Manipulating formulas: perimeter

we are told that the formula 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 and that looks like a rectangle and if this sides length is L then this sides 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 to if you have a 1w and you add it to another W that's going to give you 2 WS 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 LS so the perimeter is going to be 2 l's plus 2ws 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 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 dubbed this W right here they want it to be W is equal to a bunch of stuff with with ELLs and peas 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 have minus 2l we're doing it on both sides of the equation 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 gonna have to subtract 2l from P in order for the Equality to to keep being true so the left-hand side is going to be P minus 2l minus 2l and then that is going to be equal to well 2l minus 2l the whole reason why we subtract a 2 LS because these are going to cancel out so these cancel out and you're just left with a 2 w 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 two 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 just flop if we flip these two sides we have our W we have our W will be equal to this thing over here equals P minus 2l minus 2l all of that over 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 or 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 we get make is clear you have a to here and you have a to here so you can imagine undistributing the to 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 twos are going to cancel out 2 times anything divided by 2 is just going to be 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 P over 2 is going to be equal to L plus W L plus W and then if we want to solve for W we just subtract L from both sides and sometimes you can 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 aw 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 you could just change the order and you can write this as P over 2 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 the same colors this over here this over here is the same thing as P over over 2 P over 2 minus 2l over over to write 5 a minus B and they're both being divided by 2 I can just separate I can 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 minus L which is the exact same thing as this right there