{"id":16573,"date":"2019-10-03T19:20:04","date_gmt":"2019-10-03T19:20:04","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/chapter\/read-number-problems\/"},"modified":"2021-01-19T12:09:33","modified_gmt":"2021-01-19T12:09:33","slug":"read-number-problems","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/suny-rockland-developmentalemporium\/chapter\/read-number-problems\/","title":{"raw":"10.3.b - Value Problems","rendered":"10.3.b &#8211; Value Problems"},"content":{"raw":"<div class=\"bcc-box bcc-highlight\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Solve value problems<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Write a system of linear equations\u00a0representing\u00a0a value\u00a0problem<\/h2>\r\nSystems of equations are a very useful tool for modeling real-life situations and answering questions about them. If you can translate the application into two linear equations with two variables, then you have a system of equations that you can solve to find the solution. You can use any method to solve the system of equations.\r\n\r\nOne application of system of equations are known as value problems. Value problems are ones where each variable has a value attached to it. For example, the marketing team for an event venue wants to know how to focus their advertising based on who is attending specific events\u2014children, or adults? \u00a0They know the cost of a ticket to a basketball game is [latex]$25.00[\/latex] for children and [latex]$50.00[\/latex] for adults. Additionally, on a certain day, attendance at the game is [latex]2,000[\/latex] and the total gate revenue is [latex]$70,000[\/latex]. \u00a0How can the marketing team use this information to find out whether to spend more money on advertising directed toward children or adults?\r\n\r\nWe will use a table to help us set up and solve this value problem. The basic structure of the table is shown below:\r\n<table>\r\n<thead>\r\n<tr>\r\n<th>Number (usually what you are trying to find)<\/th>\r\n<th>Value<\/th>\r\n<th>Total<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>Item 1<\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Item 2<\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Total<\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nThe first column in the table is used for the number of things we have. Quite often, this will be our variables. The second column is used for the value each item has. The third column is used for the total value which we calculate by multiplying the number by the value.\r\n\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nFind the total number of child and adult tickets sold given that the cost of a ticket to a basketball game is [latex]$25.00[\/latex] for children and [latex]$50.00[\/latex] for adults. Additionally, on a certain day, attendance at the game is [latex]2,000[\/latex] and the total gate revenue is [latex]$70,000[\/latex].\r\n[reveal-answer q=\"181202\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"181202\"]\r\n\r\n<strong>Read and Understand:<\/strong> We want to find the number of child and adult tickets.\u00a0 We know the\u00a0total number of tickets sold, the total revenue and the cost of a child and adult ticket.\r\n\r\n<strong>Define and Translate:\u00a0<\/strong>Let <em>c<\/em> = the number of children and <em>a<\/em> = the number of adults in attendance. \u00a0Revenue comes from number of tickets sold multiplied by the price of the ticket. \u00a0We will get revenue for adults by multiplying [latex]$50.00[\/latex] times a. \u00a0[latex]$25.00[\/latex] times c will give the revenue from the number of child tickets sold.\r\n\r\n<strong>Write and Solve:\u00a0<\/strong>We can use a table as we did in the mixture problems section to organize the information we have. \u00a0Although a table is not necessary, it can help you get started. \u00a0For this problem, we labeled columns as amount, value, and total revenue because that is the information we are given.