{"id":467,"date":"2016-04-21T22:43:38","date_gmt":"2016-04-21T22:43:38","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/introstats1xmaster\/?post_type=chapter&#038;p=467"},"modified":"2021-07-06T20:29:00","modified_gmt":"2021-07-06T20:29:00","slug":"linear-equations","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/frontrange-introstats1\/chapter\/linear-equations\/","title":{"raw":"Linear Equations","rendered":"Linear Equations"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Discuss basic ideas of linear regression and correlation<\/li>\r\n<\/ul>\r\n<\/div>\r\nLinear regression for two variables is based on a linear equation with one independent variable. The equation has the form:\u00a0<span class=\"mi\" style=\"font-size: 1em;\">y\u00a0<\/span><span class=\"mo\" style=\"font-size: 1em;\">=\u00a0<\/span><em><span class=\"mi\" style=\"font-size: 1em;\">a\u00a0<\/span><\/em><span class=\"mo\" style=\"font-size: 1em;\">+\u00a0<\/span><span class=\"mtext\" style=\"font-size: 1em;\"><em>b<\/em>x,\u00a0<\/span>where <em data-effect=\"italics\">a<\/em> and <em data-effect=\"italics\">b<\/em> are constant numbers.\r\n\r\nThe variable <strong><em data-effect=\"italics\">x<\/em> is the independent variable, and <em data-effect=\"italics\">y<\/em> is the dependent variable.<\/strong> Typically, you choose a value to substitute for the independent variable and then solve for the dependent variable.\r\n<div class=\"example\" data-type=\"example\"><section>The following examples are linear equations.\r\n<div id=\"element-12495\" class=\"equation\" data-type=\"equation\">\r\n<div class=\"MathJax_Display\"><span class=\"MathJax\"><span class=\"math\"><span class=\"mrow\"><span class=\"semantics\"><span class=\"mrow\"><span class=\"mi\">y<\/span><span class=\"mo\">=<\/span><span class=\"mn\">3<\/span><span class=\"mo\">+<\/span><span class=\"mtext\">2x<\/span><\/span><\/span><\/span><\/span><\/span><\/div>\r\n<\/div>\r\n<div id=\"element-357238\" class=\"equation\" data-type=\"equation\">\r\n<div class=\"MathJax_Display\"><span class=\"MathJax\"><span class=\"math\"><span class=\"mrow\"><span class=\"semantics\"><span class=\"mrow\"><span class=\"mi\">y<\/span><span class=\"mo\">=<\/span><span class=\"mn\">\u20130.01<\/span><span class=\"mo\">+<\/span><span class=\"mtext\">1.2x<\/span><\/span><\/span><\/span><\/span><\/span><\/div>\r\n<div><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-idp88007136\" class=\"note statistics try ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"\"><header>\r\n<div class=\"title\" data-type=\"title\" data-label-parent=\"\">\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n<div id=\"fs-idp88007136\" class=\"note statistics try ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"\"><section>\r\n<div class=\"exercise\" data-type=\"exercise\"><section>\r\n<div class=\"problem\" data-type=\"problem\">\r\n\r\nIs the following an example of a linear equation?\r\n<p id=\"eip-idp75670576\"><em data-effect=\"italics\">y<\/em> = \u20130.125 \u2013 3.5<em data-effect=\"italics\">x<\/em><\/p>\r\n\r\n<\/div>\r\n<div class=\"solution\" data-type=\"solution\"><\/div>\r\n<\/section><\/div>\r\n<\/section><\/div>\r\n<p id=\"eip-498\">YES - The graph of a linear equation if of the form <em data-effect=\"italics\">y<\/em> = <em data-effect=\"italics\">a<\/em> + <em data-effect=\"italics\">bx<\/em> is a <strong>straight line<\/strong>. Any line that is not vertical can be described by this equation.<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<div id=\"element-44\" class=\"example\" data-type=\"example\"><section>\r\n<p style=\"text-align: center;\"><strong>TRY IT<\/strong><\/p>\r\n<p id=\"eip-153\">Graph the equation <em data-effect=\"italics\">y<\/em> = \u20131 + 2<em data-effect=\"italics\">x<\/em>.<\/p>\r\n[reveal-answer q=\"82736\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"82736\"]\r\n<figure id=\"linrgs_lineq1\"><span id=\"idp38882144\" data-type=\"media\" data-alt=\"Graph of the equation y = -1 + 2x. This is a straight line that crosses the y-axis at -1 and is sloped up and to the right, rising 2 units for every one unit of run.