Hey everyone and welcome back to the channel. Today we're going to be covering how to determine equilibrium quantity and price but without drawing a supply and demand graph. You'll use this if you're writing a test and you don't want to take the time to draw out the actual supply and demand curve so you simply use algebra to solve using the two equations. With that said, let's get into it. So here we have our supply and demand equations.
Quantity demanded is equal to 100 minus 3p, and the quantity supplied is equal to 2p plus 20. Now as you'll notice looking at the demand equation, there's an inverse relationship between Qd and P. And that makes sense because the law of supply says as price goes up, quantity demanded goes down. So therefore there's a negative sign in front of the coefficient for P. You'll also notice that there is a positive correlation between quantity supplied and price. And And that makes sense because as price goes up, quantity supplied will go up, which is why the coefficient on P in the quantity supplied equation is positive.
Now, let's say that you're just given these two equations, and you're asked to find the equilibrium quantity and price. So that would be where the demand curve and the supply curve intersect. However, for some reason, you don't want to actually draw out the two curves.
Well, you can do all of this algebraically, and it's quite simple. Now, remember, in equilibrium, quantity demanded is equal to quantity supplied. So The two lines are intersecting and their Q value and their P value is the same. So we can take this bit of knowledge and use it to solve for P. So we have QD is equal to QS.
Now I have quantity demanded in green and quantity supplied in red, just so you can tell them apart. However, I will lose the colors in a bit. So bear with me here. I have 100 minus 3P is equal to 2P plus 20. And as you can see, based on my colors, those are just the quantity demanded and quantity supplied equations. subbed in on either side of the equal sign since we know at equilibrium the two Q's are actually the same.
So I went from having two equations with two unknowns to having one equation with one unknown and the unknown in this case is P. So now I'm going to solve for P which is the equilibrium price. Now I'm going to collect my like terms on their respective sides of the equal sign.
So I'm going to put all of the p's on the right and all of the constants on the left. So I move that positive 20 over to the left side, which is now a negative 20. And I move that negative 3p to the right side. Now it's a positive 3p.
This results with 100 minus 20 is equal to 2p plus 3p. With more simple algebra, I'm just going to solve those two lines. 100 minus 20 is 80 and 2p plus 3p is simply 5p. Now I want to isolate p, so I'm going to divide both sides by 5. 80 divided by 5 is 16, and 5p divided by 5 is just p. So now I have 16 equals p.
And if you're like me, I like to have the letters on the left side. That's more of a mental thing than anything. So I've also written p is equal to 16. That is my equilibrium price.
Cool, but how do I use the equilibrium price to solve for the equilibrium quantity? Well we did say at the very beginning that Qd is equal to Qs, which means technically I can take that p equals 16 and sub it into either equation. And to show you that's going to work, I'm going to sub it into both equations in this video and you'll see that it yields the exact same equilibrium quantity.
So first we're going to sub it into Qd. So quantity demanded is equal to 100 minus 3p. But we know what p is, so we're going to sub it in. Quantity demanded is equal to 100 minus 3 times 16. 16 is just my equilibrium price, which I calculated earlier. Now I have that quantity demanded is equal to 100 minus 48. 48 is simply 3 times 16. And then with some simple subtraction, I'm going to have quantity demanded is equal to 52. Now that was me subbing in 16 for p in the demand equation, but I was telling you that you would get the exact same result if you subbed it into the supply equation.
So let's do that now just to make sure that we have our algebra correct. So let's pull up our quantity supplied equation. Qs is equal to 2p plus 20. Now again, we know what p is. It's 16, just like it was before.
So now we have Qs is equal to 2 times 16 plus 20. Well, 2 times 16 is simply 32. and 32 plus 20 is equal to 52. So now we've derived that quantity supplied is also equal to 52, and that's exactly what we expected since at equilibrium the two quantities quantity demanded and quantity supplied should be the same. So now we have Qs is equal to Qd which is equal to 52. And this means that Q at equilibrium must be equal to 52. And you can see my two equilibrium values on the slides in blue. So now I have my equilibrium P, equilibrium Q.
And if I wanted to, I could write them out as a point Q comma P is equal to 52 comma 16. That is my point on a graph that is my equilibrium quantity and price. However, you might've learned, and this is something that I teach my students, that you want your P and your Q to be, you want to denote them as equilibrium points. Because right now I just have P is equal to 16, but that doesn't tell me it's equilibrium price.
It doesn't tell me that it's equilibrium quantity. So I... I like to do and I teach my students to do to add a star or an asterisk which will give you p star is equal to 16 q star is equal to 52 and that just means p star and q star are your equilibrium values that way you don't confuse them for a different point that is a different p and a different q So there you have it. Now you know how to use simple algebra to solve for equilibrium price and quantity without actually having to graph the supply and demand curves yourself. If you found this video helpful, let us know by liking the video, subscribing to the channel, and of course, leave us a comment in the comment section below.
Thanks for watching this video, and we'll catch you in the next.