\r\n\r\nThe total number of people is [latex]2,000[\/latex].\r\n<table class=\" undefined\" style=\"width: 366px\">\r\n<thead>\r\n<tr class=\"border\">\r\n<th style=\"width: 82px\"><\/th>\r\n<th class=\"border\" style=\"width: 55px\">Amount<\/th>\r\n<th class=\"border\" style=\"width: 87.6719px\">Value<\/th>\r\n<th style=\"width: 97.3281px\">Total Revenue<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr class=\"border\">\r\n<td class=\"border\" style=\"width: 82px\">Child Tickets<\/td>\r\n<td class=\"border\" style=\"width: 55px\">\u00a0c<\/td>\r\n<td class=\"border\" style=\"width: 87.6719px\">\u00a0[latex]$25.00[\/latex]<\/td>\r\n<td class=\"border\" style=\"width: 97.3281px\">[latex]25c[\/latex]<\/td>\r\n<\/tr>\r\n<tr class=\"border\">\r\n<td class=\"border\" style=\"width: 82px\">Adult Tickets<\/td>\r\n<td class=\"border\" style=\"width: 55px\">\u00a0a<\/td>\r\n<td class=\"border\" style=\"width: 87.6719px\">\u00a0[latex]$50.00[\/latex]<\/td>\r\n<td style=\"width: 97.3281px\">[latex]50a[\/latex]<\/td>\r\n<\/tr>\r\n<tr class=\"border\">\r\n<td class=\"border\" style=\"text-align: left;width: 82px\">Total Tickets<\/td>\r\n<td class=\"border\" style=\"text-align: left;width: 55px\">[latex]2000[\/latex]<\/td>\r\n<td class=\"border\" style=\"text-align: left;width: 87.6719px\"><\/td>\r\n<td class=\"border\" style=\"text-align: center;width: 97.3281px\">[latex]$70,000[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nThe total revenue is [latex]$70,000[\/latex]. We can use this and the revenue from child and adult tickets to write an equation for the revenue.[latex]25c+50a=70,000[\/latex]\r\n<table class=\" undefined\">\r\n<thead>\r\n<tr class=\"border\">\r\n<th><\/th>\r\n<th class=\"border\">Amount<\/th>\r\n<th class=\"border\">Value<\/th>\r\n<th>Total Revenue<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr class=\"border\">\r\n<td class=\"border\">Child Tickets<\/td>\r\n<td class=\"border\">\u00a0c<\/td>\r\n<td class=\"border\">\u00a0[latex]$25.00[\/latex]<\/td>\r\n<td class=\"border\">[latex]25c[\/latex]<\/td>\r\n<\/tr>\r\n<tr class=\"border\">\r\n<td class=\"border\">Adult Tickets<\/td>\r\n<td class=\"border\">\u00a0a<\/td>\r\n<td class=\"border\">\u00a0[latex]$50.00[\/latex]<\/td>\r\n<td>\u00a0[latex]50a[\/latex]<\/td>\r\n<\/tr>\r\n<tr class=\"border\">\r\n<td class=\"border\" style=\"text-align: left\">Total Tickets<\/td>\r\n<td class=\"border\" style=\"text-align: left\">[latex]2000[\/latex]<\/td>\r\n<td class=\"border\" style=\"text-align: left\"><\/td>\r\n<td class=\"border\" style=\"text-align: center\">[latex]25c+50a=70,000[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nThe number of people at the game\u00a0that day is the total number of child tickets sold plus the total number of adult tickets, [latex]c+a=2,000[\/latex]\r\n<table class=\" undefined\">\r\n<thead>\r\n<tr class=\"border\">\r\n<th><\/th>\r\n<th class=\"border\">Amount<\/th>\r\n<th class=\"border\">Value<\/th>\r\n<th>Total Revenue<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr class=\"border\">\r\n<td class=\"border\">Child Tickets<\/td>\r\n<td class=\"border\">\u00a0c<\/td>\r\n<td class=\"border\">\u00a0[latex]$25.00[\/latex]<\/td>\r\n<td class=\"border\">[latex]25c[\/latex]<\/td>\r\n<\/tr>\r\n<tr class=\"border\">\r\n<td class=\"border\">Adult Tickets<\/td>\r\n<td class=\"border\">\u00a0a<\/td>\r\n<td class=\"border\">\u00a0[latex]$50.00[\/latex]<\/td>\r\n<td>\u00a0[latex]50a[\/latex]<\/td>\r\n<\/tr>\r\n<tr class=\"border\">\r\n<td class=\"border\" style=\"text-align: left\">Total Tickets<\/td>\r\n<td class=\"border\" style=\"text-align: left\">[latex]c+a=2,000[\/latex]<\/td>\r\n<td class=\"border\" style=\"text-align: left\"><\/td>\r\n<td class=\"border\" style=\"text-align: center\">[latex]25c+50a=70,000[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p style=\"text-align: center\">We now have a system of linear equations in two variables.