## Value Problems

### Learning Outcomes

• Solve value problems

## 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 $25.00$ for children and $50.00$ for adults. Additionally, on a certain day, attendance at the game is $2,000$ and the total gate revenue is $70,000$.  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 $25.00$ for children and $50.00$ for adults. Additionally, on a certain day, attendance at the game is $2,000$ and the total gate revenue is $70,000$.

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 $11$ coins that have a value of $1.85$. The coins are either quarters or dimes. How many of each kind of coin is in the jar?

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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