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so now we have a very very very interesting problem

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on the left hand side of the scale I have two different types of unknown masses

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one of these X masses and we know that they have the same identical mass, we call that identical each of them having a mass of X

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But then we have this other blue thing and that has a mass of Y, which isn't necessarily going to be the same as the mass of X.

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We have two of these X's and a Y.It seems like the total mass or it definitely is the case, their total mass balance it out to these 8 kg right over here.

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Each of these is 1kg block and balances them out. So the first question I'm going to ask you is, can you express this Mathematically?

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Can you express what we're seeing here, the fact that this total mass balances out with this total mass.

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Can you express that mathematically?

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Well let's just think about our total mass on this side.

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We have two masses of mass X so those two are gonna total at 2X, and then you have a mass of Y.

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So then you're gonna have another Y. So the total mass on the left hand side. Alright let me re-write a little bit closer to the center.

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So,it doesn't get too spread out. On the left hand side, I got 2X plus a mass of Y. That's the total mass.

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The total mass on the left hand side is 2X plus Y, the total mass on the right hand side is just 8. 1,2,3,4,5,6,7,8. It is equal to 8

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And since we see that the scale is balance, this total mass must be equal to this total mass.

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So, we can write an equal sign there.

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Now my question to you is there anything we can do just based on the information that we have here to solve for either the mass X or for the mass Y.

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Is there anything that we can do.

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Well the simple answer is just with this information here, there's actually very little.

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You might say that "Oh well, let me take the Y from both sides" You might take this Y block up.

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But if you take this Y block up you have to take away Y from this side and you don't know what Y is.

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And if you think about it algebraically you might get rid of the Y here. Subtracting Y and you're gonna subtract Y from this side too.

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So, you're not gonna get rid of the Y.

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Same thing with the X's, you actually don't have enough information. Y depends on what X is,and X depends on what Y is.

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Lucky for us however,we do have some more of these blocks laying around. And what we do is we take one of these X blocks.

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And I stack it over here,and I also take one of the Y block and I stack it right over there.

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And then I keep adding all these ones until I balance these things out.

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So, I keep adding these ones. Obviously if I just place this, this will go down cause there's nothing on that side.

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But I keep adding these blocks until it all balances out and I find that my scale balances once I have 5 kg on the right hand side

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So, once again let me ask you this information having X and Y on the left hand side and a 5kg on the right hand side

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And the fact that they are balance, how can we represent that mathematically?

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Well our total mass on the left hand side is X plus Y. And our total mass, let me right that once again a little bit closer to the center.

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It's X plus Y on the left hand side and the right hand side I have 5 kg. I have 5kg.

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I have 5 kg on the right hand side. And we know that's actually balance the scale. So these total masses must be equal to each other.

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And this information by itself, once again. There's nothing I can do with it. I don't know what X and Y.

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If Y is 4 maybe X is 1 or maybe X is 4, Y is 1. Who knows what these are.

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The interesting thing is we can actually use both of these information to figure out what X and Y actually is.

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And I'm giving you a few seconds to think about how we can approach this situation.

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Well think about it this way, we know that X plus Y is equal to 5.

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So if we were to get rid of an X and a Y on this side,on the left hand side of the equation.

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What would we have to get rid of on the right hand side of the, or if we know if we get rid of X and Y on the left hand side of the scale

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What would we get rid of the right hand side of the scale to take away the same mass?

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Well if we take away the X and Y on the left hand side, we know that an X plus Y is 5kg. So, we'll just have to take 5kg from the right hand side.

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So, lets think about what that would do. Well then I'll just have an X over here, I'll just have some of these masses left over here.Then I would what X is.

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Now lets think about how we can represent that algebraically

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Essentially for taking an X and Y from the left hand side. If I'm taking an X and Y from the left hand side.

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I'm subtracting an X, and I'm subtracting an X. Actually let me think of it this way. I'm subtracting an X plus Y. I'm subtracting an X and Y on the left hand side.

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But then what am I gonna do on the right hand side?

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Well an X and a Y we know has a mass of 5. So we can subtract 5 from the right hand side.

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And the only way I'm gonna be able to do this is because of the information that we got from the second scale.

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So I can take away 5. So this is going to be equal to taking away 5. Taking away X and a Y is equal to taking away 5.

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And we know that because an X and a Y is equal to 5kg. And if we take away an X and a Y on the left hand side, what do we left with?

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Well this is gonna be the same thing. Let me rewrite this part.

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This, taking away an X and a Y is the same thing if you distribute the negative sign as taking away an X and taking away a Y.

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And so on the left hand side, we're left with just 2X and we have taken away one of the X's, we're left just an X.

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And we had a Y and we've took away one of the Y. So we're left with no Y.

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We see that visually, we're left with just an X here.

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And what do we have on the right hand side?

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We had 8 and we know X and Y is equal to 5, so we took away 5. So to keep the scale balance. And so 8 minus 5 is going to be 3.

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8 minus 5 is equal to 3 and just like that using this extra information we're able to figure out that the mass of X is equal to 3.

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Now, one final question. We're able to figure out the mass of X, can you figure out what the mass of Y is.

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Well we can go back to either one of these scales. Probably be simpler to go back to this one.

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We know that the mass of X plus the mass of Y is equal to 5. So we could say, one thing we know that X is now is equal to 3.

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We know that this is now a 3kg mass. We can rewrite this is 3 plus Y is equal to 5

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Well now we say, we could take 3 away from both sides.

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if I take 3 away from the left hand side I just have to take 3 away from the right hand side to keep my scale balanced.

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And I'll be left with the mass of Y is balance with a mass of 2 or Y is equal to 2.

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That's an analogy of taking 3 from both sides of this equation. And on the left hand side, I'm just left with a Y and on the right hand side I'm just left with a 2.

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So, X is equal to 3kg and Y is equal to 2kg and what I encouraged you to do is verify that it made sense right up here.

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Figure out what the total mass on the left hand and the right hand or verify what the total mass right over here really was 8 to begin with.

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And you'll see that 2Xs are gonna be 6kg plus my Y is 2kg that will balance 8kg.

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And 3 plus 2 was equal to 5.