[latex]\\begin{array}{r}c+a=2,000\\,\\,\\,\\\\ 25c+50a=70,000\\end{array}[\/latex].<\/p>\r\n<p style=\"text-align: left\">We can use any method of solving systems of equations to solve this system for a and c. \u00a0Substitution looks easiest because we can \u00a0solve the first equation for either [latex]c[\/latex] or [latex]a[\/latex]. We will solve for [latex]a[\/latex].<\/p>\r\n<p style=\"text-align: center\">[latex]\\begin{array}{c}c+a=2,000\\\\ a=2,000-c\\end{array}[\/latex]<\/p>\r\n<p style=\"text-align: left\">Substitute the expression [latex]2,000-c[\/latex] in the second equation for [latex]a[\/latex] and solve for [latex]c[\/latex].<\/p>\r\n<p style=\"text-align: center\">[latex]\\begin{array}{r} 25c+50\\left(2,000-c\\right)=70,000\\,\\,\\,\\, \\\\ 25c+100,000 - 50c=70,000\\,\\,\\,\\, \\\\ -25c=-30,000 \\\\ c=1,200\\,\\,\\,\\,\\,\\,\\, \\end{array}[\/latex]<\/p>\r\n<p style=\"text-align: left\">Substitute [latex]c=1,200[\/latex] into the first equation to solve for [latex]a[\/latex].<\/p>\r\n<p style=\"text-align: center\">[latex]\\begin{array}{r}1,200+a=2,000 \\\\ a=800\\,\\,\\,\\,\\,\\, \\end{array}[\/latex]<\/p>\r\n\r\n<h4 style=\"text-align: left\">Answer<\/h4>\r\n<p style=\"text-align: left\">We find that [latex]1,200[\/latex] children and [latex]800[\/latex] adults bought tickets to the game\u00a0that day. The marketing group may want to focus their advertising toward attracting young people.<\/p>\r\n<p style=\"text-align: left\">[\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\nThis example showed you how to find two unknown values given information that connected the two unknowns. With two equations, you are able to find a solution for two unknowns. \u00a0If you were to have three unknowns, you would need three equations to find them, and so on.\r\n\r\nIn the following video, you are given an example of how to use a system of equations to find the number of children and adults admitted to an amusement park based on entrance fees and total revenue. This example shows how to write equations and solve the system without a table.\r\n\r\nhttps:\/\/youtu.be\/uH4CgUhuDv0\r\n\r\nIn our next video example, we show how to set up a system of linear equations that represents the total cost for admission to a museum.\r\n\r\nhttps:\/\/youtu.be\/euh9ksWrq0A\r\n\r\nIn the next example, we will find the number of coins in a change jar given the total amount of money in the jar and the fact that the coins are either quarters or dimes.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nIn a change jar there\u00a0are [latex]11[\/latex] coins that have a value of [latex]$1.85[\/latex]. The coins are either quarters or dimes. How many of each kind of coin is in the jar?\r\n[reveal-answer q=\"698872\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"698872\"]\r\n\r\n<strong>Read and Understand:\u00a0<\/strong>We want to find the number of quarters and the number of dimes in the jar. \u00a0We know that dimes are [latex]$0.10[\/latex] and quarters are [latex]$0.25[\/latex], and the total number of coins is [latex]11[\/latex].\r\n\r\n<strong>Define and Translate:\u00a0<\/strong>We will call the number of quarters q and the number of dimes d. The part of the total [latex]$1.85[\/latex] that comes from quarters will be determined by how many quarters and the fact that each one is worth [latex]$0.25[\/latex], so [latex]$0.25q[\/latex] represents the amount of [latex]$1.85[\/latex] that is quarters. \u00a0The same idea can be used for dimes, so [latex]$0[\/latex].10d represents the amount of [latex]$1.85[\/latex] that is dimes.