Alright, so now let's see how we can find equilibrium using the equations for the supply and demand curves. Alright, so I've got the steps listed here. The first thing we're going to want to do is make sure that for our demand equation and for our supply equation, the same variable is isolated. That means we want only the price on one side of the equation or only the quantity on one side of the equation, right. We want to isolate the same variable for both. After we do that, and that's if needed, right, you could be given 2 equations where you're already isolated the same variables like you see here. Right here in both of these equations, the one for demand and supply, price is isolated in both situations, so we are good to go with step 1 in that situation, right?
So step 2 would be to take both of the curves and set them equal to each other, alright? We'll see how to do this in a second, but the idea is to remember when we're at equilibrium, there is going to be the same quantity supplied as the quantity demanded. So quantity supplied and quantity demanded are going to be the same amount, that equilibrium quantity and the price is going to be the same too, right?
The demand and then we're going to use a little bit of algebra to solve for the remaining variable, and then we're going to use a little bit of algebra to solve for the remaining variable and then once we get that answer, we'll use it in one of our original equations to get the second variable. So let's see it all in action.
So here I have the demand and supply curves that we were using earlier in this segment. On the left I have the price isolated, on the left or excuse me on the right I have the quantity isolated, and we're gonna do it both ways just so you see we get the same answer and that it doesn't matter which variable is isolated as long as it's the same one for both. Okay, so step 1 is done already in the sense that we have price isolated here and and quantity isolated here. So we're gonna do 2 examples on the left, we'll do the example where price is isolated, on the right where quantity is isolated.
Let's start with price. Scroll down a little more. So step 2 was to set the equations equal to each other, okay? So how do we do that? Remember price is going to be the same in both, okay? So if p = 802 QD and P also = 200 + QS, that means that 802 QD = 200 + QS, alright? And one more thing is gonna be that we don't need to write QD or QS anymore, we can just use Q because the quantity demanded and quantity supplied are the same at equilibrium.
800 - 2 q = 200 + q So you see what I did there? I took this side of the equation and this side of the equation and set them equal to each other.
So let's go ahead to step 3. In step 3, we're gonna take this equation we just made and we're gonna solve for q. Right? The variable that's in there. So I'm gonna rewrite it here. 800 - 2 q = 200 + q. So the first thing we want to do is get all the q's on one side of the equation, and I like dealing with positive numbers so I'm going to move them to the right. So I'm gonna take these 2 q's and I'm gonna add 2 there, and I'm gonna add 2 here. So this is gonna cancel out and we're gonna have 800 = 200 + 1 q + 2 q's = 3 q's. Alright?
So now let's go ahead and move the 200 to the other side. So we're gonna subtract 200 from each side there. We're gonna get 600 = 3q and now we want just 1q right? We've got 3q so we gotta divide by 3 on both sides. So we're gonna divide by 3, I'll do it in blue, divide by 3, divide by 3, and what do we get? 600 ÷ 3 = 200 and 3 ÷ 3 = 1, so 200 = q. We've found our equilibrium quantity of 200. All right, so that's the step 3 and now step 4 is pretty easy. We're just going to take our equilibrium quantity that we just solved for and we're going to plug it into one of our original equations.
So I'm gonna go up here and you can pick either one, you can pick the demand equation or the supply equation. I like to look at them and pick the easier one, the one that has less math involved and in this case it looks like the supply equation is a lot easier. It's just 200 + q. So I'm gonna pick that P = 200 + q. So we've got P = 200 + our equilibrium quantity of 200. So our equilibrium price is gonna be 400. We've got our equilibrium so P* = 400 and q* = 200. We just solved that using algebra. Pretty cool.
Now let's go ahead and use the other side of the equations just so you see we get the same answers here. Right? Alright. So let's go ahead and set these equal to each other right. Quantity is isolated by itself. We're gonna set this equal to this. 400 - 12p = p - 200.
Alright? So now what we want to do is solve for P , and that's going to be step 3. I'm going to rewrite the equation 400 - 12p = p - 200. So I want to get all the p's on one side of the equation So that's going to take adding 12 p here and adding 12 p over here, and let me go back to red. So this is going to cancel and we're going to be left with 400 = p + 12 p, so p is the same as 22 p. So 2 ÷ 2 p + 1 p ÷ 2 p is going to give us 32 p, right? 1 and a half p's. We had 1 p, we added another half p, we've got 32 p. Minus 200. Alright.
And now let's go ahead and get the 200 on the other side. So we're gonna add 200 to both sides and we'll have 600 = 32p, and this will cancel out. So how do we get the p by itself now, right? We've got 600 = 32p. Well if you remember from algebra, the trick here is we're going to multiply by the reciprocal. So if we multiply 32p times 23, This 23 is gonna cancel. I'm just gonna get out of the way so I'm not dodging. The 23 is gonna cancel with the 32, and we need to multiply the other side of the equation also by 23. Okay. So the 32 oh actually I'm going to write all those in blue times 23, and this side also times 23. Okay so let's go ahead and cancel stuff out. The 2's cancel, the 3's cancel, and we're left with just p on that side of the equation, and then we'll do 23 × 600. So 2 × 600 is 1200 ÷ 3 = 400. So 23 × 600 is gonna be 400, alright? So there we go. We've gotten a price of 400, which we can confirm in step 4 the other time, we got a price of 400. So it looks like we're getting the same answer.
And let's go ahead and do the last step where we solve for quantity using this price that we have. So again, I'm gonna pick the easier formula and to me it looks like the supply formula again is easier in this situation, so I'm gonna go ahead and plug actually I'm going to use the demand one just to prove that we could use either one. So I'm going to use quantity demanded equals 400 - 12 P and I'm going to plug in our P there. So Q equals 400 - 12 O f P which was 400. So we're going to get Q = 400 - 200 is 200 and that confirms what we just got on the other side, right? We got a P* = 400 and Q* = 200. So either way, either variable was isolated and we've got the same answer there. Cool?
So let's go ahead and try some practice with this stuff.