Solve (x+y/x-y-x-y/x+y)div(x+y/x-y+1)

$$( \frac{ x+y }{ x-y } - \frac{ x-y }{ x+y } ) \div ( \frac{ x+y }{ x-y } +1)$$

Evaluate

$$\frac{2y}{x+y}$$

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}.

$$\frac{\frac{\left(x+y\right)\left(x+y\right)}{\left(x+y\right)\left(x-y\right)}-\frac{\left(x-y\right)\left(x-y\right)}{\left(x+y\right)\left(x-y\right)}}{\frac{x+y}{x-y}+1}$$

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.

$$\frac{\frac{\left(x+y\right)\left(x+y\right)-\left(x-y\right)\left(x-y\right)}{\left(x+y\right)\left(x-y\right)}}{\frac{x+y}{x-y}+1}$$

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

$$\frac{\frac{x^{2}+xy+xy+y^{2}-x^{2}+xy+xy-y^{2}}{\left(x+y\right)\left(x-y\right)}}{\frac{x+y}{x-y}+1}$$

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

$$\frac{\frac{4xy}{\left(x+y\right)\left(x-y\right)}}{\frac{x+y}{x-y}+1}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{x-y}{x-y}.

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

Since \frac{x+y}{x-y} and \frac{x-y}{x-y} have the same denominator, add them by adding their numerators.

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

Combine like terms in x+y+x-y.

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

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

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

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

$$\frac{2y}{x+y}$$

Expand

$$\frac{2y}{x+y}$$

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}.

$$\frac{\frac{\left(x+y\right)\left(x+y\right)}{\left(x+y\right)\left(x-y\right)}-\frac{\left(x-y\right)\left(x-y\right)}{\left(x+y\right)\left(x-y\right)}}{\frac{x+y}{x-y}+1}$$

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.

$$\frac{\frac{\left(x+y\right)\left(x+y\right)-\left(x-y\right)\left(x-y\right)}{\left(x+y\right)\left(x-y\right)}}{\frac{x+y}{x-y}+1}$$

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

$$\frac{\frac{x^{2}+xy+xy+y^{2}-x^{2}+xy+xy-y^{2}}{\left(x+y\right)\left(x-y\right)}}{\frac{x+y}{x-y}+1}$$

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

$$\frac{\frac{4xy}{\left(x+y\right)\left(x-y\right)}}{\frac{x+y}{x-y}+1}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{x-y}{x-y}.

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

Since \frac{x+y}{x-y} and \frac{x-y}{x-y} have the same denominator, add them by adding their numerators.

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

Combine like terms in x+y+x-y.

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

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

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

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

$$\frac{2y}{x+y}$$