\r\n\r\n<strong>Write and Solve:<\/strong> We can label a new table with the information we are given.\r\n<table class=\" undefined\">\r\n<thead>\r\n<tr class=\"border\">\r\n<th><\/th>\r\n<th class=\"border\">number<\/th>\r\n<th class=\"border\">value<\/th>\r\n<th>total<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr class=\"border\">\r\n<td class=\"border\">quarters<\/td>\r\n<td class=\"border\">\u00a0q<\/td>\r\n<td class=\"border\">\u00a0[latex]$0.25[\/latex]<\/td>\r\n<td class=\"border\">[latex]$0.25q[\/latex]<\/td>\r\n<\/tr>\r\n<tr class=\"border\">\r\n<td class=\"border\">dimes<\/td>\r\n<td class=\"border\">\u00a0d<\/td>\r\n<td class=\"border\">\u00a0[latex]$0.10[\/latex]<\/td>\r\n<td>\u00a0[latex]$0.10d[\/latex]<\/td>\r\n<\/tr>\r\n<tr class=\"border\">\r\n<td class=\"border\" style=\"text-align: left\">total number of coins<\/td>\r\n<td class=\"border\" style=\"text-align: left\">[latex]q+d=11[\/latex]<\/td>\r\n<td class=\"border\" style=\"text-align: left\"><\/td>\r\n<td class=\"border\" style=\"text-align: center\">[latex]$0.25q+$0.10d=$1.85[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nWe can write our two equations, remember that we need two to solve for two unknowns.\r\n<p style=\"text-align: center\">\u00a0[latex]\\begin{array}{r}q+d=11\\,\\,\\,\\\\0.25q+0.10d=1.85\\end{array}[\/latex]<\/p>\r\n<p style=\"text-align: left\">Substitution looks like the easiest path to a solution, solve for q.<\/p>\r\n<p style=\"text-align: center\">[latex]\\begin{array}{c}q+d=11\\\\ q=11-d\\end{array}[\/latex]<\/p>\r\n<p style=\"text-align: left\">Substitute this into the other equation, and solve for d.<\/p>\r\n<p style=\"text-align: center\">[latex]\\begin{array}{r}0.25\\left(11-d\\right)+0.10d=1.85\\,\\,\\,\\, \\\\2.75-0.25d+0.10d=1.85\\,\\,\\,\\,\\\\ 2.75-0.15d=1.85\\\\-0.15d=-0.9\\\\\\,\\,\\,\\,\\,\\,\\, d=6\\end{array}[\/latex]<\/p>\r\n<p style=\"text-align: left\">Substitute [latex]d=6[\/latex] into the first equation to solve for [latex]q[\/latex].<\/p>\r\n<p style=\"text-align: center\">[latex]\\begin{array}{r}q+6=11 \\\\q=5\\,\\,\\,\\,\\,\\, \\end{array}[\/latex]<\/p>\r\n\r\n<h4 style=\"text-align: left\">Answer<\/h4>\r\nWe have [latex]6[\/latex] dimes and [latex]5[\/latex] quarters.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<p class=\"no-indent\">In the following video, you will see an example similar to the previous one, except that the equations are written and solved without the use of a table.<\/p>\r\nhttps:\/\/youtu.be\/GZYtSP-X_is\r\n<p id=\"video1\" class=\"no-indent\"><span style=\"color: #ff0000\"><span style=\"color: #000000\">In this section, we saw two examples of writing a system of two linear equations to find two unknowns that were related to each other. \u00a0In the first, the equations were related by the sum of the number of tickets bought and the sum of the total revenue brought in by the tickets sold. \u00a0In the second problem, the relationships were similar. \u00a0The two variables were related by the sum of the number of coins, and the total value of the coins.<\/span><\/span><\/p>\r\n\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]1448[\/ohm_question]\r\n\r\n<\/div>\r\n<p class=\"no-indent\">In the next section, you will see an example of using a system of linear equations to model a cost and revenue model for a hypothetical business. Again, you will need two equations to solve for two unknowns.<\/p>","rendered":"<div class=\"bcc-box bcc-highlight\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Solve value problems<\/li>\n<\/ul>\n<\/div>\n<h2>Write a system of linear equations\u00a0representing\u00a0a value\u00a0problem<\/h2>\n<p>Systems of equations are a very useful tool for modeling real-life situations and answering questions about them. If you can translate the application into two linear equations with two variables, then you have a system of equations that you can solve to find the solution. You can use any method to solve the system of equations.