SOLUTION: Hi, I have the following question and I don't know what to do. It is the demand for goods is p^2 + q^2 = 169 where p is price and q is quantity. The supply equation is p = q + 7.

Click here to see ALL problems on Linear-systems

Question 55054: Hi, I have the following question and I don't know what to do.
It is the demand for goods is p^2 + q^2 = 169 where p is price and q is quantity. The supply equation is p = q + 7.
What are the equilibrium price and quantity?
I'm unsure but i'm assuming that equilibrium is where supply equals demand
but the demand equation has roots, so would it be p+q= square root of 169
then that equals p-q-7.
But i still from that cannot get a price for p and a price for q.
I'm extremely stuck and any help would be greatly appreciated.
Answer by Hook(36) (Show Source):
You can put this solution on YOUR website!
You have one equation that defines demand

and one that defines supply

You are right that equilibrium happens when supply equals demand.
However, what you want to do is substitute the second equation into the first one.
If we do that, we get

I'll FOIL that first term

Combine like terms

Moving the 169 to the left

Divide both sides by 2

I can solve this quadratic equation by any number of ways. I'll use factoring
Factoring it out, I get

Solving, I get

and

We can toss -12 out because a negative quantity is nonsense for this problem.
I can then put into our supply equation

Ok, equilibrium price is 7 at a quantity of 5.