{"id":15370,"date":"2021-10-11T23:05:39","date_gmt":"2021-10-11T23:05:39","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/chapter\/put-it-together-systems-of-equations-and-inequalities\/"},"modified":"2021-11-01T01:29:48","modified_gmt":"2021-11-01T01:29:48","slug":"cost-and-revenue-problems","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/chapter\/cost-and-revenue-problems\/","title":{"raw":"Cost and Revenue Problems","rendered":"Cost and Revenue Problems"},"content":{"raw":"<div class=\"bcc-box bcc-highlight\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Solve cost and revenue problems<\/li>\r\n<\/ul>\r\n<\/div>\r\nA skateboard manufacturer introduces a new line of boards. The manufacturer tracks its costs, which is the amount it spends to produce the boards, and its revenue, which is the amount it earns through sales of its boards. How can the company determine if it is making a profit with its new line? How many skateboards must be produced and sold before a profit is possible?\r\n\r\n<img class=\"size-full wp-image-5880 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/117\/2016\/08\/01202934\/CNX_Precalc_Figure_09_01_0012.jpg\" alt=\"Skateboarders at a skating rink by the beach.\" width=\"487\" height=\"252\" \/>\r\n\r\n&nbsp;\r\n\r\n<b><\/b>Using what we have learned about systems of equations, we can answer these questions.\u00a0The skateboard manufacturer\u2019s revenue equation is the equation used to calculate the amount of money that comes into the business. It can be represented as [latex]y=xp[\/latex], where [latex]x=[\/latex] quantity and [latex]p=[\/latex] price. The revenue equation is shown in orange in the graph below.\r\n\r\nThe cost equation is the equation used to calculate the costs of doing business. It includes fixed costs, such as rent and salaries, and variable costs, such as utilities. The cost equation is shown in blue in the graph below. The [latex]x[\/latex] -axis represents quantity in hundreds of units. The [latex]y-axis[\/latex] represents both\u00a0cost and revenue in hundreds of dollars.\u00a0We won't learn how to write a cost equation in this example, they will be given to you. If you take any business or economics courses, you will learn more about how to write a cost equation.\r\n\r\n<img class=\"aligncenter wp-image-5882 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/117\/2016\/08\/01203010\/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\n&nbsp;\r\n\r\nThe point at which the two lines intersect is called the break-even point, we learned that this is the solution to the system of linear equations that in this case comprise the cost and revenue equations.\r\n\r\nRead the axes of the graph carefully, note that quantity is in hundreds, and money is in thousands. The solution to the graphed system is [latex](7, 33)[\/latex]. This means\u00a0that if [latex]700[\/latex] units are produced, the cost to make them is [latex]$3,300[\/latex] and the revenue is also [latex]$3,300[\/latex]. In other words, the company breaks even if they produce and sell [latex]700[\/latex] 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.\r\n\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nA business wants to manufacture bike frames. Before they start production, they need to make sure they can make a profit with the materials and labor force they have. Their accountant has given them a cost equation of [latex]y=0.85x+35,000[\/latex] and a revenue equation of [latex]y=1.55x[\/latex]:\r\n<ol>\r\n \t<li>Interpret [latex]x[\/latex] and [latex]y[\/latex] for the cost equation<\/li>\r\n \t<li>Interpret [latex]x[\/latex] and [latex]y[\/latex] for the revenue equation<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"86281\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"86281\"]\r\n\r\nCost: [latex]y=0.85x+35,000[\/latex]\r\n\r\nRevenue:[latex]y=1.55x[\/latex]\r\n\r\nThe cost equation represents money leaving the company, namely how much it costs to produce a given number of bike frames. If we use the skateboard example as a model, [latex]x[\/latex] would represent the number of frames produced (instead of skateboards) and y would represent the amount of money it would cost to produce them (the same as the skateboard problem).