<\/p>\n<p>One application of system of equations are known as value problems. Value problems are ones where each variable has a value attached to it. For example, the marketing team for an event venue wants to know how to focus their advertising based on who is attending specific events\u2014children, or adults? \u00a0They know the cost of a ticket to a basketball game is [latex]$25.00[\/latex] for children and [latex]$50.00[\/latex] for adults. Additionally, on a certain day, attendance at the game is [latex]2,000[\/latex] and the total gate revenue is [latex]$70,000[\/latex]. \u00a0How can the marketing team use this information to find out whether to spend more money on advertising directed toward children or adults?<\/p>\n<p>We will use a table to help us set up and solve this value problem. The basic structure of the table is shown below:<\/p>\n<table>\n<thead>\n<tr>\n<th>Number (usually what you are trying to find)<\/th>\n<th>Value<\/th>\n<th>Total<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>Item 1<\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Item 2<\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Total<\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>The first column in the table is used for the number of things we have. Quite often, this will be our variables. The second column is used for the value each item has. The third column is used for the total value which we calculate by multiplying the number by the value.<\/p>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Find the total number of child and adult tickets sold given that the cost of a ticket to a basketball game is [latex]$25.00[\/latex] for children and [latex]$50.00[\/latex] for adults. Additionally, on a certain day, attendance at the game is [latex]2,000[\/latex] and the total gate revenue is [latex]$70,000[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q181202\">Show Solution<\/span><\/p>\n<div id=\"q181202\" class=\"hidden-answer\" style=\"display: none\">\n<p><strong>Read and Understand:<\/strong> We want to find the number of child and adult tickets.\u00a0 We know the\u00a0total number of tickets sold, the total revenue and the cost of a child and adult ticket.<\/p>\n<p><strong>Define and Translate:\u00a0<\/strong>Let <em>c<\/em> = the number of children and <em>a<\/em> = the number of adults in attendance. \u00a0Revenue comes from number of tickets sold multiplied by the price of the ticket. \u00a0We will get revenue for adults by multiplying [latex]$50.00[\/latex] times a. \u00a0[latex]$25.00[\/latex] times c will give the revenue from the number of child tickets sold.<\/p>\n<p><strong>Write and Solve:\u00a0<\/strong>We can use a table as we did in the mixture problems section to organize the information we have. \u00a0Although a table is not necessary, it can help you get started. \u00a0For this problem, we labeled columns as amount, value, and total revenue because that is the information we are given.<\/p>\n<p>The total number of people is [latex]2,000[\/latex].<\/p>\n<table class=\"undefined\" style=\"width: 366px\">\n<thead>\n<tr class=\"border\">\n<th style=\"width: 82px\"><\/th>\n<th class=\"border\" style=\"width: 55px\">Amount<\/th>\n<th class=\"border\" style=\"width: 87.6719px\">Value<\/th>\n<th style=\"width: 97.3281px\">Total Revenue<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr class=\"border\">\n<td class=\"border\" style=\"width: 82px\">Child Tickets<\/td>\n<td class=\"border\" style=\"width: 55px\">\u00a0c<\/td>\n<td class=\"border\" style=\"width: 87.6719px\">\u00a0[latex]$25.00[\/latex]<\/td>\n<td class=\"border\" style=\"width: 97.3281px\">[latex]25c[\/latex]<\/td>\n<\/tr>\n<tr class=\"border\">\n<td class=\"border\" style=\"width: 