Solve for both price and quantities and show your work. Two linear equations are given: P = 22 - 4QD P = -2 + 2QS
Answer (1)
To solve for both price P and quantities Q D and Q S using the provided linear equations, we need to determine the equilibrium point where the quantity demanded equals the quantity supplied. The given equations for demand and supply are:
P = 22 − 4 Q D
P = − 2 + 2 Q S
Since equilibrium occurs when the quantity demanded Q D is equal to the quantity supplied Q S , we can set Q D = Q S = Q and solve the equations.
Step 1: Set the equations equal to each other.
22 − 4 Q = − 2 + 2 Q
Step 2: Solve for Q .
Add 4 Q to both sides:
22 = − 2 + 6 Q
Add 2 to both sides:
24 = 6 Q
Divide both sides by 6:
Q = 4
So, the equilibrium quantity is 4.
Step 3: Find the equilibrium price P .
Substitute Q = 4 back into either the demand or the supply equation to find the equilibrium price. Using the demand equation:
P = 22 − 4 ( 4 )
P = 22 − 16
P = 6
Conclusion:
The equilibrium price P is 6, and the equilibrium quantity Q for both demand and supply is 4. This means that at a price of 6, the quantity demanded by consumers equals the quantity supplied by producers, achieving market equilibrium.