\"><img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/132\/2016\/04\/21214813\/fig-ch12_02_01.jpg\" alt=\"Graph of the equation y = -1 + 2x. This is a straight line that crosses the y-axis at -1 and is sloped up and to the right, rising 2 units for every one unit of run.\" width=\"516\" height=\"320\" data-media-type=\"image\/jpeg\" \/><\/span><\/figure>\r\nNote that the line would pass through the points (5, 9) and (10, 19)\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section><\/div>\r\n<\/div>\r\n<\/div>\r\n<\/header><\/div>\r\n<div id=\"fs-idp3088096\" class=\"note statistics try ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"\"><header>\r\n<div class=\"title\" data-type=\"title\" data-label-parent=\"\">\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n<div id=\"fs-idp3088096\" class=\"note statistics try ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"\"><section>\r\n<div class=\"exercise\" data-type=\"exercise\"><section>\r\n<div id=\"eip-373\" class=\"problem\" data-type=\"problem\">\r\n\r\nIs the following an example of a linear equation? Why or why not?\r\n\r\n<span id=\"eip-idp139727523885744\" data-type=\"media\" data-alt=\"This is a graph of an equation. The x-axis is labeled in intervals of 2 from 0 - 14; the y-axis is labeled in intervals of 2 from 0 - 12. The equation's graph is a curve that crosses the y-axis at 2 and curves upward and to the right.\" data-display=\"block\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/132\/2016\/04\/21214814\/CNX_Stats_C012_M02_tryit001.png\" alt=\"This is a graph of an equation. The x-axis is labeled in intervals of 2 from 0 - 14; the y-axis is labeled in intervals of 2 from 0 - 12. The equation's graph is a curve that crosses the y-axis at 2 and curves upward and to the right.\" width=\"250\" data-media-type=\"image\/png\" \/><\/span>\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><\/div>\r\n<\/div>\r\n<div id=\"fs-idp3088096\" class=\"note statistics try ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"\"><header>\r\n<div class=\"title\" data-type=\"title\" data-label-parent=\"\">\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n<div id=\"fs-idp3088096\" class=\"note statistics try ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"\"><section>\r\n<div class=\"exercise\" data-type=\"exercise\"><section>\r\n<div id=\"eip-373\" class=\"problem\" data-type=\"problem\">\r\n\r\nAaron's Word Processing Service (AWPS) does word processing. The rate for services is $32 per hour plus a $31.50 one-time charge. The total cost to a customer depends on the number of hours it takes to complete the job.\r\n\r\nFind the equation that expresses the total cost in terms of the number of hours\u00a0required to complete the job.\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><\/div>\r\n<\/div>\r\n<\/div>\r\n<\/header><\/div>\r\n<\/div>\r\n<\/header><\/div>\r\n<div id=\"fs-idm27179040\" class=\"note statistics try ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"\"><header>\r\n<div class=\"title\" data-type=\"title\" data-label-parent=\"\">\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n<section>\r\n<div id=\"eip-977\" class=\"exercise\" data-type=\"exercise\"><section>\r\n<div class=\"problem\" data-type=\"problem\">\r\n\r\nEmma\u2019s Extreme Sports hires hang-gliding instructors and pays them a fee of $50 per class as well as $20 per student in the class. The total cost Emma pays depends on the number of students in a class.\r\n\r\nFind the equation that expresses the total cost in terms of the number of students in a class.\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><\/div>\r\n<\/div>\r\n<\/header><\/div>\r\n<section id=\"fs-idm35899664\" data-depth=\"1\">\r\n<h1 data-type=\"title\">Slope and <em data-effect=\"italics\">Y<\/em>-Intercept of a Linear Equation<\/h1>\r\n<p id=\"fs-idp15969440\">For the linear equation <em data-effect=\"italics\">y<\/em> = <em data-effect=\"italics\">a<\/em> + <em data-effect=\"italics\">bx<\/em>, <em data-effect=\"italics\">b<\/em> = slope and <em data-effect=\"italics\">a<\/em> = <em data-effect=\"italics\">y<\/em>-intercept. From algebra recall that the slope is a number that describes the steepness of a line, and the <em data-effect=\"italics\">y<\/em>-intercept is the <em data-effect=\"italics\">y<\/em> coordinate of the point (0, <em data-effect=\"italics\">a<\/em>) where the line crosses the <em data-effect=\"italics\">y<\/em>-axis.