs from -1 to 7. A line passes through the points (0, 5), (3, 3) and (6, 1). A line is drawn from both (0, 5) and (3, 3) to the point (0, 3).\" width=\"262\" height=\"225\" data-media-type=\"image\/png\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Count the rise \u2013 it is negative.<\/td>\r\n<td>The rise is [latex]-2[\/latex].<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Count the run.<\/td>\r\n<td>The run is [latex]3[\/latex].<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Use the slope formula.<\/td>\r\n<td>[latex]m=\\Large\\frac{\\text{rise}}{\\text{run}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Substitute the values of the rise and run.<\/td>\r\n<td>[latex]m={\\Large\\frac{-2}{3}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]m=-{\\Large\\frac{2}{3}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>The slope of the line is [latex]-{\\Large\\frac{2}{3}}[\/latex].<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nNotice that the slope is negative since the line slants downward from left to right.\r\n\r\nWhat if we had chosen different points? Let\u2019s find the slope of the line again, this time using different points. We will use the points [latex]\\left(-3,7\\right)[\/latex] and [latex]\\left(6,1\\right)[\/latex].\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/25224606\/CNX_BMath_Figure_11_04_043_img.png\" alt=\"The graph shows the x y-coordinate plane. The x-axis runs from -1 to 9. The y-axis runs from -1 to 7. A line passes through the points (6, 1) and (negative 3, 7).\" width=\"344\" height=\"353\" data-media-type=\"image\/png\" \/>\r\nStarting at [latex]\\left(-3,7\\right)[\/latex], sketch a right triangle to [latex]\\left(6,1\\right)[\/latex].\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/25224607\/CNX_BMath_Figure_11_04_044_img.png\" alt=\"The graph shows the x y-coordinate plane. The x-axis runs from -1 to 9. The y-axis runs from -1 to 7. A line passes through the points (6, 1) and (negative 3, 7). Lines are drawn from both points to the point (negative 3, 1). The line connecting (6, 1) and (negative 3, 1) is labeled run, and the line connecting (negative 3, 7) and (negative 3, 1) is labeled rise.\" width=\"344\" height=\"353\" data-media-type=\"image\/png\" \/>\r\n<table id=\"eip-id1168469716067\" class=\"unnumbered unstyled\" summary=\".\" data-label=\"\">\r\n<tbody>\r\n<tr>\r\n<td>Count the rise.<\/td>\r\n<td>The rise is [latex]-6[\/latex].<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Count the run.<\/td>\r\n<td>The run is [latex]9[\/latex].<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Use the slope formula.<\/td>\r\n<td>[latex]m=\\Large\\frac{\\text{rise}}{\\text{run}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Substitute the values of the rise and run.<\/td>\r\n<td>[latex]m={\\Large\\frac{-6}{9}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify the fraction.<\/td>\r\n<td>[latex]m=-{\\Large\\frac{2}{3}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>The slope of the line is [latex]-{\\Large\\frac{2}{3}}[\/latex].<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nIt does not matter which points you use\u2014the slope of the line is always the same. The slope of a line is constant!\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]147015[\/ohm_question]\r\n\r\n<\/div>\r\nThe lines in the previous examples had [latex]y[\/latex] -intercepts with integer values, so it was convenient to use the <em data-effect=\"italics\">y<\/em>-intercept as one of the points we used to find the slope. In the next example, the [latex]y[\/latex] -intercept is a fraction. The calculations are easier if we use two points with integer coordinates.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nFind the slope of the line shown:\r\n\r\n<img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/25224611\/CNX_BMath_Figure_11_04_045_img.png\" alt=\"The graph shows the x y-coordinate plane. The x-axis runs from 0 to 7. The y-axis runs from 0 to 8. A line passes through the points (2, 3) and (7, 6).\" width=\"238\" height=\"224\" data-media-type=\"image\/png\" \/>\r\n[reveal-answer q=\"439279\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"439279\"]\r\n\r\nSolution\r\n<table id=\"eip-id1170321819050\" class=\"unnumbered unstyled\" summary=\"...\" data-label=\"\">\r\n<tbody>\r\n<tr>\r\n<td>Locate two points on the graph whose coordinates are integers.<\/td>\r\n<td>[latex]\\left(2,3\\right)[\/latex] and [latex]\\left(7,6\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Which point is on the left?<\/td>\r\n<td>[latex]\\left(2,3\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Starting at [latex]\\left(2,3\\right)[\/latex] , sketch a right angle to [latex]\\left(7,6\\right)[\/latex] as shown below.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table id=\"eip-id1168467128258\" class=\"unnumbered unstyled\" summary=\".\" data-label=\"\">\r\n<tbody>\r\n<tr>\r\n<td><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/25224612\/CNX_BMath_Figure_11_04_046_img.png\" alt=\"The graph shows the x y-coordinate plane. The x-axis runs from 0 to 7. The y-axis runs from 0 to 8. A line passes through the points (2, 3) and (7, 6). Lines are drawn from both points to the point (2, 6). The line that connects (2, 3) and (2, 6) is labeled rise, and the line that connects (7, 6) to (2, 6) is labeled run.\" width=\"238\" height=\"224\" data-media-type=\"image\/png\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Count the rise.<\/td>\r\n<td>The rise is [latex]3[\/latex].<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Count the run.<\/td>\r\n<td>The run is [latex]5[\/latex].<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Use the slope formula.<\/td>\r\n<td>[latex]m=\\Large\\frac{\\text{rise}}{\\text{run}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Substitute the values of the rise and run.<\/td>\r\n<td>[latex]m={\\Large\\frac{3}{5}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>The slope of the line is [latex]{\\Large\\frac{3}{5}}[\/latex].<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]147016[\/ohm_question]\r\n\r\n<\/div>\r\nIn the following video we show another example of how to find the slope of a line given a graph. This graph has a positive slope.\r\n\r\nhttps:\/\/youtu.be\/zPognXmmaEo\r\n\r\nIn the following video we show another example of how to find the slope of a line given a graph. This graph has a negative slope.\r\n\r\nhttps:\/\/youtu.be\/dmla9Lj4rqg\r\n<p data-type=\"title\">Sometimes we need to find the slope of a line between two points and we might not have a graph to count out the rise and the run. We could plot the points on grid paper, then count out the rise and the run, but there is a way to find the slope without graphing.<\/p>\r\n<p data-type=\"title\">Before we get to it, we need to introduce some new algebraic notation. We have seen that an ordered pair [latex]\\left(x,y\\right)[\/latex] gives the coordinates of a point. But when we work with slopes, we use two points. How can the same symbol [latex]\\left(x,y\\right)[\/latex] be used to represent two different points?<\/p>\r\n<p data-type=\"title\">Mathematicians use subscripts to distinguish between the points. A subscript is a small number written to the right of, and a little lower than, a variable.