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Sagot :
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]
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]
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