simpify expression

**jvignacio** · June 16th 2008, 09:41 AM

Hey guys, need help simplifing this expression..
$ \frac{x}{y-x} - \frac{y}{y+x} - \frac{2xy}{y^2-x^2} $

thank u

**TheEmptySet** · June 16th 2008, 10:00 AM

Originally Posted by jvignacio

Hey guys, need help simplifing this expression..
$ \frac{x}{y-x} - \frac{y}{y+x} - \frac{2xy}{y^2-x^2} $

thank u

First we need to find the LCD of all three fractions

Lets start with $y^2-x^2=(y-x)(y+x)$ This is the lowest common denominator

so now we build each fraction to have the same denomniantor

$ \frac{x}{y-x}\left( \frac{x+y}{x+y} \right) - \frac{y}{y+x}\left( \frac{y-x}{y-x} \right) - \frac{2xy}{(y-x)(y+x)} $

Now lets multiply out the numerators and simplify

$\frac{x^2+xy-(y^2-xy)-(2xy)}{(y-x)(y+x)}=\frac{x^2+xy+xy-y^2-2xy}{(y-x)(y+x)}=\frac{x^2-y^2}{(y-x)(y+x)}=$

$\frac{(x-y)(x+y)}{(y-x)(y+x)}=-1 \mbox{ if } x \ne y$

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