{"id":3176,"date":"2025-08-15T22:28:43","date_gmt":"2025-08-15T22:28:43","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=3176"},"modified":"2026-03-20T16:02:03","modified_gmt":"2026-03-20T16:02:03","slug":"systems-of-linear-equations-two-variables-apply-it","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/systems-of-linear-equations-two-variables-apply-it\/","title":{"raw":"Systems of Linear Equations: Two Variables: Apply It","rendered":"Systems of Linear Equations: Two Variables: Apply It"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Solve systems of equations by graphing.<\/li>\r\n \t<li>Solve systems of equations algebraically.<\/li>\r\n \t<li>Identify inconsistent and dependent systems of equations containing two variables.<\/li>\r\n \t<li>Solve applied systems.<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Using Systems of Equations to Investigate Profits<\/h2>\r\nUsing what we have learned about systems of equations, we can return to the skateboard manufacturing problem at the beginning of the section. The skateboard manufacturer\u2019s <strong>revenue function<\/strong> is the function used to calculate the amount of money that comes into the business. It can be represented by the equation [latex]R=xp[\/latex], where [latex]x=[\/latex] quantity and [latex]p=[\/latex] price.\r\n\r\nThe <strong>cost function<\/strong> is the function used to calculate the costs of doing business. It includes fixed costs, such as rent and salaries, and variable costs, such as utilities. The [latex]x[\/latex] -axis represents quantity in hundreds of units. The <em>y<\/em>-axis represents either cost or revenue in hundreds of dollars.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27181315\/CNX_Precalc_Figure_09_01_0092.jpg\" alt=\"A graph showing money in hundreds of dollars on the y axis and quantity in hundreds of units on the x axis. A line representing cost and a line representing revenue cross at the point (7,33), which is marked break-even. The shaded space between the two lines to the right of the break-even point is labeled profit.\" width=\"488\" height=\"347\" \/>\r\n\r\nThe point at which the two lines intersect is called the <strong>break-even point<\/strong>. We can see from the graph that if 700 units are produced, the cost is $3,300 and the revenue is also $3,300. In other words, the company breaks even if they produce and sell 700 units. They neither make money nor lose money.\r\n\r\nThe shaded region to the right of the break-even point represents quantities for which the company makes a profit. The shaded region to the left represents quantities for which the company suffers a loss. The <strong>profit function<\/strong> is the revenue function minus the cost function, written as [latex]P\\left(x\\right)=R\\left(x\\right)-C\\left(x\\right)[\/latex]. Clearly, knowing the quantity for which the cost equals the revenue is of great importance to businesses.\r\n\r\n<section class=\"textbox example\" aria-label=\"Example\">Given the cost function [latex]C\\left(x\\right)=0.85x+35,000[\/latex] and the revenue function [latex]R\\left(x\\right)=1.55x[\/latex], find the break-even point and the profit function.[reveal-answer q=\"173283\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"173283\"]Write the system of equations using [latex]y[\/latex] to replace function notation.\r\n<p style=\"text-align: center;\">[latex]\\begin{align} y&amp;=0.85x+35,000 \\\\ y&amp;=1.55x \\end{align}[\/latex]<\/p>\r\nSubstitute the expression [latex]0.85x+35,000[\/latex] from the first equation into the second equation and solve for [latex]x[\/latex].\r\n<p style=\"text-align: center;\">[latex]\\begin{gathered}0.85x+35,000=1.55x\\\\ 35,000=0.7x\\\\ 50,000=x\\end{gathered}[\/latex]<\/p>\r\nThen, we substitute [latex]x=50,000[\/latex] into either the cost function or the revenue function.\r\n<p style=\"text-align: center;\">[latex]1.55\\left(50,000\\right)=77,500[\/latex]<\/p>\r\nThe break-even point is [latex]\\left(50,000,77,500\\right)[\/latex].\r\n\r\nThe profit function is found using the formula [latex]P\\left(x\\right)=R\\left(x\\right)-C\\left(x\\right)[\/latex].