# How do you simplify the expression (y/x-x/y)/((x+y)/(xy))?

Apr 5, 2017

$\frac{\frac{y}{x} - \frac{x}{y}}{\frac{x + y}{x y}} = y - x$

#### Explanation:

$\frac{\frac{y}{x} - \frac{x}{y}}{\frac{x + y}{x y}}$

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

=$\frac{{y}^{2} - {x}^{2}}{x y} \div \frac{x + y}{x y}$

= $\frac{{y}^{2} - {x}^{2}}{x y} \times \frac{x y}{x + y}$

= $\frac{\left(y + x\right) \left(y - x\right)}{\cancel{x y}} \times \frac{\cancel{x y}}{x + y}$

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

= $\frac{\cancel{\left(x + y\right)} \left(y - x\right)}{\cancel{\left(x + y\right)}}$

= $y - x$