# Simplify the complex rational expression completely. Show all work: (1/x+1/y)/(x/y-y/x)

Simplify the complex rational expression completely. Show all work:
$\frac{\frac{1}{x}+\frac{1}{y}}{\frac{x}{y}-\frac{y}{x}}$
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Anna Juarez
The given expression is
$\frac{\frac{1}{x}+\frac{1}{y}}{\frac{x}{y}-\frac{y}{x}}$
So,
$\frac{\frac{1}{x}+\frac{1}{y}}{\frac{x}{y}-\frac{y}{x}}=\frac{\frac{y+x}{xy}}{\frac{{x}^{2}-{y}^{2}}{yx}}\phantom{\rule{0ex}{0ex}}=\frac{x+y}{xy}×\frac{xy}{{x}^{2}-{y}^{2}}\phantom{\rule{0ex}{0ex}}=\frac{x+y}{{x}^{2}-{y}^{2}}\phantom{\rule{0ex}{0ex}}=\frac{\left(x+y\right)}{\left(x+y\right)\left(x-y\right)}\phantom{\rule{0ex}{0ex}}=\frac{\frac{1}{x}+\frac{1}{y}}{\frac{x}{y}-\frac{y}{x}}=\frac{1}{x-y}$