{"id":16520,"date":"2019-10-03T17:08:44","date_gmt":"2019-10-03T17:08:44","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/chapter\/read-or-watch-house-value\/"},"modified":"2020-10-22T09:19:29","modified_gmt":"2020-10-22T09:19:29","slug":"read-or-watch-house-value","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/suny-rockland-developmentalemporium\/chapter\/read-or-watch-house-value\/","title":{"raw":"9.6.a - Interpreting Slope in Equations and Graphs","rendered":"9.6.a &#8211; Interpreting Slope in Equations and Graphs"},"content":{"raw":"<div class=\"bcc-box bcc-highlight\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Solve slope applications using equations and graphs<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Solve Slope Applications<\/h2>\r\nAt the beginning of this section, we said there are many applications of slope in the real world. Let\u2019s look at a few now.\u00a0 But first, let's review the different kinds of slopes possible in a linear equation.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064320\/image042.gif\" alt=\"Uphill line with positive slope has a line that starts at the bottom-left and goes into the top-right of the graph. Downhill line with negative slope starts in the top-left and ends in the bottom-right part of the graph. Horizontal lines have a slope of 0. Vertical lines have an undefined slope.\" width=\"456\" height=\"183\" \/>\r\n\r\nWe often use specific words to describe the different types of slopes when we are using lines and equations to represent \"real\" situations. The following table pairs the type of slope with the common language used to describe it both verbally and visually.\r\n<table style=\"height: 145px\" width=\"533\">\r\n<tbody>\r\n<tr>\r\n<td><strong>Type of Slope<\/strong><\/td>\r\n<td><strong>Visual Description\u00a0<\/strong><\/td>\r\n<td><strong>Verbal Description<\/strong><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>positive<\/td>\r\n<td>uphill<\/td>\r\n<td>increasing<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>negative<\/td>\r\n<td>downhill<\/td>\r\n<td>decreasing<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]0[\/latex]<\/td>\r\n<td>horizontal<\/td>\r\n<td>constant<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>undefined<\/td>\r\n<td>vertical<\/td>\r\n<td>N\/A<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<h3>Visual Interpretations of Slope<\/h3>\r\nOne of the ways that we can interpret slope is through a visual interpretation.\u00a0 You have heard us describe slope as going \"uphill\" or \"downhill\".\u00a0 In the following application problems, we are interpreting slope visually in real-world contexts.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nThe pitch of a building\u2019s roof is the slope of the roof. Knowing the pitch is important in climates where there is heavy snowfall. If the roof is too flat, the weight of the snow may cause it to collapse. What is the slope of the roof shown?\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/25224646\/CNX_BMath_Figure_11_04_039.png\" alt=\"This figure shows a house with a sloped roof. The roof on one half of the building is labeled \" \/>\r\n[reveal-answer q=\"473512\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"473512\"]\r\n\r\nSolution\r\n<table id=\"eip-id1168469670624\" class=\"unnumbered unstyled\" summary=\".\">\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>Substitute the values for rise and run.<\/td>\r\n<td>[latex]m={\\Large\\frac{\\text{9 ft}}{\\text{18 ft}}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]m={\\Large\\frac{1}{2}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>The slope of the roof is [latex]{\\Large\\frac{1}{2}}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]147027[\/ohm_question]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nHave you ever thought about the sewage pipes going from your house to the street? Their slope is an important factor in how they take waste away from your house.\r\n\r\nSewage pipes must slope down [latex]{\\Large\\frac{1}{4}}[\/latex] inch per foot in order to drain properly. What is the required slope?\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/25224647\/CNX_BMath_Figure_11_04_042.png\" alt=\"This figure shows a right triangle. The short leg is vertical and is labeled \" \/>\r\n[reveal-answer q=\"947126\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"947126\"]\r\n\r\nSolution\r\n<table id=\"eip-id1168469343232\" class=\"unnumbered unstyled\" summary=\".