\r\n<p style=\"text-align: center;\">[latex]\\begin{align}P\\left(x\\right)&amp;=1.55x-\\left(0.85x+35,000\\right) \\\\ &amp;=0.7x - 35,000 \\end{align}[\/latex]<\/p>\r\nThe profit function is [latex]P\\left(x\\right)=0.7x - 35,000[\/latex].\r\n<h4>Analysis of the Solution<\/h4>\r\nThe cost to produce 50,000 units is $77,500, and the revenue from the sales of 50,000 units is also $77,500. To make a profit, the business must produce and sell more than 50,000 units.\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27181317\/CNX_Precalc_Figure_09_01_0102.jpg\" alt=\"A graph showing money in dollars on the y axis and quantity on the x axis. A line representing cost and a line representing revenue cross at the break-even point of fifty thousand, seventy-seven thousand five hundred. The cost line's equation is C(x)=0.85x+35,000. The revenue line's equation is R(x)=1.55x. The shaded space between the two lines to the right of the break-even point is labeled profit.\" width=\"487\" height=\"390\" \/>\r\n\r\nWe see from the graph that the profit function has a negative value until [latex]x=50,000[\/latex], when the graph crosses the <em>x<\/em>-axis. Then, the graph emerges into positive <em>y<\/em>-values and continues on this path as the profit function is a straight line. This illustrates that the break-even point for businesses occurs when the profit function is 0. The area to the left of the break-even point represents operating at a loss.\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27181320\/CNX_Precalc_Figure_09_01_0112.jpg\" alt=\"A graph showing dollars profit on the y axis and quantity on the x axis. The profit line crosses the break-even point at fifty thousand, zero. The profit line's equation is P(x)=0.7x-35,000.\" width=\"731\" height=\"507\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">The cost of a ticket to the circus is $25.00 for children and $50.00 for adults. On a certain day, attendance at the circus is 2,000 and the total gate revenue is $70,000. How many children and how many adults bought tickets?[reveal-answer q=\"694801\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"694801\"]Let <em>c<\/em> = the number of children and <em>a<\/em> = the number of adults in attendance.The total number of people is [latex]2,000[\/latex]. We can use this to write an equation for the number of people at the circus that day.\r\n<p style=\"text-align: center;\">[latex]c+a=2,000[\/latex]<\/p>\r\nThe revenue from all children can be found by multiplying $25.00 by the number of children, [latex]25c[\/latex]. The revenue from all adults can be found by multiplying $50.00 by the number of adults, [latex]50a[\/latex]. The total revenue is $70,000. We can use this to write an equation for the revenue.\r\n<p style=\"text-align: center;\">[latex]25c+50a=70,000[\/latex]<\/p>\r\nWe now have a system of linear equations in two variables.\r\n<p style=\"text-align: center;\">[latex]\\begin{gathered}c+a=2,000\\\\ 25c+50a=70,000\\end{gathered}[\/latex]<\/p>\r\nIn the first equation, the coefficient of both variables is 1. We can quickly solve the first equation for either [latex]c[\/latex] or [latex]a[\/latex]. We will solve for [latex]a[\/latex].\r\n<p style=\"text-align: center;\">[latex]\\begin{gathered}c+a=2,000\\\\ a=2,000-c\\end{gathered}[\/latex]<\/p>\r\nSubstitute the expression [latex]2,000-c[\/latex] in the second equation for [latex]a[\/latex] and solve for [latex]c[\/latex].\r\n<p style=\"text-align: center;\">[latex]\\begin{gathered} 25c+50\\left(2,000-c\\right)=70,000 \\\\ 25c+100,000 - 50c=70,000\\\\-25c=-30,000 \\\\ c=1,200 \\end{gathered}[\/latex]<\/p>\r\nSubstitute [latex]c=1,200[\/latex] into the first equation to solve for [latex]a[\/latex].\r\n<p style=\"text-align: center;\">[latex]\\begin{gathered}1,200+a=2,000 \\\\ a=800 \\end{gathered}[\/latex]<\/p>\r\nWe find that [latex]1,200[\/latex] children and [latex]800[\/latex] adults bought tickets to the circus that day.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]321615[\/ohm_question]<\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Solve systems of equations by graphing.<\/li>\n<li>Solve systems of equations algebraically.<\/li>\n<li>Identify inconsistent and dependent systems of equations containing two variables.<\/li>\n<li>Solve applied systems.