\r\n\r\nThe revenue equation represents money coming into the company, so in this context [latex]x[\/latex] still represents the number of bike frames manufactured, and [latex]y[\/latex] now represents the amount of money made from selling them. \u00a0Let's organize this information in a table:\r\n<table>\r\n<thead>\r\n<tr>\r\n<td>Equation Type<\/td>\r\n<td>[latex]x[\/latex] represents<\/td>\r\n<td>[latex]y[\/latex] represents<\/td>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>Revenue Eqn.<\/td>\r\n<td>number of frames<\/td>\r\n<td>amount of money made selling frames<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Cost Eqn.<\/td>\r\n<td>number of frames<\/td>\r\n<td>cost for making frames<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nGiven the same cost and revenue equations from the previous example, find the break-even point for the bike manufacturer. \u00a0Interpret the solution with words.\r\n\r\nCost: [latex]y=0.85x+35,000[\/latex]\r\n\r\nRevenue: [latex]y=1.55x[\/latex]\r\n[reveal-answer q=\"145700\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"145700\"]\r\n\r\n<strong>Read and Understand:\u00a0<\/strong>We want the break even point for this system that represents cost and revenue. \u00a0This means we want to find where the two lines cross, and we have learned a few different methods for doing this because this is the solution to the system of equations! \u00a0Substitution looks like the easiest method since the revenue equation is already solved for [latex]y[\/latex],\u00a0[latex]y=1.55x[\/latex].\r\n\r\n<strong>Define and Translate:\u00a0<\/strong>Write the system of equations.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}\\\\ y=0.85x+35,000\\hfill \\\\ y=1.55x\\hfill \\end{array}\\\\[\/latex]<\/p>\r\n&nbsp;\r\n\r\n<strong>Write and Solve:\u00a0<\/strong>The equations are already written for us, so we just need to solve the system using substitution.\r\n\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[latex]\\begin{array}{r}0.85x+35,000=1.55x\\\\ 35,000=0.7x\\,\\,\\,\\\\ 50,000=x\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\end{array}\\\\[\/latex]\r\nThen, we substitute [latex]x=50,000[\/latex] into either the cost equation or the revenue equation.\r\n[latex]1.55\\left(50,000\\right)=77,500\\\\[\/latex]\r\nThe break-even point is [latex]\\left(50,000,77,500\\right)[\/latex].\r\n<h4><strong>Check and Interpret:<\/strong><\/h4>\r\nThe solution to this system is[latex]\\left(50,000,77,500\\right)[\/latex], but what does that mean? Think of a point as [latex]\\left(x,y\\right)[\/latex], where in this case x is the quantity of bikes manufactured and [latex]y[\/latex] is an amount of money. For our system [latex]y[\/latex] represents two different things and [latex]x[\/latex] represents one thing. \u00a0Refer to the table we made in the first example, shown below:\r\n<table>\r\n<thead>\r\n<tr>\r\n<td>Equation Type<\/td>\r\n<td>[latex]x[\/latex] represents<\/td>\r\n<td>[latex]y[\/latex] represents<\/td>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>Revenue Eqn.<\/td>\r\n<td>number of frames<\/td>\r\n<td>amount of money made selling frames<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Cost Eqn.<\/td>\r\n<td>number of frames<\/td>\r\n<td>cost for making frames<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nLet's interpret the solution with respect to the Cost equation first. [latex]x = 50,000[\/latex] and [latex]y = 77,500[\/latex]. \u00a0Using our table, we can translate this as \"the cost for producing [latex]50,000[\/latex] bike frames is [latex]$75,500[\/latex]\".\r\n\r\nIn the same way, the Revenue equation can be interpreted as \"the amount of money the company makes from selling [latex]50,000[\/latex] bike frames is [latex]$77,500[\/latex]\".\r\n\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]150438[\/ohm_question]\r\n\r\n<\/div>\r\nIn the video you will see an example of how to find the break even point for a small snow cone business.\r\n\r\nhttps:\/\/youtu.be\/qey3FmE8saQ\r\n\r\nWe have seen that systems of linear equations and inequalities can help to define market behaviors that are very helpful to businesses. \u00a0The intersection of cost and revenue equations gives the break even point, and also helps define the region for which a company will make a profit.\r\n\r\nAnother situation that calls for system of equations is comparison of offers available to customers.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nThere are two cell phone companies that offer different packages. Company A charges a monthly service fee of $34 plus $.05\/min talk-time. Company B charges a monthly service fee of $40 plus $.04\/min talk-time.