\">\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>[latex]m={\\Large\\frac{-\\frac{1}{4}\\text{in}\\text{.}}{1\\text{ft}}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]m={\\Large\\frac{-\\frac{1}{4}\\text{in}\\text{.}}{1\\text{ft}}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Convert [latex]1[\/latex] foot to [latex]12[\/latex] inches.<\/td>\r\n<td>[latex]m={\\Large\\frac{-\\frac{1}{4}\\text{in}\\text{.}}{12\\text{in.}}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]m=-{\\Large\\frac{1}{48}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>The slope of the pipe is [latex]-{\\Large\\frac{1}{48}}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]147028[\/ohm_question]\r\n\r\n<\/div>\r\n<h2><\/h2>\r\n<h2>Verify Slope From a Dataset<\/h2>\r\nMassive amounts of data is being collected every day by a wide range of institutions and groups. \u00a0This data is used for many purposes including business decisions about location and marketing, government decisions about allocation of resources and infrastructure, and personal decisions about where to live or where to buy food.\r\n\r\nIn the following example, you will see how a dataset can be used to define\u00a0the slope of a linear equation.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nGiven the dataset, verify the values of the slopes of each equation.\r\n\r\nLinear equations describing the change in median home values between [latex]1950[\/latex] and [latex]2000[\/latex] in Mississippi and Hawaii are as follows:\r\n\r\n<strong>Hawaii:\u00a0<\/strong> [latex]y=3966x+74,400[\/latex]\r\n\r\n<strong>Mississippi:\u00a0\u00a0<\/strong>[latex]y=924x+25,200[\/latex]\r\n\r\nThe equations are based on the following dataset.\r\n\r\n[latex]x[\/latex] = the number of years since [latex]1950[\/latex], and y = the median value of a house in the given state.\r\n<table id=\"Table_04_02_03\" summary=\"This table shows three rows and three columns. The first column is labeled: \u201cYear\u201d, the second: \u201cMississippi\u201d and the third: \u201cHawaii\u201d. The two year entries are: \u201c1950\u201d and \u201c2000\u201d. The two Mississippi entries are: \u201c$25,200\u201d and \u201c$71,400\u201d. The two Hawaii entries are: \u201c$74,400\u201d and \u201c$272,700\u201d.\">\r\n<thead>\r\n<tr>\r\n<th scope=\"col\">Year (<em>x<\/em>)<\/th>\r\n<th scope=\"col\">Mississippi House Value (<em>y<\/em>)<\/th>\r\n<th scope=\"col\">Hawaii House Value (<em>y<\/em>)<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>[latex]0[\/latex]<\/td>\r\n<td>[latex]$25,200[\/latex]<\/td>\r\n<td>[latex]$74,400[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]50[\/latex]<\/td>\r\n<td>[latex]$71,400[\/latex]<\/td>\r\n<td>[latex]$272,700[\/latex]<strong>\u00a0\u00a0<\/strong><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nThe slopes of each equation can be calculated with the formula you learned in the section on slope.\r\n<p style=\"text-align: center\">[latex] \\displaystyle m=\\frac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}[\/latex]<\/p>\r\n<strong>Mississippi:<\/strong>\r\n<table>\r\n<thead>\r\n<tr>\r\n<th>Name<\/th>\r\n<th>Ordered Pair<\/th>\r\n<th>Coordinates<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>Point 1<\/td>\r\n<td>[latex](0, 25200)[\/latex]<\/td>\r\n<td>[latex]\\begin{array}{l}x_{1}=0\\\\y_{1}=25200\\end{array}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Point 2<\/td>\r\n<td>[latex](50, 71400)[\/latex]<\/td>\r\n<td>[latex]\\begin{array}{l}x_{2}=50\\\\y_{2}=71400\\end{array}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p style=\"text-align: center\">[latex] \\displaystyle m=\\frac{{71,400}-{25,200}}{{50}-{0}}=\\frac{{46,200}}{{50}} = 924[\/latex]<\/p>\r\nWe have verified that the slope [latex] \\displaystyle m = 924[\/latex] matches the dataset provided.