82px\">Adult Tickets<\/td>\n<td class=\"border\" style=\"width: 55px\">\u00a0a<\/td>\n<td class=\"border\" style=\"width: 87.6719px\">\u00a0[latex]$50.00[\/latex]<\/td>\n<td style=\"width: 97.3281px\">[latex]50a[\/latex]<\/td>\n<\/tr>\n<tr class=\"border\">\n<td class=\"border\" style=\"text-align: left;width: 82px\">Total Tickets<\/td>\n<td class=\"border\" style=\"text-align: left;width: 55px\">[latex]2000[\/latex]<\/td>\n<td class=\"border\" style=\"text-align: left;width: 87.6719px\"><\/td>\n<td class=\"border\" style=\"text-align: center;width: 97.3281px\">[latex]$70,000[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>The total revenue is [latex]$70,000[\/latex]. We can use this and the revenue from child and adult tickets to write an equation for the revenue.[latex]25c+50a=70,000[\/latex]<\/p>\n<table class=\"undefined\">\n<thead>\n<tr class=\"border\">\n<th><\/th>\n<th class=\"border\">Amount<\/th>\n<th class=\"border\">Value<\/th>\n<th>Total Revenue<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr class=\"border\">\n<td class=\"border\">Child Tickets<\/td>\n<td class=\"border\">\u00a0c<\/td>\n<td class=\"border\">\u00a0[latex]$25.00[\/latex]<\/td>\n<td class=\"border\">[latex]25c[\/latex]<\/td>\n<\/tr>\n<tr class=\"border\">\n<td class=\"border\">Adult Tickets<\/td>\n<td class=\"border\">\u00a0a<\/td>\n<td class=\"border\">\u00a0[latex]$50.00[\/latex]<\/td>\n<td>\u00a0[latex]50a[\/latex]<\/td>\n<\/tr>\n<tr class=\"border\">\n<td class=\"border\" style=\"text-align: left\">Total Tickets<\/td>\n<td class=\"border\" style=\"text-align: left\">[latex]2000[\/latex]<\/td>\n<td class=\"border\" style=\"text-align: left\"><\/td>\n<td class=\"border\" style=\"text-align: center\">[latex]25c+50a=70,000[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>The number of people at the game\u00a0that day is the total number of child tickets sold plus the total number of adult tickets, [latex]c+a=2,000[\/latex]<\/p>\n<table class=\"undefined\">\n<thead>\n<tr class=\"border\">\n<th><\/th>\n<th class=\"border\">Amount<\/th>\n<th class=\"border\">Value<\/th>\n<th>Total Revenue<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr class=\"border\">\n<td class=\"border\">Child Tickets<\/td>\n<td class=\"border\">\u00a0c<\/td>\n<td class=\"border\">\u00a0[latex]$25.00[\/latex]<\/td>\n<td class=\"border\">[latex]25c[\/latex]<\/td>\n<\/tr>\n<tr class=\"border\">\n<td class=\"border\">Adult Tickets<\/td>\n<td class=\"border\">\u00a0a<\/td>\n<td class=\"border\">\u00a0[latex]$50.00[\/latex]<\/td>\n<td>\u00a0[latex]50a[\/latex]<\/td>\n<\/tr>\n<tr class=\"border\">\n<td class=\"border\" style=\"text-align: left\">Total Tickets<\/td>\n<td class=\"border\" style=\"text-align: left\">[latex]c+a=2,000[\/latex]<\/td>\n<td class=\"border\" style=\"text-align: left\"><\/td>\n<td class=\"border\" style=\"text-align: center\">[latex]25c+50a=70,000[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: center\">We now have a system of linear equations in two variables.[latex]\\begin{array}{r}c+a=2,000\\,\\,\\,\\\\ 25c+50a=70,000\\end{array}[\/latex].<\/p>\n<p style=\"text-align: left\">We can use any method of solving systems of equations to solve this system for a and c. \u00a0Substitution looks easiest because we can \u00a0solve 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{array}{c}c+a=2,000\\\\ a=2,000-c\\end{array}[\/latex]<\/p>\n<p style=\"text-align: left\">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{array}{r} 25c+50\\left(2,000-c\\right)=70,000\\,\\,\\,\\, \\\\ 25c+100,000 - 50c=70,000\\,\\,\\,\\, \\\\ -25c=-30,000 \\\\ c=1,200\\,\\,\\,\\,\\,\\,\\, \\end{array}[\/latex]<\/p>\n<p style=\"text-align: left\">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{array}{r}1,200+a=2,000 \\\\ a=800\\,\\,\\,\\,\\,\\, \\end{array}[\/latex]<\/p>\n<h4 style=\"text-align: left\">Answer<\/h4>\n<p style=\"text-align: left\">We find that [latex]1,200[\/latex] children and [latex]800[\/latex] adults bought tickets to the game\u00a0that day. The marketing group may want to focus their advertising toward attracting young people.