To find the equilibrium point for Max's T-shirt business, we need to determine the point where the demand and supply functions intersect. This means finding the values of [tex]\(Q\)[/tex] (quantity) and [tex]\(P\)[/tex] (price) that satisfy both equations simultaneously.
The demand function is given by:
[tex]\[ P = -Q + 34 \][/tex]
And the supply function is given by:
[tex]\[ P = Q - 10 \][/tex]
To find the equilibrium point, we equate the two expressions for [tex]\(P\)[/tex] from the demand and supply functions and solve for [tex]\(Q\)[/tex]:
[tex]\[ -Q + 34 = Q - 10 \][/tex]
Now, let's solve this equation step-by-step:
1. Add [tex]\(Q\)[/tex] to both sides to move the [tex]\(Q\)[/tex] term to one side:
[tex]\[ 34 = 2Q - 10 \][/tex]
2. Add 10 to both sides to isolate the [tex]\(2Q\)[/tex] term:
[tex]\[ 34 + 10 = 2Q \][/tex]
[tex]\[ 44 = 2Q \][/tex]
3. Divide both sides by 2 to solve for [tex]\(Q\)[/tex]:
[tex]\[ Q = \frac{44}{2} \][/tex]
[tex]\[ Q = 22 \][/tex]
Now that we have the quantity [tex]\(Q = 22\)[/tex], we need to find the corresponding price [tex]\(P\)[/tex]. We can use either the demand or supply function to do this. Let's use the supply function:
[tex]\[ P = Q - 10 \][/tex]
Substitute [tex]\(Q = 22\)[/tex] into the supply function:
[tex]\[ P = 22 - 10 \][/tex]
[tex]\[ P = 12 \][/tex]
Therefore, the equilibrium point for Max's T-shirt business is [tex]\((22, 12)\)[/tex].
The correct answer is:
A. [tex]\((22, 12)\)[/tex]