\r\n\r\n&nbsp;\r\n\r\n<strong>Hawaii:<\/strong>\r\n<table>\r\n<thead>\r\n<tr>\r\n<th>Name<\/th>\r\n<th>Ordered Pair<\/th>\r\n<th>Coordinates<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>Point 1<\/td>\r\n<td>[latex](0, 74400)[\/latex]<\/td>\r\n<td>[latex]\\begin{array}{l}x_{1}=1950\\\\y_{1}=74400\\end{array}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Point 2<\/td>\r\n<td>[latex](50, 272700)[\/latex]<\/td>\r\n<td>[latex]\\begin{array}{l}x_{2}=2000\\\\y_{2}=272700\\end{array}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p style=\"text-align: center\">[latex]\\displaystyle m=\\frac{{272,700}-{74,400}}{{50}-{0}}=\\frac{{198,300}}{{50}} = 3966[\/latex]<\/p>\r\nWe have verified that the slope [latex] \\displaystyle m = 3966[\/latex] matches the dataset provided.\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nGiven the dataset, verify the values of the slopes of the\u00a0equation.\r\n\r\nA linear equation describing the change in the number of high school students who smoke, in\u00a0a group of [latex]100[\/latex], between [latex]2011[\/latex] and [latex]2015[\/latex] is given as:\r\n<p style=\"text-align: center\">\u00a0[latex]y = -1.75x+16[\/latex]<\/p>\r\nAnd is based on the data from this table, provided by the Centers for Disease Control.\r\n\r\n[latex]x[\/latex] = the number of years since [latex]2011[\/latex], and [latex]y[\/latex] = the number of high school smokers per [latex]100[\/latex] students.\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>Year<\/td>\r\n<td>Number of \u00a0High School Students Smoking\u00a0Cigarettes (per 100)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]0[\/latex]<\/td>\r\n<td>[latex]16[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]4[\/latex]<\/td>\r\n<td>[latex]9[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table>\r\n<thead>\r\n<tr>\r\n<th>Name<\/th>\r\n<th>Ordered Pair<\/th>\r\n<th>Coordinates<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>Point 1<\/td>\r\n<td>[latex](0, 16)[\/latex]<\/td>\r\n<td>[latex]\\begin{array}{l}x_{1}=0\\\\y_{1}=16\\end{array}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Point 2<\/td>\r\n<td>[latex](4, 9)[\/latex]<\/td>\r\n<td>[latex]\\begin{array}{l}x_{2}=4\\\\y_{2}=9\\end{array}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p style=\"text-align: center\">[latex] \\displaystyle m=\\frac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}=\\frac{{9-16}}{{4-0}} =\\frac{{-7}}{{4}}=-1.75[\/latex]<\/p>\r\nWe have verified that the slope [latex] \\displaystyle{m=-1.75}[\/latex] matches the dataset provided.\r\n\r\n<\/div>\r\n<h2>Interpret the Slope of \u00a0Linear Equation<\/h2>\r\nOkay, now we have verified that data can provide us with the slope of a linear equation. So what? We can use this information to describe how something changes using words.\r\n\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nInterpret the slope of each equation for house values using words.\r\n\r\n<strong>Hawaii:\u00a0<\/strong> [latex]y = 3966x+74,400[\/latex]\r\n\r\n<strong>Mississippi:\u00a0\u00a0<\/strong>[latex]y = 924x+25,200[\/latex]\r\n\r\n[reveal-answer q=\"871726\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"871726\"]It helps to apply the units to the points that we used to define slope. \u00a0The <em>x<\/em>-values represent years, and the <em>y<\/em>-values represent dollar amounts.\r\n\r\nFor Mississippi:\r\n<p style=\"text-align: center\">[latex] \\displaystyle m=\\frac{{71,400}-{25,200}}{{0}-{50}}=\\frac{{46,200\\text{ dollars}}}{{50\\text{ year}}} = 924\\frac{\\text{dollars}}{\\text{year}}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\nThe slope for the Mississippi home prices equation is <strong>positive<\/strong>, so each year the price of a home in Mississippi\u00a0<strong>increases<\/strong> by [latex]924[\/latex] dollars.\r\n\r\nWe can apply the same thinking for Hawaii home prices. The slope for the Hawaii\u00a0home prices equation tells us that each year, the price of a home increases by [latex]3966[\/latex]\u00a0dollars.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2 class=\"yt watch-title-container\"><span id=\"eow-title\" class=\"watch-title\" dir=\"ltr\" title=\"Intepret the Meaning of the Slope Given a Linear Equation - Median Home Values\">Interpret the Meaning of the Slope Given a Linear Equation\u2014Median Home Values<\/span><\/h2>\r\nhttps:\/\/youtu.be\/JT0WX5KOkJ8\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nInterpret the slope of the line describing the change in the number of high school smokers using words.\r\n\r\nApply units to the formula for slope. The [latex]x[\/latex] values represent years, and the y values represent the number of smokers. Remember that this dataset is per [latex]100[\/latex] high school students.