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.
