Write a system of linear equations representing a value problem
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.
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—children, or adults? They 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]. How can the marketing team use this information to find out whether to spend more money on advertising directed toward children or adults?
We will use a table to help us set up and solve this value problem. The basic structure of the table is shown below:
Number (usually what you are trying to find)
Value
Total
Item 1
Item 2
Total
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.
Example
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].
Show Solution
Read and Understand: We want to find the number of child and adult tickets. We know the total number of tickets sold, the total revenue and the cost of a child and adult ticket.
Define and Translate: Let c = the number of children and a = the number of adults in attendance. Revenue comes from number of tickets sold multiplied by the price of the ticket. We will get revenue for adults by multiplying [latex]$50.00[/latex] times a. [latex]$25.00[/latex] times c will give the revenue from the number of child tickets sold.
Write and Solve: We can use a table as we did in the mixture problems section to organize the information we have. Although a table is not necessary, it can help you get started. For this problem, we labeled columns as amount, value, and total revenue because that is the information we are given.
The total number of people is [latex]2,000[/latex].
Amount
Value
Total Revenue
Child Tickets
c
[latex]$25.00[/latex]
[latex]25c[/latex]
Adult Tickets
a
[latex]$50.00[/latex]
[latex]50a[/latex]
Total Tickets
[latex]2000[/latex]
[latex]$70,000[/latex]
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]
Amount
Value
Total Revenue
Child Tickets
c
[latex]$25.00[/latex]
[latex]25c[/latex]
Adult Tickets
a
[latex]$50.00[/latex]
[latex]50a[/latex]
Total Tickets
[latex]2000[/latex]
[latex]25c+50a=70,000[/latex]
The number of people at the game that day is the total number of child tickets sold plus the total number of adult tickets, [latex]c+a=2,000[/latex]
Amount
Value
Total Revenue
Child Tickets
c
[latex]$25.00[/latex]
[latex]25c[/latex]
Adult Tickets
a
[latex]$50.00[/latex]
[latex]50a[/latex]
Total Tickets
[latex]c+a=2,000[/latex]
[latex]25c+50a=70,000[/latex]
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].
We can use any method of solving systems of equations to solve this system for a and c. Substitution looks easiest because we can solve the first equation for either [latex]c[/latex] or [latex]a[/latex]. We will solve for [latex]a[/latex].
We find that [latex]1,200[/latex] children and [latex]800[/latex] adults bought tickets to the game that day. The marketing group may want to focus their advertising toward attracting young people.
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. If you were to have three unknowns, you would need three equations to find them, and so on.
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.
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.
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.
Example
In a change jar there are [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?
Show Solution
Read and Understand: We want to find the number of quarters and the number of dimes in the jar. We 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].
Define and Translate: 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. The same idea can be used for dimes, so [latex]$0[/latex].10d represents the amount of [latex]$1.85[/latex] that is dimes.
Write and Solve: We can label a new table with the information we are given.
number
value
total
quarters
q
[latex]$0.25[/latex]
[latex]$0.25q[/latex]
dimes
d
[latex]$0.10[/latex]
[latex]$0.10d[/latex]
total number of coins
[latex]q+d=11[/latex]
[latex]$0.25q+$0.10d=$1.85[/latex]
We can write our two equations, remember that we need two to solve for two unknowns.
We have [latex]6[/latex] dimes and [latex]5[/latex] quarters.
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.
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. In 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. In the second problem, the relationships were similar. The two variables were related by the sum of the number of coins, and the total value of the coins.
Try It
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.
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