\r\n<p style=\"text-align: center\">[latex] \\displaystyle m=\\frac{{9-16}}{{2015-2011}} =\\frac{{-7 \\text{ smokers}}}{{4\\text{ year}}}=-1.75\\frac{\\text{ smokers}}{\\text{ year}}[\/latex]<\/p>\r\nThe slope of this linear equation is <strong>negative<\/strong>, so this tells us that there is a <strong>decrease<\/strong> in the number of high school age smokers each year.\r\n\r\nThe number of high schoolers that smoke decreases by [latex]1.75[\/latex] per [latex]100[\/latex] each year.\r\n\r\n<\/div>\r\n<h2>Interpret the Meaning of the Slope of a Linear Equation\u2014Smokers<\/h2>\r\nhttps:\/\/youtu.be\/aHLw5FcMjdc\r\n\r\n&nbsp;\r\n\r\nOn the next page, we will see how to interpret the <em>y<\/em>-intercept of a linear equation, and make a prediction based on a linear equation.","rendered":"<div class=\"bcc-box bcc-highlight\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Solve slope applications using equations and graphs<\/li>\n<\/ul>\n<\/div>\n<h2>Solve Slope Applications<\/h2>\n<p>At the beginning of this section, we said there are many applications of slope in the real world. Let\u2019s look at a few now.\u00a0 But first, let&#8217;s review the different kinds of slopes possible in a linear equation.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064320\/image042.gif\" alt=\"Uphill line with positive slope has a line that starts at the bottom-left and goes into the top-right of the graph. Downhill line with negative slope starts in the top-left and ends in the bottom-right part of the graph. Horizontal lines have a slope of 0. Vertical lines have an undefined slope.\" width=\"456\" height=\"183\" \/><\/p>\n<p>We often use specific words to describe the different types of slopes when we are using lines and equations to represent &#8220;real&#8221; situations. The following table pairs the type of slope with the common language used to describe it both verbally and visually.<\/p>\n<table style=\"height: 145px; width: 533px;\">\n<tbody>\n<tr>\n<td><strong>Type of Slope<\/strong><\/td>\n<td><strong>Visual Description\u00a0<\/strong><\/td>\n<td><strong>Verbal Description<\/strong><\/td>\n<\/tr>\n<tr>\n<td>positive<\/td>\n<td>uphill<\/td>\n<td>increasing<\/td>\n<\/tr>\n<tr>\n<td>negative<\/td>\n<td>downhill<\/td>\n<td>decreasing<\/td>\n<\/tr>\n<tr>\n<td>[latex]0[\/latex]<\/td>\n<td>horizontal<\/td>\n<td>constant<\/td>\n<\/tr>\n<tr>\n<td>undefined<\/td>\n<td>vertical<\/td>\n<td>N\/A<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h3>Visual Interpretations of Slope<\/h3>\n<p>One of the ways that we can interpret slope is through a visual interpretation.\u00a0 You have heard us describe slope as going &#8220;uphill&#8221; or &#8220;downhill&#8221;.\u00a0 In the following application problems, we are interpreting slope visually in real-world contexts.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>The pitch of a building\u2019s roof is the slope of the roof. Knowing the pitch is important in climates where there is heavy snowfall. If the roof is too flat, the weight of the snow may cause it to collapse. What is the slope of the roof shown?<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/25224646\/CNX_BMath_Figure_11_04_039.png\" alt=\"This figure shows a house with a sloped roof. The roof on one half of the building is labeled\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q473512\">Show Solution<\/span><\/p>\n<div id=\"q473512\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-id1168469670624\" class=\"unnumbered unstyled\" summary=\".\">\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>Substitute the values for rise and run.<\/td>\n<td>[latex]m={\\Large\\frac{\\text{9 ft}}{\\text{18 ft}}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]m={\\Large\\frac{1}{2}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>The slope of the roof is [latex]{\\Large\\frac{1}{2}}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm147027\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=147027&theme=oea&iframe_resize_id=ohm147027&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Have you ever thought about the sewage pipes going from your house to the street? Their slope is an important factor in how they take waste away from your house.<\/p>\n<p>Sewage pipes must slope down [latex]{\\Large\\frac{1}{4}}[\/latex] inch per foot in order to drain properly. What is the required slope?