Manipulating expressions using structure (example 2) | High School Math | Khan Academy

We're told, suppose ( a + b ) is equal to ( 2a ). Which of these expressions equals ( b - a )?

All right, I encourage you to pause the video and see if you can figure that out. Which of these expressions would be equal to ( b - a )? It's going to just involve some algebraic manipulation.

All right, let's work through this together. So we are told that ( a + b ) is equal to ( 2a ). The first thing I would want to do is get all my ( a )'s in one place, and one way I could do that is I could subtract ( a ) from both sides.

So if I subtract ( a ) from both sides, I am going to be left with just ( b ) on the left-hand side, and on the right-hand side, I'm going to be left with ( 2a - a ). Well, that's just going to be ( a ). If I have two of something, and I subtract one of them, take away one of them, I'm going to have just one of those something—it's equal to ( 1a ).

So, we want to figure out what ( b - a ) is. Well, luckily, I can figure that out if I subtract ( a ) from both sides. So if I subtract ( a ) from both sides, well then I'm going to get on the left-hand side ( b - a ), which is what we want to figure out, is equal to ( a - a ), which is equal to zero.

So ( b - a ) is equal to ( 0 ), which is not one of the choices. All right, so let's see if we can figure out some other things over here. So ( b - a ) is equal to zero, but that is not one of the choices.

All right, is there any other way to manipulate this? No?

I could just go straight ahead and subtract ( 2a ) from both sides, and I would get ( b - a ) is equal to zero. Oh, this is interesting; this is a tricky one.

So ( b - a ) is zero. Well, if ( b - a ) is equal to zero, if we take the negative of both sides of this... If we take the negative of both sides, if we multiply both sides by -1—well, on the left-hand side, we get ( a - b ), and on the right-hand side, we still get zero.

If ( b - a ) is zero, then the negative of it, which is ( a - b ), is also going to be equal to zero. And that's this choice. Let me do that in a little darker color. That is this choice right over there. That was a good one!