Solve x-y/x+y-x+y/x-y/frac{x{x-y}-x/x+y}

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

Evaluate

$$-2$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x+y and x-y is \left(x+y\right)\left(x-y\right). Multiply \frac{x-y}{x+y} times \frac{x-y}{x-y}. Multiply \frac{x+y}{x-y} times \frac{x+y}{x+y}.

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

Since \frac{\left(x-y\right)\left(x-y\right)}{\left(x+y\right)\left(x-y\right)} and \frac{\left(x+y\right)\left(x+y\right)}{\left(x+y\right)\left(x-y\right)} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in \left(x-y\right)\left(x-y\right)-\left(x+y\right)\left(x+y\right).

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

Combine like terms in x^{2}-xy-xy+y^{2}-x^{2}-xy-xy-y^{2}.

$$\frac{\frac{-4xy}{\left(x+y\right)\left(x-y\right)}}{\frac{x}{x-y}-\frac{x}{x+y}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x-y and x+y is \left(x+y\right)\left(x-y\right). Multiply \frac{x}{x-y} times \frac{x+y}{x+y}. Multiply \frac{x}{x+y} times \frac{x-y}{x-y}.

$$\frac{\frac{-4xy}{\left(x+y\right)\left(x-y\right)}}{\frac{x\left(x+y\right)}{\left(x+y\right)\left(x-y\right)}-\frac{x\left(x-y\right)}{\left(x+y\right)\left(x-y\right)}}$$

Since \frac{x\left(x+y\right)}{\left(x+y\right)\left(x-y\right)} and \frac{x\left(x-y\right)}{\left(x+y\right)\left(x-y\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{-4xy}{\left(x+y\right)\left(x-y\right)}}{\frac{x\left(x+y\right)-x\left(x-y\right)}{\left(x+y\right)\left(x-y\right)}}$$

Do the multiplications in x\left(x+y\right)-x\left(x-y\right).

$$\frac{\frac{-4xy}{\left(x+y\right)\left(x-y\right)}}{\frac{x^{2}+xy-x^{2}+xy}{\left(x+y\right)\left(x-y\right)}}$$

Combine like terms in x^{2}+xy-x^{2}+xy.

$$\frac{\frac{-4xy}{\left(x+y\right)\left(x-y\right)}}{\frac{2xy}{\left(x+y\right)\left(x-y\right)}}$$

Divide \frac{-4xy}{\left(x+y\right)\left(x-y\right)} by \frac{2xy}{\left(x+y\right)\left(x-y\right)} by multiplying \frac{-4xy}{\left(x+y\right)\left(x-y\right)} by the reciprocal of \frac{2xy}{\left(x+y\right)\left(x-y\right)}.

$$\frac{-4xy\left(x+y\right)\left(x-y\right)}{\left(x+y\right)\left(x-y\right)\times 2xy}$$

Cancel out 2xy\left(x+y\right)\left(x-y\right) in both numerator and denominator.

$$-2$$

Factor

$$-2$$