Everyone's favorite:algebra!

1

concept

## Equation of a Demand Curve

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now let's learn how to find equilibrium. Using just a little bit of algebra and I know I scared you there with that nasty a word I just dropped but I promise you that algebra is not going to get so intense in this segment alright, there's just gonna be a little bit of formula rearranging and solving so it's gonna be pretty basic and I'll we'll do some reviews as we go along. So hopefully you'll get a good hang of the kind of algebra will need. So let's start with the demand curve here. Um sometimes you can be given a an equation for a line like you see here p equals 800 -2, quantity demanded, right? And the easiest way to take this information and and be able to put it on a graph is to just pick some values for the quantity here and we'll solve for price, right? We'll pick values for quantity because it's on the right hand side and prices already by itself. So it'll be a lot easier to just solve for price in that situation. So let's go ahead and start with a very easy value for quantity. Why don't we start with the value of zero? What if the quantity is zero? Let's go ahead and see what happens in our equation. P equals 800 minus two. And we're gonna plug in zero for our quantity, right? Two times zero, that's gonna disappear. So we're gonna end up with P equals 800. Right? That is going to be our price when the quantity is zero, the price is going to be 800. So if people if the if the market is charging a price of 800 there will be zero quantity demanded. Let's Go ahead to our graph and let's plot this point. So we've got our price axis over here, our quantity access over here. Alright. And let's take that point. So we have a quantity of zero which is right here along the line and let's go ahead and mark the price of 800. So at a price of 800 up here I'm gonna use I'll use blue price of 800. We're gonna have a quantity demanded of zero. Alright, let's go ahead and pick some other quantities. So how do we go go about picking the quantity we're gonna use? Well, you want it to be something that's gonna be easy to put on your graph. Right? Our graph is already quantities of 100. 200. 300. So I'm gonna go ahead and just pick a quantity of 200 now. Okay. Because I know I'll be able to find that easily on the graph. Okay? And you can pick any number, you could pick six right now, but I think it makes it a lot more difficult to graph. So I'm gonna pick a number 200. Let's go ahead and plug that in our equation. So quantity of 200 P equals 800 minus two times 200. So that's gonna be p equals 800 minus two times 200 is 400 And P is Gonna Equal 400. So with a price of 400, quantity demanded will be 200. Let's go ahead and get that on our graph. Price 400 quantity demanded 200 will be somewhere right here. Alright, let's and by the way right now with these two points we can make our demand curve, right? It's gonna be a straight line. So we know that our curve is gonna be like this, right? Something like that. It's gonna pass through both of those points. I guess. I could draw a little better. Let me try one more time here. I miss the other way now, one more. All right, that was a little better. So there we go. That's what our demand curve is gonna look like. But it doesn't hurt to do an extra one. We can always just confirm what we're doing here. So let's go ahead and pick a quantity of 300 right? It looks like at 300 we're gonna be at this 200 price level, right? That's where it looks like it's gonna cross and erase those. Um So let's go ahead and see if that's the answer. We get price or quantity of 300. Let's solve for price P equals 800 minus two times 300 P equals 800 minus two times 300 is 600. And sure enough we got a price of 200 here, right? So there it is. We got that answer. I feel pretty good about our lines, so that is what our demand curve is gonna look like here. Alright, so now I want to do a quick recap of how we can isolate different variables and let's go ahead and do that in the next video.

2

concept

## Isolating Variables in the Demand Equation

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Alright, so you'll notice in our equation price is isolated by itself on the left hand side of the equation right? But sometimes we might want to rearrange the equation just so quantity is instead the isolated variable? Right? So sometimes it could be the case that we want quantity to be just by itself on one side of the equation. So how do we do that? We're gonna use just a little bit of our here to rearrange the equation. So let's go ahead and do that. I'm gonna rewrite it. P equals 800 minus two Q. D. So we want to get the quantity by itself. So the first thing we want to do is move the 800 to the other side, right minus 800 minus 800 on this side. And we are gonna have p minus 800 equals and this cancels out right here and we're left with negative two Q. D. On the right hand side of the equation. Right? So the next thing is to get the Q. D. All the way by itself, right? So what we need to do to get rid of that coefficient, the negative two in front of the queue D. Is right now it's being multiplied negative two times Q. D. So we need to divide to get it out of there. So we're gonna divide by negative two. And that's gonna get rid of that coefficient there for the quantity. And but if we do that side by negative two, we also have to do the other side by negative two as well. So this will cancel out these negative two's and we're gonna be left with just quantity demanded on the right hand side of the equation. But how do we do this? P minus 800 over negative two. Well you might not remember exactly how to do that. So one thing I suggest is just take the denominator the negative two and just put it under each of them. So what we're gonna have is P divided by negative two minus 800 divided by negative two. And that's exactly what is happening here. Right P over negative two minus 800 over negative two. So let's go ahead and finish this up. 800 divided by negative two is gonna give us negative 400. Right? And P over negative two. Well that's the same thing as saying negative half P. Right. P over two is the same as half of a P. So negative half P minus a negative 400. Right? So one more thing that you guys remember from algebra is when we have two negatives negative and a negative makes a positive and we're gonna end up with this formula right here, negative half P plus 400 equals quantity demanded. Right? That is how we got the quantity demanded by itself. We can also rewrite this. I'm gonna do an arrow. It's also acceptable to have this as our final answer. Um 400 minus half P equals Q. D. Right. I just rearrange the terms so that the negative term wasn't first. I think it's just easier to read like that. 400 minus half P. And of course just in case you didn't remember, we could also do this. These are all interchangeable. Um Q. D equals 400 minus half P. Right? So I just took the right hand side and I flipped it. I just put the Q. T. On the left and the little equation stuff on the right. So that is how we switch which variable isolated with just a little bit of algebra. There's more than one way to do this. So if you have another way you're more comfortable with to switch the variables around, go ahead and do it your way. This is just the way I would have done it. All right. We're going to do the same thing now with supply and we'll have another example of of isolating variables there as well. Alright, let's move on to supply.

