The supply and demand for a product are given by
2p − q = 60 and pq = 100 + 25q,
respectively. Find the market equilibrium point
To find the market equilibrium point, we need to determine the price (p) and quantity (q) at which the supply and demand equations intersect. This is where the quantity demanded (q) equals the quantity supplied (q).
We are given the supply equation: 2p - q = 60 and the demand equation: pq = 100 + 25q. Let's solve this system of equations to find the equilibrium point.
Step 1: Solve the supply equation for q.
2p - q = 60
q = 2p - 60
Step 2: Substitute this value of q into the demand equation.
p(2p - 60) = 100 + 25q
Step 3: Simplify and rearrange the equation.
2p^2 - 60p = 100 + 25(2p - 60)
2p^2 - 60p = 100 + 50p - 1500
2p^2 - 110p + 1400 = 0
Step 4: Solve the quadratic equation.
Since it does not factor easily, we can use the quadratic formula:
p = (-b ± √(b^2 - 4ac)) / (2a)
For our equation, a = 2, b = -110, and c = 1400
p = (-(-110) ± √((-110)^2 - 4 * 2 * 1400)) / (2 * 2)
p = (110 ± √(12100 - 11200)) / 4
p = (110 ± √900) / 4
p = (110 ± 30) / 4
This gives us two potential values for p:
p1 = (110 + 30) / 4 = 140 / 4 = 35
p2 = (110 - 30) / 4 = 80 / 4 = 20
Step 5: Substitute the values of p back into the supply equation to find the corresponding values of q.
For p = 35:
q1 = 2p - 60 = 2 * 35 - 60 = 70 - 60 = 10
For p = 20:
q2 = 2p - 60 = 2 * 20 - 60 = 40 - 60 = -20
Step 6: Determine the valid equilibrium point.
Since q represents quantity, negative values do not make sense in this context. Thus, the only valid equilibrium point is (p1, q1) = (35, 10).
Therefore, the market equilibrium point is p = 35 and q = 10.