Basic algebra problem: $ \frac{\frac{1}{x}+\frac{1}{y}}{\frac{1}{x^2}-\frac{1}{y^2}} $ Basic algebra problem I can't seem to figure out: $$ \frac{\frac{1}{x}+\frac{1}{y}}{\frac{1}{x^2}-\frac{1}{y^2}} $$ $x,y \in \mathbb{R}, x^2 \neq y^2, xy\neq0$.
Now I know the result is: $\frac{xy}{y-x}$, but I am not sure how to get it, I get into a mess like this: $=x+\frac{x^2}{y}-\frac{y^2}{x}-y=\frac{x(xy)+x^3-y^3-y(xy)}{xy}=?$ which doesn't seem to help me much. Halp please.

By using the rule for difference of two squares; i.e. $a^2 - b^2 = (a+b) (a-b)$, we have that $\frac{1}{x^2} - \frac{1}{y^2} = (\frac{1}{x})^2 - (\frac{1}{y})^2 = (\frac{1}{x} + \frac{1}{y})(\frac{1}{x} - \frac{1}{y})$. And so in the case of your question by using factorization and simplification we have
$\frac{(\frac{1}{x} + \frac{1}{y})}{\frac{1}{x^2} - \frac{1}{y^2}} = \frac{(\frac{1}{x} + \frac{1}{y})}{(\frac{1}{x} + \frac{1}{y})(\frac{1}{x} - \frac{1}{y})} = \frac{1}{\frac{1}{x} - \frac{1}{y}} = \frac{1}{\frac{y - x}{x y}} = \frac{\frac{1}{1}}{\frac{y - x}{x y}} = \frac{1 \times (xy)}{1 \times (y - x)} = \frac{xy}{y - x}.$
And we are done.
