How to find the equilibrium price and quantity mathematically.

In a previous article, we talked about how we could find the market equilibrium by graphing the supply and demand curves and see where they intersect and then our equilibrium price was the price where the quantity demanded equaled the quantity supplied. There's another way that we can do it and we could use mathematical equations to solve for what our market equilibrium is.

So let's say we're talking about the market for chocolate bars and I've got a demand schedule here and a supply schedule to show the amount of the quantity demanded at the different prices between $1 and $5 and so what we did before we graph out our supply curve and then we graphed out our demand curve and we found that our equilibrium was a price of $3 and in a quantity of $9 or a quantity of 9 chocolate bars.

The question is can we find that without having to draw those graphs? and the answer is "yes". Let's say that our demand instead of thinking about it as a line we could think about it as an equation that represents the line, and so the equation that represents the demand curve, in this case, would be Price = (18 - Quantity Demanded) divided by 3. Then our supply curve is represented by Price = Quantity Supplied divided by 3.

Just to show you how that's true, take our quantity supplied, let's say it's 3 so that'd be 3 for Quantity supplied and then we divide that by 3 that gives us 1 and that's a price of 1. So I'm not gonna talk too much about how we derive these and something maybe I'll make a different article on that but I just want to show you that we can find the equilibrium price and quantity which again was (3, 9) that we found using these graphs we can find that with these equations.

So what we do is we set the equations equal to each other and then we solve. So we're gonna have (18 - Quantity Demanded) divided by 3 and when we just set that equal to Quantity Supplied divided by 3 and we just use a generic Q because we assume that at the equilibrium point the quantity supplied is can be equal to quantity demanded. 

So what we can do is there's a number of ways you could solve this but let's multiply each side by 3. If we multiply this side of the equation by 3, (3 X Q) over 3 is going to give us Q and if we multiply the left side equation by 3 then that's gonna have {3(18 - Q)} over 3. 

Now, what's going to happen in the left equation is that the 3 in the numerator and 3 and denominator will cancel out and that leaves us with 18 - Q = Q. Now we can add Q to both sides, so we add Q to the left side that gives us 18, and add Q to the right side that gives us 2Q, and then we could divide each side by 2. We divide the right side by the 2 that gives us Q, divide the left side 18 by 2 which gives us 9. So we know that our equilibrium quantity is going to be 9 but now you might say "Well, what is the equilibrium price? How do we find the price? 

We can do is take this Q = 9 and we could plug it back into one of our equations, so let's plug it in our supply equation that'll be easier. So now we're gonna have well I'll just put here's our P = Q over 3 and then we're gonna plug in that 9 for Q so P = 9 over 3, and 9 divided by 3 is 3. 

So we see that 3 is at our equilibrium which is where our demand or demand equals supply, so we just set the equations equal to each other and solve for Q, and then we just plug Q back in. Now we see that we have an equilibrium price of $3 and an equilibrium quantity of 9 and that matches up exactly with what we found in our graph here.

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