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

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

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 xy and y\left(x+y\right) is xy\left(x+y\right). Multiply \frac{x^{2}+y^{2}}{xy} times \frac{x+y}{x+y}. Multiply \frac{x^{2}}{y\left(x+y\right)} times \frac{x}{x}.

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

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

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

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

$$\frac{x^{3}+x^{2}y+y^{2}x+y^{3}-x^{3}}{xy\left(x+y\right)}-\frac{y^{2}}{x\left(x+y\right)}$$

Combine like terms in x^{3}+x^{2}y+y^{2}x+y^{3}-x^{3}.

$$\frac{y^{2}x+x^{2}y+y^{3}}{xy\left(x+y\right)}-\frac{y^{2}}{x\left(x+y\right)}$$

Factor the expressions that are not already factored in \frac{y^{2}x+x^{2}y+y^{3}}{xy\left(x+y\right)}.

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

Cancel out y in both numerator and denominator.

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

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

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

Combine like terms in x^{2}+xy+y^{2}-y^{2}.

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

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

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

Cancel out x\left(x+y\right) in both numerator and denominator.

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Factor

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