If the supply and demand functions for a commodity are given by
p − q = 10
and
q(2p − 10) = 2100,
what is the equilibrium price and what is the corresponding number of units supplied and demanded?
equilibrium price $
number of units
units
The equilibrium price is the place where the two lines cross.
You might want to re-arrange the first one to obtain p = 10 + q then sub it into the second equation (every place you see the p) and solve for q : )
To find the equilibrium price and the corresponding number of units supplied and demanded, we need to solve the system of equations formed by the supply and demand functions.
Supply Function: p - q = 10 -- Equation (1)
Demand Function: q(2p - 10) = 2100 -- Equation (2)
We can solve this system of equations by substitution or elimination method. Let's use the substitution method:
Step 1: Solve Equation (1) for q.
p - q = 10
q = p - 10 -- Equation (3)
Step 2: Substitute Equation (3) into Equation (2).
(p - 10)(2p - 10) = 2100
Simplify and solve for p:
2p^2 - 20p - 20p + 100 = 2100
2p^2 - 40p + 100 - 2100 = 0
2p^2 - 40p - 2000 = 0
p^2 - 20p - 1000 = 0 -- Equation (4)
Step 3: Solve Equation (4) for p using the quadratic formula:
p = (-b ± √(b^2 - 4ac)) / (2a)
where a = 1, b = -20, c = -1000
p = (-(-20) ± √((-20)^2 - 4(1)(-1000))) / (2(1))
p = (20 ± √(400 + 4000)) / 2
p = (20 ± √4400) / 2
Since p represents price, it cannot be negative. Therefore, we only consider the positive root:
p = (20 + √4400) / 2
p = (20 + 2√1100) / 2
p = 10 + √1100
So, the equilibrium price is $10 + √1100.
Step 4: Substitute the equilibrium price back into Equation (3) to find q:
q = p - 10
q = (10 + √1100) - 10
q = √1100
Hence, the equilibrium price is $10 + √1100, and the corresponding number of units supplied and demanded is √1100 units.