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

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

Evaluate

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Short Solution Steps

Factor xy+y^{2}.

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

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

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

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

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

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

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

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

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

Cancel out y in both numerator and denominator.

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

Factor x^{2}+xy.

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

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

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

Combine like terms in y-y.

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

Zero divided by any non-zero term gives zero.

$$0$$

Factor

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