Solve x+y/x-y-x-y/x+y/frac{x+y{x}-x+2y/x+y}

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

Evaluate

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

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}.

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

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.

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

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

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

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

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x and x+y is x\left(x+y\right). Multiply \frac{x+y}{x} times \frac{x+y}{x+y}. Multiply \frac{x+2y}{x+y} times \frac{x}{x}.

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

Since \frac{\left(x+y\right)\left(x+y\right)}{x\left(x+y\right)} and \frac{\left(x+2y\right)x}{x\left(x+y\right)} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in \left(x+y\right)\left(x+y\right)-\left(x+2y\right)x.

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

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

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

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

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

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

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

Multiply x and x to get x^{2}.

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

Use the distributive property to multiply y by x-y.

$$\frac{4x^{2}}{yx-y^{2}}$$

Expand

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

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}.

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

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.

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

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

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

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

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x and x+y is x\left(x+y\right). Multiply \frac{x+y}{x} times \frac{x+y}{x+y}. Multiply \frac{x+2y}{x+y} times \frac{x}{x}.

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

Since \frac{\left(x+y\right)\left(x+y\right)}{x\left(x+y\right)} and \frac{\left(x+2y\right)x}{x\left(x+y\right)} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in \left(x+y\right)\left(x+y\right)-\left(x+2y\right)x.

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

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

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

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

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

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

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

Multiply x and x to get x^{2}.

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

Use the distributive property to multiply y by x-y.

$$\frac{4x^{2}}{yx-y^{2}}$$