<\/p>\n<p style=\"text-align: left\"><\/div>\n<\/div>\n<\/div>\n<p>This example showed you how to find two unknown values given information that connected the two unknowns. With two equations, you are able to find a solution for two unknowns. \u00a0If you were to have three unknowns, you would need three equations to find them, and so on.<\/p>\n<p>In the following video, you are given an example of how to use a system of equations to find the number of children and adults admitted to an amusement park based on entrance fees and total revenue. This example shows how to write equations and solve the system without a table.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Ex:  System of Equations Application - Entrance Fees\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/uH4CgUhuDv0?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>In our next video example, we show how to set up a system of linear equations that represents the total cost for admission to a museum.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Ex: Solve an Application Problem Using a System of Linear Equations (09x-43)\" width=\"500\" height=\"375\" src=\"https:\/\/www.youtube.com\/embed\/euh9ksWrq0A?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>In the next example, we will find the number of coins in a change jar given the total amount of money in the jar and the fact that the coins are either quarters or dimes.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>In a change jar there\u00a0are [latex]11[\/latex] coins that have a value of [latex]$1.85[\/latex]. The coins are either quarters or dimes. How many of each kind of coin is in the jar?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q698872\">Show Solution<\/span><\/p>\n<div id=\"q698872\" class=\"hidden-answer\" style=\"display: none\">\n<p><strong>Read and Understand:\u00a0<\/strong>We want to find the number of quarters and the number of dimes in the jar. \u00a0We know that dimes are [latex]$0.10[\/latex] and quarters are [latex]$0.25[\/latex], and the total number of coins is [latex]11[\/latex].<\/p>\n<p><strong>Define and Translate:\u00a0<\/strong>We will call the number of quarters q and the number of dimes d. The part of the total [latex]$1.85[\/latex] that comes from quarters will be determined by how many quarters and the fact that each one is worth [latex]$0.25[\/latex], so [latex]$0.25q[\/latex] represents the amount of [latex]$1.85[\/latex] that is quarters. \u00a0The same idea can be used for dimes, so [latex]$0[\/latex].10d represents the amount of [latex]$1.85[\/latex] that is dimes.<\/p>\n<p><strong>Write and Solve:<\/strong> We can label a new table with the information we are given.<\/p>\n<table class=\"undefined\">\n<thead>\n<tr class=\"border\">\n<th><\/th>\n<th class=\"border\">number<\/th>\n<th class=\"border\">value<\/th>\n<th>total<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr class=\"border\">\n<td class=\"border\">quarters<\/td>\n<td class=\"border\">\u00a0q<\/td>\n<td class=\"border\">\u00a0[latex]$0.25[\/latex]<\/td>\n<td class=\"border\">[latex]$0.25q[\/latex]<\/td>\n<\/tr>\n<tr class=\"border\">\n<td class=\"border\">dimes<\/td>\n<td class=\"border\">\u00a0d<\/td>\n<td class=\"border\">\u00a0[latex]$0.10[\/latex]<\/td>\n<td>\u00a0[latex]$0.10d[\/latex]<\/td>\n<\/tr>\n<tr class=\"border\">\n<td class=\"border\" style=\"text-align: left\">total number of coins<\/td>\n<td class=\"border\" style=\"text-align: left\">[latex]q+d=11[\/latex]<\/td>\n<td class=\"border\" style=\"text-align: left\"><\/td>\n<td class=\"border\" style=\"text-align: center\">[latex]$0.25q+$0.10d=$1.85[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>We can write our two equations, remember that we need two to solve for two unknowns.