3

concept

## Equation of a Supply Curve

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So just like we did with demand, let's go ahead and put a supply curve on the graph. So here we have a equation given to us. P equals 200 plus Q. S Okay. So piece by itself on the left hand side of the equation. So we'll pick quant we'll pick numbers for quantity and then it'll be easy to just find out what the price would be at that quantity. Alright, so let's go ahead and pick let's start with an easy number zero. Right? So at a quantity of zero let's solve for price P equals 200 plus zero. Hopefully you guys can do this math right here, P equals 200. Right? So the quantity of zero, we're gonna have a price of 200. Let's go ahead and put that in there. We've got our price axis quantity access. It's gonna be right here. Alright, let's go ahead and pick other numbers. Um I believe in the other video we did 203 100. Um Let's go ahead and just pick we'll do 200 as well here I guess. Why not? So P equals 200 plus and quantity supplied is going to be 200. So P equals 400 right there. Alright, so we'll say at a quantity of 200 price will be 400 and will be something like that. So let's go ahead and pick one more. Right. Right now we could go ahead and make our whole line because we have the two points. But just for fun let's do one more point and we did 300 before. So let's do it here. P. Equals 200 plus quantity supplied of 300 we're gonna get a P equal to 500 here. So at 300 we have a price of 500 right here and it looks like it's gonna be along the same line. Here's my attempt at the line attempt. Number one. Oh it started so good. Alright let's go again. Oh my God people alright that we're just gonna have to settle for that. It almost goes through that last point but there we go, that is gonna be our supply curve right there. Okay so we've taken those points, we just plug them into the equation and then we put it onto the graph. Let's do one more about isolating variables.

4

concept

## Isolating Variables in the Supply Equation

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So just like we did with demand, we can also isolate different variables with the supply curve, right? There could be a situation that we want to get the quantity supplied by itself on one side of the of the equation. So we're gonna have to rearrange, and luckily this equation is pretty simple. So we're gonna have 200 plus Q. S. So we want to get Qs by itself, We need to subtract this 200 out of here and we'll subtract 200. Let me do it in Blue, subtract 200 from here. Subtract 200 from here, and we will get P minus 200 equals Q. S. So this equation was simpler than the other one. Um We were able to isolate Q just by moving the 200 from one side to the other. Alright, so now let's see how these work together.