\r\n<ol>\r\n \t<li>Write a linear equation that models the packages offered by both companies.<\/li>\r\n \t<li>If the average number of minutes used each month is 1,160, which company offers the better plan?<\/li>\r\n \t<li>If the average number of minutes used each month is 420, which company offers the better plan?<\/li>\r\n \t<li>How many minutes of talk-time would yield equal monthly statements from both companies?<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"785384\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"785384\"]\r\n<ol>\r\n \t<li>The model for Company <em>A<\/em> can be written as [latex]A=0.05x+34[\/latex]. This includes the variable cost of [latex]0.05x[\/latex] plus the monthly service charge of $34. Company <em>B<\/em>\u2019s package charges a higher monthly fee of $40, but a lower variable cost of [latex]0.04x[\/latex]. Company <em>B<\/em>\u2019s model can be written as [latex]B=0.04x+40[\/latex].<\/li>\r\n \t<li>If the average number of minutes used each month is 1,160, we have the following:\r\n<div>[latex]\\begin{array}{l}\\text{Company }A\\hfill&amp;=0.05\\left(1,160\\right)+34\\hfill \\\\ \\hfill&amp;=58+34\\hfill \\\\ \\hfill&amp;=92\\hfill \\\\ \\hfill \\\\ \\text{Company }B\\hfill&amp;=0.04\\left(1,160\\right)+40\\hfill \\\\ \\hfill&amp;=46.4+40\\hfill \\\\ \\hfill&amp;=86.4\\hfill \\end{array}[\/latex]<\/div>\r\nSo, Company <em>B<\/em> offers the lower monthly cost of $86.40 as compared with the $92 monthly cost offered by Company <em>A<\/em> when the average number of minutes used each month is 1,160.<\/li>\r\n \t<li>If the average number of minutes used each month is 420, we have the following:\r\n<div>[latex]\\begin{array}{l}\\text{Company }A\\hfill&amp;=0.05\\left(420\\right)+34\\hfill \\\\ \\hfill&amp;=21+34\\hfill \\\\ \\hfill&amp;=55\\hfill \\\\ \\hfill \\\\ \\text{Company }B\\hfill&amp;=0.04\\left(420\\right)+40\\hfill \\\\ \\hfill&amp;=16.8+40\\hfill \\\\ \\hfill&amp;=56.8\\hfill \\end{array}[\/latex]<\/div>\r\nIf the average number of minutes used each month is 420, then Company <em>A <\/em>offers a lower monthly cost of $55 compared to Company <em>B<\/em>\u2019s monthly cost of $56.80.<\/li>\r\n \t<li>To answer the question of how many talk-time minutes would yield the same bill from both companies, we should think about the problem in terms of [latex]\\left(x,y\\right)[\/latex] coordinates: At what point are both the <em>x-<\/em>value and the <em>y-<\/em>value equal? We can find this point by setting the equations equal to each other and solving for <em>x.<\/em>\r\n<div style=\"text-align: center;\">[latex]\\begin{array}{l}0.05x+34=0.04x+40\\hfill \\\\ 0.01x=6\\hfill \\\\ x=600\\hfill \\end{array}[\/latex]<\/div>\r\nCheck the <em>x-<\/em>value in each equation.\r\n<div style=\"text-align: left;\">[latex]\\begin{array}{l}0.05\\left(600\\right)+34=64\\hfill \\\\ 0.04\\left(600\\right)+40=64\\hfill \\end{array}[\/latex]<\/div>\r\nTherefore, a monthly average of 600 talk-time minutes renders the plans equal.<\/li>\r\n<\/ol>\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/09\/25200339\/CNX_CAT_Figure_02_03_002.jpg\" alt=\"Coordinate plane with the x-axis ranging from 0 to 1200 in intervals of 100 and the y-axis ranging from 0 to 90 in intervals of 10. The functions A = 0.05x + 34 and B = 0.04x + 40 are graphed on the same plot\" width=\"731\" height=\"420\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<iframe id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=92426&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"500\" data-mce-fragment=\"1\"><\/iframe>\r\n\r\n<\/div>\r\n<h2><\/h2>\r\n<div id=\"Example_02_03_04\" class=\"example\">\r\n<div id=\"fs-id1165137714280\" class=\"exercise\">\r\n<div id=\"fs-id1165137676055\" class=\"problem textbox shaded\">\r\n<h3 style=\"text-align: center;\">Example<\/h3>\r\n<p id=\"fs-id1165137423863\">Jamal is choosing between two truck-rental companies. The first, Keep on Trucking, Inc., charges an up-front fee of $20, then 59 cents a mile. The second, Move It Your Way, charges an up-front fee of $16, then 63 cents a mile<a href=\"#footnote1\" name=\"footnote-ref1\"><sup>1<\/sup><\/a>. When will Keep on Trucking, Inc. be the better choice for Jamal?<\/p>\r\n[reveal-answer q=\"957679\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"957679\"]\r\n<p id=\"fs-id1165137653015\">The two important quantities in this problem are the cost and the number of miles driven. Because we have two companies to consider, we will define two functions.