<\/p>\r\n\r\n<figure id=\"linrgs_slope1\" class=\"ui-has-child-figcaption\"><span id=\"idp66522560a\" data-type=\"media\" data-alt=\"Three possible graphs of the equation y = a + bx. For the first graph, (a), b &gt; 0 and so the line slopes upward to the right. For the second, b = 0 and the graph of the equation is a horizontal line. In the third graph, (c), b &lt; 0 and the line slopes downward to the right.\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/132\/2016\/04\/21214816\/fig-ch12_03_01.jpg\" alt=\"Three possible graphs of the equation y = a + bx. For the first graph, (a), b &gt; 0 and so the line slopes upward to the right. For the second, b = 0 and the graph of the equation is a horizontal line. In the third graph, (c), b &lt; 0 and the line slopes downward to the right.\" width=\"480\" data-media-type=\"image\/jpg\" \/><\/span><figcaption>Three possible graphs of <em data-effect=\"italics\">y<\/em> = <em data-effect=\"italics\">a<\/em> + <em data-effect=\"italics\">bx<\/em>. (a) If <em data-effect=\"italics\">b<\/em> &gt; 0, the line slopes upward to the right. (b) If <em data-effect=\"italics\">b<\/em> = 0, the line is horizontal. (c) If <em data-effect=\"italics\">b<\/em> &lt; 0, the line slopes downward to the right.<\/figcaption><\/figure>\r\n<div id=\"element-201\" class=\"example\" data-type=\"example\"><section>Svetlana tutors to make extra money for college. For each tutoring session, she charges a one-time fee of $25 plus $15 per hour of tutoring. A linear equation that expresses the total amount of money Svetlana earns for each session she tutors is <em data-effect=\"italics\">y<\/em> = 25 + 15<em data-effect=\"italics\">x<\/em>.\r\n<div id=\"element-00112\" class=\"exercise\" data-type=\"exercise\"><section>\r\n<div id=\"idp139982000\" class=\"problem\" data-type=\"problem\">\r\n<p id=\"element-71\">What are the independent and dependent variables? What is the <em data-effect=\"italics\">y<\/em>-intercept and what is the slope? Interpret them using complete sentences.<\/p>\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-idp5590144\" class=\"note statistics try ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"\"><header>\r\n<div class=\"title\" data-type=\"title\" data-label-parent=\"\">\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n<section>\r\n<div class=\"exercise\" data-type=\"exercise\"><section>\r\n<div class=\"problem\" data-type=\"problem\">\r\n<p id=\"eip-379\">Ethan repairs household appliances like dishwashers and refrigerators. For each visit, he charges $25 plus $20 per hour of work. A linear equation that expresses the total amount of money Ethan earns per visit is <em data-effect=\"italics\">y<\/em> = 25 + 20<em data-effect=\"italics\">x<\/em>.<\/p>\r\n<p id=\"eip-idm158116832\">What are the independent and dependent variables? What is the <em data-effect=\"italics\">y<\/em>-intercept and what is the slope? Interpret them using complete sentences.<\/p>\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><\/div>\r\n<\/div>\r\n<\/header><\/div>\r\n<\/section><section class=\"references\" data-depth=\"1\">\r\n<h1 data-type=\"title\">References<\/h1>\r\n<p id=\"eip-idp123435600\">Data from the Centers for Disease Control and Prevention.<\/p>\r\n<p id=\"eip-idp89356864\">Data from the National Center for HIV, STD, and TB Prevention.