<\/li>\n<\/ul>\n<\/section>\n<h2>Using Systems of Equations to Investigate Profits<\/h2>\n<p>Using what we have learned about systems of equations, we can return to the skateboard manufacturing problem at the beginning of the section. The skateboard manufacturer\u2019s <strong>revenue function<\/strong> is the function used to calculate the amount of money that comes into the business. It can be represented by the equation [latex]R=xp[\/latex], where [latex]x=[\/latex] quantity and [latex]p=[\/latex] price.<\/p>\n<p>The <strong>cost function<\/strong> is the function used to calculate the costs of doing business. It includes fixed costs, such as rent and salaries, and variable costs, such as utilities. The [latex]x[\/latex] -axis represents quantity in hundreds of units. The <em>y<\/em>-axis represents either cost or revenue in hundreds of dollars.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27181315\/CNX_Precalc_Figure_09_01_0092.jpg\" alt=\"A graph showing money in hundreds of dollars on the y axis and quantity in hundreds of units on the x axis. A line representing cost and a line representing revenue cross at the point (7,33), which is marked break-even. The shaded space between the two lines to the right of the break-even point is labeled profit.\" width=\"488\" height=\"347\" \/><\/p>\n<p>The point at which the two lines intersect is called the <strong>break-even point<\/strong>. We can see from the graph that if 700 units are produced, the cost is $3,300 and the revenue is also $3,300. In other words, the company breaks even if they produce and sell 700 units. They neither make money nor lose money.<\/p>\n<p>The shaded region to the right of the break-even point represents quantities for which the company makes a profit. The shaded region to the left represents quantities for which the company suffers a loss. The <strong>profit function<\/strong> is the revenue function minus the cost function, written as [latex]P\\left(x\\right)=R\\left(x\\right)-C\\left(x\\right)[\/latex]. Clearly, knowing the quantity for which the cost equals the revenue is of great importance to businesses.<\/p>\n<section class=\"textbox example\" aria-label=\"Example\">Given the cost function [latex]C\\left(x\\right)=0.85x+35,000[\/latex] and the revenue function [latex]R\\left(x\\right)=1.55x[\/latex], find the break-even point and the profit function.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q173283\">Show Solution<\/button><\/p>\n<div id=\"q173283\" class=\"hidden-answer\" style=\"display: none\">Write the system of equations using [latex]y[\/latex] to replace function notation.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align} y&=0.85x+35,000 \\\\ y&=1.55x \\end{align}[\/latex]<\/p>\n<p>Substitute the expression [latex]0.85x+35,000[\/latex] from the first equation into the second equation and solve for [latex]x[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{gathered}0.85x+35,000=1.55x\\\\ 35,000=0.7x\\\\ 50,000=x\\end{gathered}[\/latex]<\/p>\n<p>Then, we substitute [latex]x=50,000[\/latex] into either the cost function or the revenue function.<\/p>\n<p style=\"text-align: center;\">[latex]1.55\\left(50,000\\right)=77,500[\/latex]<\/p>\n<p>The break-even point is [latex]\\left(50,000,77,500\\right)[\/latex].<\/p>\n<p>The profit function is found using the formula [latex]P\\left(x\\right)=R\\left(x\\right)-C\\left(x\\right)[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}P\\left(x\\right)&=1.55x-\\left(0.85x+35,000\\right) \\\\ &=0.7x - 35,000 \\end{align}[\/latex]<\/p>\n<p>The profit function is [latex]P\\left(x\\right)=0.7x - 35,000[\/latex].<\/p>\n<h4>Analysis of the Solution<\/h4>\n<p>The cost to produce 50,000 units is $77,500, and the revenue from the sales of 50,000 units is also $77,500. To make a profit, the business must produce and sell more than 50,000 units.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27181317\/CNX_Precalc_Figure_09_01_0102.jpg\" alt=\"A graph showing money in dollars on the y axis and quantity on the x axis. A line representing cost and a line representing revenue cross at the break-even point of fifty thousand, seventy-seven thousand five hundred. The cost line's equation is C(x)=0.85x+35,000. The revenue line's equation is R(x)=1.55x. The shaded space between the two lines to the right of the break-even point is labeled profit.