<\/p>\r\n\r\n<table id=\"fs-id1165137566638\" summary=\"Three rows and three columns. In the first column, are the years 1950 and 2000. In the second columns are the house values for Indiana, which are 37700 for 1950 and 94300 for 2000. In the third columns are the house values for Alabama, which are 27100 for 1950 and 85100 for 2000.\">\r\n<tbody>\r\n<tr>\r\n<td>Input<\/td>\r\n<td><em>d<\/em>, distance driven in miles<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Outputs<\/td>\r\n<td><em>K<\/em>(<em>d<\/em>): cost, in dollars, for renting from Keep on Trucking<em>M<\/em>(<em>d<\/em>) cost, in dollars, for renting from Move It Your Way<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Initial Value<\/td>\r\n<td>Up-front fee: <em>K<\/em>(0) = 20 and <em>M<\/em>(0) = 16<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Rate of Change<\/td>\r\n<td><em>K<\/em>(<em>d<\/em>) = $0.59\/mile and <em>P<\/em>(<em>d<\/em>) = $0.63\/mile<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p id=\"fs-id1165135209833\">A linear function is of the form [latex]f\\left(x\\right)=mx+b[\/latex]. Using the rates of change and initial charges, we can write the equations<\/p>\r\n\r\n<div id=\"fs-id1165137435497\" class=\"equation unnumbered\">[latex]\\begin{cases}K\\left(d\\right)=0.59d+20\\\\ M\\left(d\\right)=0.63d+16\\end{cases}[\/latex]<\/div>\r\n<p id=\"fs-id1165137726240\">Using these equations, we can determine when Keep on Trucking, Inc., will be the better choice. Because all we have to make that decision from is the costs, we are looking for when Move It Your Way, will cost less, or when [latex]K\\left(d\\right)&lt;M\\left(d\\right)[\/latex]. The solution pathway will lead us to find the equations for the two functions, find the intersection, and then see where the [latex]K\\left(d\\right)[\/latex] function is smaller.<\/p>\r\n\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"731\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010659\/CNX_Precalc_Figure_02_03_0072.jpg\" alt=\"\" width=\"731\" height=\"340\" \/> <b>Figure 7<\/b>[\/caption]\r\n<p id=\"fs-id1165137874768\">These graphs are sketched in Figure 7, with <em>K<\/em>(<em>d<\/em>)\u00a0in blue.<span id=\"fs-id1165137526514\">\r\n<\/span><\/p>\r\n<p id=\"fs-id1165137731943\">To find the intersection, we set the equations equal and solve:<\/p>\r\n\r\n<div id=\"fs-id1165137448488\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\begin{cases}K\\left(d\\right)=M\\left(d\\right)\\hfill \\\\ 0.59d+20=0.63d+16\\hfill \\\\ 4=0.04d\\hfill \\\\ 100=d\\hfill \\\\ d=100\\hfill \\end{cases}[\/latex]<\/div>\r\n<p id=\"fs-id1165137628623\">This tells us that the cost from the two companies will be the same if 100 miles are driven. Either by looking at the graph, or noting that [latex]K\\left(d\\right)[\/latex]\u00a0is growing at a slower rate, we can conclude that Keep on Trucking, Inc. will be the cheaper price when more than 100 miles are driven, that is [latex]d&gt;100[\/latex].<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<h2>Contribute!<\/h2>\r\n<div style=\"margin-bottom: 8px;\">Did you have an idea for improving this content? We\u2019d love your input.<\/div>\r\n<a style=\"font-size: 10pt; font-weight: 600; color: #077fab; text-decoration: none; border: 2px solid #077fab; border-radius: 7px; padding: 5px 25px; text-align: center; cursor: pointer; line-height: 1.5em;\" href=\"https:\/\/docs.google.com\/document\/d\/1bU8bOB4dgrDD7gy3-7mp70KTqAMLJW4a2j4sja7Yr8c\" target=\"_blank\" rel=\"noopener\">Improve this page<\/a><a style=\"margin-left: 16px;\" href=\"https:\/\/docs.google.com\/document\/d\/1vy-T6DtTF-BbMfpVEI7VP_R7w2A4anzYZLXR8Pk4Fu4\" target=\"_blank\" rel=\"noopener\">Learn More<\/a>","rendered":"<div class=\"bcc-box bcc-highlight\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Solve cost and revenue problems<\/li>\n<\/ul>\n<\/div>\n<p>A skateboard manufacturer introduces a new line of boards. The manufacturer tracks its costs, which is the amount it spends to produce the boards, and its revenue, which is the amount it earns through sales of its boards. How can the company determine if it is making a profit with its new line? How many skateboards must be produced and sold before a profit is possible?<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-5880 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/117\/2016\/08\/01202934\/CNX_Precalc_Figure_09_01_0012.jpg\" alt=\"Skateboarders at a skating rink by the beach.