<\/p>\r\n\r\n<\/section><section class=\"summary\" data-depth=\"1\">\r\n<h1 data-type=\"title\">Concept Review<\/h1>\r\n<p id=\"fs-idm110721248\">The most basic type of association is a linear association. This type of relationship can be defined algebraically by the equations used, numerically with actual or predicted data values, or graphically from a plotted curve (lines are classified as straight curves). Algebraically, a linear equation typically takes the form <strong><em data-effect=\"italics\">y = mx + b<\/em><\/strong>, where <strong><em data-effect=\"italics\">m<\/em><\/strong> and <strong><em data-effect=\"italics\">b<\/em><\/strong> are constants, <strong><em data-effect=\"italics\">x<\/em><\/strong> is the independent variable, <strong><em data-effect=\"italics\">y<\/em><\/strong> is the dependent variable. In a statistical context, a linear equation is written in the form <strong><em data-effect=\"italics\">y = a + bx<\/em><\/strong>, where <strong><em data-effect=\"italics\">a\u00a0<\/em><\/strong>and <strong><em data-effect=\"italics\">b<\/em><\/strong> are the constants. This form is used to help readers distinguish the statistical context from the algebraic context. In the equation <em data-effect=\"italics\">y = a + bx<\/em>, the constant <em data-effect=\"italics\">b<\/em> that multiplies the <strong><em data-effect=\"italics\">x<\/em><\/strong> variable (<em data-effect=\"italics\">b<\/em> is called a coefficient) is called as the <strong>slope<\/strong>. The slope describes the rate of change between the independent and dependent variables; in other words, the rate of change describes the change that occurs in the dependent variable as the independent variable is changed. In the equation <em data-effect=\"italics\">y = a + bx<\/em>, the constant a is called as the <em data-effect=\"italics\">y<\/em>-intercept. Graphically, the <em data-effect=\"italics\">y<\/em>-intercept is the <em data-effect=\"italics\">y<\/em> coordinate of the point where the graph of the line crosses the <em data-effect=\"italics\">y<\/em> axis. At this point <em data-effect=\"italics\">x<\/em> = 0.<\/p>\r\nThe <strong>slope of a line<\/strong> is a value that describes the rate of change between the independent and dependent variables. The <strong>slope<\/strong> tells us how the dependent variable (<em data-effect=\"italics\">y<\/em>) changes for every one unit increase in the independent (<em data-effect=\"italics\">x<\/em>) variable, on average. The <strong><em data-effect=\"italics\">y<\/em>-intercept<\/strong> is used to describe the dependent variable when the independent variable equals zero. Graphically, the slope is represented by three line types in elementary statistics.\r\n\r\n<\/section><section class=\"formula-review\" data-depth=\"1\">\r\n<h1 data-type=\"title\">Formula Review<\/h1>\r\n<em data-effect=\"italics\">y<\/em> = <em data-effect=\"italics\">a<\/em> + <em data-effect=\"italics\">bx<\/em> where <em data-effect=\"italics\">a<\/em> is the <em data-effect=\"italics\">y<\/em>-intercept and <em data-effect=\"italics\">b<\/em> is the slope. The variable <em data-effect=\"italics\">x<\/em> is the independent variable and\u00a0<em data-effect=\"italics\">y<\/em> is the dependent variable.\r\n\r\n<\/section><section class=\"practice\" data-depth=\"1\"><\/section><section class=\"formula-review\" data-depth=\"1\"><\/section>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Discuss basic ideas of linear regression and correlation<\/li>\n<\/ul>\n<\/div>\n<p>Linear regression for two variables is based on a linear equation with one independent variable. The equation has the form:\u00a0<span class=\"mi\" style=\"font-size: 1em;\">y\u00a0<\/span><span class=\"mo\" style=\"font-size: 1em;\">=\u00a0<\/span><em><span class=\"mi\" style=\"font-size: 1em;\">a\u00a0<\/span><\/em><span class=\"mo\" style=\"font-size: 1em;\">+\u00a0<\/span><span class=\"mtext\" style=\"font-size: 1em;\"><em>b<\/em>x,\u00a0<\/span>where <em data-effect=\"italics\">a<\/em> and <em data-effect=\"italics\">b<\/em> are constant numbers.<\/p>\n<p>The variable <strong><em data-effect=\"italics\">x<\/em> is the independent variable, and <em data-effect=\"italics\">y<\/em> is the dependent variable.<\/strong> Typically, you choose a value to substitute for the independent variable and then solve for the dependent variable.<\/p>\n<div class=\"example\" data-type=\"example\">\n<section>The following examples are linear equations.