The supply function for a product is
2p − q − 50 = 0,
while the demand function for the same product is
(p + 10)(q + 30) = 9600.
Find the market equilibrium point.
that is, of course, when supply = demand.
So solve the system
2p - q = 50
(p+10)(q+30) = 9600
Since q = 2p-50, that gives you
(p+10)(2p-50+30) = 9600
p = 70
so, q = 90
To find the market equilibrium point, we need to solve the system of equations formed by the supply and demand functions.
Step 1: Solve the supply equation for q:
2p − q − 50 = 0
Rearrange the equation:
q = 2p - 50
Step 2: Substitute the value of q from Step 1 into the demand equation:
(p + 10)(q + 30) = 9600
Substitute q = 2p - 50:
(p + 10)((2p - 50) + 30) = 9600
Simplify:
(p + 10)(2p - 20) = 9600
Expand the equation:
2p^2 - 20p + 20p - 200 = 9600
Simplify:
2p^2 - 200 = 9600
Rearrange the equation:
2p^2 = 9800
Divide by 2:
p^2 = 4900
Take the square root of both sides:
p = ±70
Step 3: Substitute the value of p into the supply equation to find the corresponding q values:
For p = 70:
q = 2p - 50
q = 2(70) - 50
q = 90
For p = -70:
q = 2p - 50
q = 2(-70) - 50
q = -190
Therefore, the market equilibrium point occurs at (p, q) = (70, 90).
To find the market equilibrium point, we need to find the values of p and q that satisfy both the supply and demand functions.
First, let's solve the supply function:
2p - q - 50 = 0
We can rearrange this equation to isolate q:
q = 2p - 50
Next, let's solve the demand function:
(p + 10)(q + 30) = 9600
We can substitute the value of q from the supply function into the demand function:
(p + 10)((2p - 50) + 30) = 9600
Simplifying this equation:
(p + 10)(2p - 20) = 9600
Expanding the brackets:
2p^2 - 20p + 20p - 200 = 9600
Simplifying further:
2p^2 - 200 = 9600
Now, let's isolate the p term:
2p^2 = 9600 + 200
2p^2 = 9800
Dividing both sides by 2:
p^2 = 4900
Taking the square root of both sides:
p = ±70
Now that we have the possible values of p, let's substitute them back into the supply function to find the corresponding values of q.
For p = 70:
q = 2p - 50
q = 2(70) - 50
q = 90
For p = -70:
q = 2p - 50
q = 2(-70) - 50
q = -190
Thus, the market equilibrium point is (p, q) = (70, 90) or (-70, -190).
This means that at the equilibrium point, the price (p) of the product is 70 (or -70) and the quantity (q) demanded and supplied is 90 (or -190).