To determine the cost of one T-shirt [tex]\(t\)[/tex] and one pair of shorts [tex]\(s\)[/tex], we will set up and solve a system of linear equations based on the given information.
Let's denote:
- [tex]\(t\)[/tex] as the cost of one T-shirt.
- [tex]\(s\)[/tex] as the cost of one pair of shorts.
We have two customers and the following information:
1. The first customer buys 5 T-shirts and 3 pairs of shorts for a total of \[tex]$66.
2. The second customer buys 15 T-shirts and 2 pairs of shorts for a total of \$[/tex]114.
We can translate this information into two equations:
[tex]\[
5t + 3s = 66 \tag{1}
\][/tex]
[tex]\[
15t + 2s = 114 \tag{2}
\][/tex]
We will solve this system of equations step-by-step.
### Step 1: Solve one equation for one variable in terms of the other
First, we solve equation (1) for [tex]\(s\)[/tex]:
[tex]\[
5t + 3s = 66
\][/tex]
Isolate [tex]\(s\)[/tex] by subtracting [tex]\(5t\)[/tex] from both sides:
[tex]\[
3s = 66 - 5t
\][/tex]
Divide by 3:
[tex]\[
s = 22 - \frac{5}{3}t \tag{3}
\][/tex]
### Step 2: Substitute the expression from equation (3) into equation (2)
Substitute [tex]\(s = 22 - \frac{5}{3}t\)[/tex] into the second equation [tex]\(15t + 2s = 114\)[/tex]:
[tex]\[
15t + 2\left(22 - \frac{5}{3}t\right) = 114
\][/tex]
Distribute 2 in the equation:
[tex]\[
15t + 44 - \frac{10}{3}t = 114
\][/tex]
To combine like terms, convert [tex]\(15t\)[/tex] to a common denominator of 3:
[tex]\[
\frac{45}{3}t + 44 - \frac{10}{3}t = 114
\][/tex]
Combine the [tex]\(t\)[/tex] terms:
[tex]\[
\frac{35}{3}t + 44 = 114
\][/tex]
Subtract 44 from both sides:
[tex]\[
\frac{35}{3}t = 70
\][/tex]
Multiply both sides by 3 to clear the fraction:
[tex]\[
35t = 210
\][/tex]
Divide by 35:
[tex]\[
t = 6
\][/tex]
### Step 3: Substitute [tex]\(t = 6\)[/tex] back into equation (3) to find [tex]\(s\)[/tex]
Using [tex]\(t = 6\)[/tex] in equation (3):
[tex]\[
s = 22 - \frac{5}{3}(6)
\][/tex]
Calculate:
[tex]\[
s = 22 - 10
\][/tex]
[tex]\[
s = 12
\][/tex]
Therefore, the cost of one T-shirt ([tex]\(t\)[/tex]) is \[tex]$6, and the cost of one pair of shorts (\(s\)) is \$[/tex]12.
So, the correct answer is:
A. [tex]\(t = \$6, s = \$12\)[/tex]