5

concept

## Finding Equilibrium Using Algebra

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Alright. So now let's see how we can find equilibrium using the equations for the supply and demand curves. All right. So, I've got the steps listed here. Um The first thing we're gonna wanna 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 two 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 one in that situation, right? Um So step two would be to take both of the curve and set them equal to each other. All right. We'll see how to do this in a second. But the idea is remember um when we're at equilibrium where there is gonna be the same quantity supplied as the quantity demanded. So quantity supplied and quantity demanded are gonna be the same amount that equilibrium quantity and the price is gonna be the same to write the demand and the supply are at the same price at equilibrium. Alright. So that's why we can set them equal to each other and then we're gonna 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. Um 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 name. One for both. Okay, so step one is done already in the sense that we have price isolated here and quantity isolated here. So we're gonna do two examples on the left. We'll do the example where prices isolated on the right where quantity is isolated. Let's start with price scroll down a little more. So step two was to set the equations equal to each other. Okay, so how do we do that remember? Price is gonna be the same in both. Okay, so if p is equal to 800 minus two Q. D. And P is also equal to 200 plus Q. S. That means that 800 minus two Q. D. Is equal to 200 plus Qs. Alright. And one more thing is gonna be that we don't need to write Q. D. Or Q. S anymore. We can just use Q. Because the quantity demanded and quantity supplied are the same at equilibrium. 200 plus 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. 800 minus two. Q equals 200 plus Q. And again we just use Q. Because quantity supplied and chronic demanded is the same at equilibrium. So let's go ahead to step three and step three. 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 minus two. Q equals 200 plus Q. So the first thing we wanna do is get all the cues on one side of the equation and I like dealing with positive numbers so I'm gonna move them to the right so I'm gonna take these two cues and I'm gonna add to there and I'm gonna add two here. So this is gonna cancel out. And we're gonna have 800 equals 200 plus one Q. Plus two cues is three cues. 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 equals and this cancels out three Q. And now we want just one queue right? We've got three Q. So we gotta divide by three on both sides. So we're gonna divide by three. I'll do it in blue divide by three divide by three. And what do we got 600 divided by three is 203 divided three is one. So 200 equals Q. We found our equilibrium quantity of 200. All right, so that's the step three And now step four is pretty easy. We're just gonna take our equilibrium quantity that we just solved for and we're gonna 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 plus que. So I'm gonna pick that p equals 200 plus Q. So we've got p equals 200 plus que. And we know what Q is. Right. We just solved for the equilibrium quantity. So let's plug it in. P equals 200 plus. Our equilibrium quantity of 200 me redo that plus sign got away from me plus 200. So our equilibrium price is gonna be 400. We've got our equilibrium soapy star. Remember P star is our equilibrium price is 400 Q. Star? Our equilibrium quantity is 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 minus half P equals p minus 200. All right, so now what we wanna do is solve for P. That's gonna be step three. I'm gonna rewrite the equation 400 minus half P equals p minus 200. So I want to get all the peas on one side of the equation. So that's gonna take adding half a P. Here and adding half a pea over here. And let me go back to red. So this is gonna cancel and we're gonna be left with 400 equals P plus half P. So P. Is the same as two P. Over to write to P over two. So two P over two plus one, P over two is gonna give us 3/2 P. Right, 1.5 P. S. We had one P. We added another half P. We've got three over to p minus 200. Right? 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 equals 3/2 P. And this will cancel out. So how do we get the P. By itself now? Right. We've got 600 equals 3/2 P. Well if you remember from algebra, the trick here is we're gonna multiply by the reciprocal. So if we multiply three over to p times two thirds this two thirds is gonna cancel. I'm just gonna get out of the way so I'm not dodging the two thirds is gonna cancel with the 3/2 and we need to multiply the other side of the equation also by two thirds. Okay so the three over to actually I'm gonna write all those in blue times two thirds and this side also times two thirds. Okay so let's go ahead and cancel stuff out the twos, cancel the three's cancel and we're left with just pee on that side of the equation and then we'll do two thirds times 600. So two times 600 is 1200 divided by three is 400. So two thirds times 600 is going to be 400. Alright so there we go we've gotten a price of 400 which we can confirm in step four. 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 sold 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. Um Actually I'm gonna use the demand one just to prove that we could use either one. So I'm gonna use quantity demanded equals 400 minus half P. And I'm gonna plug in R. P. There. So Q. Equals 400 minus half P. And P. Was 400. So we're gonna get Q. Equals 400 minus half of 400 is 200. So Q. Equals 200. And that confirms what we just got what we just got on the other side. Right? We got a P. Star here of 400 a Q. Star of 200. So either way either variable was isolated and we got the same answer there. Cool. So let's go ahead and try some practice with this stuff.

