Question

the equation P=2(L +w) can be used to find the perimeter P of a rectangle with length L and width W. Solve the equation for w to produce a formula for finding the width of a rectangle given its perimeter and length

Answer 1

P = 2( L + W )
-divide both sides by 2
P/2 = L + W
-subtract L from both sides
P/2 - L = W
or
(P - L) / 2 = W

Answer 2

Hello, 

We know that if we make the same operation on both sides, we don't affect the equation:

For example:  
$2=2 \\ 2\bold{+2}=2\bold{+2} \\ 4=4$

Do you see? we add 2 on both sides and we don't affect the equality

We can do the same with the multiplication, for example

4=4   (i'll multiply both sides by 1/2) 
$4*\bold{ \frac{1}{2}}=4*\bold{ \frac{1}{2}} \\ 2=2$

Now, we do exactly the same in your excercise:

$P=2(L+W) \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,[multiply\,\,by\,\,1/2] \\ \\ P*\bold{ \frac{1}{2}}=2(L+W)*\bold{ \frac{1}{2}} \\ \\ \frac{P}{2}=L+W\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,[Subtract\,\,L]\\ \\ \frac{P}{2}-\bold{L}=L+W-\bold{L}\\ \\ \frac{P}{2}-\bold{L}=L-\bold{L}+W \\ \\ \frac{P}{2}-L=W\\ \\ \boxed{ W= \frac{P}{2}-L }$