Solve (x-y/x^2-2xy+y^2-x/x^2-2xy)divy/x-2y

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

Evaluate

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

Short Solution Steps

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

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

Cancel out x-y in both numerator and denominator.

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

Factor the expressions that are not already factored in \frac{x}{x^{2}-2xy}.

$$\frac{\frac{1}{x-y}-\frac{x}{x\left(x-2y\right)}}{\frac{y}{x-2y}}$$

Cancel out x in both numerator and denominator.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x-y and x-2y is \left(x-2y\right)\left(x-y\right). Multiply \frac{1}{x-y} times \frac{x-2y}{x-2y}. Multiply \frac{1}{x-2y} times \frac{x-y}{x-y}.

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

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

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

Do the multiplications in x-2y-\left(x-y\right).

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

Combine like terms in x-2y-x+y.

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

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

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

Cancel out y\left(x-2y\right) 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 in \frac{x-y}{x^{2}-2xy+y^{2}}.

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

Cancel out x-y in both numerator and denominator.

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

Factor the expressions that are not already factored in \frac{x}{x^{2}-2xy}.

$$\frac{\frac{1}{x-y}-\frac{x}{x\left(x-2y\right)}}{\frac{y}{x-2y}}$$

Cancel out x in both numerator and denominator.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x-y and x-2y is \left(x-2y\right)\left(x-y\right). Multiply \frac{1}{x-y} times \frac{x-2y}{x-2y}. Multiply \frac{1}{x-2y} times \frac{x-y}{x-y}.

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

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

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

Do the multiplications in x-2y-\left(x-y\right).

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

Combine like terms in x-2y-x+y.

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

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

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

Cancel out y\left(x-2y\right) in both numerator and denominator.

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