{"id":305,"date":"2021-07-14T15:59:11","date_gmt":"2021-07-14T15:59:11","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/chapter\/linear-equations\/"},"modified":"2023-12-05T09:45:34","modified_gmt":"2023-12-05T09:45:34","slug":"linear-equations","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/chapter\/linear-equations\/","title":{"raw":"What is a Linear Equation?","rendered":"What is a Linear Equation?"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Determine if a graph or equation is linear<\/li>\r\n \t<li>Create a linear equation for a given scenario<\/li>\r\n \t<li>For a given linear equation, identify the independent and dependent variables<\/li>\r\n \t<li>Interpret the slope and y-intercept in context for a given linear equation<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>Recall:\u00a0GRAPHING INDEPENDENT &amp; DEPENDENT VARIABLE<\/h3>\r\nWhen both the input (independent variable) and the output (dependent variable) are real numbers, a function can be represented by a coordinate graph. The input is plotted on the horizontal x-axis and the output is plotted on the vertical y-axis.\r\n\r\n<\/div>\r\n&nbsp;\r\n\r\nLinear regression for two variables is based on a linear equation with one independent variable. The equation has the form:\r\n<p style=\"text-align: center;\">[latex]y = a + bx[\/latex]<\/p>\r\nwhere <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=\"textbox exercises\">\r\n<h3>Example 1<\/h3>\r\nThe following examples are linear equations.\r\n<p style=\"text-align: center;\">[latex]y = 3 + 2x[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]y = -0.01 + 1.2x[\/latex]<\/p>\r\n\r\n<\/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 1<\/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\" style=\"text-align: center;\">[latex]y = \u20130.125 \u2013 3.5x[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><\/div>\r\n<\/div>\r\nThe graph of a linear equation 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.\r\n<div class=\"textbox exercises\">\r\n<h3>Example 2<\/h3>\r\n<div id=\"element-44\" class=\"example\" data-type=\"example\"><section>\r\n<p id=\"eip-153\">Graph the equation [latex]y = \u20131 + 2x[\/latex].<\/p>\r\n<img class=\"aligncenter wp-image-2288 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5668\/2021\/07\/11145019\/58c6fc49ff4d159b3739185a0969b76ec243ea94.jpeg\" 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=\"703\" height=\"401\" \/>\r\n\r\n<\/section><\/div>\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>Recall:\u00a0GRAPHING A LINEAR EQUATION OF THE FORM [latex]y = a + bx[\/latex]<\/h3>\r\nTo graph an equation in the form of [latex]y = a + bx[\/latex], the a is where the line crosses the [latex]y[\/latex]-axis and the [latex]b[\/latex] is the slope. Slope is the [latex]\\frac{\\mathrm{rise}}{\\mathrm{run}}[\/latex]. The rise goes up for a positive number and down for negative numbers, the run goes right with a positive number and left with negative numbers. So to graph the line in Example 2, put a dot where the equation crosses the [latex]y[\/latex]-axis (at -1) and count up 2 and right 1.\r\n\r\n<\/div>\r\n&nbsp;\r\n\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 2<\/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<img class=\"aligncenter wp-image-2289 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5668\/2021\/07\/11145119\/4daa281de84a53ce9a4a47f1375057b29ceb86de.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=\"488\" height=\"333\" \/>\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=\"\"><section>\r\n<div class=\"exercise\" data-type=\"exercise\"><section>\r\n<div id=\"eip-373\" class=\"problem\" data-type=\"problem\">\r\n<div class=\"textbox exercises\">\r\n<h3>Example 3<\/h3>\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<div id=\"element-455\" class=\" unnumbered\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"idp59356256\" data-type=\"problem\">\r\n<div class=\"os-problem-container \">\r\n<p id=\"element-60\" class=\" \">Find the equation that expresses the\u00a0<strong>total cost<\/strong>\u00a0in terms of the\u00a0<strong>number of hours<\/strong>\u00a0required to complete the job.\r\n[reveal-answer q=\"935381\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"935381\"]<\/p>\r\nLet <em>x<\/em> = the number of hours it takes to get the job done.\r\nLet <em>y<\/em> = the total cost to the customer.