The supply and demand for a product are given by
2p − q = 60 and pq = 1500 + 25q,
respectively. Find the market equilibrium point.
equilibrium means supply = demand, so just eliminate q and solve
2p-60 = 1500/(p-25)
Well, to find the market equilibrium point, we need to find the values of p and q that satisfy both equations simultaneously. Let's solve it in a funny way.
First, let's rewrite the first equation as:
q = 2p - 60
Now, substitute this value of q into the second equation:
p(2p - 60) = 1500 + 25(2p - 60)
Simplifying, we get:
2p^2 - 60p = 1500 + 50p - 1500
After some rearranging, we have:
2p^2 - 110p = 0
Factor out a p:
p(2p - 110) = 0
So, we have two possible solutions: p = 0 or 2p - 110 = 0.
If p = 0, then substitute it back into the first equation:
2(0) - q = 60
q = -60
But since we're talking about a market, we can't have negative quantities. So let's disregard this solution.
If 2p - 110 = 0, then:
2p = 110
p = 55
Now, substitute this value of p back into the first equation:
2(55) - q = 60
110 - q = 60
q = 50
So, the market equilibrium point is p = 55 and q = 50.
And there you have it! The market equilibrium has been found. Don't worry, I didn't clown around with the answer!
To find the market equilibrium point, we need to determine the values of p and q that satisfy both supply and demand equations simultaneously.
Given:
Supply equation: 2p - q = 60
Demand equation: pq = 1500 + 25q
We can start by solving the supply equation for q:
2p - q = 60
q = 2p - 60
Now substitute this value of q into the demand equation:
p(2p - 60) = 1500 + 25(2p - 60)
2p^2 - 60p = 1500 + 50p - 1500
2p^2 - 110p = 0
Divide through by 2:
p^2 - 55p = 0
Factor out p:
p(p - 55) = 0
So p = 0 or p = 55.
If p = 0, then q = 2(0) - 60 = -60. However, since q represents quantity, it cannot be negative, so we discard this solution.
If p = 55, then q = 2(55) - 60 = 110 - 60 = 50.
Therefore, the market equilibrium point is (p, q) = (55, 50).
To find the market equilibrium point, we need to solve the system of equations formed by the supply and demand equations:
Supply Equation: 2p - q = 60
Demand Equation: pq = 1500 + 25q
To solve the system of equations, we can use the substitution method. Here's the step-by-step breakdown:
Step 1: Solve one equation for one variable in terms of the other variable.
From the supply equation, we can solve for p in terms of q:
2p = q + 60
p = (q + 60)/2
Step 2: Substitute the expression for the variable found in Step 1 into the other equation.
Substitute p = (q + 60)/2 into the demand equation:
(q + 60)/2 * q = 1500 + 25q
Step 3: Simplify and solve the resulting equation for q.
Multiply both sides of the equation by 2 to get rid of the fraction:
q(q + 60) = 3000 + 50q
Expand and rearrange the equation:
q^2 + 60q = 3000 + 50q
Move all terms to one side of the equation:
q^2 + 10q - 3000 = 0
Step 4: Solve the quadratic equation using factoring or the quadratic formula.
Since the quadratic equation doesn't factor easily, we'll use the quadratic formula:
q = (-b ± sqrt(b^2 - 4ac)) / 2a
In our equation, a = 1, b = 10, and c = -3000.
Substituting these values into the quadratic formula, we have:
q = (-10 ± sqrt(10^2 - 4(1)(-3000))) / 2(1)
q = (-10 ± sqrt(100 + 12000)) / 2
q = (-10 ± sqrt(12100)) / 2
q = (-10 ± 110) / 2
Simplifying further, we have two possible values for q:
q1 = (-10 + 110) / 2 = 100 / 2 = 50
q2 = (-10 - 110) / 2 = -120 / 2 = -60
Step 5: Substitute the values of q into either of the original equations to find the corresponding values of p.
Using the supply equation:
When q = 50:
2p - 50 = 60
2p = 110
p = 55
When q = -60:
2p - (-60) = 60
2p + 60 = 60
2p = 0
p = 0
So, the market equilibrium points are (p, q) = (55, 50) and (p, q) = (0, -60).