<\/p>\n<p style=\"text-align: center\">\u00a0[latex]\\begin{array}{r}q+d=11\\,\\,\\,\\\\0.25q+0.10d=1.85\\end{array}[\/latex]<\/p>\n<p style=\"text-align: left\">Substitution looks like the easiest path to a solution, solve for q.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{c}q+d=11\\\\ q=11-d\\end{array}[\/latex]<\/p>\n<p style=\"text-align: left\">Substitute this into the other equation, and solve for d.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{r}0.25\\left(11-d\\right)+0.10d=1.85\\,\\,\\,\\, \\\\2.75-0.25d+0.10d=1.85\\,\\,\\,\\,\\\\ 2.75-0.15d=1.85\\\\-0.15d=-0.9\\\\\\,\\,\\,\\,\\,\\,\\, d=6\\end{array}[\/latex]<\/p>\n<p style=\"text-align: left\">Substitute [latex]d=6[\/latex] into the first equation to solve for [latex]q[\/latex].<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{r}q+6=11 \\\\q=5\\,\\,\\,\\,\\,\\, \\end{array}[\/latex]<\/p>\n<h4 style=\"text-align: left\">Answer<\/h4>\n<p>We have [latex]6[\/latex] dimes and [latex]5[\/latex] quarters.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p class=\"no-indent\">In the following video, you will see an example similar to the previous one, except that the equations are written and solved without the use of a table.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-3\" title=\"Ex:  System of Equations Application - Coin Problem\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/GZYtSP-X_is?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p id=\"video1\" class=\"no-indent\"><span style=\"color: #ff0000\"><span style=\"color: #000000\">In this section, we saw two examples of writing a system of two linear equations to find two unknowns that were related to each other. \u00a0In the first, the equations were related by the sum of the number of tickets bought and the sum of the total revenue brought in by the tickets sold. \u00a0In the second problem, the relationships were similar. \u00a0The two variables were related by the sum of the number of coins, and the total value of the coins.<\/span><\/span><\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm1448\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=1448&theme=oea&iframe_resize_id=ohm1448&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p class=\"no-indent\">In the next section, you will see an example of using a system of linear equations to model a cost and revenue model for a hypothetical business. Again, you will need two equations to solve for two unknowns.<\/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-16573\">\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>Ex: System of Equations Application - Entrance Fees. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/uH4CgUhuDv0\">https:\/\/youtu.be\/uH4CgUhuDv0<\/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><li>Ex: System of Equations Application - Coin Problem. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/GZYtSP-X_is\">https:\/\/youtu.be\/GZYtSP-X_is<\/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":169554,"menu_order":13,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Ex: System of Equations Application - Entrance Fees\",\"author\":\"James Sousa (Mathispower4u.com) for Lumen Learning\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/uH4CgUhuDv0\",\"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 Program\",\"author\":\"\",\"organization\":\"Monterey Institute of Technology and Education\",\"url\":\" http:\/\/nrocnetwork.org\/resources\/downloads\/nroc-math-open-textbook-units-1-12-pdf-and-word-formats\/\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Ex: System of Equations Application - 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