\r\n\r\nThe $31.50 is a fixed cost. If it takes <em>x<\/em> hours to complete the job, then (32)(<em>x<\/em>) is the cost of the word processing only. The total cost is: <em>y<\/em> = 31.50 + 32<em>x<\/em>\r\n<p class=\" \">[\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><\/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 3<\/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. Find 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 class=\"aligncenter\" 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\" height=\"229\" data-media-type=\"image\/jpg\" \/><\/span><\/figure>\r\n<\/section>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.\r\n<div id=\"element-201\" class=\"example\" data-type=\"example\"><section>\r\n<div class=\"textbox exercises\">\r\n<h3>Example 4<\/h3>\r\nSvetlana 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.\r\n[reveal-answer q=\"221156\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"221156\"]<\/p>\r\nThe independent variable (<em>x<\/em>) is the number of hours Svetlana tutors each session. The dependent variable (<em>y<\/em>) is the amount, in dollars, Svetlana earns for each session.\r\n\r\nThe y-intercept is 25 (<em>a<\/em> = 25). At the start of the tutoring session, Svetlana charges a one-time fee of $25 (this is when <em>x<\/em> = 0). The slope is 15 (<em>b<\/em> = 15). For each session, Svetlana earns $15 for each hour she tutors.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/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 4<\/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 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>Determine if a graph or equation is linear<\/li>\n<li>Create a linear equation for a given scenario<\/li>\n<li>For a given linear equation, identify the independent and dependent variables<\/li>\n<li>Interpret the slope and y-intercept in context for a given linear equation<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox examples\">\n<h3>Recall:\u00a0GRAPHING INDEPENDENT &amp; DEPENDENT VARIABLE<\/h3>\n<p>When both the input (independent variable) and the output (dependent variable) are real numbers, a function can be represented by a coordinate graph. The input is plotted on the horizontal x-axis and the output is plotted on the vertical y-axis.<\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<p>Linear regression for two variables is based on a linear equation with one independent variable. The equation has the form:<\/p>\n<p style=\"text-align: center;\">[latex]y = a + bx[\/latex]<\/p>\n<p>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=\"textbox exercises\">\n<h3>Example 1<\/h3>\n<p>The following examples are linear equations.<\/p>\n<p style=\"text-align: center;\">[latex]y = 3 + 2x[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]y = -0.01 + 1.2x[\/latex]<\/p>\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 1<\/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\" style=\"text-align: center;\">[latex]y = \u20130.125 \u2013 3.5x[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<\/div>\n<\/div>\n<p>The graph of a linear equation 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 class=\"textbox exercises\">\n<h3>Example 2<\/h3>\n<div id=\"element-44\" class=\"example\" data-type=\"example\">\n<section>\n<p id=\"eip-153\">Graph the equation [latex]y = \u20131 + 2x[\/latex].<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-2288 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5668\/2021\/07\/11145019\/58c6fc49ff4d159b3739185a0969b76ec243ea94.jpeg\" 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=\"703\" height=\"401\" \/><\/p>\n<\/section>\n<\/div>\n<\/div>\n<div class=\"textbox examples\">\n<h3>Recall:\u00a0GRAPHING A LINEAR EQUATION OF THE FORM [latex]y = a + bx[\/latex]<\/h3>\n<p>To graph an equation in the form of [latex]y = a + bx[\/latex], the a is where the line crosses the [latex]y[\/latex]-axis and the [latex]b[\/latex] is the slope. Slope is the [latex]\\frac{\\mathrm{rise}}{\\mathrm{run}}[\/latex]. The rise goes up for a positive number and down for negative numbers, the run goes right with a positive number and left with negative numbers. So to graph the line in Example 2, put a dot where the equation crosses the [latex]y[\/latex]-axis (at -1) and count up 2 and right 1.<\/p>\n<\/div>\n<p>&nbsp;<\/p>\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 2<\/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><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-2289 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5668\/2021\/07\/11145119\/4daa281de84a53ce9a4a47f1375057b29ceb86de.