Solve (1-2x/x+y)/frac{x^2-2x+y{2x+2y}+x^2-xy/x^2-y^2}

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

Evaluate

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

Short Solution Steps

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

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

Since \frac{x+y}{x+y} and \frac{2x}{x+y} have the same denominator, subtract them by subtracting their numerators.

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

Combine like terms in x+y-2x.

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

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

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

Cancel out x-y in both numerator and denominator.

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

Factor 2x+2y.

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

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

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

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

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

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

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

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

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

Cancel out x+y in both numerator and denominator.

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

Use the distributive property to multiply 2 by -x+y.

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

Expand

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

Short Solution Steps

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

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

Since \frac{x+y}{x+y} and \frac{2x}{x+y} have the same denominator, subtract them by subtracting their numerators.

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

Combine like terms in x+y-2x.

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

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

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

Cancel out x-y in both numerator and denominator.

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

Factor 2x+2y.

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

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

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

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

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

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

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

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

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

Cancel out x+y in both numerator and denominator.

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

Use the distributive property to multiply 2 by -x+y.

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