Manipulating formulas: perimeter

We are told that the formula for finding the perimeter of a rectangle is P is equal to 2l plus 2w, where P is the perimeter, l is the length, and w is the width. And just to visualize what they're saying, and you might already be familiar with this, let me draw a rectangle. That looks like a rectangle. And if this side's length is l, then this side's length is also going to be l. And if this width is w, then this width up here is w. And the perimeter is just how, what is the distance if you were to go around this rectangle? And so, that distance is going to be this w plus this l, plus this w-- or that width-- plus this length. And if you have 1 w and you add it to another w, that's going to give you 2 w's. So that's 2 w's. And then if you have 1 l, and then you have another l, that's going to give you, if you add them together, that's going to give you 2 l's. So the perimeter is going to be 2 l's plus 2 w's. They just wrote it in a different order than the way I wrote it. But the same thing, so hopefully that makes sense. Now, their question is, rewrite the formula so that it solves for width. So the formula, the way it's written now, it says P is equal to something. They want us to write it so it's, this w, right here, they want it to be w is equal to a bunch of stuff with l's and P's in it, and maybe some numbers there. So let's think about how we can do this. So they tell us that P is equal to 2 times l, plus 2 times w. We want to solve for w. Well, a good starting point might be to get rid of the l on this side of the equation. And to get rid of it on that side of the equation, we could subtract the 2l from both sides of the equation. So let's do it this way. So you subtract 2l over here. Minus 2l. You're also going to have to do that on the left-hand side. So you're going to have minus 2l. We're doing it on both sides of the equation. And remember, an equation says P is equal to that, so if you do anything to that, you have to do it to P. So if you subtract 2l from this, you're going to have to subtract 2l from P in order for the equality to keep being true. So the left-hand side is going to be P minus 2l, and then that is going to be equal to-- well, 2l minus 2l, the whole reason why we subtracted 2l is because these are going to cancel out. So these cancel out, and you're just left with a 2w here. You're just left with a 2w. We're almost there. We've almost solved for w. To finish it up, we just have to divide both sides of this equation by 2. And the whole reason why I'm dividing both sides of this equation by 2 is to get rid of this 2 coefficient, this 2 that's multiplying w. So if you divide both sides of this equation by 2, once again, if you do something to one side of the equation, you do it to the other side. The whole reason why I divided the right-hand side by 2 is 2 times anything divided by 2 is just going to be that anything, so this is going to be a w. And then we have our left-hand side. So we're done. If we flip these two sides, we have our w will be equal to this thing over here-- equals P minus 2l, all of that over 2. Now, this is the correct answer. There's other ways to write it, though. You might want to rewrite this, so let me square this off, because this is completely the correct answer. This is the correct answer, but there's other ways that you might have been able to get this answer, other expressions for this answer. You might have also, you know, another completely legitimate way to do this problem-- let me write it this way-- so our original problem is P is equal to 2l plus 2w-- is on this right-hand side, what if we factor out a 2? So let me make this clear. You have a 2 here, and you have a 2 here. So you could imagine undistributing the 2. So we would get P is equal to 2 times l plus w. This is an equally legitimate way to do this problem. Now, we can divide both sides of this equation by 2, so that we get rid of this 2 on the right-hand side. So if you divide both sides of this equation by 2, these 2's are going to cancel out-- 2 times anything divided by 2 is just going to be the anything-- is equal to P over 2. So let me just rewrite this over here. Let me just rewrite this. So we will get P over 2 is going to be equal to l plus w. And then if we want to solve for w, we just subtract l from both sides. And sometimes, you know, you could write it in a separate line like this. Sometimes you could just write it like this. You could say, I'm going to subtract an l on that side. If I do it on that side, I have to do it on this side, too. That's the same thing as adding a negative l. And so the right-hand side, you're just left with a w. And then the left-hand side, you're going to have, it could be a negative l plus P over 2, or you could just change the order, and you can write this as P over 2 minus l. And this is also an equally legitimate answer. And you're probably saying, hey Sal, wait. These things look different. P minus 2l over 2, that looks different than P over 2 minus l. And they're not. Think about this. We could rewrite this as P-- let me do this with the same colors-- this over here is the same thing as P over 2 minus 2l over 2. Right? If I have a minus b and they're both being divided by 2, I can just separate-- you can imagine I'm distributing the division by 2 right over here. And over here, 2 times l divided by 2, that's just an l. So this is going to be equal to P over 2 minus l, which is the exact same thing as this right there.