<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/25224647\/CNX_BMath_Figure_11_04_042.png\" alt=\"This figure shows a right triangle. The short leg is vertical and is labeled\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q947126\">Show Solution<\/span><\/p>\n<div id=\"q947126\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-id1168469343232\" class=\"unnumbered unstyled\" summary=\".\">\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>[latex]m={\\Large\\frac{-\\frac{1}{4}\\text{in}\\text{.}}{1\\text{ft}}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]m={\\Large\\frac{-\\frac{1}{4}\\text{in}\\text{.}}{1\\text{ft}}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Convert [latex]1[\/latex] foot to [latex]12[\/latex] inches.<\/td>\n<td>[latex]m={\\Large\\frac{-\\frac{1}{4}\\text{in}\\text{.}}{12\\text{in.}}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]m=-{\\Large\\frac{1}{48}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>The slope of the pipe is [latex]-{\\Large\\frac{1}{48}}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm147028\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=147028&theme=oea&iframe_resize_id=ohm147028&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<h2><\/h2>\n<h2>Verify Slope From a Dataset<\/h2>\n<p>Massive amounts of data is being collected every day by a wide range of institutions and groups. \u00a0This data is used for many purposes including business decisions about location and marketing, government decisions about allocation of resources and infrastructure, and personal decisions about where to live or where to buy food.<\/p>\n<p>In the following example, you will see how a dataset can be used to define\u00a0the slope of a linear equation.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Given the dataset, verify the values of the slopes of each equation.<\/p>\n<p>Linear equations describing the change in median home values between [latex]1950[\/latex] and [latex]2000[\/latex] in Mississippi and Hawaii are as follows:<\/p>\n<p><strong>Hawaii:\u00a0<\/strong> [latex]y=3966x+74,400[\/latex]<\/p>\n<p><strong>Mississippi:\u00a0\u00a0<\/strong>[latex]y=924x+25,200[\/latex]<\/p>\n<p>The equations are based on the following dataset.<\/p>\n<p>[latex]x[\/latex] = the number of years since [latex]1950[\/latex], and y = the median value of a house in the given state.<\/p>\n<table id=\"Table_04_02_03\" summary=\"This table shows three rows and three columns. The first column is labeled: \u201cYear\u201d, the second: \u201cMississippi\u201d and the third: \u201cHawaii\u201d. The two year entries are: \u201c1950\u201d and \u201c2000\u201d. The two Mississippi entries are: \u201c$25,200\u201d and \u201c$71,400\u201d. The two Hawaii entries are: \u201c$74,400\u201d and \u201c$272,700\u201d.\">\n<thead>\n<tr>\n<th scope=\"col\">Year (<em>x<\/em>)<\/th>\n<th scope=\"col\">Mississippi House Value (<em>y<\/em>)<\/th>\n<th scope=\"col\">Hawaii House Value (<em>y<\/em>)<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>[latex]0[\/latex]<\/td>\n<td>[latex]$25,200[\/latex]<\/td>\n<td>[latex]$74,400[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]50[\/latex]<\/td>\n<td>[latex]$71,400[\/latex]<\/td>\n<td>[latex]$272,700[\/latex]<strong>\u00a0\u00a0<\/strong><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>The slopes of each equation can be calculated with the formula you learned in the section on slope.<\/p>\n<p style=\"text-align: center\">[latex]\\displaystyle m=\\frac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}[\/latex]<\/p>\n<p><strong>Mississippi:<\/strong><\/p>\n<table>\n<thead>\n<tr>\n<th>Name<\/th>\n<th>Ordered Pair<\/th>\n<th>Coordinates<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>Point 1<\/td>\n<td>[latex](0, 25200)[\/latex]<\/td>\n<td>[latex]\\begin{array}{l}x_{1}=0\\\\y_{1}=25200\\end{array}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Point 2<\/td>\n<td>[latex](50, 71400)[\/latex]<\/td>\n<td>[latex]\\begin{array}{l}x_{2}=50\\\\y_{2}=71400\\end{array}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: center\">[latex]\\displaystyle m=\\frac{{71,400}-{25,200}}{{50}-{0}}=\\frac{{46,200}}{{50}} = 924[\/latex]<\/p>\n<p>We have verified that the slope [latex]\\displaystyle m = 924[\/latex] matches the dataset provided.