{"id":2373,"date":"2021-10-12T16:04:07","date_gmt":"2021-10-12T16:04:07","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/?post_type=chapter&#038;p=2373"},"modified":"2022-04-23T00:12:51","modified_gmt":"2022-04-23T00:12:51","slug":"slope-intercept-form-for-the-equation-of-a-line","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/chapter\/slope-intercept-form-for-the-equation-of-a-line\/","title":{"raw":"Slope-Intercept Form for the Equation of a Line","rendered":"Slope-Intercept Form for the Equation of a Line"},"content":{"raw":"<div class=\"bcc-box bcc-highlight\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Use slope-intercept form to plot equations of lines<\/li>\r\n \t<li>Interpret the slope and y-intercept for the equation of a line<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Slope-Intercept Form<\/h2>\r\nPerhaps the most familiar form of a linear equation is slope-intercept form written as [latex]y=mx+b[\/latex], where [latex]m=\\text{slope}[\/latex] and [latex]b=y\\text{-intercept}[\/latex]. Let us begin with the slope.\r\n<h3>The Slope of a Line<\/h3>\r\nThe <strong>slope<\/strong> of a line refers to the ratio of the vertical change in <em>y<\/em> over the horizontal change in <em>x<\/em> between any two points on a line. It indicates the direction in which a line slants as well as its steepness. Slope is sometimes described as rise over run.\r\n<div style=\"text-align: center;\">[latex]m=\\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}[\/latex]<\/div>\r\nIf the slope is positive, the line slants upward to the right. If the slope is negative, the line slants downward to the right. As the slope increases, the line becomes steeper. Some examples are shown below. The lines indicate the following slopes: [latex]m=-3[\/latex], [latex]m=2[\/latex], and [latex]m=\\frac{1}{3}[\/latex].\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/11185922\/CNX_CAT_Figure_02_02_002.jpg\" alt=\"Coordinate plane with the x and y axes ranging from negative 10 to 10. Three linear functions are plotted: y = negative 3 times x minus 2; y = 2 times x plus 1; and y = x over 3 plus 2.\" width=\"487\" height=\"442\" \/>\r\n<div class=\"textbox\">\r\n<h3>A General Note: The Slope of a Line<\/h3>\r\nThe slope of a line, <em>m<\/em>, represents the change in <em>y<\/em> over the change in <em>x.<\/em> Given two points, [latex]\\left({x}_{1},{y}_{1}\\right)[\/latex] and [latex]\\left({x}_{2},{y}_{2}\\right)[\/latex], the following formula determines the slope of a line containing these points:\r\n<div style=\"text-align: center;\">[latex]m=\\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}[\/latex]<\/div>\r\n<\/div>\r\n<div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Finding the Slope of a Line Given Two Points<\/h3>\r\nFind the slope of a line that passes through the points [latex]\\left(2,-1\\right)[\/latex] and [latex]\\left(-5,3\\right)[\/latex].\r\n[reveal-answer q=\"688301\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"688301\"]\r\n\r\nWe substitute the <em>y-<\/em>values and the <em>x-<\/em>values into the formula.\r\n<div style=\"text-align: center;\">[latex]\\begin{array}{l}m\\hfill&amp;=\\frac{3-\\left(-1\\right)}{-5 - 2}\\hfill \\\\ \\hfill&amp;=\\frac{4}{-7}\\hfill \\\\ \\hfill&amp;=-\\frac{4}{7}\\hfill \\end{array}[\/latex]<\/div>\r\nThe slope is [latex]-\\frac{4}{7}[\/latex].\r\n<h4>Analysis of the Solution<\/h4>\r\nIt does not matter which point is called [latex]\\left({x}_{1},{y}_{1}\\right)[\/latex] or [latex]\\left({x}_{2},{y}_{2}\\right)[\/latex]. As long as we are consistent with the order of the <em>y<\/em> terms and the order of the <em>x<\/em> terms in the numerator and denominator, the calculation will yield the same result.\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]2100[\/ohm_question]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Identifying the Slope and <em>y-<\/em>intercept of a Line Given an Equation<\/h3>\r\nIdentify the slope and <em>y-<\/em>intercept given the equation [latex]y=-\\frac{3}{4}x - 4[\/latex].\r\n[reveal-answer q=\"757424\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"757424\"]\r\n\r\nAs the line is in [latex]y=mx+b[\/latex] form, the given line has a slope of [latex]m=-\\frac{3}{4}[\/latex]. The <em>y-<\/em>intercept is [latex]b=-4[\/latex].\r\n<h4>Analysis of the Solution<\/h4>\r\nThe <em>y<\/em>-intercept is the point at which the line crosses the <em>y-<\/em>axis. On the <em>y-<\/em>axis, [latex]x=0[\/latex]. We can always identify the <em>y-<\/em>intercept when the line is in slope-intercept form, as it will always equal <em>b.<\/em> Or, just substitute [latex]x=0[\/latex] and solve for <em>y.<\/em>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h3>INTERPRETING THE SLOPE AND Y-INTERCEPT<\/h3>\r\nThe <em>y<\/em>-intercept, <em>b<\/em>, is the point where the line crosses the <em>y<\/em>-axis. The y-intercept also tells us the value of <em>y<\/em> when\u00a0[latex]x\u00a0= 0[\/latex].