<\/p>\n<div id=\"element-12495\" class=\"equation\" data-type=\"equation\">\n<div class=\"MathJax_Display\"><span class=\"MathJax\"><span class=\"math\"><span class=\"mrow\"><span class=\"semantics\"><span class=\"mrow\"><span class=\"mi\">y<\/span><span class=\"mo\">=<\/span><span class=\"mn\">3<\/span><span class=\"mo\">+<\/span><span class=\"mtext\">2x<\/span><\/span><\/span><\/span><\/span><\/span><\/div>\n<\/div>\n<div id=\"element-357238\" class=\"equation\" data-type=\"equation\">\n<div class=\"MathJax_Display\"><span class=\"MathJax\"><span class=\"math\"><span class=\"mrow\"><span class=\"semantics\"><span class=\"mrow\"><span class=\"mi\">y<\/span><span class=\"mo\">=<\/span><span class=\"mn\">\u20130.01<\/span><span class=\"mo\">+<\/span><span class=\"mtext\">1.2x<\/span><\/span><\/span><\/span><\/span><\/span><\/div>\n<div><\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-idp88007136\" class=\"note statistics try ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"\">\n<header>\n<div class=\"title\" data-type=\"title\" data-label-parent=\"\">\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<div id=\"fs-idp88007136\" class=\"note statistics try ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"\">\n<section>\n<div class=\"exercise\" data-type=\"exercise\">\n<section>\n<div class=\"problem\" data-type=\"problem\">\n<p>Is the following an example of a linear equation?<\/p>\n<p id=\"eip-idp75670576\"><em data-effect=\"italics\">y<\/em> = \u20130.125 \u2013 3.5<em data-effect=\"italics\">x<\/em><\/p>\n<\/div>\n<div class=\"solution\" data-type=\"solution\"><\/div>\n<\/section>\n<\/div>\n<\/section>\n<\/div>\n<p id=\"eip-498\">YES &#8211; The graph of a linear equation if of the form <em data-effect=\"italics\">y<\/em> = <em data-effect=\"italics\">a<\/em> + <em data-effect=\"italics\">bx<\/em> is a <strong>straight line<\/strong>. Any line that is not vertical can be described by this equation.<\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<div id=\"element-44\" class=\"example\" data-type=\"example\">\n<section>\n<p style=\"text-align: center;\"><strong>TRY IT<\/strong><\/p>\n<p id=\"eip-153\">Graph the equation <em data-effect=\"italics\">y<\/em> = \u20131 + 2<em data-effect=\"italics\">x<\/em>.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q82736\">Show Answer<\/span><\/p>\n<div id=\"q82736\" class=\"hidden-answer\" style=\"display: none\">\n<figure id=\"linrgs_lineq1\"><span id=\"idp38882144\" data-type=\"media\" data-alt=\"Graph of the equation y = -1 + 2x. This is a straight line that crosses the y-axis at -1 and is sloped up and to the right, rising 2 units for every one unit of run.\"><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/132\/2016\/04\/21214813\/fig-ch12_02_01.jpg\" alt=\"Graph of the equation y = -1 + 2x. This is a straight line that crosses the y-axis at -1 and is sloped up and to the right, rising 2 units for every one unit of run.\" width=\"516\" height=\"320\" data-media-type=\"image\/jpeg\" \/><\/span><\/figure>\n<p>Note that the line would pass through the points (5, 9) and (10, 19)<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/div>\n<\/div>\n<\/header>\n<\/div>\n<div id=\"fs-idp3088096\" class=\"note statistics try ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"\">\n<header>\n<div class=\"title\" data-type=\"title\" data-label-parent=\"\">\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<div id=\"fs-idp3088096\" class=\"note statistics try ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"\">\n<section>\n<div class=\"exercise\" data-type=\"exercise\">\n<section>\n<div id=\"eip-373\" class=\"problem\" data-type=\"problem\">\n<p>Is the following an example of a linear equation? Why or why not?<\/p>\n<p><span id=\"eip-idp139727523885744\" data-type=\"media\" data-alt=\"This is a graph of an equation. The x-axis is labeled in intervals of 2 from 0 - 14; the y-axis is labeled in intervals of 2 from 0 - 12. The equation's graph is a curve that crosses the y-axis at 2 and curves upward and to the right.