<\/p>\n<p>&nbsp;<\/p>\n<p><strong>Hawaii:<\/strong><\/p>\n<table>\n<thead>\n<tr>\n<th>Name<\/th>\n<th>Ordered Pair<\/th>\n<th>Coordinates<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>Point 1<\/td>\n<td>[latex](0, 74400)[\/latex]<\/td>\n<td>[latex]\\begin{array}{l}x_{1}=1950\\\\y_{1}=74400\\end{array}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Point 2<\/td>\n<td>[latex](50, 272700)[\/latex]<\/td>\n<td>[latex]\\begin{array}{l}x_{2}=2000\\\\y_{2}=272700\\end{array}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: center\">[latex]\\displaystyle m=\\frac{{272,700}-{74,400}}{{50}-{0}}=\\frac{{198,300}}{{50}} = 3966[\/latex]<\/p>\n<p>We have verified that the slope [latex]\\displaystyle m = 3966[\/latex] matches the dataset provided.<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Given the dataset, verify the values of the slopes of the\u00a0equation.<\/p>\n<p>A linear equation describing the change in the number of high school students who smoke, in\u00a0a group of [latex]100[\/latex], between [latex]2011[\/latex] and [latex]2015[\/latex] is given as:<\/p>\n<p style=\"text-align: center\">\u00a0[latex]y = -1.75x+16[\/latex]<\/p>\n<p>And is based on the data from this table, provided by the Centers for Disease Control.<\/p>\n<p>[latex]x[\/latex] = the number of years since [latex]2011[\/latex], and [latex]y[\/latex] = the number of high school smokers per [latex]100[\/latex] students.<\/p>\n<table>\n<tbody>\n<tr>\n<td>Year<\/td>\n<td>Number of \u00a0High School Students Smoking\u00a0Cigarettes (per 100)<\/td>\n<\/tr>\n<tr>\n<td>[latex]0[\/latex]<\/td>\n<td>[latex]16[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]4[\/latex]<\/td>\n<td>[latex]9[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table>\n<thead>\n<tr>\n<th>Name<\/th>\n<th>Ordered Pair<\/th>\n<th>Coordinates<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>Point 1<\/td>\n<td>[latex](0, 16)[\/latex]<\/td>\n<td>[latex]\\begin{array}{l}x_{1}=0\\\\y_{1}=16\\end{array}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Point 2<\/td>\n<td>[latex](4, 9)[\/latex]<\/td>\n<td>[latex]\\begin{array}{l}x_{2}=4\\\\y_{2}=9\\end{array}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: center\">[latex]\\displaystyle m=\\frac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}=\\frac{{9-16}}{{4-0}} =\\frac{{-7}}{{4}}=-1.75[\/latex]<\/p>\n<p>We have verified that the slope [latex]\\displaystyle{m=-1.75}[\/latex] matches the dataset provided.<\/p>\n<\/div>\n<h2>Interpret the Slope of \u00a0Linear Equation<\/h2>\n<p>Okay, now we have verified that data can provide us with the slope of a linear equation. So what? We can use this information to describe how something changes using words.<\/p>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Interpret the slope of each equation for house values using words.<\/p>\n<p><strong>Hawaii:\u00a0<\/strong> [latex]y = 3966x+74,400[\/latex]<\/p>\n<p><strong>Mississippi:\u00a0\u00a0<\/strong>[latex]y = 924x+25,200[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q871726\">Show Solution<\/span><\/p>\n<div id=\"q871726\" class=\"hidden-answer\" style=\"display: none\">It helps to apply the units to the points that we used to define slope. \u00a0The <em>x<\/em>-values represent years, and the <em>y<\/em>-values represent dollar amounts.<\/p>\n<p>For Mississippi:<\/p>\n<p style=\"text-align: center\">[latex]\\displaystyle m=\\frac{{71,400}-{25,200}}{{0}-{50}}=\\frac{{46,200\\text{ dollars}}}{{50\\text{ year}}} = 924\\frac{\\text{dollars}}{\\text{year}}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>The slope for the Mississippi home prices equation is <strong>positive<\/strong>, so each year the price of a home in Mississippi\u00a0<strong>increases<\/strong> by [latex]924[\/latex] dollars.<\/p>\n<p>We can apply the same thinking for Hawaii home prices. The slope for the Hawaii\u00a0home prices equation tells us that each year, the price of a home increases by [latex]3966[\/latex]\u00a0dollars.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h2 class=\"yt watch-title-container\"><span id=\"eow-title\" class=\"watch-title\" dir=\"ltr\" title=\"Intepret the Meaning of the Slope Given a Linear Equation - Median Home Values\">Interpret the Meaning of the Slope Given a Linear Equation\u2014Median Home Values<\/span><\/h2>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Interpret the Meaning of the Slope Given a Linear Equation - Median Home Values\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/JT0WX5KOkJ8?