<\/p>\r\n\r\n<ul id=\"fs-id1707109\" data-bullet-style=\"none\">\r\n \t<li>[latex]\\left({x}_{1},{y}_{1}\\right)\\text{ read }x\\text{ sub }1,y\\text{ sub }1[\/latex]<\/li>\r\n \t<li>[latex]\\left({x}_{2},{y}_{2}\\right)\\text{ read }x\\text{ sub }2,y\\text{ sub }2[\/latex]<\/li>\r\n<\/ul>\r\nWe will use [latex]\\left({x}_{1},{y}_{1}\\right)[\/latex] to identify the first point and [latex]\\left({x}_{2},{y}_{2}\\right)[\/latex] to identify the second point. If we had more than two points, we could use [latex]\\left({x}_{3},{y}_{3}\\right),\\left({x}_{4},{y}_{4}\\right)[\/latex], and so on.\r\n\r\nTo see how the rise and run relate to the coordinates of the two points, let\u2019s take another look at the slope of the line between the points [latex]\\left(2,3\\right)[\/latex] and [latex]\\left(7,6\\right)[\/latex] below.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/25224620\/CNX_BMath_Figure_11_04_030.png\" alt=\"The graph shows the x y-coordinate plane. The x-axis runs from 0 to 7. The y-axis runs from 0 to 7. A line runs through the labeled points 2, 3 and 7, 6. A line segment runs from the point 2, 3 to the unlabeled point 2, 6. It is labeled y sub 2 minus y sub 1, 6 minus 3, 3. A line segment runs from the point 7, 6 to the unlabeled point 2, 6. It os labeled x sub 2 minus x sub 1, 7 minus 2, 5. \" width=\"292\" height=\"285\" data-media-type=\"image\/png\" \/>\r\nSince we have two points, we will use subscript notation.\r\n<p style=\"text-align: center;\">[latex]\\stackrel{{x}_{1},{y}_{1}}{\\left(2,3\\right)}\\stackrel{{x}_{2},{y}_{2}}{\\left(7,6\\right)}[\/latex]<\/p>\r\nOn the graph, we counted the rise of [latex]3[\/latex]. The rise can also be found by subtracting the [latex]y\\text{-coordinates}[\/latex] of the points.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}{y}_{2}-{y}_{1}\\\\ 6 - 3\\\\ 3\\end{array}[\/latex]<\/p>\r\nWe counted a run of [latex]5[\/latex]. The run can also be found by subtracting the [latex]x\\text{-coordinates}[\/latex].\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}{x}_{2}-{x}_{1}\\\\ 7 - 2\\\\ 5\\end{array}[\/latex]<\/p>\r\n\r\n<table id=\"eip-id1168468520883\" class=\"unnumbered unstyled\" summary=\".\" data-label=\"\">\r\n<tbody>\r\n<tr>\r\n<td>We know<\/td>\r\n<td>[latex]m={\\Large\\frac{\\text{rise}}{\\text{run}}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>So<\/td>\r\n<td>[latex]m={\\Large\\frac{3}{5}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>We rewrite the rise and run by putting in the coordinates.<\/td>\r\n<td>[latex]m={\\Large\\frac{6 - 3}{7 - 2}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>But [latex]6[\/latex] is the [latex]y[\/latex] -coordinate of the second point, [latex]{y}_{2}[\/latex]\r\n\r\nand [latex]3[\/latex] is the [latex]y[\/latex] -coordinate of the first point [latex]{y}_{1}[\/latex] .\r\n\r\nSo we can rewrite the rise using subscript notation.<\/td>\r\n<td>[latex]m={\\Large\\frac{{y}_{2}-{y}_{1}}{7 - 2}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Also [latex]7[\/latex] is the [latex]x[\/latex] -coordinate of the second point, [latex]{x}_{2}[\/latex]\r\n\r\nand [latex]2[\/latex] is the [latex]x[\/latex] -coordinate of the first point [latex]{x}_{2}[\/latex] .\r\n\r\nSo we rewrite the run using subscript notation.<\/td>\r\n<td>[latex]m={\\Large\\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nWe\u2019ve shown that [latex]m={\\Large\\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}}[\/latex] is really another version of [latex]m={\\Large\\frac{\\text{rise}}{\\text{run}}}[\/latex]. We can use this formula to find the slope of a line when we have two points on the line.\r\n<div class=\"textbox shaded\">\r\n<h3>Slope Formula or rate of Change formula<\/h3>\r\nThe slope of the line between two points [latex]\\left({x}_{1},{y}_{1}\\right)[\/latex] and [latex]\\left({x}_{2},{y}_{2}\\right)[\/latex] is\r\n<p style=\"text-align: center;\">[latex]m={\\Large\\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}}[\/latex]<\/p>\r\nSay the formula to yourself to help you remember it:\r\n<p style=\"text-align: center;\">[latex]\\text{Slope is }y\\text{ of the second point minus }y\\text{ of the first point}[\/latex]\r\n[latex]\\text{over}[\/latex]\r\n[latex]x\\text{ of the second point minus }x\\text{ of the first point.}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nFind the slope of the line between the points [latex]\\left(1,2\\right)[\/latex] and [latex]\\left(4,5\\right)[\/latex].\r\n\r\nSolution\r\n<table id=\"eip-id1168468461864\" class=\"unnumbered unstyled\" summary=\".\" data-label=\"\">\r\n<tbody>\r\n<tr>\r\n<td>We\u2019ll call [latex]\\left(1,2\\right)[\/latex] point #1 and [latex]\\left(4,5\\right)[\/latex] point #2.<\/td>\r\n<td>[latex]\\stackrel{{x}_{1},{y}_{1}}{\\left(1,2\\right)}\\text{and}\\stackrel{{x}_{2},{y}_{2}}{\\left(4,5\\right)}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Use the slope formula.<\/td>\r\n<td>[latex]m={\\Large\\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Substitute the values in the slope formula:<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]y[\/latex] of the second point minus [latex]y[\/latex] of the first point<\/td>\r\n<td>[latex]m={\\Large\\frac{5 - 2}{{x}_{2}-{x}_{1}}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]x[\/latex] of the second point minus [latex]x[\/latex] of the first point<\/td>\r\n<td>[latex]m={\\Large\\frac{5 - 2}{4 - 1}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify the numerator and the denominator.<\/td>\r\n<td>[latex]m={\\Large\\frac{3}{3}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]m=1[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nLet\u2019s confirm this by counting out the slope on the graph.\r\n\r\n<img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/25224621\/CNX_BMath_Figure_11_04_031.png\" alt=\"The graph shows the x y-coordinate plane. The x-axis runs from -1 to 7. The y-axis runs from -1 to 7. A line passes through two points labeled (1, 2) and (4, 5). Lines are drawn connecting both points to the point (1, 5). The line connecting (1, 2) and (1, 5) is labeled rise, and the line connecting (4, 5) and (1, 5) is labeled run.\" width=\"219\" height=\"224\" data-media-type=\"image\/png\" \/>\r\nThe rise is [latex]3[\/latex] and the run is [latex]3[\/latex], so\r\n[latex]\\begin{array}{}\\\\ m=\\frac{\\text{rise}}{\\text{run}}\\hfill \\\\ m={\\Large\\frac{3}{3}}\\hfill \\\\ m=1\\hfill \\end{array}[\/latex]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]147021[\/ohm_question]\r\n\r\n<\/div>\r\nHow do we know which point to call #1 and which to call #2? Let\u2019s find the slope again, this time switching the names of the points to see what happens. Since we will now be counting the run from right to left, it will be negative.\r\n<table id=\"eip-id1168465988183\" class=\"unnumbered unstyled\" summary=\".\" data-label=\"\">\r\n<tbody>\r\n<tr>\r\n<td>We\u2019ll call [latex]\\left(4,5\\right)[\/latex] point #1 and [latex]\\left(1,2\\right)[\/latex] point #2.<\/td>\r\n<td>[latex]\\stackrel{{x}_{1},{y}_{1}}{\\left(4,5\\right)}\\text{and}\\stackrel{{x}_{2},{y}_{2}}{\\left(1,2\\right)}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Use the slope formula.<\/td>\r\n<td>[latex]m={\\Large\\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Substitute the values in the slope formula:<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]y[\/latex] of the second point minus [latex]y[\/latex] of the first point<\/td>\r\n<td>[latex]m={\\Large\\frac{2 - 5}{{x}_{2}-{x}_{1}}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]x[\/latex] of the second point minus [latex]x[\/latex] of the first point<\/td>\r\n<td>[latex]m={\\Large\\frac{2 - 5}{1 - 4}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify the numerator and the denominator.