\" width=\"487\" height=\"252\" \/><\/p>\n<p>&nbsp;<\/p>\n<p><b><\/b>Using what we have learned about systems of equations, we can answer these questions.\u00a0The skateboard manufacturer\u2019s revenue equation is the equation used to calculate the amount of money that comes into the business. It can be represented as [latex]y=xp[\/latex], where [latex]x=[\/latex] quantity and [latex]p=[\/latex] price. The revenue equation is shown in orange in the graph below.<\/p>\n<p>The cost equation is the equation used to calculate the costs of doing business. It includes fixed costs, such as rent and salaries, and variable costs, such as utilities. The cost equation is shown in blue in the graph below. The [latex]x[\/latex] -axis represents quantity in hundreds of units. The [latex]y-axis[\/latex] represents both\u00a0cost and revenue in hundreds of dollars.\u00a0We won&#8217;t learn how to write a cost equation in this example, they will be given to you. If you take any business or economics courses, you will learn more about how to write a cost equation.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-5882 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/117\/2016\/08\/01203010\/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>&nbsp;<\/p>\n<p>The point at which the two lines intersect is called the break-even point, we learned that this is the solution to the system of linear equations that in this case comprise the cost and revenue equations.<\/p>\n<p>Read the axes of the graph carefully, note that quantity is in hundreds, and money is in thousands. The solution to the graphed system is [latex](7, 33)[\/latex]. This means\u00a0that if [latex]700[\/latex] units are produced, the cost to make them is [latex]$3,300[\/latex] and the revenue is also [latex]$3,300[\/latex]. In other words, the company breaks even if they produce and sell [latex]700[\/latex] 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.<\/p>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>A business wants to manufacture bike frames. Before they start production, they need to make sure they can make a profit with the materials and labor force they have. Their accountant has given them a cost equation of [latex]y=0.85x+35,000[\/latex] and a revenue equation of [latex]y=1.55x[\/latex]:<\/p>\n<ol>\n<li>Interpret [latex]x[\/latex] and [latex]y[\/latex] for the cost equation<\/li>\n<li>Interpret [latex]x[\/latex] and [latex]y[\/latex] for the revenue equation<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q86281\">Show Solution<\/span><\/p>\n<div id=\"q86281\" class=\"hidden-answer\" style=\"display: none\">\n<p>Cost: [latex]y=0.85x+35,000[\/latex]<\/p>\n<p>Revenue:[latex]y=1.55x[\/latex]<\/p>\n<p>The cost equation represents money leaving the company, namely how much it costs to produce a given number of bike frames. If we use the skateboard example as a model, [latex]x[\/latex] would represent the number of frames produced (instead of skateboards) and y would represent the amount of money it would cost to produce them (the same as the skateboard problem).<\/p>\n<p>The revenue equation represents money coming into the company, so in this context [latex]x[\/latex] still represents the number of bike frames manufactured, and [latex]y[\/latex] now represents the amount of money made from selling them. \u00a0Let&#8217;s organize this information in a table:<\/p>\n<table>\n<thead>\n<tr>\n<td>Equation Type<\/td>\n<td>[latex]x[\/latex] represents<\/td>\n<td>[latex]y[\/latex] represents<\/td>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>Revenue Eqn.<\/td>\n<td>number of frames<\/td>\n<td>amount of money made selling frames<\/td>\n<\/tr>\n<tr>\n<td>Cost Eqn.<\/td>\n<td>number of frames<\/td>\n<td>cost for making frames<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Given the same cost and revenue equations from the previous example, find the break-even point for the bike manufacturer. \u00a0Interpret the solution with words.<\/p>\n<p>Cost: [latex]y=0.85x+35,000[\/latex]<\/p>\n<p>Revenue: [latex]y=1.55x[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q145700\">Show Solution<\/span><\/p>\n<div id=\"q145700\" class=\"hidden-answer\" style=\"display: none\">\n<p><strong>Read and Understand:\u00a0<\/strong>We want the break even point for this system that represents cost and revenue. \u00a0This means we want to find where the two lines cross, and we have learned a few different methods for doing this because this is the solution to the system of equations! \u00a0Substitution looks like the easiest method since the revenue equation is already solved for [latex]y[\/latex],\u00a0[latex]y=1.55x[\/latex].