The demand equation for a certain brand of GPS Navigator is x + 3p - 565 = 0, where x is the quantity demanded per week and p is the wholesale unit price in dollars.
The supply equation is x - 16p + 480 = 0, where x is the quantity the supplier will make available in the market when the wholesale price is p dollars each. Find the equilibrium quantity and the equilibrium price for the GPS Navigators.
a. equilibrium quantity 2,000 units; equilibrium price $55
b. equilibrium quantity 2,000 units; equilibrium price $40
c. equilibrium quantity 400 units; equilibrium price $55
d. equilibrium quantity 400 units; equilibrium price $40
$5500
To find the equilibrium quantity and equilibrium price for the GPS Navigators, we need to solve the system of equations formed by the demand and supply equations.
Demand Equation: x + 3p - 565 = 0
Supply Equation: x - 16p + 480 = 0
To solve this system, we can use the method of substitution.
Step 1: Solve one equation for x in terms of p.
From the supply equation, we have x = 16p - 480.
Step 2: Substitute the expression for x in terms of p into the other equation.
Substituting x = 16p - 480 into the demand equation, we get:
16p - 480 + 3p - 565 = 0
Simplifying the equation, we have:
19p - 1045 = 0
Step 3: Solve for p.
Adding 1045 to both sides of the equation, we get:
19p = 1045
Dividing both sides by 19, we find:
p = 55
Step 4: Substitute the value of p into one of the equations to find x.
Using the supply equation, we have:
x - 16p + 480 = 0
x - 16(55) + 480 = 0
x - 880 + 480 = 0
x - 400 = 0
x = 400
Therefore, the equilibrium quantity is 400 units and the equilibrium price is $55 (option c)