\" width=\"487\" height=\"390\" \/><\/p>\n<p>We see from the graph that the profit function has a negative value until [latex]x=50,000[\/latex], when the graph crosses the <em>x<\/em>-axis. Then, the graph emerges into positive <em>y<\/em>-values and continues on this path as the profit function is a straight line. This illustrates that the break-even point for businesses occurs when the profit function is 0. The area to the left of the break-even point represents operating at a loss.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27181320\/CNX_Precalc_Figure_09_01_0112.jpg\" alt=\"A graph showing dollars profit on the y axis and quantity on the x axis. The profit line crosses the break-even point at fifty thousand, zero. The profit line's equation is P(x)=0.7x-35,000.\" width=\"731\" height=\"507\" \/><\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">The cost of a ticket to the circus is $25.00 for children and $50.00 for adults. On a certain day, attendance at the circus is 2,000 and the total gate revenue is $70,000. How many children and how many adults bought tickets?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q694801\">Show Solution<\/button><\/p>\n<div id=\"q694801\" class=\"hidden-answer\" style=\"display: none\">Let <em>c<\/em> = the number of children and <em>a<\/em> = the number of adults in attendance.The total number of people is [latex]2,000[\/latex]. We can use this to write an equation for the number of people at the circus that day.<\/p>\n<p style=\"text-align: center;\">[latex]c+a=2,000[\/latex]<\/p>\n<p>The revenue from all children can be found by multiplying $25.00 by the number of children, [latex]25c[\/latex]. The revenue from all adults can be found by multiplying $50.00 by the number of adults, [latex]50a[\/latex]. The total revenue is $70,000. We can use this to write an equation for the revenue.<\/p>\n<p style=\"text-align: center;\">[latex]25c+50a=70,000[\/latex]<\/p>\n<p>We now have a system of linear equations in two variables.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{gathered}c+a=2,000\\\\ 25c+50a=70,000\\end{gathered}[\/latex]<\/p>\n<p>In the first equation, the coefficient of both variables is 1. We can quickly solve the first equation for either [latex]c[\/latex] or [latex]a[\/latex]. We will solve for [latex]a[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{gathered}c+a=2,000\\\\ a=2,000-c\\end{gathered}[\/latex]<\/p>\n<p>Substitute the expression [latex]2,000-c[\/latex] in the second equation for [latex]a[\/latex] and solve for [latex]c[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{gathered} 25c+50\\left(2,000-c\\right)=70,000 \\\\ 25c+100,000 - 50c=70,000\\\\-25c=-30,000 \\\\ c=1,200 \\end{gathered}[\/latex]<\/p>\n<p>Substitute [latex]c=1,200[\/latex] into the first equation to solve for [latex]a[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{gathered}1,200+a=2,000 \\\\ a=800 \\end{gathered}[\/latex]<\/p>\n<p>We find that [latex]1,200[\/latex] children and [latex]800[\/latex] adults bought tickets to the circus that day.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm321615\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=321615&theme=lumen&iframe_resize_id=ohm321615&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n","protected":false},"author":67,"menu_order":12,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":131,"module-header":"apply_it","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/3176"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/users\/67"}],"version-history":[{"count":6,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/3176\/revisions"}],"predecessor-version":[{"id":5935,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/3176\/revisions\/5935"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/parts\/131"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/3176\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=3176"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=3176"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=3176"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=3176"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}