Question

A store manager buys 8 T-Shirts and 5 hats for $220. The next day, the manager buys 5 T-shirts and 1 hat for $112.

Part A

Select the system of equations that models the situation.
A. 8t + 5h = 220
t + 5h = 112
B. 13(t – h) = 220
6(t – h) = 112
C. 8t + 5h = 220
5t + h = 112
Part B

How much does each T-shirt and each hat cost?

T-shirt : $
Hat : $

Answer 1

To solve the problem, we will first identify the correct system of equations and then solve for the costs of the T-shirts and hats.

### Part A

Let's denote:
- \( t \) = cost of one T-shirt
- \( h \) = cost of one hat

From the problem:

1. The store manager buys 8 T-shirts and 5 hats for $220:
\[
8t + 5h = 220
\]

2. The next day, the manager buys 5 T-shirts and 1 hat for $112:
\[
5t + 1h = 112
\]

Therefore, the correct system of equations is:
- \( 8t + 5h = 220 \)
- \( 5t + h = 112 \)

So the answer to Part A is:
**C.** \( 8t + 5h = 220\)
\( 5t + h = 112\)

### Part B

Now let’s solve the system of equations for \( t \) (the cost of the T-shirt) and \( h \) (the cost of the hat).

We have:
1. \( 8t + 5h = 220 \) (Equation 1)
2. \( 5t + h = 112 \) (Equation 2)

From Equation 2, we can express \( h \) in terms of \( t \):
\[
h = 112 - 5t
\]

Now, substitute this expression for \( h \) into Equation 1:
\[
8t + 5(112 - 5t) = 220
\]
Distributing the 5:
\[
8t + 560 - 25t = 220
\]
Combine like terms:
\[
-17t + 560 = 220
\]
Subtract 560 from both sides:
\[
-17t = 220 - 560
\]
\[
-17t = -340
\]
Divide by -17:
\[
t = 20
\]

Now that we have \( t = 20 \), we can find \( h \) using Equation 2:
\[
h = 112 - 5(20)
\]
\[
h = 112 - 100
\]
\[
h = 12
\]

Thus, the costs are:
- T-shirt: **\$20**
- Hat: **\$12**

## Summary
### Part B answers:
- **T-shirt:** \$20
- **Hat:** \$12