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

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

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{1}{x-1} times \frac{-y+1}{-y+1}.

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

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

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

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

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

Combine like terms in xy-x+y-yx+y-1.

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

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{1}{1-y} times \frac{x-1}{x-1}.

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

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

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

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

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

Combine like terms in -x+2y-1-x+1.

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

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{1}{x-1} times \frac{-y+1}{-y+1}.

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

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

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

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

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

Combine like terms in xy-x+y-yx+y-1.

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

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{1}{1-y} times \frac{x-1}{x-1}.

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

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

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

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

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

Combine like terms in -x+2y-1-x+1.

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

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

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