The supply and demand for a product are given by
2p − q = 100 and pq = 300 + 25q, respectively. Find the market equilibrium point.
To find the market equilibrium point, we need to find the values of p and q that satisfy both the supply and demand equations.
Given:
Supply equation: 2p - q = 100
Demand equation: pq = 300 + 25q
Step 1: Solve one of the equations for one variable in terms of the other. Let's solve the supply equation for q in terms of p.
2p - q = 100
q = 2p - 100
Step 2: Substitute the expression for q in terms of p into the demand equation.
p(2p - 100) = 300 + 25(2p - 100)
Simplify the equation:
2p^2 - 100p = 300 + 50p - 2500
2p^2 - 100p - 50p - 300 = 0
Combine like terms:
2p^2 - 150p - 300 = 0
Step 3: Solve this quadratic equation using factoring, completing the square, or the quadratic formula.
Alternatively, you can use an online quadratic equation solver or a graphing calculator to find the solutions.
Solving the quadratic equation, we find two possible values for p:
p = -5 or p = 30
Step 4: Substitute the values of p back into the supply equation (2p - q = 100) to find the corresponding values of q.
For p = -5:
2(-5) - q = 100
-10 - q = 100
q = -110
For p = 30:
2(30) - q = 100
60 - q = 100
q = -40
Step 5: Check if the solutions for (p, q) satisfy both the supply and demand equations.
For (p, q) = (-5, -110):
Supply equation: 2(-5) - (-110) = 100 => 10 + 110 = 100 (not satisfied)
Demand equation: (-5)(-110) = 300 + 25(-110) => 550 = 550 (satisfied)
For (p, q) = (30, -40):
Supply equation: 2(30) - (-40) = 100 => 60 + 40 = 100 (satisfied)
Demand equation: (30)(-40) = 300 + 25(-40) => -1200 = -1200 (satisfied)
The only solution that satisfies both equations is (p, q) = (30, -40).
Therefore, the market equilibrium point is where the price (p) is 30 and the quantity (q) is -40.