Solve x/x-y-x^2+y^2/x^2-y^2+y/x+y

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

Evaluate

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

Short Solution Steps

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

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

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

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

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

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

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

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

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

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

Factor the expressions that are not already factored in \frac{-y^{2}+xy}{\left(x+y\right)\left(x-y\right)}.

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

Cancel out x-y in both numerator and denominator.

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

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

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

Combine like terms in y+y.

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

Expand

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

Short Solution Steps

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

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

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

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

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

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

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

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

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

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

Factor the expressions that are not already factored in \frac{-y^{2}+xy}{\left(x+y\right)\left(x-y\right)}.

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

Cancel out x-y in both numerator and denominator.

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

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

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

Combine like terms in y+y.

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