Solve 1/x-1/x+y/frac{y{x-y}-y/x+frac{y}{x+y}}=

$$\frac { \frac { 1 } { x } - \frac { 1 } { x + y } } { \frac { y } { x - y } - \frac { y } { x } + \frac { y } { x + y } } =$$

Evaluate

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

Short Solution Steps

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{1}{x} times \frac{x+y}{x+y}. Multiply \frac{1}{x+y} times \frac{x}{x}.

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

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

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

Combine like terms in x+y-x.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Divide \frac{y}{x\left(x+y\right)} by \frac{yx^{2}+y^{3}}{x\left(x+y\right)\left(x-y\right)} by multiplying \frac{y}{x\left(x+y\right)} by the reciprocal of \frac{yx^{2}+y^{3}}{x\left(x+y\right)\left(x-y\right)}.

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

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

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

Factor the expressions that are not already factored.

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

Cancel out y in both numerator and denominator.

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

Expand

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

Short Solution Steps

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{1}{x} times \frac{x+y}{x+y}. Multiply \frac{1}{x+y} times \frac{x}{x}.

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

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

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

Combine like terms in x+y-x.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Divide \frac{y}{x\left(x+y\right)} by \frac{yx^{2}+y^{3}}{x\left(x+y\right)\left(x-y\right)} by multiplying \frac{y}{x\left(x+y\right)} by the reciprocal of \frac{yx^{2}+y^{3}}{x\left(x+y\right)\left(x-y\right)}.

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

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

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

Factor the expressions that are not already factored.

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

Cancel out y in both numerator and denominator.

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