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

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

Evaluate

$$\frac{3xy}{2\left(x^{2}-y^{2}\right)}$$

Short Solution Steps

Factor x^{2}-y^{2}.

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

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

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

Since \frac{x^{2}}{\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.

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

Do the multiplications in x^{2}-x\left(x-y\right).

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

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

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

Factor 2x-2y.

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

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

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

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

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

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

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

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

$$\frac{y^{2}+3xy}{2\left(x+y\right)\left(x-y\right)}-\frac{y^{2}}{2x^{2}-2y^{2}}$$

Factor 2x^{2}-2y^{2}.

$$\frac{y^{2}+3xy}{2\left(x+y\right)\left(x-y\right)}-\frac{y^{2}}{2\left(x+y\right)\left(x-y\right)}$$

Since \frac{y^{2}+3xy}{2\left(x+y\right)\left(x-y\right)} and \frac{y^{2}}{2\left(x+y\right)\left(x-y\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{y^{2}+3xy-y^{2}}{2\left(x+y\right)\left(x-y\right)}$$

Combine like terms in y^{2}+3xy-y^{2}.

$$\frac{3xy}{2\left(x+y\right)\left(x-y\right)}$$

Expand 2\left(x+y\right)\left(x-y\right).

$$\frac{3xy}{2x^{2}-2y^{2}}$$

Factor

$$\frac{3xy}{2\left(x^{2}-y^{2}\right)}$$