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
thank you
demand=supply
is x + 3p - 565=x - 16p + 480
19p=1045
p=$55
Now for quanitity
x - 16p + 480 = 0
x=16(55)-480= 400 units
To find the equilibrium quantity and price, we need to solve the system of equations formed by the demand and supply equations:
x + 3p - 565 = 0 (Demand equation)
x - 16p + 480 = 0 (Supply equation)
1. Multiply the first equation by 16 to eliminate x:
16x + 48p - 9040 = 0
2. Subtract the second equation from the result above:
(16x + 48p - 9040) - (x - 16p + 480) = 0
Simplifying, we get:
15x + 64p - 9520 = 0
3. Solve the equation above for p:
15x + 64p - 9520 = 0
64p = 9520 - 15x
p = (9520 - 15x) / 64
4. Substitute the value of p into one of the original equations to solve for x, for example, the demand equation:
x + 3p - 565 = 0
x + 3((9520 - 15x) / 64) - 565 = 0
Simplifying the equation above, we get:
64x + 3(9520 - 15x) - 64(565) = 0
64x + 28560 - 45x - 36160 = 0
19x - 7600 = 0
19x = 7600
x = 7600 / 19
x = 400
So the equilibrium quantity is 400 units.
5. Substitute the value of x into either the demand or supply equation to find the equilibrium price, for example, using the demand equation:
x + 3p - 565 = 0
400 + 3p - 565 = 0
3p - 165 = 0
3p = 165
p = 165 / 3
p = 55
So the equilibrium price is $55.
Therefore, the correct answer is:
c. equilibrium quantity 400 units; equilibrium price $55
To find the equilibrium quantity and equilibrium price, we need to solve the given demand and supply equations simultaneously.
The demand equation is given as: x + 3p - 565 = 0
The supply equation is given as: x - 16p + 480 = 0
We can solve this system of equations using the method of substitution or elimination. Let's use the method of substitution to solve this system:
1. First, solve one of the equations for one variable in terms of the other variable. From the supply equation, solve for x in terms of p:
x = 16p - 480
2. Substitute this expression for x in the demand equation:
16p - 480 + 3p - 565 = 0
3. Simplify and solve for p:
19p - 1045 = 0
19p = 1045
p = 1045 / 19
p ≈ 55
4. Substitute the obtained value of p back into either the demand or supply equation to solve for x:
x = 16(55) - 480
x = 880 - 480
x = 400
The equilibrium quantity is 400 units, and the equilibrium price is $55.
Therefore, the correct answer is option c: equilibrium quantity 400 units; equilibrium price $55.