\" data-display=\"block\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/132\/2016\/04\/21214814\/CNX_Stats_C012_M02_tryit001.png\" alt=\"This is a graph of an equation. The x-axis is labeled in intervals of 2 from 0 - 14; the y-axis is labeled in intervals of 2 from 0 - 12. The equation's graph is a curve that crosses the y-axis at 2 and curves upward and to the right.\" width=\"250\" data-media-type=\"image\/png\" \/><\/span><\/p>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<\/div>\n<\/div>\n<div id=\"fs-idp3088096\" class=\"note statistics try ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"\"><\/div>\n<\/div>\n<\/header>\n<header>\n<div class=\"title\" data-type=\"title\" data-label-parent=\"\">\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<div id=\"fs-idp3088096\" class=\"note statistics try ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"\">\n<section>\n<div class=\"exercise\" data-type=\"exercise\">\n<section>\n<div id=\"eip-373\" class=\"problem\" data-type=\"problem\">\n<p>Aaron&#8217;s Word Processing Service (AWPS) does word processing. The rate for services is $32 per hour plus a $31.50 one-time charge. The total cost to a customer depends on the number of hours it takes to complete the job.<\/p>\n<p>Find the equation that expresses the total cost in terms of the number of hours\u00a0required to complete the job.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<\/div>\n<\/div>\n<\/div>\n<\/header>\n<\/div>\n<div id=\"fs-idm27179040\" class=\"note statistics try ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"\">\n<header>\n<div class=\"title\" data-type=\"title\" data-label-parent=\"\">\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<section>\n<div id=\"eip-977\" class=\"exercise\" data-type=\"exercise\">\n<section>\n<div class=\"problem\" data-type=\"problem\">\n<p>Emma\u2019s Extreme Sports hires hang-gliding instructors and pays them a fee of $50 per class as well as $20 per student in the class. The total cost Emma pays depends on the number of students in a class.<\/p>\n<p>Find the equation that expresses the total cost in terms of the number of students in a class.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<\/div>\n<\/div>\n<\/header>\n<\/div>\n<section id=\"fs-idm35899664\" data-depth=\"1\">\n<h1 data-type=\"title\">Slope and <em data-effect=\"italics\">Y<\/em>-Intercept of a Linear Equation<\/h1>\n<p id=\"fs-idp15969440\">For the linear equation <em data-effect=\"italics\">y<\/em> = <em data-effect=\"italics\">a<\/em> + <em data-effect=\"italics\">bx<\/em>, <em data-effect=\"italics\">b<\/em> = slope and <em data-effect=\"italics\">a<\/em> = <em data-effect=\"italics\">y<\/em>-intercept. From algebra recall that the slope is a number that describes the steepness of a line, and the <em data-effect=\"italics\">y<\/em>-intercept is the <em data-effect=\"italics\">y<\/em> coordinate of the point (0, <em data-effect=\"italics\">a<\/em>) where the line crosses the <em data-effect=\"italics\">y<\/em>-axis.<\/p>\n<figure id=\"linrgs_slope1\" class=\"ui-has-child-figcaption\"><span id=\"idp66522560a\" data-type=\"media\" data-alt=\"Three possible graphs of the equation y = a + bx. For the first graph, (a), b &gt; 0 and so the line slopes upward to the right. For the second, b = 0 and the graph of the equation is a horizontal line. In the third graph, (c), b &lt; 0 and the line slopes downward to the right.\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/132\/2016\/04\/21214816\/fig-ch12_03_01.jpg\" alt=\"Three possible graphs of the equation y = a + bx. For the first graph, (a), b &gt; 0 and so the line slopes upward to the right. For the second, b = 0 and the graph of the equation is a horizontal line. In the third graph, (c), b &lt; 0 and the line slopes downward to the right.\" width=\"480\" data-media-type=\"image\/jpg\" \/><\/span><figcaption>Three possible graphs of <em data-effect=\"italics\">y<\/em> = <em data-effect=\"italics\">a<\/em> + <em data-effect=\"italics\">bx<\/em>. (a) If <em data-effect=\"italics\">b<\/em> &gt; 0, the line slopes upward to the right. (b) If <em data-effect=\"italics\">b<\/em> = 0, the line is horizontal. (c) If <em data-effect=\"italics\">b<\/em> &lt; 0, the line slopes downward to the right.