<\/p>\n<p><strong>Define and Translate:\u00a0<\/strong>Write the system of equations.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}\\\\ y=0.85x+35,000\\hfill \\\\ y=1.55x\\hfill \\end{array}\\\\[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p><strong>Write and Solve:\u00a0<\/strong>The equations are already written for us, so we just need to solve the system using substitution.<\/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].<br \/>\n[latex]\\begin{array}{r}0.85x+35,000=1.55x\\\\ 35,000=0.7x\\,\\,\\,\\\\ 50,000=x\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\end{array}\\\\[\/latex]<br \/>\nThen, we substitute [latex]x=50,000[\/latex] into either the cost equation or the revenue equation.<br \/>\n[latex]1.55\\left(50,000\\right)=77,500\\\\[\/latex]<br \/>\nThe break-even point is [latex]\\left(50,000,77,500\\right)[\/latex].<\/p>\n<h4><strong>Check and Interpret:<\/strong><\/h4>\n<p>The solution to this system is[latex]\\left(50,000,77,500\\right)[\/latex], but what does that mean? Think of a point as [latex]\\left(x,y\\right)[\/latex], where in this case x is the quantity of bikes manufactured and [latex]y[\/latex] is an amount of money. For our system [latex]y[\/latex] represents two different things and [latex]x[\/latex] represents one thing. \u00a0Refer to the table we made in the first example, shown below:<\/p>\n<table>\n<thead>\n<tr>\n<td>Equation Type<\/td>\n<td>[latex]x[\/latex] represents<\/td>\n<td>[latex]y[\/latex] represents<\/td>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>Revenue Eqn.<\/td>\n<td>number of frames<\/td>\n<td>amount of money made selling frames<\/td>\n<\/tr>\n<tr>\n<td>Cost Eqn.<\/td>\n<td>number of frames<\/td>\n<td>cost for making frames<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Let&#8217;s interpret the solution with respect to the Cost equation first. [latex]x = 50,000[\/latex] and [latex]y = 77,500[\/latex]. \u00a0Using our table, we can translate this as &#8220;the cost for producing [latex]50,000[\/latex] bike frames is [latex]$75,500[\/latex]&#8220;.<\/p>\n<p>In the same way, the Revenue equation can be interpreted as &#8220;the amount of money the company makes from selling [latex]50,000[\/latex] bike frames is [latex]$77,500[\/latex]&#8220;.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm150438\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=150438&theme=oea&iframe_resize_id=ohm150438&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>In the video you will see an example of how to find the break even point for a small snow cone business.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"System of Equations App:  Break-Even Point\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/qey3FmE8saQ?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>We have seen that systems of linear equations and inequalities can help to define market behaviors that are very helpful to businesses. \u00a0The intersection of cost and revenue equations gives the break even point, and also helps define the region for which a company will make a profit.<\/p>\n<p>Another situation that calls for system of equations is comparison of offers available to customers.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>There are two cell phone companies that offer different packages. Company A charges a monthly service fee of $34 plus $.05\/min talk-time. Company B charges a monthly service fee of $40 plus $.04\/min talk-time.<\/p>\n<ol>\n<li>Write a linear equation that models the packages offered by both companies.<\/li>\n<li>If the average number of minutes used each month is 1,160, which company offers the better plan?<\/li>\n<li>If the average number of minutes used each month is 420, which company offers the better plan?<\/li>\n<li>How many minutes of talk-time would yield equal monthly statements from both companies?<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q785384\">Show Solution<\/span><\/p>\n<div id=\"q785384\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>The model for Company <em>A<\/em> can be written as [latex]A=0.05x+34[\/latex]. This includes the variable cost of [latex]0.05x[\/latex] plus the monthly service charge of $34. Company <em>B<\/em>\u2019s package charges a higher monthly fee of $40, but a lower variable cost of [latex]0.04x[\/latex]. Company <em>B<\/em>\u2019s model can be written as [latex]B=0.04x+40[\/latex].