<\/td>\r\n<td>[latex]m={\\Large\\frac{-3}{-3}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]m=1[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nThe slope is the same no matter which order we use the points.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nFind the slope of the line through the points [latex]\\left(-2,-3\\right)[\/latex] and [latex]\\left(-7,4\\right)[\/latex].\r\n[reveal-answer q=\"265532\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"265532\"]\r\n\r\nSolution\r\n<table id=\"eip-id1168469701671\" class=\"unnumbered unstyled\" summary=\".\" data-label=\"\">\r\n<tbody>\r\n<tr>\r\n<td>We\u2019ll call [latex]\\left(-2,-3\\right)[\/latex] point #1 and [latex]\\left(-7,4\\right)[\/latex] point #2.<\/td>\r\n<td>[latex]\\stackrel{{x}_{1},{y}_{1}}{\\left(-2,-3\\right)}\\text{and}\\stackrel{{x}_{2},{y}_{2}}{\\left(-7,4\\right)}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Use the slope formula.<\/td>\r\n<td>[latex]m={\\Large\\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Substitute the values<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]y[\/latex] of the second point minus [latex]y[\/latex] of the first point<\/td>\r\n<td>[latex]m={\\Large\\frac{4-\\left(-3\\right)}{{x}_{2}-{x}_{1}}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]x[\/latex] of the second point minus [latex]x[\/latex] of the first point<\/td>\r\n<td>[latex]m={\\Large\\frac{4-\\left(-3\\right)}{-7-\\left(-2\\right)}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]m={\\Large\\frac{7}{-5}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]m=-{\\Large\\frac{7}{5}}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nLet\u2019s confirm this on the graph shown.\r\n\r\n<img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/25224622\/CNX_BMath_Figure_11_04_032.png\" alt=\"The graph shows the x y-coordinate plane. The x-axis runs from -8 to 2. The y-axis runs from -6 to 5. A line passes though the points labeled (negative 7, 4) and (negative 2, negative 3). Lines are drawn from both points to the point (negative 7, negative 3). The line connecting (negative 7, 4) and (negative 7, negative 3) is labeled rise, and the line connecting (negative 2, negative 3) and (negative 7, negative 3) is labeled run.\" width=\"262\" height=\"290\" data-media-type=\"image\/png\" \/>\r\n[latex]\\begin{array}{}\\\\ \\\\ \\\\ m=\\frac{\\text{rise}}{\\text{run}}\\\\ m={\\Large\\frac{-7}{5}}\\\\ m=-{\\Large\\frac{7}{5}}\\end{array}[\/latex]\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[ohm_question]147022[\/ohm_question]\r\n\r\n<\/div>\r\nWatch this video to see more examples of how to determine slope given two points on a line.\r\n\r\nhttps:\/\/youtu.be\/6qONExlVGgc","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning outcome<\/h3>\n<ul>\n<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Calculate the rate of change using data points and graphical representations&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:13185,&quot;3&quot;:[null,0],&quot;10&quot;:2,&quot;11&quot;:4,&quot;12&quot;:0,&quot;15&quot;:&quot;Calibri&quot;,&quot;16&quot;:10}\">Calculate the rate of change using data points and graphical representations<\/span><\/li>\n<\/ul>\n<\/div>\n<p>Frequently we want to compare two different types of measurements, such as miles to gallons. To make this comparison, we use a rate. Examples of rates are [latex]120[\/latex] miles in [latex]2[\/latex] hours, [latex]160[\/latex] words in [latex]4[\/latex] minutes, and [latex]\\text{\\$5}[\/latex] dollars per [latex]64[\/latex] ounces.<\/p>\n<div class=\"textbox shaded\">\n<h3>Rate<\/h3>\n<p>A rate compares two quantities of different units. A rate is usually written as a fraction.<\/p>\n<\/div>\n<p>When writing a fraction as a rate, we put the first given amount with its units in the numerator and the second amount with its units in the denominator. When rates are simplified, the units remain in the numerator and denominator.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Bob drove his car [latex]525[\/latex] miles in [latex]9[\/latex] hours. Write this rate as a fraction.<\/p>\n<p>Solution<\/p>\n<table id=\"eip-id1168466124872\" class=\"unnumbered unstyled\" summary=\".\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]\\text{525 miles in 9 hours}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Write as a fraction, with [latex]525[\/latex] miles in the numerator and [latex]9[\/latex] hours in the denominator.<\/td>\n<td>[latex]{\\Large\\frac{\\text{525 miles}}{\\text{9 hours}}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]{\\Large\\frac{\\text{175 miles}}{\\text{3 hours}}}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>So [latex]525[\/latex] miles in [latex]9[\/latex] hours is equivalent to [latex]{\\Large\\frac{\\text{175 miles}}{\\text{3 hours}}}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146615\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146615&theme=oea&iframe_resize_id=ohm146615&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Rates and Unit Rates\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/jlEJU-l5DWw?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p data-type=\"title\">Let&#8217;s examine how a rate is represented on a graph and determine how to identify it.<\/p>\n<p data-type=\"title\">Using rubber bands on a geoboard gives a concrete way to model lines on a coordinate grid. By stretching a rubber band between two pegs on a geoboard, we can discover how to find the slope of a line. And when you ride a bicycle, you <u data-effect=\"underline\">feel<\/u> the slope as you pump uphill or coast downhill.<\/p>\n<p>We\u2019ll start by stretching a rubber band between two pegs to make a line as shown in the image below.<\/p>\n<p><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/25224529\/CNX_BMath_Figure_11_04_001.png\" alt=\"The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style loop connecting the point in column 1 row 4 and the point in column 4 row 2.\" data-media-type=\"image\/png\" \/><br \/>\nDoes it look like a line?<\/p>\n<p>Now we stretch one part of the rubber band straight up from the left peg and around a third peg to make the sides of a right triangle as shown in the image below. We carefully make a [latex]90^ \\circ[\/latex] angle around the third peg, so that one side is vertical and the other is horizontal.<\/p>\n<p><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/25224530\/CNX_BMath_Figure_11_04_002.png\" alt=\"The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style triangle connecting three of the three points at column 1 row 2, column 1 row 4,and column 4 row 2.\" data-media-type=\"image\/png\" \/><br \/>\nTo find the slope of the line, we measure the distance along the vertical and horizontal legs of the triangle. The vertical distance is called the <em data-effect=\"italics\">rise<\/em> and the horizontal distance is called the <em data-effect=\"italics\">run<\/em>, as shown below.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/25224531\/CNX_BMath_Figure_11_04_003.png\" alt=\"This figure shows two arrows. The first arrow is vertical and is labeled rise. The second arrow is horizontal and labeled run. The second arrow points from the end of the first arrow.