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

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

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 x-1 is \left(x-1\right)\left(-y+1\right). Multiply \frac{x}{x-1} times \frac{-y+1}{-y+1}.

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

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

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

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

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

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

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

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

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

Since \frac{xy-2x+y}{\left(x-1\right)\left(-y+1\right)} and \frac{y\left(x-1\right)}{\left(x-1\right)\left(-y+1\right)} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in xy-2x+y-y\left(x-1\right).

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

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

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

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

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

Expand

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

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 x-1 is \left(x-1\right)\left(-y+1\right). Multiply \frac{x}{x-1} times \frac{-y+1}{-y+1}.

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

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

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

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

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

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

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

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

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

Since \frac{xy-2x+y}{\left(x-1\right)\left(-y+1\right)} and \frac{y\left(x-1\right)}{\left(x-1\right)\left(-y+1\right)} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in xy-2x+y-y\left(x-1\right).

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

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

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

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

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