<\/li>\n<li>If the average number of minutes used each month is 1,160, we have the following:\n<div>[latex]\\begin{array}{l}\\text{Company }A\\hfill&=0.05\\left(1,160\\right)+34\\hfill \\\\ \\hfill&=58+34\\hfill \\\\ \\hfill&=92\\hfill \\\\ \\hfill \\\\ \\text{Company }B\\hfill&=0.04\\left(1,160\\right)+40\\hfill \\\\ \\hfill&=46.4+40\\hfill \\\\ \\hfill&=86.4\\hfill \\end{array}[\/latex]<\/div>\n<p>So, Company <em>B<\/em> offers the lower monthly cost of $86.40 as compared with the $92 monthly cost offered by Company <em>A<\/em> when the average number of minutes used each month is 1,160.<\/li>\n<li>If the average number of minutes used each month is 420, we have the following:\n<div>[latex]\\begin{array}{l}\\text{Company }A\\hfill&=0.05\\left(420\\right)+34\\hfill \\\\ \\hfill&=21+34\\hfill \\\\ \\hfill&=55\\hfill \\\\ \\hfill \\\\ \\text{Company }B\\hfill&=0.04\\left(420\\right)+40\\hfill \\\\ \\hfill&=16.8+40\\hfill \\\\ \\hfill&=56.8\\hfill \\end{array}[\/latex]<\/div>\n<p>If the average number of minutes used each month is 420, then Company <em>A <\/em>offers a lower monthly cost of $55 compared to Company <em>B<\/em>\u2019s monthly cost of $56.80.<\/li>\n<li>To answer the question of how many talk-time minutes would yield the same bill from both companies, we should think about the problem in terms of [latex]\\left(x,y\\right)[\/latex] coordinates: At what point are both the <em>x-<\/em>value and the <em>y-<\/em>value equal? We can find this point by setting the equations equal to each other and solving for <em>x.<\/em>\n<div style=\"text-align: center;\">[latex]\\begin{array}{l}0.05x+34=0.04x+40\\hfill \\\\ 0.01x=6\\hfill \\\\ x=600\\hfill \\end{array}[\/latex]<\/div>\n<p>Check the <em>x-<\/em>value in each equation.<\/p>\n<div style=\"text-align: left;\">[latex]\\begin{array}{l}0.05\\left(600\\right)+34=64\\hfill \\\\ 0.04\\left(600\\right)+40=64\\hfill \\end{array}[\/latex]<\/div>\n<p>Therefore, a monthly average of 600 talk-time minutes renders the plans equal.<\/li>\n<\/ol>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/09\/25200339\/CNX_CAT_Figure_02_03_002.jpg\" alt=\"Coordinate plane with the x-axis ranging from 0 to 1200 in intervals of 100 and the y-axis ranging from 0 to 90 in intervals of 10. The functions A = 0.05x + 34 and B = 0.04x + 40 are graphed on the same plot\" width=\"731\" height=\"420\" \/><\/p>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=92426&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"500\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<\/div>\n<h2><\/h2>\n<div id=\"Example_02_03_04\" class=\"example\">\n<div id=\"fs-id1165137714280\" class=\"exercise\">\n<div id=\"fs-id1165137676055\" class=\"problem textbox shaded\">\n<h3 style=\"text-align: center;\">Example<\/h3>\n<p id=\"fs-id1165137423863\">Jamal is choosing between two truck-rental companies. The first, Keep on Trucking, Inc., charges an up-front fee of $20, then 59 cents a mile. The second, Move It Your Way, charges an up-front fee of $16, then 63 cents a mile<a href=\"#footnote1\" name=\"footnote-ref1\"><sup>1<\/sup><\/a>. When will Keep on Trucking, Inc. be the better choice for Jamal?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q957679\">Solution<\/span><\/p>\n<div id=\"q957679\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137653015\">The two important quantities in this problem are the cost and the number of miles driven. Because we have two companies to consider, we will define two functions.<\/p>\n<table id=\"fs-id1165137566638\" summary=\"Three rows and three columns. In the first column, are the years 1950 and 2000. In the second columns are the house values for Indiana, which are 37700 for 1950 and 94300 for 2000. In the third columns are the house values for Alabama, which are 27100 for 1950 and 85100 for 2000.