Why we do the same thing to both sides of an equation
So once again, we have three equal, or we say three identical objects. They all have the same mass, but we don't know what the mass is of each of them. But what we do know is that if you total up their mass, it's the same exact mass as these nine objects right over here. And each of these nine objects have a mass of 1 kilograms. So in total, you have 9 kilograms on this side. And over here, you have three objects. They all have the same mass. And we don't know what it is. We're just calling that mass x. And what I want to do here is try to tackle this a little bit more symbolically. In the last video, we said, hey, why don't we just multiply 1/3 of this and multiply 1/3 of this? And then, essentially, we're going to keep things balanced, because we're taking 1/3 of the same mass. This total is the same as this total. That's why the scale is balanced. Now, let's think about how we can represent this symbolically. So the first thing I want you to think about is, can we set up an equation that expresses that we have these three things of mass x, and that in total, their mass is equal to the total mass over here? Can we express that as an equation? And I'll give you a few seconds to do it. Well, let's think about it. Over here, we have three things with mass x. So their total mass, we could write as-- we could write their total mass as x plus x plus x. And over here, we have nine things with mass of 1 kilogram. I guess we could write 1 plus 1 plus 1. That's 3. Plus 1 plus 1 plus 1 plus 1. How many is that? 1, 2, 3, 4, 5, 6, 7, 8, 9. And actually, this is a mathematical representation. If we set it up as an equation, it's an algebraic representation. It's not the simplest possible way we can do it, but it is a reasonable way to do it. If we want, we can say, well, if I have an x plus another x plus another x, I have three x's. So I could rewrite this as 3x. And 3x will be equal to? Well, if I sum up all of these 1's right over here-- 1 plus 1 plus 1. We're doing that. We have 9 of them, so we get 3x is equal to 9. And let me make sure I did that. 1, 2, 3, 4, 5, 6, 7, 8, 9. So that's how we would set it up. And so the next question is, what would we do? What can we do mathematically? Actually, to either one of these equations, but we'll focus on this one right now. What can we do mathematically in order to essentially solve for the x? In order to figure out what that mystery mass actually is? And I'll give you another second or two to think about it. Well, when we did it the last time with just the scales we said, OK, we've got three of these x's here. We want to have just one x here. So we can say, whatever this x is, if the scale stays balanced, it's going to be the same as whatever we have there. There might be a temptation to subtract two of the x's maybe from this side, but that won't help us. And we can even see it mathematically over here. If we subtract two x's from both sides, on the left-hand side you're going to have 3x minus 2x. And on the right-hand side, you're going to have 9 minus 2x. And you're just going to be left with 3 of something minus 2 something is just 1 of something. So you will just have an x there if you get rid of two of them. But on the right-hand side, you're going to get 9 minus 2 x's. So the x's still didn't help you out. You still have a mystery mass on the right-hand side. So that doesn't help. So instead, what we say is-- and we did this the last time. We said, well, what if we took 1/3 of these things? If we take 1/3 of these things and take 1/3 of these things, we should still get the same mass on both sides because the original things had the same mass. And the equivalent of doing that mathematically is to say, why don't we multiply both sides by 1/3? Or another way to say it is we could divide both sides by 3. Multiplying by 1/3 is the same thing as dividing by 3. So we're going to multiply both sides by 1/3. When you multiply both sides by 1/3-- visually over here, if you had three x's, you multiply it by 1/3, you're only going to have one x left. If you have nine of these one-kilogram boxes, you multiply it by 1/3, you're only going to have three left. And over here, you can even visually-- if you divide by 3, which is the same thing as multiplying by 1/3, you divide by 3. So you divide by 3. You have an x is equal to a 1 plus 1 plus 1. An x is equal to 3. Or you see here, an x is equal to 3. Over here you do the math. 1/3 times 3 is 1. You're left with 1x. So you're left with x is equal to 9 times 1/3. Or you could even view it as 9 divided by 3, which is equal to 3.