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The volleyball team at West View High School is comparing T-shirt companies where they can purchase their practice shirts. The two companies, Shirt Box and Just Tees, are represented by this system of equations where [tex]\(x\)[/tex] is the number of T-shirts and [tex]\(y\)[/tex] is the total cost of the T-shirts.

[tex]\[
\begin{array}{l}
y = 10.5x \\
y = 7.5x + 30
\end{array}
\][/tex]

How many T-shirts would the volleyball team need to purchase from each company for the total cost to be equal?

For the total cost to be the same for both companies, the volleyball team would need to purchase [tex]\(\square\)[/tex] T-shirts from each company for a total of [tex]\(\$\square\)[/tex] for each company.



Answer :

GinnyAnswer
To determine the number of T-shirts [tex]\(x\)[/tex] the volleyball team should purchase such that the total cost is the same for both companies, we need to set the equations equal to each other and solve for [tex]\(x\)[/tex].

Given the equations:
[tex]\[ y = 10.5x \][/tex]
[tex]\[ y = 7.5x + 30 \][/tex]

1. Set the two equations equal to each other to find [tex]\(x\)[/tex]:
[tex]\[ 10.5x = 7.5x + 30 \][/tex]

2. Subtract [tex]\(7.5x\)[/tex] from both sides to combine like terms:
[tex]\[ 10.5x - 7.5x = 30 \][/tex]
[tex]\[ 3x = 30 \][/tex]

3. Solve for [tex]\(x\)[/tex] by dividing both sides by 3:
[tex]\[ x = \frac{30}{3} \][/tex]
[tex]\[ x = 10 \][/tex]

So, the volleyball team would need to purchase 10 T-shirts for the total cost to be equal between the two companies.

Next, calculate the total cost for each company when purchasing 10 T-shirts:

For Shirt Box:
[tex]\[ y = 10.5x \][/tex]
[tex]\[ y = 10.5 \times 10 \][/tex]
[tex]\[ y = 105 \][/tex]

For Just Tees:
[tex]\[ y = 7.5x + 30 \][/tex]
[tex]\[ y = 7.5 \times 10 + 30 \][/tex]
[tex]\[ y = 75 + 30 \][/tex]
[tex]\[ y = 105 \][/tex]

Thus, to ensure the total cost is the same for both companies, the volleyball team would need to purchase 10 T-shirts, costing a total of \$105 for each company.

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