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Interpret the slope of the line describing the change in the number of high school smokers using words.<\/p>\n<p>Apply units to the formula for slope. The [latex]x[\/latex] values represent years, and the y values represent the number of smokers. Remember that this dataset is per [latex]100[\/latex] high school students.<\/p>\n<p style=\"text-align: center\">[latex]\\displaystyle m=\\frac{{9-16}}{{2015-2011}} =\\frac{{-7 \\text{ smokers}}}{{4\\text{ year}}}=-1.75\\frac{\\text{ smokers}}{\\text{ year}}[\/latex]<\/p>\n<p>The slope of this linear equation is <strong>negative<\/strong>, so this tells us that there is a <strong>decrease<\/strong> in the number of high school age smokers each year.<\/p>\n<p>The number of high schoolers that smoke decreases by [latex]1.75[\/latex] per [latex]100[\/latex] each year.<\/p>\n<\/div>\n<h2>Interpret the Meaning of the Slope of a Linear Equation\u2014Smokers<\/h2>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Interpret the Meaning of the Slope of a Linear Equation - Smokers\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/aHLw5FcMjdc?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>&nbsp;<\/p>\n<p>On the next page, we will see how to interpret the <em>y<\/em>-intercept of a linear equation, and make a prediction based on a linear equation.<\/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-16520\">\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>Intepret the Meaning of the Slope Given a Linear Equation - Median Home Values. <strong>Authored by<\/strong>: Mathispower4u. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/JT0WX5KOkJ8\">https:\/\/youtu.be\/JT0WX5KOkJ8<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/about\/pdm\">Public Domain: No Known Copyright<\/a><\/em><\/li><li>Intepret the Meaning of the Slope of a Linear Equation - Smokers. <strong>Authored by<\/strong>: Mathispower4u. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/aHLw5FcMjdc\">https:\/\/youtu.be\/aHLw5FcMjdc<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/about\/pdm\">Public Domain: No Known Copyright<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">Public domain content<\/div><ul class=\"citation-list\"><li>Youth and Tobacco Use. <strong>Authored by<\/strong>: Centers for Disease Control and Prevention. <strong>Provided by<\/strong>: U.S. Department of Health and Human Services. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/www.cdc.gov\/tobacco\/data_statistics\/fact_sheets\/youth_data\/tobacco_use\/index.htm\">http:\/\/www.cdc.gov\/tobacco\/data_statistics\/fact_sheets\/youth_data\/tobacco_use\/index.htm<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/about\/pdm\">Public Domain: No Known Copyright<\/a><\/em><\/li><li>Historical Census of Housing Tables Home Values. <strong>Authored by<\/strong>: United States Census Bureau. <strong>Provided by<\/strong>: U.S. Dept. of Housing. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/www.census.gov\/hhes\/www\/housing\/census\/historic\/values.html\">https:\/\/www.census.gov\/hhes\/www\/housing\/census\/historic\/values.html<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/about\/pdm\">Public Domain: No Known Copyright<\/a><\/em><\/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":169554,"menu_order":27,"template":"","meta":{"_candela_citation":"[{\"type\":\"pd\",\"description\":\"Youth and Tobacco Use\",\"author\":\"Centers for Disease Control and Prevention\",\"organization\":\"U.S. Department of Health and Human Services\",\"url\":\"http:\/\/www.cdc.gov\/tobacco\/data_statistics\/fact_sheets\/youth_data\/tobacco_use\/index.htm\",\"project\":\"\",\"license\":\"pd\",\"license_terms\":\"\"},{\"type\":\"pd\",\"description\":\"Historical Census of Housing Tables Home Values\",\"author\":\"United States Census Bureau\",\"organization\":\"U.S. Dept. of Housing\",\"url\":\"https:\/\/www.census.gov\/hhes\/www\/housing\/census\/historic\/values.html\",\"project\":\"\",\"license\":\"pd\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Intepret the Meaning of the Slope Given a Linear Equation - 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