To determine the cost of one T-shirt and one pair of shorts, we need to solve the system of equations based on the given information.
Let's define:
- [tex]\( t \)[/tex] as the cost of one T-shirt.
- [tex]\( s \)[/tex] as the cost of one pair of shorts.
We are given the following two pieces of information:
1. A customer buys 3 T-shirts and 4 pairs of shorts for [tex]$72:
\[
3t + 4s = 72
\]
2. Another customer buys 4 T-shirts and 1 pair of shorts for $[/tex]44:
[tex]\[
4t + s = 44
\][/tex]
We now have the system of equations:
[tex]\[
\begin{cases}
3t + 4s = 72 \\
4t + s = 44
\end{cases}
\][/tex]
To solve this system, we can use the substitution or elimination method. Here, we'll use the elimination method.
First, let's eliminate [tex]\( s \)[/tex] by multiplying the second equation by 4 to align the coefficients of [tex]\( s \)[/tex] in both equations:
[tex]\[
4t + s = 44 \quad \text{multiplied by 4} \quad \Rightarrow \quad 16t + 4s = 176
\][/tex]
Now we rewrite the system with the modified second equation:
[tex]\[
\begin{cases}
3t + 4s = 72 \\
16t + 4s = 176
\end{cases}
\][/tex]
Next, subtract the first equation from the second equation to eliminate [tex]\( s \)[/tex]:
[tex]\[
(16t + 4s) - (3t + 4s) = 176 - 72
\][/tex]
Simplifying, we get:
[tex]\[
16t + 4s - 3t - 4s = 104
\][/tex]
[tex]\[
13t = 104
\][/tex]
Solving for [tex]\( t \)[/tex]:
[tex]\[
t = \frac{104}{13}
\][/tex]
[tex]\[
t = 8
\][/tex]
Now that we have [tex]\( t \)[/tex], we can substitute it back into the second original equation to find [tex]\( s \)[/tex]:
[tex]\[
4t + s = 44
\][/tex]
Substituting [tex]\( t = 8 \)[/tex]:
[tex]\[
4(8) + s = 44
\][/tex]
[tex]\[
32 + s = 44
\][/tex]
Solving for [tex]\( s \)[/tex]:
[tex]\[
s = 44 - 32
\][/tex]
[tex]\[
s = 12
\][/tex]
Thus, the cost of one T-shirt and one pair of shorts is:
[tex]\[
t = 8 \quad \text{and} \quad s = 12
\][/tex]
So the correct answer is:
C. [tex]\( t = \$ 8 \)[/tex], [tex]\( s = \$ 12 \)[/tex]
