So now I want to draw our curves on the graph and find equilibrium on the graph, right? Just like we had been doing before, when we were doing this without any math and we were just setting up our equation, how did we know where equilibrium was? It was where they crossed, right? That was the equilibrium point. So, we can do the same thing when we start plotting these lines that we're given, these equations. If we plot them on the graph and find where they intersect, that's going to be our equilibrium. So we can find them on the graph as the intersection, right. They're going to be the intersection of the supply curve and the demand curve just like we saw when we weren't using math. So let's go ahead and take these equations, and we are going to pick some values for price, and then we will find the quantity demanded, the quantity supplied at those prices, and we'll go ahead and plot them on the graph. So let's start with these equations here. I'm going to pull down the equation where quantity supplied is isolated; we're going to be picking prices and solving for quantity, so it's easier to do that when the quantity is already isolated in the graph, or in the equation. So let me go ahead and grab these equations, the first one being Qd=400−12P, and the other one was Qs=P−200. So QS equals P minus 200, and you could do this with either one, you could use either equation, price isolated, quantity isolated, you just have to do a little more algebra when selecting prices here.
So let's go ahead and start with a really easy price, a price of 0. At a price of 0, we're going to find a quantity demanded, \(400 - \frac{1}{2} \times 0\). We're going to get quantity demanded equals 400. Let's do the same thing with supply. Quantity supplied equals 0 minus 200. So we're actually going to get a negative number here, negative 200. At a price of 0, this is kind of meaningless, right? They're not going to produce anything with the price of 0. They can't produce a negative supply, so this we can't even really plot this on our graph because we don't have negative numbers on the graph; this isn't going to be so useful for us.
So let's try another number that might give us something that we can plot. Remember on our graph, our axes are going up 200, 400, 600, 800, it's going up pretty quickly there for price. So I think it'd be safe to pick a price, let's say around 400. Let's see what we get at a price of 400. So we're going to get \(Q_d = 400 - \frac{1}{2} \times 400\). We'll get a quantity demanded of 200. Let's do the same thing with supply. Quantity supplied equals the price of 400 minus 200, and we will get a quantity supplied of 200, right. So we just plugged in 400 into that equation, and we got 200. So notice what we got here. We got a quantity demanded of 200 and a quantity supplied of 200. What did we learn about equilibrium? That's when quantity demanded equals quantity supplied. We actually found it by mistake here, so this is actually going to be pretty helpful once we're making our graph, but let's go ahead and find one more point so that we can make our supply curve, and I'm going to pick a price even a little higher. Let's pick a price of 600 and see what we get. So quantity demanded equals 400 minus half of our price which is 600. So quantity demanded equals \(400 - \frac{1}{2} \times 600\) is 300. Our quantity demanded is going to equal 100. All right, let's do that over here with supply. Quantity supplied equals our price of 600 minus, our equation was \(P - 200\), price minus 200, quantity supplied is going to equal 400.
Alright, so now we have some points that we will be able to graph, right? So for demand, they all worked. For supply, this first one didn't really work for us, right? And that was because we got a negative number. So let's go ahead and plot these other points. So demand at a price of 0, quantity demanded was 400. So this will be our price axis, quantity axis, and I will use blue for demand here and red for supply because we're not going to be shifting or anything, so I think it's safe to use 2 different colors. We're going to be at that point there. Let's just go ahead and do all of the demand ones first. At a price of 400, they're going to demand 200. So price 400, demand 200. And last but not least, at a price of 600, demand is 100. So we'll be right there. So our demand line is going to look something like this. Oh, wow. First try. That's about as good as I'm going to get. Alright. Let's do the same thing with supply. So we can't plot that first one. Let's go to the second one. Remember, you only need 2 points to be able to make a line. You just connect the line and keep going. So at a price of 400, quantity supplied is 200, and you can kind of see where we're going to end up here, where I'm going to draw that red right on top of the blue, and at a price of 600 quantity supplied is 400. Price of 600, quantity supplied 400. We'll end up somewhere like that. So let me try and draw that. Wow. I'm improving. Okay. So what have we found? We found that they intersect on the graph right there at that point. 402100, right? So that is going to be our equilibrium. Price of 400 and a quantity of 200. So we've now solved that algebraically just using algebra, we've been able to find it on the graph as well. Alright, so that's about as deep as this algebra stuff is probably going to go, so let's go ahead and move on.