So basically what you're looking for when you multiply the numerators only on both sides, is when you know that it will cancel out all the denominators?

An example of what i'm stuck on would be:

\(\displaystyle \frac{9}{y+2} - \frac{7}{y-2} = \frac{8}{y^2-4}\)

that one multiplies only the top by (y-2)(y+2)

\(\displaystyle \frac{1}{5} - \frac{1}{8} = \frac{1}{x}\)

while this one multiplies both top and bottom to get 40 as a denominator

Hi fb280,

you do have a choice of continuations,

here's how you can solve these in alternative ways...

\(\displaystyle \frac{9}{y+2}-\frac{7}{y-2}=\frac{8}{y^2-4}\)

Add the two fractions on the left, first obtaining a common denominator to make them compatible

(so that we may add or subtract the numerators)

\(\displaystyle \frac{y-2}{y-2}\ \frac{9}{y+2}-\frac{y+2}{y+2}\ \frac{7}{y-2}=\frac{8}{y^2-4}\)

Notice that \(\displaystyle (y-2)(y+2)=y^2+2y-2y-4=y^2-4\)

Hence the denominators on both sides are the same

\(\displaystyle \frac{9(y-2)}{y^2-4}-\frac{7(y+2)}{y^2-4}=\frac{8}{y^2-4}\)

\(\displaystyle \frac{9(y-2)-7(y+2)}{y^2-4}=\frac{8}{y^2-4}\)

If these are equal, since the denominators are the same,

the numerators must also be equal.
\(\displaystyle 9(y-2)-7(y+2)=8\)

Alternatively, we could have multiplied both sides of the fraction by (y-2)(y+2) as both sides will still be equal, since they were equal to begin with.

That will remove the denominators.

\(\displaystyle \frac{(y+2)(y-2)9}{y+2}-\frac{(y+2)(y-2)7}{y-2}=\frac{8(y+2)(y-2)}{y^2-4}\)

This gives \(\displaystyle 9(y-2)-7(y+2)=8\)

which is the same result as before.

\(\displaystyle \frac{1}{5}-\frac{1}{8}=\frac{1}{x}\)

If you wish you may multiply both sides by 5, 8 and x to get

\(\displaystyle \frac{40x}{5}-\frac{40x}{8}=\frac{40x}{x}\)

\(\displaystyle 8x-5x=40\ \Rightarrow\ 3x=40\ \Rightarrow\ x=\frac{40}{3}\)

Alternatively

\(\displaystyle \frac{1}{5}-\frac{1}{8}=\frac{8}{8}\ \frac{1}{5}-\frac{5}{5}\ \frac{1}{8}=\frac{8}{40}-\frac{5}{40}=\frac{8-5}{40}=\frac{3}{40}\)

\(\displaystyle \frac{3}{40}=\frac{1}{x}\ \Rightarrow\ \frac{40}{3}=\frac{x}{1}=x\)