To determine the equilibrium point for John's T-shirt business, we need to solve the system of equations given by the demand and supply functions. The demand function is [tex]\( P = -Q + 26 \)[/tex] and the supply function is [tex]\( P = Q - 10 \)[/tex].
1. Set the two equations equal to each other since at equilibrium, the price [tex]\( P \)[/tex] in the demand function should equal the price [tex]\( P \)[/tex] in the supply function:
[tex]\[
-Q + 26 = Q - 10
\][/tex]
2. Solve for [tex]\( Q \)[/tex]:
- First, add [tex]\( Q \)[/tex] to both sides to combine the [tex]\( Q \)[/tex] terms:
[tex]\[
26 = 2Q - 10
\][/tex]
- Next, add 10 to both sides to isolate the term with [tex]\( Q \)[/tex]:
[tex]\[
36 = 2Q
\][/tex]
- Finally, divide both sides by 2:
[tex]\[
Q = 18
\][/tex]
3. Find the price [tex]\( P \)[/tex] corresponding to this quantity [tex]\( Q = 18 \)[/tex]:
- Substitute [tex]\( Q = 18 \)[/tex] back into either the original demand or supply function. Let's use the demand function [tex]\( P = -Q + 26 \)[/tex]:
[tex]\[
P = -18 + 26
\][/tex]
- Simplify to find [tex]\( P \)[/tex]:
[tex]\[
P = 8
\][/tex]
4. State the equilibrium point:
- The equilibrium point [tex]\((P, Q)\)[/tex] is therefore:
[tex]\[
(P, Q) = (8, 18)
\][/tex]
Thus, the equilibrium point for John's T-shirt business is [tex]\((P, Q) = (8, 18)\)[/tex].
The correct answer is:
C. [tex]\((8, 18)\)[/tex]
