Solve x/(x-y)(x-2)+y/(y)(y-x)+x/(2-x)(2-y)

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

Evaluate

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

Short Solution Steps

Cancel out y in both numerator and denominator.

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

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

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

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

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

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

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

Combine like terms in -x+x-2.

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

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

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

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

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

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

$$\frac{2y-4+x^{2}-xy}{\left(-x+2\right)\left(-y+2\right)\left(x-y\right)}$$

Factor the expressions that are not already factored in \frac{2y-4+x^{2}-xy}{\left(-x+2\right)\left(-y+2\right)\left(x-y\right)}.

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

Cancel out -x+2 in both numerator and denominator.

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

Expand \left(-y+2\right)\left(x-y\right).

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

Expand

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

Short Solution Steps

Cancel out y in both numerator and denominator.

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

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

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

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

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

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

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

Combine like terms in -x+x-2.

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

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

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

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

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

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

$$\frac{2y-4+x^{2}-xy}{\left(-x+2\right)\left(-y+2\right)\left(x-y\right)}$$

Factor the expressions that are not already factored in \frac{2y-4+x^{2}-xy}{\left(-x+2\right)\left(-y+2\right)\left(x-y\right)}.

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

Cancel out -x+2 in both numerator and denominator.

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

Expand \left(-y+2\right)\left(x-y\right).

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