\r\n\r\nThe slope, <em>m<\/em>, is the change in <em>y<\/em> divided by the change in <em>x<\/em>. Then if <em>x<\/em> increases by 1 unit, <em>y<\/em> changes by <em>m<\/em> units.\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Identifying the Slope and <em>y-<\/em>intercept of a Line Given an Equation<\/h3>\r\nA restaurant charges $200 to rent a private dining room and $15 per person for a luncheon. Then the cost y, in dollars, if x persons attend the luncheon is described by\r\n<p style=\"text-align: center;\">[latex]y=15x+200[\/latex].<\/p>\r\nThe line is in slope-intercept form, [latex]y=mx+b[\/latex], the given line has a slope of [latex]m=15[\/latex], and the <em>y<\/em>-intercept is [latex]b=200[\/latex]. The <em>y<\/em>-intercept tells us that the restaurant will charge $200 even if no one attends the luncheon. The slope, 15, tells us the cost will increase by $15 for every additional guest at the luncheon.\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Identifying the Slope and <em>y-<\/em>intercept of a Line Given an Equation<\/h3>\r\nA student begins driving home for a holiday break. The distance, <em>y<\/em>, from home after <em>x<\/em> hours of driving is described by\r\n<p style=\"text-align: center;\">[latex]y=-45x+150[\/latex].<\/p>\r\nThe line is in slope-intercept form, [latex]y=mx+b[\/latex], the given line has a slope of [latex]m=-45[\/latex], and the <em>y<\/em>-intercept is [latex]b=150[\/latex]. The <em>y<\/em>-intercept tells us that when the student begins their trip, they are 150 miles from home. The slope, -45, tells us the distance from home decreases by 45 miles for every hour the student drives.\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]3212[\/ohm_question]\r\n\r\n<\/div>\r\n<\/div>","rendered":"<div class=\"bcc-box bcc-highlight\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Use slope-intercept form to plot equations of lines<\/li>\n<li>Interpret the slope and y-intercept for the equation of a line<\/li>\n<\/ul>\n<\/div>\n<h2>Slope-Intercept Form<\/h2>\n<p>Perhaps the most familiar form of a linear equation is slope-intercept form written as [latex]y=mx+b[\/latex], where [latex]m=\\text{slope}[\/latex] and [latex]b=y\\text{-intercept}[\/latex]. Let us begin with the slope.<\/p>\n<h3>The Slope of a Line<\/h3>\n<p>The <strong>slope<\/strong> of a line refers to the ratio of the vertical change in <em>y<\/em> over the horizontal change in <em>x<\/em> between any two points on a line. It indicates the direction in which a line slants as well as its steepness. Slope is sometimes described as rise over run.<\/p>\n<div style=\"text-align: center;\">[latex]m=\\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}[\/latex]<\/div>\n<p>If the slope is positive, the line slants upward to the right. If the slope is negative, the line slants downward to the right. As the slope increases, the line becomes steeper. Some examples are shown below. The lines indicate the following slopes: [latex]m=-3[\/latex], [latex]m=2[\/latex], and [latex]m=\\frac{1}{3}[\/latex].<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/11185922\/CNX_CAT_Figure_02_02_002.jpg\" alt=\"Coordinate plane with the x and y axes ranging from negative 10 to 10. Three linear functions are plotted: y = negative 3 times x minus 2; y = 2 times x plus 1; and y = x over 3 plus 2.\" width=\"487\" height=\"442\" \/><\/p>\n<div class=\"textbox\">\n<h3>A General Note: The Slope of a Line<\/h3>\n<p>The slope of a line, <em>m<\/em>, represents the change in <em>y<\/em> over the change in <em>x.<\/em> Given two points, [latex]\\left({x}_{1},{y}_{1}\\right)[\/latex] and [latex]\\left({x}_{2},{y}_{2}\\right)[\/latex], the following formula determines the slope of a line containing these points:<\/p>\n<div style=\"text-align: center;\">[latex]m=\\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}[\/latex]<\/div>\n<\/div>\n<div>\n<div class=\"textbox exercises\">\n<h3>Example: Finding the Slope of a Line Given Two Points<\/h3>\n<p>Find the slope of a line that passes through the points [latex]\\left(2,-1\\right)[\/latex] and [latex]\\left(-5,3\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q688301\">Show Solution<\/span><\/p>\n<div id=\"q688301\" class=\"hidden-answer\" style=\"display: none\">\n<p>We substitute the <em>y-<\/em>values and the <em>x-<\/em>values into the formula.<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{l}m\\hfill&=\\frac{3-\\left(-1\\right)}{-5 - 2}\\hfill \\\\ \\hfill&=\\frac{4}{-7}\\hfill \\\\ \\hfill&=-\\frac{4}{7}\\hfill \\end{array}[\/latex]<\/div>\n<p>The slope is [latex]-\\frac{4}{7}[\/latex].