<\/figcaption><\/figure>\n<div id=\"element-201\" class=\"example\" data-type=\"example\">\n<section>Svetlana tutors to make extra money for college. For each tutoring session, she charges a one-time fee of $25 plus $15 per hour of tutoring. A linear equation that expresses the total amount of money Svetlana earns for each session she tutors is <em data-effect=\"italics\">y<\/em> = 25 + 15<em data-effect=\"italics\">x<\/em>.<\/p>\n<div id=\"element-00112\" class=\"exercise\" data-type=\"exercise\">\n<section>\n<div id=\"idp139982000\" class=\"problem\" data-type=\"problem\">\n<p id=\"element-71\">What are the independent and dependent variables? What is the <em data-effect=\"italics\">y<\/em>-intercept and what is the slope? Interpret them using complete sentences.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-idp5590144\" class=\"note statistics try ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"\">\n<header>\n<div class=\"title\" data-type=\"title\" data-label-parent=\"\">\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<section>\n<div class=\"exercise\" data-type=\"exercise\">\n<section>\n<div class=\"problem\" data-type=\"problem\">\n<p id=\"eip-379\">Ethan repairs household appliances like dishwashers and refrigerators. For each visit, he charges $25 plus $20 per hour of work. A linear equation that expresses the total amount of money Ethan earns per visit is <em data-effect=\"italics\">y<\/em> = 25 + 20<em data-effect=\"italics\">x<\/em>.<\/p>\n<p id=\"eip-idm158116832\">What are the independent and dependent variables? What is the <em data-effect=\"italics\">y<\/em>-intercept and what is the slope? Interpret them using complete sentences.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<\/div>\n<\/div>\n<\/header>\n<\/div>\n<\/section>\n<section class=\"references\" data-depth=\"1\">\n<h1 data-type=\"title\">References<\/h1>\n<p id=\"eip-idp123435600\">Data from the Centers for Disease Control and Prevention.<\/p>\n<p id=\"eip-idp89356864\">Data from the National Center for HIV, STD, and TB Prevention.<\/p>\n<\/section>\n<section class=\"summary\" data-depth=\"1\">\n<h1 data-type=\"title\">Concept Review<\/h1>\n<p id=\"fs-idm110721248\">The most basic type of association is a linear association. This type of relationship can be defined algebraically by the equations used, numerically with actual or predicted data values, or graphically from a plotted curve (lines are classified as straight curves). Algebraically, a linear equation typically takes the form <strong><em data-effect=\"italics\">y = mx + b<\/em><\/strong>, where <strong><em data-effect=\"italics\">m<\/em><\/strong> and <strong><em data-effect=\"italics\">b<\/em><\/strong> are constants, <strong><em data-effect=\"italics\">x<\/em><\/strong> is the independent variable, <strong><em data-effect=\"italics\">y<\/em><\/strong> is the dependent variable. In a statistical context, a linear equation is written in the form <strong><em data-effect=\"italics\">y = a + bx<\/em><\/strong>, where <strong><em data-effect=\"italics\">a\u00a0<\/em><\/strong>and <strong><em data-effect=\"italics\">b<\/em><\/strong> are the constants. This form is used to help readers distinguish the statistical context from the algebraic context. In the equation <em data-effect=\"italics\">y = a + bx<\/em>, the constant <em data-effect=\"italics\">b<\/em> that multiplies the <strong><em data-effect=\"italics\">x<\/em><\/strong> variable (<em data-effect=\"italics\">b<\/em> is called a coefficient) is called as the <strong>slope<\/strong>. The slope describes the rate of change between the independent and dependent variables; in other words, the rate of change describes the change that occurs in the dependent variable as the independent variable is changed. In the equation <em data-effect=\"italics\">y = a + bx<\/em>, the constant a is called as the <em data-effect=\"italics\">y<\/em>-intercept. Graphically, the <em data-effect=\"italics\">y<\/em>-intercept is the <em data-effect=\"italics\">y<\/em> coordinate of the point where the graph of the line crosses the <em data-effect=\"italics\">y<\/em> axis. At this point <em data-effect=\"italics\">x<\/em> = 0.