Solve x^-1+y^-1/x^-1y-y^-1x

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

Evaluate

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

Short Solution Steps

Factor the expressions that are not already factored.

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

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

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

Expand the expression.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Cancel out y in both numerator and denominator.

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

Factor the expressions that are not already factored.

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

Extract the negative sign in y+x.

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

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

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

Expand

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

Short Solution Steps

Factor the expressions that are not already factored.

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

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

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

Expand the expression.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Cancel out y in both numerator and denominator.

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

Factor the expressions that are not already factored.

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

Extract the negative sign in y+x.

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

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

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