\" width=\"154\" height=\"111\" data-media-type=\"image\/png\" \/><br \/>\nTo help remember the terms, it may help to think of the images shown below.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/25224533\/CNX_BMath_Figure_11_04_004.png\" alt=\"A hot air balloon goes straight up, as if along the y-axis. Rise. A jogger runs straight across, as if along the x-axis. Run.\" width=\"364\" height=\"279\" data-media-type=\"image\/png\" \/><br \/>\nOn our geoboard, the rise is [latex]2[\/latex] units because the rubber band goes up [latex]2[\/latex] spaces on the vertical leg. See the image below.<\/p>\n<p>What is the run? Be sure to count the spaces between the pegs rather than the pegs themselves! The rubber band goes across [latex]3[\/latex] spaces on the horizontal leg, so the run is [latex]3[\/latex] units.<\/p>\n<p><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/25224536\/CNX_BMath_Figure_11_04_005.png\" alt=\"The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style triangle connecting three of the three points at column 1 row 2, column 1 row 4, and column 4 row 2. The triangle has a rise of 2 units and a run of 3 units.\" data-media-type=\"image\/png\" \/><br \/>\nThe slope of a line is the ratio of the rise to the run. So the slope of our line is [latex]{\\Large\\frac{2}{3}}[\/latex]. In mathematics, the slope is always represented by the letter [latex]m[\/latex].<\/p>\n<div class=\"textbox shaded\">\n<h3>Slope of a line or rate of change<\/h3>\n<p>The slope of a line is [latex]m={\\Large\\frac{\\text{rise}}{\\text{run}}}[\/latex].<\/p>\n<p>The rise measures the vertical change and the run measures the horizontal change.<\/p>\n<\/div>\n<p>What is the slope of the line on the geoboard in the image above?<\/p>\n<p style=\"text-align: center;\">[latex]m={\\Large\\frac{\\text{rise}}{\\text{run}}}[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]m={\\Large\\frac{2}{3}}[\/latex]<\/p>\n<p style=\"text-align: center;\">The line has slope [latex]{\\Large\\frac{2}{3}}[\/latex]<\/p>\n<p>If we start by going up the rise is positive, and if we stretch it down the rise is negative. We will count the run from left to right, just like you read this paragraph, so the run will be positive.<\/p>\n<p>Since the slope formula has rise over run, it may be easier to always count out the rise first and then the run.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>What is the slope of the line on the geoboard shown?<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/25224537\/CNX_BMath_Figure_11_04_006.png\" alt=\"The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style loop connecting the point in column 1 row 5 and the point in column 5 row 2.\" data-media-type=\"image\/png\" \/><\/p>\n<p>Solution<br \/>\nUse the definition of slope.<br \/>\n[latex]m={\\Large\\frac{\\text{rise}}{\\text{run}}}[\/latex]<\/p>\n<p>Start at the left peg and make a right triangle by stretching the rubber band up and to the right to reach the second peg.<br \/>\nCount the rise and the run as shown.<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/25224538\/CNX_BMath_Figure_11_04_007.png\" alt=\"The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style triangle connecting three of the three points at column 1 row 2, column 1 row 5,and column 5 row 2.\" data-media-type=\"image\/png\" \/><br \/>\n[latex]\\begin{array}{cccc}\\text{The rise is }3\\text{ units}.\\hfill & & & m={\\Large\\frac{3}{\\text{run}}}\\hfill \\\\ \\text{The run is}4\\text{ units}.\\hfill & & & m={\\Large\\frac{3}{4}}\\hfill \\\\ & & & \\text{The slope is }{\\Large\\frac{3}{4}}\\hfill \\end{array}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>What is the slope of the line on the geoboard shown?<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/25224542\/CNX_BMath_Figure_11_04_010.png\" alt=\"The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style loop connecting the point in column 1 row 3 and the point in column 4 row 4.\" data-media-type=\"image\/png\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q698295\">Show Answer<\/span><\/p>\n<div id=\"q698295\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<br \/>\nUse the definition of slope.<br \/>\n[latex]m={\\Large\\frac{\\text{rise}}{\\text{run}}}[\/latex]<\/p>\n<p>Start at the left peg and make a right triangle by stretching the rubber band to the peg on the right. This time we need to stretch the rubber band down to make the vertical leg, so the rise is negative.<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/25224543\/CNX_BMath_Figure_11_04_011.png\" alt=\"The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style triangle connecting three of the three points at column 1 row 3, column 1 row 4,and column 4 row 4.\" data-media-type=\"image\/png\" \/><br \/>\n[latex]\\begin{array}{cccc}\\text{The rise is }-1.\\hfill & & & m={\\Large\\frac{-1}{\\text{run}}}\\hfill \\\\ \\text{The run is}3.\\hfill & & & m={\\Large\\frac{-1}{3}}\\hfill \\\\ & & & m=-{\\Large\\frac{1}{3}}\\hfill \\\\ & & & \\text{The slope is }-{\\Large\\frac{1}{3}}\\hfill \\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm147013\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=147013&theme=oea&iframe_resize_id=ohm147013&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>Notice that in the first example, the slope is positive and in the second example the slope is negative. Do you notice any difference in the two lines shown in the images below.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/25224547\/CNX_BMath_Figure_11_04_059.png\" alt=\"Two grids of 5 by 5 pegs. The first grid is labeled A. Below a, there is the equation m equals 3 fourths. Within the grid, a line is drawn from the bottom-right corner peg to the peg in the second row and fifth column. Below b, there is the equation m equals negative one third. Within the grid, a line is drawn from the first column of the third row to the fourth column of the fourth row.\" width=\"529\" height=\"328\" data-media-type=\"image\/png\" \/><br \/>\nAs you read from left to right, the line in Figure A, is going up; it has positive slope. The line Figure B is going down; it has negative slope.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/25224549\/CNX_BMath_Figure_11_04_060_img.png\" alt=\"A diagnol arrow points up and to the right, and it is labeled positive slope. Another diagnol arrow points down and to the right, and it is labeled negative slope.\" width=\"427\" height=\"122\" data-media-type=\"image\/png\" \/><\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Use a geoboard to model a line with slope [latex]{\\Large\\frac{1}{2}}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q225850\">Show Answer<\/span><\/p>\n<div id=\"q225850\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<br \/>\nTo model a line with a specific slope on a geoboard, we need to know the rise and the run.<\/p>\n<table id=\"eip-id1172468198215\" class=\"unnumbered unstyled\" summary=\".\" data-label=\"\">\n<tbody>\n<tr>\n<td>Use the slope formula.<\/td>\n<td>[latex]m=\\Large\\frac{\\text{rise}}{\\text{run}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Replace [latex]m[\/latex] with [latex]{\\Large\\frac{1}{2}}[\/latex] .<\/td>\n<td>[latex]\\Large\\frac{1}{2}\\normalsize =\\Large\\frac{\\text{rise}}{\\text{run}}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>So, the rise is [latex]1[\/latex] unit and the run is [latex]2[\/latex] units.