<\/p>\n<p>The <strong>slope of a line<\/strong> is a value that describes the rate of change between the independent and dependent variables. The <strong>slope<\/strong> tells us how the dependent variable (<em data-effect=\"italics\">y<\/em>) changes for every one unit increase in the independent (<em data-effect=\"italics\">x<\/em>) variable, on average. The <strong><em data-effect=\"italics\">y<\/em>-intercept<\/strong> is used to describe the dependent variable when the independent variable equals zero. Graphically, the slope is represented by three line types in elementary statistics.<\/p>\n<\/section>\n<section class=\"formula-review\" data-depth=\"1\">\n<h1 data-type=\"title\">Formula Review<\/h1>\n<p><em data-effect=\"italics\">y<\/em> = <em data-effect=\"italics\">a<\/em> + <em data-effect=\"italics\">bx<\/em> where <em data-effect=\"italics\">a<\/em> is the <em data-effect=\"italics\">y<\/em>-intercept and <em data-effect=\"italics\">b<\/em> is the slope. The variable <em data-effect=\"italics\">x<\/em> is the independent variable and\u00a0<em data-effect=\"italics\">y<\/em> is the dependent variable.<\/p>\n<\/section>\n<section class=\"practice\" data-depth=\"1\"><\/section>\n<section class=\"formula-review\" data-depth=\"1\"><\/section>\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-467\">\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>Introductory Statistics . <strong>Authored by<\/strong>: Barbara Illowski, Susan Dean. <strong>Provided by<\/strong>: Open Stax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/30189442-6998-4686-ac05-ed152b91b9de@17.44\">http:\/\/cnx.org\/contents\/30189442-6998-4686-ac05-ed152b91b9de@17.44<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download for free at http:\/\/cnx.org\/contents\/30189442-6998-4686-ac05-ed152b91b9de@17.44<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":21,"menu_order":3,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Introductory Statistics \",\"author\":\"Barbara Illowski, Susan Dean\",\"organization\":\"Open Stax\",\"url\":\"http:\/\/cnx.org\/contents\/30189442-6998-4686-ac05-ed152b91b9de@17.44\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download for free at http:\/\/cnx.org\/contents\/30189442-6998-4686-ac05-ed152b91b9de@17.44\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-467","chapter","type-chapter","status-publish","hentry"],"part":457,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/frontrange-introstats1\/wp-json\/pressbooks\/v2\/chapters\/467","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/frontrange-introstats1\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/frontrange-introstats1\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/frontrange-introstats1\/wp-json\/wp\/v2\/users\/21"}],"version-history":[{"count":14,"href":"https:\/\/courses.lumenlearning.com\/frontrange-introstats1\/wp-json\/pressbooks\/v2\/chapters\/467\/revisions"}],"predecessor-version":[{"id":2383,"href":"https:\/\/courses.lumenlearning.com\/frontrange-introstats1\/wp-json\/pressbooks\/v2\/chapters\/467\/revisions\/2383"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/frontrange-introstats1\/wp-json\/pressbooks\/v2\/parts\/457"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/frontrange-introstats1\/wp-json\/pressbooks\/v2\/chapters\/467\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/frontrange-introstats1\/wp-json\/wp\/v2\/media?parent=467"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/frontrange-introstats1\/wp-json\/pressbooks\/v2\/chapter-type?post=467"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/frontrange-introstats1\/wp-json\/wp\/v2\/contributor?post=467"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/frontrange-introstats1\/wp-json\/wp\/v2\/license?post=467"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}