<\/p>\n<h4>Analysis of the Solution<\/h4>\n<p>It does not matter which point is called [latex]\\left({x}_{1},{y}_{1}\\right)[\/latex] or [latex]\\left({x}_{2},{y}_{2}\\right)[\/latex]. As long as we are consistent with the order of the <em>y<\/em> terms and the order of the <em>x<\/em> terms in the numerator and denominator, the calculation will yield the same result.\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm2100\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=2100&theme=oea&iframe_resize_id=ohm2100&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Identifying the Slope and <em>y-<\/em>intercept of a Line Given an Equation<\/h3>\n<p>Identify the slope and <em>y-<\/em>intercept given the equation [latex]y=-\\frac{3}{4}x - 4[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q757424\">Show Solution<\/span><\/p>\n<div id=\"q757424\" class=\"hidden-answer\" style=\"display: none\">\n<p>As the line is in [latex]y=mx+b[\/latex] form, the given line has a slope of [latex]m=-\\frac{3}{4}[\/latex]. The <em>y-<\/em>intercept is [latex]b=-4[\/latex].<\/p>\n<h4>Analysis of the Solution<\/h4>\n<p>The <em>y<\/em>-intercept is the point at which the line crosses the <em>y-<\/em>axis. On the <em>y-<\/em>axis, [latex]x=0[\/latex]. We can always identify the <em>y-<\/em>intercept when the line is in slope-intercept form, as it will always equal <em>b.<\/em> Or, just substitute [latex]x=0[\/latex] and solve for <em>y.<\/em><\/p>\n<\/div>\n<\/div>\n<\/div>\n<h3>INTERPRETING THE SLOPE AND Y-INTERCEPT<\/h3>\n<p>The <em>y<\/em>-intercept, <em>b<\/em>, is the point where the line crosses the <em>y<\/em>-axis. The y-intercept also tells us the value of <em>y<\/em> when\u00a0[latex]x\u00a0= 0[\/latex].<\/p>\n<p>The slope, <em>m<\/em>, is the change in <em>y<\/em> divided by the change in <em>x<\/em>. Then if <em>x<\/em> increases by 1 unit, <em>y<\/em> changes by <em>m<\/em> units.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example: Identifying the Slope and <em>y-<\/em>intercept of a Line Given an Equation<\/h3>\n<p>A restaurant charges $200 to rent a private dining room and $15 per person for a luncheon. Then the cost y, in dollars, if x persons attend the luncheon is described by<\/p>\n<p style=\"text-align: center;\">[latex]y=15x+200[\/latex].<\/p>\n<p>The line is in slope-intercept form, [latex]y=mx+b[\/latex], the given line has a slope of [latex]m=15[\/latex], and the <em>y<\/em>-intercept is [latex]b=200[\/latex]. The <em>y<\/em>-intercept tells us that the restaurant will charge $200 even if no one attends the luncheon. The slope, 15, tells us the cost will increase by $15 for every additional guest at the luncheon.<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Identifying the Slope and <em>y-<\/em>intercept of a Line Given an Equation<\/h3>\n<p>A student begins driving home for a holiday break. The distance, <em>y<\/em>, from home after <em>x<\/em> hours of driving is described by<\/p>\n<p style=\"text-align: center;\">[latex]y=-45x+150[\/latex].<\/p>\n<p>The line is in slope-intercept form, [latex]y=mx+b[\/latex], the given line has a slope of [latex]m=-45[\/latex], and the <em>y<\/em>-intercept is [latex]b=150[\/latex]. The <em>y<\/em>-intercept tells us that when the student begins their trip, they are 150 miles from home. The slope, -45, tells us the distance from home decreases by 45 miles for every hour the student drives.<\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm3212\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=3212&theme=oea&iframe_resize_id=ohm3212&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<\/div>\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-2373\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Revision and Adaptation. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>College Algebra. <strong>Authored by<\/strong>: Abramson, Jay et al.. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/openstax.org\/books\/college-algebra\/pages\/1-introduction-to-prerequisites\">https:\/\/openstax.org\/books\/college-algebra\/pages\/1-introduction-to-prerequisites<\/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\/college-algebra\/pages\/1-introduction-to-prerequisites<\/li><li>Question ID 2100, 3212. <strong>Authored by<\/strong>: Lippman, D. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC- BY + GPL<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":169134,"menu_order":5,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Revision and Adaptation\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"College Algebra\",\"author\":\"Abramson, Jay et al.\",\"organization\":\"OpenStax\",\"url\":\"https:\/\/openstax.org\/books\/college-algebra\/pages\/1-introduction-to-prerequisites\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Access for free at https:\/\/openstax.org\/books\/college-algebra\/pages\/1-introduction-to-prerequisites\"},{\"type\":\"cc\",\"description\":\"Question ID 2100, 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