<br \/>\nStart at a peg in the lower left of the geoboard. Stretch the rubber band up [latex]1[\/latex] unit, and then right [latex]2[\/latex] units.<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/25224550\/CNX_BMath_Figure_11_04_014.png\" alt=\"The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style triangle connecting three of the three points at column 1 row 3, column 1 row 4,and column 3 row 3.\" data-media-type=\"image\/png\" \/><br \/>\nThe hypotenuse of the right triangle formed by the rubber band represents a line with a slope of [latex]{\\Large\\frac{1}{2}}[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p>Use a geoboard to model a line with the given slope: [latex]m=\\Large\\frac{1}{3}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q152695\">Show Answer<\/span><\/p>\n<div id=\"q152695\" class=\"hidden-answer\" style=\"display: none\">\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/25224551\/CNX_BMath_Figure_11_04_015_img.png\" alt=\"The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style triangle connecting three of the three points at column 2 row 3, column 2 row 4,and column 5 row 3.\" data-media-type=\"image\/png\" \/><\/p>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<p>Use a geoboard to model a line with the given slope: [latex]m=\\Large\\frac{3}{2}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q64394\">Show Answer<\/span><\/p>\n<div id=\"q64394\" class=\"hidden-answer\" style=\"display: none\">\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/25224552\/CNX_BMath_Figure_11_04_016_img.png\" alt=\"The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style triangle connecting three of the three points at column 1 row 1, column 1 row 4,and column 3 row 1.\" data-media-type=\"image\/png\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Use a geoboard to model a line with slope [latex]{\\Large\\frac{-1}{4}}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q953715\">Show Answer<\/span><\/p>\n<div id=\"q953715\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-id1172466948346\" class=\"unnumbered unstyled\" summary=\".\" data-label=\"\">\n<tbody>\n<tr>\n<td>Use the slope formula.<\/td>\n<td>[latex]m=\\Large\\frac{\\text{rise}}{\\text{run}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Replace [latex]m[\/latex] with [latex]-{\\Large\\frac{1}{4}}[\/latex] .<\/td>\n<td>[latex]-\\Large\\frac{1}{4}\\normalsize =\\Large\\frac{\\text{rise}}{\\text{run}}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>So, the rise is [latex]-1[\/latex] and the run is [latex]4[\/latex].<br \/>\nSince the rise is negative, we choose a starting peg on the upper left that will give us room to count down. We stretch the rubber band down [latex]1[\/latex] unit, then to the right [latex]4[\/latex] units.<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/25224553\/CNX_BMath_Figure_11_04_017_img.png\" alt=\"The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style triangle connecting three of the three points at column 1 row 2, column 1 row 3,and column 5 row 3.\" data-media-type=\"image\/png\" \/><br \/>\nThe hypotenuse of the right triangle formed by the rubber band represents a line whose slope is [latex]-{\\Large\\frac{1}{4}}[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p>Use a geoboard to model a line with the given slope: [latex]m={\\Large\\frac{-3}{2}}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q99402\">Show Answer<\/span><\/p>\n<div id=\"q99402\" class=\"hidden-answer\" style=\"display: none\">\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/25224555\/CNX_BMath_Figure_11_04_018_img.png\" alt=\"The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style triangle connecting three of the three points at column 2 row 3, column 2 row 5,and column 3 row 5.\" data-media-type=\"image\/png\" \/><\/p>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<p>Use a geoboard to model a line with the given slope: [latex]m={\\Large\\frac{-1}{3}}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q714001\">Show Answer<\/span><\/p>\n<div id=\"q714001\" class=\"hidden-answer\" style=\"display: none\">\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/25224556\/CNX_BMath_Figure_11_04_019_img.png\" alt=\"The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style triangle connecting three of the three points at column 1 row 1, column 1 row 2,and column 4 row 2.\" data-media-type=\"image\/png\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p data-type=\"title\">Now we\u2019ll look at some graphs on a coordinate grid to find their slopes. The method will be very similar to what we just modeled on our geoboards.<\/p>\n<p>To find the slope, we must count out the rise and the run. But where do we start?<\/p>\n<p>We locate any two points on the line. We try to choose points with coordinates that are integers to make our calculations easier. We then start with the point on the left and sketch a right triangle, so we can count the rise and run.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Find the slope of the line shown:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/25224557\/CNX_BMath_Figure_11_04_020.png\" alt=\"The graph shows the x y-coordinate plane. The x-axis runs from -1 to 6. The y-axis runs from -4 to 2. A line passes through the points (0, negative 3) and (5, 1).\" width=\"192\" height=\"180\" data-media-type=\"image\/png\" \/><\/p>\n<p>Solution<br \/>\nLocate two points on the graph, choosing points whose coordinates are integers. We will use [latex]\\left(0,-3\\right)[\/latex] and [latex]\\left(5,1\\right)[\/latex].<\/p>\n<p>Starting with the point on the left, [latex]\\left(0,-3\\right)[\/latex], sketch a right triangle, going from the first point to the second point, [latex]\\left(5,1\\right)[\/latex].<\/p>\n<table id=\"eip-id1168466130951\" class=\"unnumbered unstyled\" summary=\".\" data-label=\"\">\n<tbody>\n<tr>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/25224558\/CNX_BMath_Figure_11_04_021.png\" alt=\"The graph shows the x y-coordinate plane. The x-axis runs from -1 to 6. The y-axis runs from -4 to 2. A line passes through the points (0, negative 3) and (5, 1). A line is drawn from each of the points to the point (0, 1).\" width=\"197\" height=\"181\" data-media-type=\"image\/png\" \/><\/td>\n<\/tr>\n<tr>\n<td>Count the rise on the vertical leg of the triangle.<\/td>\n<td>The rise is [latex]4[\/latex] units.<\/td>\n<\/tr>\n<tr>\n<td>Count the run on the horizontal leg.<\/td>\n<td>The run is [latex]5[\/latex] units.<\/td>\n<\/tr>\n<tr>\n<td>Use the slope formula.<\/td>\n<td>[latex]m={\\Large\\frac{\\text{rise}}{\\text{run}}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Substitute the values of the rise and run.<\/td>\n<td>[latex]m={\\Large\\frac{4}{5}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>The slope of the line is [latex]{\\Large\\frac{4}{5}}[\/latex] .<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p>Notice that the slope is positive since the line slants upward from left to right.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm147014\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=147014&theme=oea&iframe_resize_id=ohm147014&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Find the slope from a graph<\/h3>\n<ol id=\"eip-id1168469837806\" class=\"stepwise\" data-number-style=\"arabic\">\n<li>Locate two points on the line whose coordinates are integers.