\">\n<tbody>\n<tr>\n<td>Input<\/td>\n<td><em>d<\/em>, distance driven in miles<\/td>\n<\/tr>\n<tr>\n<td>Outputs<\/td>\n<td><em>K<\/em>(<em>d<\/em>): cost, in dollars, for renting from Keep on Trucking<em>M<\/em>(<em>d<\/em>) cost, in dollars, for renting from Move It Your Way<\/td>\n<\/tr>\n<tr>\n<td>Initial Value<\/td>\n<td>Up-front fee: <em>K<\/em>(0) = 20 and <em>M<\/em>(0) = 16<\/td>\n<\/tr>\n<tr>\n<td>Rate of Change<\/td>\n<td><em>K<\/em>(<em>d<\/em>) = $0.59\/mile and <em>P<\/em>(<em>d<\/em>) = $0.63\/mile<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1165135209833\">A linear function is of the form [latex]f\\left(x\\right)=mx+b[\/latex]. Using the rates of change and initial charges, we can write the equations<\/p>\n<div id=\"fs-id1165137435497\" class=\"equation unnumbered\">[latex]\\begin{cases}K\\left(d\\right)=0.59d+20\\\\ M\\left(d\\right)=0.63d+16\\end{cases}[\/latex]<\/div>\n<p id=\"fs-id1165137726240\">Using these equations, we can determine when Keep on Trucking, Inc., will be the better choice. Because all we have to make that decision from is the costs, we are looking for when Move It Your Way, will cost less, or when [latex]K\\left(d\\right)<M\\left(d\\right)[\/latex]. The solution pathway will lead us to find the equations for the two functions, find the intersection, and then see where the [latex]K\\left(d\\right)[\/latex] function is smaller.<\/p>\n<div style=\"width: 741px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010659\/CNX_Precalc_Figure_02_03_0072.jpg\" alt=\"\" width=\"731\" height=\"340\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 7<\/b><\/p>\n<\/div>\n<p id=\"fs-id1165137874768\">These graphs are sketched in Figure 7, with <em>K<\/em>(<em>d<\/em>)\u00a0in blue.<span id=\"fs-id1165137526514\"><br \/>\n<\/span><\/p>\n<p id=\"fs-id1165137731943\">To find the intersection, we set the equations equal and solve:<\/p>\n<div id=\"fs-id1165137448488\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\begin{cases}K\\left(d\\right)=M\\left(d\\right)\\hfill \\\\ 0.59d+20=0.63d+16\\hfill \\\\ 4=0.04d\\hfill \\\\ 100=d\\hfill \\\\ d=100\\hfill \\end{cases}[\/latex]<\/div>\n<p id=\"fs-id1165137628623\">This tells us that the cost from the two companies will be the same if 100 miles are driven. Either by looking at the graph, or noting that [latex]K\\left(d\\right)[\/latex]\u00a0is growing at a slower rate, we can conclude that Keep on Trucking, Inc. will be the cheaper price when more than 100 miles are driven, that is [latex]d>100[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<h2>Contribute!<\/h2>\n<div style=\"margin-bottom: 8px;\">Did you have an idea for improving this content? We\u2019d love your input.<\/div>\n<p><a style=\"font-size: 10pt; font-weight: 600; color: #077fab; text-decoration: none; border: 2px solid #077fab; border-radius: 7px; padding: 5px 25px; text-align: center; cursor: pointer; line-height: 1.5em;\" href=\"https:\/\/docs.google.com\/document\/d\/1bU8bOB4dgrDD7gy3-7mp70KTqAMLJW4a2j4sja7Yr8c\" target=\"_blank\" rel=\"noopener\">Improve this page<\/a><a style=\"margin-left: 16px;\" href=\"https:\/\/docs.google.com\/document\/d\/1vy-T6DtTF-BbMfpVEI7VP_R7w2A4anzYZLXR8Pk4Fu4\" target=\"_blank\" rel=\"noopener\">Learn More<\/a><\/p>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-15370\">\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>System of Equations App: Break-Even Point. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/qey3FmE8saQ\">https:\/\/youtu.be\/qey3FmE8saQ<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><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>: Jay Abrams, et al.. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/openstaxcollege.org\/textbooks\/college-algebra.\">https:\/\/openstaxcollege.org\/textbooks\/college-algebra.<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Unit 14: Systems of Equations and Inequalities, from Developmental Math: An Open Program. <strong>Provided by<\/strong>: Monterey Institute of Technology and Education. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/nrocnetwork.org\/resources\/downloads\/nroc-math-open-textbook-units-1-12-pdf-and-word-formats\/\">http:\/\/nrocnetwork.org\/resources\/downloads\/nroc-math-open-textbook-units-1-12-pdf-and-word-formats\/<\/a>. <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>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":167848,"menu_order":29,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"System of Equations App: Break-Even Point\",\"author\":\"James Sousa (Mathispower4u.com) for Lumen Learning\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/qey3FmE8saQ\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"College Algebra\",\"author\":\"Jay Abrams, et al.\",\"organization\":\"OpenStax\",\"url\":\" https:\/\/openstaxcollege.org\/textbooks\/college-algebra.\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Revision and Adaptation\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Unit 14: Systems of Equations and Inequalities, from Developmental Math: An Open 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