6

example

## Finding Equilibrium Using Algebra

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Alright let's go ahead and try this one here. So it gives us a couple of curves here we've got a supply curve and a demand curve and it's asking us to find equilibrium price and quantity. First thing you should notice is that different variables are isolated in each equation. Right? In our demand equation, the first equation price is isolated and in the second equation the supply equation, quantity is isolated and how do I know which is which is because the first one has quantity demanded in it. The second one has quantity supplied in it. Okay so the first thing we want to do is we want to isolate the same variable. So I don't like the looks of the first equation. I kind of that that fraction is just messing with me. I'm gonna rearrange that so that I have quantity supplied by the quantity demanded by itself. But remember you can also rearrange the other one and we'll get to the same answer. This is the way I'm gonna do it. So we've got P equals six minus 1/50 Q. D. Right? And I want to get that Q. D. By itself. So the first thing we need to do is move that six from one side to the other and we're gonna get p minus six equals negative 1/50 Q. D. So how do we get rid of that? Pesky fraction? We need to multiply by the reciprocal. Right? So I'm gonna multiply by 50/1. Right? So just negative 50. So multiply by negative 50 and negative to get rid of the negative sign in the front there. So I want to do negative times negative to get rid of the negative and then the 50 times the 1/50 to cancel out that fraction. So if I'm gonna multiply this side by negative 50 I need to multiply the other side by negative 50 as well. Let's go ahead and isolate this variable. So this negative and this negative cancel. The 50 and the 1/50 cancel. And we're left with just quantity demanded on this side of the equation. And let's expand this out. We've got negative 50 times P. So negative 50 P. And negative 50 times negative six. Those negatives are gonna cancel out. We're gonna get a positive uh six times 50 is 300. Alright so negative 50 times P. And the negative 50 times negative six. So there we go. That is gonna be the same equation with quantity demanded isolated. Right so now we can go on to the next step where we set the two equations equal to each other based on that isolated variable. Alright so I'm gonna take this side of this equation right here and this side of this equation right here our supply and demand equation. Let's go ahead and set them equal. So negative 50 P plus 300 equals 1 50 P minus 100. Alright so now I want to isolate my ps. So I'm gonna get them all on one side plus 50 P plus 50 P. And I will have 200 P. Over here minus 100. And on this side the p is canceled. And I'll have 300. 300 equals 200 p minus 100. Let's go ahead and add 100 to each side. So 400 equals 200 P. That cancels will divide both sides by 200 and we'll get an answer of P equals two. Alright? So we've figured out what P is now. It's just a matter of plugging this into either of our original equations and we will get our quantity. So this will be our equilibrium price and based on this alone will know which answer to the question. It is right. It's gonna be be it's the only one with an equilibrium price of two. So on a test you're crunched for time. Hey, this is enough information to get this right, but let's go ahead and finish it up. So with the price of two, I'm gonna go ahead and use the supply equation. It looks easier to just plug a number in. So quantity supplied or just quantity. Right? At equilibrium, They're going to be the same for demand and supply quantity equals 150 times our price of two minus 100. Right? So I just took this equation up here, Q. S equals 150 P minus 100 plugged in two for P. And let's find out what Q. E. Q equals 301 50 times two is 300 minus 100 Q equals 200. Just like we see in that answer. So the answer is gonna be be all right, let's move on.

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Problem

The supply and demand curves for a product are as follows:

Q_{d} = 10560 − 80P

P = 1 40Q_{s} + 6

What is the equilibrium price and quantity of the product?

A

P* = $60, Q* = 5760

B

P* = $70, Q* = 4960

C

P* = $80, Q* = 4160

D

P* = $90, Q* = 3360

8

concept

## Finding Equilibrium with Equations and a Graph

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So now I want to draw our curves on the graph and find equilibrium on the graph. Right? So just like we had been doing before, right 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. Um when we start plotting these these lines that were 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 there, 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 easy with these equations here, I'm gonna pull down the equation where quantity supplied is isolated, right? We're gonna be picking prices and solving for quantity. So it's easier to do that when the quantity is already isolated in the graph R. In the equation. So let me go ahead and grab these um These equations. The first one being quantity demanded equals 400 minus half P. And the other one was quantity supplied equals P minus 200. So Q. S equals p minus 200. And you could do this with either one, you could use either equation. Um price isolated quantity isolated. You just have to do a little more algebra um When selecting prices here. So let's go ahead and start with a really easy price. A price of zero. So at a price of zero we're gonna find a quantity demanded 400 minus half time zero. We're gonna get quantity demanded equals 400. To the same thing with supply, quantity supplied equals zero minus 200. So we're actually gonna get a negative number here negative 200 at a price of zero. This is kind of meaningless, right? They're not going to produce anything. Um With the price of zero they can't produce negative supply. So this this we can't even really plot this on our graph because we don't have negative numbers on the graph. This isn't gonna be so useful for us. So let's try another number that that might give us something that we can plot so we might want to try something little bigger. 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. Um let's say around 400. Let's see what we get at a price of 400. So we're going to get um Q. D. Equals 400 minus half. And our price in this case is 400. So we'll get 400 minus half of 400 is 200. We'll get a quantity demanded of 200. Do the same thing with supply. Quality supply equals price of 400 -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 gonna 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 gonna 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 minus half of 600 is 300 quantity demanded is gonna equal 100. Alright let's do that over here. With supply quantity supplied equals our price of 600 minus. Our equation was p minus 200. Price minus 200. Quality supply is gonna 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? Um 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 zero, quantity demanded was 400. So this will be our price axis, Quantity Axis and I will use blue for demand here and read for supply because we're not gonna be shifting or anything. So I think it's safe to use two different colors just to keep that easy to see. So at a price of zero, they are going to demand 400. So we're gonna 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 gonna 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 gonna look something like this. Oh wow. First try. That's about as good as I'm gonna 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 two 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 gonna end up here where I'm gonna 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 will 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 .400 and 200. Right, so that is gonna be our equilibrium price of 400 and a quantity of 200. So we've now solved that algebraic Lee, 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 gonna go. Um So let's go ahead and move on.