<\/li>\n<li>Starting with the point on the left, sketch a right triangle, going from the first point to the second point.<\/li>\n<li>Count the rise and the run on the legs of the triangle.<\/li>\n<li>Take the ratio of rise to run to find the slope. [latex]m={\\Large\\frac{\\text{rise}}{\\text{run}}}[\/latex]<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Find the slope of the line shown:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/25224602\/CNX_BMath_Figure_11_04_024.png\" alt=\"The graph shows the x y-coordinate plane. The x-axis runs from -1 to 9. The y-axis runs from -1 to 7. A line passes through the points (0, 5) and (6, 1).\" width=\"259\" height=\"221\" data-media-type=\"image\/png\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q966343\">Show Answer<\/span><\/p>\n<div id=\"q966343\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<br \/>\nLocate two points on the graph. Look for points with coordinates that are integers. We can choose any points, but we will use [latex](0, 5)[\/latex] and [latex](3, 3)[\/latex]. Starting with the point on the left, sketch a right triangle, going from the first point to the second point.<\/p>\n<table id=\"eip-id1168465988432\" class=\"unnumbered unstyled\" summary=\".\" data-label=\"\">\n<tbody>\n<tr>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/25224603\/CNX_BMath_Figure_11_04_025.png\" alt=\"The graph shows the x y-coordinate plane. The x-axis runs from -1 to 9. The y-axis runs from -1 to 7. A line passes through the points (0, 5), (3, 3) and (6, 1). A line is drawn from both (0, 5) and (3, 3) to the point (0, 3).\" width=\"262\" height=\"225\" data-media-type=\"image\/png\" \/><\/td>\n<\/tr>\n<tr>\n<td>Count the rise \u2013 it is negative.<\/td>\n<td>The rise is [latex]-2[\/latex].<\/td>\n<\/tr>\n<tr>\n<td>Count the run.<\/td>\n<td>The run is [latex]3[\/latex].<\/td>\n<\/tr>\n<tr>\n<td>Use the slope formula.<\/td>\n<td>[latex]m=\\Large\\frac{\\text{rise}}{\\text{run}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Substitute the values of the rise and run.<\/td>\n<td>[latex]m={\\Large\\frac{-2}{3}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]m=-{\\Large\\frac{2}{3}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>The slope of the line is [latex]-{\\Large\\frac{2}{3}}[\/latex].<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>Notice that the slope is negative since the line slants downward from left to right.<\/p>\n<p>What if we had chosen different points? Let\u2019s find the slope of the line again, this time using different points. We will use the points [latex]\\left(-3,7\\right)[\/latex] and [latex]\\left(6,1\\right)[\/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\/277\/2017\/04\/25224606\/CNX_BMath_Figure_11_04_043_img.png\" alt=\"The graph shows the x y-coordinate plane. The x-axis runs from -1 to 9. The y-axis runs from -1 to 7. A line passes through the points (6, 1) and (negative 3, 7).\" width=\"344\" height=\"353\" data-media-type=\"image\/png\" \/><br \/>\nStarting at [latex]\\left(-3,7\\right)[\/latex], sketch a right triangle to [latex]\\left(6,1\\right)[\/latex].<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/25224607\/CNX_BMath_Figure_11_04_044_img.png\" alt=\"The graph shows the x y-coordinate plane. The x-axis runs from -1 to 9. The y-axis runs from -1 to 7. A line passes through the points (6, 1) and (negative 3, 7). Lines are drawn from both points to the point (negative 3, 1). The line connecting (6, 1) and (negative 3, 1) is labeled run, and the line connecting (negative 3, 7) and (negative 3, 1) is labeled rise.\" width=\"344\" height=\"353\" data-media-type=\"image\/png\" \/><\/p>\n<table id=\"eip-id1168469716067\" class=\"unnumbered unstyled\" summary=\".\" data-label=\"\">\n<tbody>\n<tr>\n<td>Count the rise.<\/td>\n<td>The rise is [latex]-6[\/latex].<\/td>\n<\/tr>\n<tr>\n<td>Count the run.<\/td>\n<td>The run is [latex]9[\/latex].<\/td>\n<\/tr>\n<tr>\n<td>Use the slope formula.<\/td>\n<td>[latex]m=\\Large\\frac{\\text{rise}}{\\text{run}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Substitute the values of the rise and run.<\/td>\n<td>[latex]m={\\Large\\frac{-6}{9}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify the fraction.<\/td>\n<td>[latex]m=-{\\Large\\frac{2}{3}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>The slope of the line is [latex]-{\\Large\\frac{2}{3}}[\/latex].<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>It does not matter which points you use\u2014the slope of the line is always the same. The slope of a line is constant!<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm147015\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=147015&theme=oea&iframe_resize_id=ohm147015&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>The lines in the previous examples had [latex]y[\/latex] -intercepts with integer values, so it was convenient to use the <em data-effect=\"italics\">y<\/em>-intercept as one of the points we used to find the slope. In the next example, the [latex]y[\/latex] -intercept is a fraction. The calculations are easier if we use two points with integer coordinates.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Find the slope of the line shown:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/25224611\/CNX_BMath_Figure_11_04_045_img.png\" alt=\"The graph shows the x y-coordinate plane. The x-axis runs from 0 to 7. The y-axis runs from 0 to 8. A line passes through the points (2, 3) and (7, 6).\" width=\"238\" height=\"224\" data-media-type=\"image\/png\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q439279\">Show Answer<\/span><\/p>\n<div id=\"q439279\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-id1170321819050\" class=\"unnumbered unstyled\" summary=\"...\" data-label=\"\">\n<tbody>\n<tr>\n<td>Locate two points on the graph whose coordinates are integers.<\/td>\n<td>[latex]\\left(2,3\\right)[\/latex] and [latex]\\left(7,6\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Which point is on the left?<\/td>\n<td>[latex]\\left(2,3\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Starting at [latex]\\left(2,3\\right)[\/latex] , sketch a right angle to [latex]\\left(7,6\\right)[\/latex] as shown below.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table id=\"eip-id1168467128258\" class=\"unnumbered unstyled\" summary=\".\" data-label=\"\">\n<tbody>\n<tr>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/25224612\/CNX_BMath_Figure_11_04_046_img.png\" alt=\"The graph shows the x y-coordinate plane. The x-axis runs from 0 to 7. The y-axis runs from 0 to 8. A line passes through the points (2, 3) and (7, 6). Lines are drawn from both points to the point (2, 6). The line that connects (2, 3) and (2, 6) is labeled rise, and the line that connects (7, 6) to (2, 6) is labeled run.\" width=\"238\" height=\"224\" data-media-type=\"image\/png\" \/><\/td>\n<\/tr>\n<tr>\n<td>Count the rise.<\/td>\n<td>The rise is [latex]3[\/latex].<\/td>\n<\/tr>\n<tr>\n<td>Count the run.<\/td>\n<td>The run is [latex]5[\/latex].<\/td>\n<\/tr>\n<tr>\n<td>Use the slope formula.<\/td>\n<td>[latex]m=\\Large\\frac{\\text{rise}}{\\text{run}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Substitute the values of the rise and run.<\/td>\n<td>[latex]m={\\Large\\frac{3}{5}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>The slope of the line is [latex]{\\Large\\frac{3}{5}}[\/latex].