Solve xleft(y-1right)+yleft(1-xright)/left(x-1right)left(1-yright)-y-x/(x-1)(y-1)

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

Evaluate

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

Short Solution Steps

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

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

Since \frac{-\left(x\left(y-1\right)+y\left(1-x\right)\right)}{\left(x-1\right)\left(y-1\right)} and \frac{y-x}{\left(x-1\right)\left(y-1\right)} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in -\left(x\left(y-1\right)+y\left(1-x\right)\right)-\left(y-x\right).

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

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

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

Expand \left(x-1\right)\left(y-1\right).

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

Expand

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

Short Solution Steps

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

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

Since \frac{-\left(x\left(y-1\right)+y\left(1-x\right)\right)}{\left(x-1\right)\left(y-1\right)} and \frac{y-x}{\left(x-1\right)\left(y-1\right)} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in -\left(x\left(y-1\right)+y\left(1-x\right)\right)-\left(y-x\right).

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

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

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

Expand \left(x-1\right)\left(y-1\right).

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