Solve x^-1y-xy^-1/x^-1-y^-1

$$\frac{ { x }^{ -1 } y-x { y }^{ -1 } }{ { x }^{ -1 } - { y }^{ -1 } }$$

Evaluate

$$x+y$$

Short Solution Steps

Factor the expressions that are not already factored.

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

Cancel out \frac{1}{x} in both numerator and denominator.

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

Expand the expression.

$$\frac{-\frac{1}{y}x^{2}+y}{-\frac{1}{y}x+1}$$

Express \frac{1}{y}x^{2} as a single fraction.

$$\frac{-\frac{x^{2}}{y}+y}{-\frac{1}{y}x+1}$$

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

$$\frac{-\frac{x^{2}}{y}+\frac{yy}{y}}{-\frac{1}{y}x+1}$$

Since -\frac{x^{2}}{y} and \frac{yy}{y} have the same denominator, add them by adding their numerators.

$$\frac{\frac{-x^{2}+yy}{y}}{-\frac{1}{y}x+1}$$

Do the multiplications in -x^{2}+yy.

$$\frac{\frac{-x^{2}+y^{2}}{y}}{-\frac{1}{y}x+1}$$

Express \frac{1}{y}x as a single fraction.

$$\frac{\frac{-x^{2}+y^{2}}{y}}{-\frac{x}{y}+1}$$

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

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

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

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

Divide \frac{-x^{2}+y^{2}}{y} by \frac{-x+y}{y} by multiplying \frac{-x^{2}+y^{2}}{y} by the reciprocal of \frac{-x+y}{y}.

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

Cancel out y in both numerator and denominator.

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

Factor the expressions that are not already factored.

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

Cancel out -x+y in both numerator and denominator.

$$x+y$$

Expand

$$x+y$$

Short Solution Steps

Factor the expressions that are not already factored.

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

Cancel out \frac{1}{x} in both numerator and denominator.

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

Expand the expression.

$$\frac{-\frac{1}{y}x^{2}+y}{-\frac{1}{y}x+1}$$

Express \frac{1}{y}x^{2} as a single fraction.

$$\frac{-\frac{x^{2}}{y}+y}{-\frac{1}{y}x+1}$$

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

$$\frac{-\frac{x^{2}}{y}+\frac{yy}{y}}{-\frac{1}{y}x+1}$$

Since -\frac{x^{2}}{y} and \frac{yy}{y} have the same denominator, add them by adding their numerators.

$$\frac{\frac{-x^{2}+yy}{y}}{-\frac{1}{y}x+1}$$

Do the multiplications in -x^{2}+yy.

$$\frac{\frac{-x^{2}+y^{2}}{y}}{-\frac{1}{y}x+1}$$

Express \frac{1}{y}x as a single fraction.

$$\frac{\frac{-x^{2}+y^{2}}{y}}{-\frac{x}{y}+1}$$

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

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

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

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

Divide \frac{-x^{2}+y^{2}}{y} by \frac{-x+y}{y} by multiplying \frac{-x^{2}+y^{2}}{y} by the reciprocal of \frac{-x+y}{y}.

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

Cancel out y in both numerator and denominator.

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

Factor the expressions that are not already factored.

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

Cancel out -x+y in both numerator and denominator.

$$x+y$$