<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm147016\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=147016&theme=oea&iframe_resize_id=ohm147016&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>In the following video we show another example of how to find the slope of a line given a graph. This graph has a positive slope.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Ex 1: Determine the Slope Given the Graph of a Line (positive slope)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/zPognXmmaEo?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>In the following video we show another example of how to find the slope of a line given a graph. This graph has a negative slope.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-3\" title=\"Ex 2: Determine the Slope Given the Graph of a Line (negative slope)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/dmla9Lj4rqg?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p data-type=\"title\">Sometimes we need to find the slope of a line between two points and we might not have a graph to count out the rise and the run. We could plot the points on grid paper, then count out the rise and the run, but there is a way to find the slope without graphing.<\/p>\n<p data-type=\"title\">Before we get to it, we need to introduce some new algebraic notation. We have seen that an ordered pair [latex]\\left(x,y\\right)[\/latex] gives the coordinates of a point. But when we work with slopes, we use two points. How can the same symbol [latex]\\left(x,y\\right)[\/latex] be used to represent two different points?<\/p>\n<p data-type=\"title\">Mathematicians use subscripts to distinguish between the points. A subscript is a small number written to the right of, and a little lower than, a variable.<\/p>\n<ul id=\"fs-id1707109\" data-bullet-style=\"none\">\n<li>[latex]\\left({x}_{1},{y}_{1}\\right)\\text{ read }x\\text{ sub }1,y\\text{ sub }1[\/latex]<\/li>\n<li>[latex]\\left({x}_{2},{y}_{2}\\right)\\text{ read }x\\text{ sub }2,y\\text{ sub }2[\/latex]<\/li>\n<\/ul>\n<p>We will use [latex]\\left({x}_{1},{y}_{1}\\right)[\/latex] to identify the first point and [latex]\\left({x}_{2},{y}_{2}\\right)[\/latex] to identify the second point. If we had more than two points, we could use [latex]\\left({x}_{3},{y}_{3}\\right),\\left({x}_{4},{y}_{4}\\right)[\/latex], and so on.<\/p>\n<p>To see how the rise and run relate to the coordinates of the two points, let\u2019s take another look at the slope of the line between the points [latex]\\left(2,3\\right)[\/latex] and [latex]\\left(7,6\\right)[\/latex] below.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/25224620\/CNX_BMath_Figure_11_04_030.png\" alt=\"The graph shows the x y-coordinate plane. The x-axis runs from 0 to 7. The y-axis runs from 0 to 7. A line runs through the labeled points 2, 3 and 7, 6. A line segment runs from the point 2, 3 to the unlabeled point 2, 6. It is labeled y sub 2 minus y sub 1, 6 minus 3, 3. A line segment runs from the point 7, 6 to the unlabeled point 2, 6. It os labeled x sub 2 minus x sub 1, 7 minus 2, 5.\" width=\"292\" height=\"285\" data-media-type=\"image\/png\" \/><br \/>\nSince we have two points, we will use subscript notation.<\/p>\n<p style=\"text-align: center;\">[latex]\\stackrel{{x}_{1},{y}_{1}}{\\left(2,3\\right)}\\stackrel{{x}_{2},{y}_{2}}{\\left(7,6\\right)}[\/latex]<\/p>\n<p>On the graph, we counted the rise of [latex]3[\/latex]. The rise can also be found by subtracting the [latex]y\\text{-coordinates}[\/latex] of the points.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}{y}_{2}-{y}_{1}\\\\ 6 - 3\\\\ 3\\end{array}[\/latex]<\/p>\n<p>We counted a run of [latex]5[\/latex]. The run can also be found by subtracting the [latex]x\\text{-coordinates}[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}{x}_{2}-{x}_{1}\\\\ 7 - 2\\\\ 5\\end{array}[\/latex]<\/p>\n<table id=\"eip-id1168468520883\" class=\"unnumbered unstyled\" summary=\".\" data-label=\"\">\n<tbody>\n<tr>\n<td>We know<\/td>\n<td>[latex]m={\\Large\\frac{\\text{rise}}{\\text{run}}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>So<\/td>\n<td>[latex]m={\\Large\\frac{3}{5}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>We rewrite the rise and run by putting in the coordinates.<\/td>\n<td>[latex]m={\\Large\\frac{6 - 3}{7 - 2}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>But [latex]6[\/latex] is the [latex]y[\/latex] -coordinate of the second point, [latex]{y}_{2}[\/latex]<\/p>\n<p>and [latex]3[\/latex] is the [latex]y[\/latex] -coordinate of the first point [latex]{y}_{1}[\/latex] .<\/p>\n<p>So we can rewrite the rise using subscript notation.<\/td>\n<td>[latex]m={\\Large\\frac{{y}_{2}-{y}_{1}}{7 - 2}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Also [latex]7[\/latex] is the [latex]x[\/latex] -coordinate of the second point, [latex]{x}_{2}[\/latex]<\/p>\n<p>and [latex]2[\/latex] is the [latex]x[\/latex] -coordinate of the first point [latex]{x}_{2}[\/latex] .<\/p>\n<p>So we rewrite the run using subscript notation.<\/td>\n<td>[latex]m={\\Large\\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>We\u2019ve shown that [latex]m={\\Large\\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}}[\/latex] is really another version of [latex]m={\\Large\\frac{\\text{rise}}{\\text{run}}}[\/latex]. We can use this formula to find the slope of a line when we have two points on the line.<\/p>\n<div class=\"textbox shaded\">\n<h3>Slope Formula or rate of Change formula<\/h3>\n<p>The slope of the line between two points [latex]\\left({x}_{1},{y}_{1}\\right)[\/latex] and [latex]\\left({x}_{2},{y}_{2}\\right)[\/latex] is<\/p>\n<p style=\"text-align: center;\">[latex]m={\\Large\\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}}[\/latex]<\/p>\n<p>Say the formula to yourself to help you remember it:<\/p>\n<p style=\"text-align: center;\">[latex]\\text{Slope is }y\\text{ of the second point minus }y\\text{ of the first point}[\/latex]<br \/>\n[latex]\\text{over}[\/latex]<br \/>\n[latex]x\\text{ of the second point minus }x\\text{ of the first point.}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Find the slope of the line between the points [latex]\\left(1,2\\right)[\/latex] and [latex]\\left(4,5\\right)[\/latex].<\/p>\n<p>Solution<\/p>\n<table id=\"eip-id1168468461864\" class=\"unnumbered unstyled\" summary=\".\" data-label=\"\">\n<tbody>\n<tr>\n<td>We\u2019ll call [latex]\\left(1,2\\right)[\/latex] point #1 and [latex]\\left(4,5\\right)[\/latex] point #2.<\/td>\n<td>[latex]\\stackrel{{x}_{1},{y}_{1}}{\\left(1,2\\right)}\\text{and}\\stackrel{{x}_{2},{y}_{2}}{\\left(4,5\\right)}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Use the slope formula.<\/td>\n<td>[latex]m={\\Large\\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Substitute the values in the slope formula:<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>[latex]y[\/latex] of the second point minus [latex]y[\/latex] of the first point<\/td>\n<td>[latex]m={\\Large\\frac{5 - 2}{{x}_{2}-{x}_{1}}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]x[\/latex] of the second point minus [latex]x[\/latex] of the first point<\/td>\n<td>[latex]m={\\Large\\frac{5 - 2}{4 - 1}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify the numerator and the denominator.<\/td>\n<td>[latex]m={\\Large\\frac{3}{3}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]m=1[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Let\u2019s confirm this by counting out the slope on the graph.