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

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

Evaluate

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Short Solution Steps

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

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

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

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

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

$$\frac{-xy+xz+yx-yz}{\left(x-y\right)\left(x-z\right)\left(-y+z\right)}+\frac{z}{\left(z-x\right)\left(z-y\right)}$$

Combine like terms in -xy+xz+yx-yz.

$$\frac{xz-yz}{\left(x-y\right)\left(x-z\right)\left(-y+z\right)}+\frac{z}{\left(z-x\right)\left(z-y\right)}$$

Factor the expressions that are not already factored in \frac{xz-yz}{\left(x-y\right)\left(x-z\right)\left(-y+z\right)}.

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

Cancel out x-y in both numerator and denominator.

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

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

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

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

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

Combine like terms in -z+z.

$$\frac{0}{\left(-x+z\right)\left(-y+z\right)}$$

Zero divided by any non-zero term gives zero.

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Factor

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