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=\"488\" height=\"333\" \/><\/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=\"\">\n<section>\n<div class=\"exercise\" data-type=\"exercise\">\n<section>\n<div id=\"eip-373\" class=\"problem\" data-type=\"problem\">\n<div class=\"textbox exercises\">\n<h3>Example 3<\/h3>\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<div id=\"element-455\" class=\"unnumbered\" data-type=\"exercise\"><\/div>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<\/div>\n<\/div>\n<\/header>\n<header><\/header>\n<section>\n<div id=\"idp59356256\" data-type=\"problem\">\n<div class=\"os-problem-container\">\n<p id=\"element-60\" class=\"\">Find the equation that expresses the\u00a0<strong>total cost<\/strong>\u00a0in terms of the\u00a0<strong>number of hours<\/strong>\u00a0required to complete the job.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q935381\">Show Answer<\/span><\/p>\n<div id=\"q935381\" class=\"hidden-answer\" style=\"display: none\">\n<p>Let <em>x<\/em> = the number of hours it takes to get the job done.<br \/>\nLet <em>y<\/em> = the total cost to the customer.<\/p>\n<p>The $31.50 is a fixed cost. If it takes <em>x<\/em> hours to complete the job, then (32)(<em>x<\/em>) is the cost of the word processing only. The total cost is: <em>y<\/em> = 31.50 + 32<em>x<\/em><\/p>\n<p class=\"\"><\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\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 3<\/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. 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 loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" 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\" height=\"229\" data-media-type=\"image\/jpg\" \/><\/span><\/figure>\n<\/section>\n<p>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.<\/p>\n<div id=\"element-201\" class=\"example\" data-type=\"example\">\n<section>\n<div class=\"textbox exercises\">\n<h3>Example 4<\/h3>\n<p>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 class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q221156\">Show Answer<\/span><\/p>\n<div id=\"q221156\" class=\"hidden-answer\" style=\"display: none\">\n<p>The independent variable (<em>x<\/em>) is the number of hours Svetlana tutors each session. The dependent variable (<em>y<\/em>) is the amount, in dollars, Svetlana earns for each session.<\/p>\n<p>The y-intercept is 25 (<em>a<\/em> = 25). At the start of the tutoring session, Svetlana charges a one-time fee of $25 (this is when <em>x<\/em> = 0). The slope is 15 (<em>b<\/em> = 15). For each session, Svetlana earns $15 for each hour she tutors.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\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 4<\/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 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-305\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li><strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Introductory Statistics. <strong>Authored by<\/strong>: Barbara Illowsky, Susan Dean. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/openstax.org\/books\/introductory-statistics\/pages\/1-introduction\">https:\/\/openstax.org\/books\/introductory-statistics\/pages\/1-introduction<\/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>: Access for free at https:\/\/openstax.org\/books\/introductory-statistics\/pages\/1-introduction<\/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":169134,"menu_order":8,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Introductory Statistics\",\"author\":\"Barbara Illowsky, Susan Dean\",\"organization\":\"OpenStax\",\"url\":\"https:\/\/openstax.org\/books\/introductory-statistics\/pages\/1-introduction\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Access for free at https:\/\/openstax.org\/books\/introductory-statistics\/pages\/1-introduction\"},{\"type\":\"original\",\"description\":\"\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-305","chapter","type-chapter","status-publish","hentry"],"part":303,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/305","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/users\/169134"}],"version-history":[{"count":16,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/305\/revisions"}],"predecessor-version":[{"id":3970,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/305\/revisions\/3970"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/parts\/303"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/305\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/media?parent=305"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapter-type?post=305"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/contributor?post=305"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/license?post=305"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}