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/25224621\/CNX_BMath_Figure_11_04_031.png\" alt=\"The graph shows the x y-coordinate plane. The x-axis runs from -1 to 7. The y-axis runs from -1 to 7. A line passes through two points labeled (1, 2) and (4, 5). Lines are drawn connecting both points to the point (1, 5). The line connecting (1, 2) and (1, 5) is labeled rise, and the line connecting (4, 5) and (1, 5) is labeled run.\" width=\"219\" height=\"224\" data-media-type=\"image\/png\" \/><br \/>\nThe rise is [latex]3[\/latex] and the run is [latex]3[\/latex], so<br \/>\n[latex]\\begin{array}{}\\\\ m=\\frac{\\text{rise}}{\\text{run}}\\hfill \\\\ m={\\Large\\frac{3}{3}}\\hfill \\\\ m=1\\hfill \\end{array}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm147021\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=147021&theme=oea&iframe_resize_id=ohm147021&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>How do we know which point to call #1 and which to call #2? Let\u2019s find the slope again, this time switching the names of the points to see what happens. Since we will now be counting the run from right to left, it will be negative.<\/p>\n<table id=\"eip-id1168465988183\" class=\"unnumbered unstyled\" summary=\".\" data-label=\"\">\n<tbody>\n<tr>\n<td>We\u2019ll call [latex]\\left(4,5\\right)[\/latex] point #1 and [latex]\\left(1,2\\right)[\/latex] point #2.<\/td>\n<td>[latex]\\stackrel{{x}_{1},{y}_{1}}{\\left(4,5\\right)}\\text{and}\\stackrel{{x}_{2},{y}_{2}}{\\left(1,2\\right)}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Use the slope formula.<\/td>\n<td>[latex]m={\\Large\\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Substitute the values in the slope formula:<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>[latex]y[\/latex] of the second point minus [latex]y[\/latex] of the first point<\/td>\n<td>[latex]m={\\Large\\frac{2 - 5}{{x}_{2}-{x}_{1}}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]x[\/latex] of the second point minus [latex]x[\/latex] of the first point<\/td>\n<td>[latex]m={\\Large\\frac{2 - 5}{1 - 4}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify the numerator and the denominator.<\/td>\n<td>[latex]m={\\Large\\frac{-3}{-3}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]m=1[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>The slope is the same no matter which order we use the points.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Find the slope of the line through the points [latex]\\left(-2,-3\\right)[\/latex] and [latex]\\left(-7,4\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q265532\">Show Answer<\/span><\/p>\n<div id=\"q265532\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-id1168469701671\" class=\"unnumbered unstyled\" summary=\".\" data-label=\"\">\n<tbody>\n<tr>\n<td>We\u2019ll call [latex]\\left(-2,-3\\right)[\/latex] point #1 and [latex]\\left(-7,4\\right)[\/latex] point #2.<\/td>\n<td>[latex]\\stackrel{{x}_{1},{y}_{1}}{\\left(-2,-3\\right)}\\text{and}\\stackrel{{x}_{2},{y}_{2}}{\\left(-7,4\\right)}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Use the slope formula.<\/td>\n<td>[latex]m={\\Large\\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Substitute the values<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>[latex]y[\/latex] of the second point minus [latex]y[\/latex] of the first point<\/td>\n<td>[latex]m={\\Large\\frac{4-\\left(-3\\right)}{{x}_{2}-{x}_{1}}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]x[\/latex] of the second point minus [latex]x[\/latex] of the first point<\/td>\n<td>[latex]m={\\Large\\frac{4-\\left(-3\\right)}{-7-\\left(-2\\right)}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]m={\\Large\\frac{7}{-5}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]m=-{\\Large\\frac{7}{5}}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Let\u2019s confirm this on the graph shown.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/25224622\/CNX_BMath_Figure_11_04_032.png\" alt=\"The graph shows the x y-coordinate plane. The x-axis runs from -8 to 2. The y-axis runs from -6 to 5. A line passes though the points labeled (negative 7, 4) and (negative 2, negative 3). Lines are drawn from both points to the point (negative 7, negative 3). The line connecting (negative 7, 4) and (negative 7, negative 3) is labeled rise, and the line connecting (negative 2, negative 3) and (negative 7, negative 3) is labeled run.\" width=\"262\" height=\"290\" data-media-type=\"image\/png\" \/><br \/>\n[latex]\\begin{array}{}\\\\ \\\\ \\\\ m=\\frac{\\text{rise}}{\\text{run}}\\\\ m={\\Large\\frac{-7}{5}}\\\\ m=-{\\Large\\frac{7}{5}}\\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm147022\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=147022&theme=oea&iframe_resize_id=ohm147022&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>Watch this video to see more examples of how to determine slope given two points on a line.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-4\" title=\"Ex 2:  Determine the Slope a Line Given Two Points on a Line\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/6qONExlVGgc?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-379\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Prealgebra. <strong>Provided by<\/strong>: OpenStax. <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\/caa57dab-41c7-455e-bd6f-f443cda5519c@9.757<\/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":17533,"menu_order":17,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Prealgebra\",\"author\":\"\",\"organization\":\"OpenStax\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download for free at http:\/\/cnx.org\/contents\/caa57dab-41c7-455e-bd6f-f443cda5519c@9.757\"}]","CANDELA_OUTCOMES_GUID":"b6c59980-0701-4248-a23a-f0a739f6282e, 52bfa41e-73d5-4b4f-a87a-e8cf29da2c97","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-379","chapter","type-chapter","status-publish","hentry"],"part":26,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/wm-accountingformanagers\/wp-json\/pressbooks\/v2\/chapters\/379","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/wm-accountingformanagers\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/wm-accountingformanagers\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/wm-accountingformanagers\/wp-json\/wp\/v2\/users\/17533"}],"version-history":[{"count":18,"href":"https:\/\/courses.lumenlearning.com\/wm-accountingformanagers\/wp-json\/pressbooks\/v2\/chapters\/379\/revisions"}],"predecessor-version":[{"id":4006,"href":"https:\/\/courses.lumenlearning.com\/wm-accountingformanagers\/wp-json\/pressbooks\/v2\/chapters\/379\/revisions\/4006"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/wm-accountingformanagers\/wp-json\/pressbooks\/v2\/parts\/26"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/wm-accountingformanagers\/wp-json\/pressbooks\/v2\/chapters\/379\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/wm-accountingformanagers\/wp-json\/wp\/v2\/media?parent=379"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/wm-accountingformanagers\/wp-json\/pressbooks\/v2\/chapter-type?post=379"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/wm-accountingformanagers\/wp-json\/wp\/v2\/contributor?post=379"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/wm-accountingformanagers\/wp-json\/wp\/v2\/license?post=379"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}