If the supply and demand functions for a commodity are given by
p − q = 10
and
q(2p − 10) = 5500,
what is the equilibrium price and what is the corresponding number of units supplied and demanded
Well, this is quite a tricky situation we have here. Let's see if I can clown my way through this problem!
To find the equilibrium price, we need to solve the given system of equations. So, let's start with the first equation:
p - q = 10
Now, we can rearrange this equation to solve for q in terms of p:
q = p - 10
Now, let's move on to the second equation:
q(2p - 10) = 5500
To simplify things, let's substitute the value of q from the first equation into the second equation:
(p - 10)(2p - 10) = 5500
Expanding the equation, we get:
2p^2 - 20p - 10p + 100 = 5500
2p^2 - 30p + 100 - 5500 = 0
2p^2 - 30p - 5400 = 0
Now, we can solve this quadratic equation. Let me just grab my clown calculator...
*Clown calculator appears out of thin air*
After doing some clown math, I have found that the roots of this equation are p = 90 and p = -30. Now, since we're talking about prices here, I don't think it makes sense to have a negative price. So, let's ignore the p = -30 solution.
Therefore, the equilibrium price is p = 90.
To find the corresponding number of units supplied and demanded, we can substitute this value of p back into the first equation:
q = p - 10
q = 90 - 10
q = 80
So, at the equilibrium price of $90, the corresponding number of units supplied and demanded is 80 units.
And there you have it! The equilibrium price is $90, and the corresponding number of units supplied and demanded is 80 units.
To find the equilibrium price and corresponding number of units supplied and demanded, we need to solve the given system of equations.
The supply function is given by p - q = 10.
The demand function is given by q(2p - 10) = 5500.
Step 1: Solve the supply function for q in terms of p.
p - q = 10
q = p - 10
Step 2: Substitute the expression for q from step 1 into the demand function.
(p - 10)(2p - 10) = 5500
Step 3: Expand and simplify the equation.
2p^2 - 20p - 10p + 100 = 5500
2p^2 - 30p + 100 = 5500
2p^2 - 30p - 5400 = 0
Step 4: Solve the quadratic equation for p. We can use factoring, completing the square, or the quadratic formula. In this case, let's use factoring.
2(p - 60)(p + 45) = 0
Setting each factor equal to zero, we get:
p - 60 = 0 or p + 45 = 0
Solving these equations, we find:
p = 60 or p = -45
Since the price cannot be negative, we discard p = -45 as an extraneous solution.
Step 5: Substitute the value of p into one of the supply or demand functions to solve for q. Let's use the supply function.
q = p - 10
q = 60 - 10
q = 50
Therefore, the equilibrium price is 60 units and the corresponding number of units supplied and demanded is 50.
To find the equilibrium price and the corresponding number of units supplied and demanded, we need to solve the given system of equations:
Equation 1: p - q = 10
Equation 2: q(2p - 10) = 5500
Step 1: Solve Equation 1 for q:
p - q = 10
q = p - 10
Step 2: Substitute the expression for q from Step 1 into Equation 2:
(p - 10)(2p - 10) = 5500
Step 3: Simplify and solve Equation 2:
2p^2 - 20p - 10p + 100 = 5500
2p^2 - 30p - 5400 = 0
Step 4: Solve the quadratic equation using factoring, completing the square, or the quadratic formula. Alternatively, you can use an online quadratic equation solver or graphing calculator to find the solutions.
Assuming we use the quadratic formula, which is widely applicable,
p = (-b ± √(b^2 - 4ac)) / (2a)
For our equation, a = 2, b = -30, c = -5400:
p = (-(-30) ± √((-30)^2 - 4*2*(-5400))) / (2*2)
p = (30 ± √(900 + 43200)) / 4
p = (30 ± √44100) / 4
p = (30 ± 210) / 4
Now we have two potential values for p. Let's calculate both:
Case 1: p = (30 + 210) / 4 = 240 / 4 = 60
Case 2: p = (30 - 210) / 4 = -180 / 4 = -45
Since price cannot be negative, we discard the negative solution (-45). Therefore, the potential equilibrium price is 60.
Step 5: Substitute the value of p = 60 into Equation 1 to find q:
p - q = 10
60 - q = 10
q = 60 - 10
q = 50
Therefore, the equilibrium price is 60, and the corresponding number of units supplied and demanded is 50.
Since you did not define p and q, all we can do is solve the two equations
for p and q, and then it is up to you to decide who is who.
from p − q = 10 --> p = q+10
sub that into the other equation:
q(2p − 10) = 5500
q(2q+20 - 10) = 5500
q^2 + 10q - 5500 = 0
q = (-10 ± √(100 + 4(5500)) )/2
= ...
clearly you would reject the negative answer, sub the positive into
p = q+10 to find p