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

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

Evaluate

$$-\frac{2}{x}$$

Short Solution Steps

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

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

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

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

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

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

Do the multiplications in yy-xx.

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

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

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

Extract the negative sign in y-x.

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

Cancel out x-y in both numerator and denominator.

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

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

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

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

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

Combine like terms in -x-y+x-y.

$$\frac{-2y}{xy}$$

Cancel out y in both numerator and denominator.

$$\frac{-2}{x}$$

Expand

$$-\frac{2}{x}$$

Short Solution Steps

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

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

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

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

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

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

Do the multiplications in yy-xx.

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

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

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

Extract the negative sign in y-x.

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

Cancel out x-y in both numerator and denominator.

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

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

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

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

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

Combine like terms in -x-y+x-y.

$$\frac{-2y}{xy}$$

Cancel out y in both numerator and denominator.

$$\frac{-2}{x}$$