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

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

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, add them by adding 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.

$$\frac{2x^{2}+y^{2}+4xy}{xy\left(x+y\right)}-\frac{y}{x^{2}+xy}$$

Factor x^{2}+xy.

$$\frac{2x^{2}+y^{2}+4xy}{xy\left(x+y\right)}-\frac{y}{x\left(x+y\right)}$$

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

$$\frac{2x^{2}+y^{2}+4xy}{xy\left(x+y\right)}-\frac{yy}{xy\left(x+y\right)}$$

Since \frac{2x^{2}+y^{2}+4xy}{xy\left(x+y\right)} and \frac{yy}{xy\left(x+y\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{2x^{2}+y^{2}+4xy-yy}{xy\left(x+y\right)}$$

Do the multiplications in 2x^{2}+y^{2}+4xy-yy.

$$\frac{2x^{2}+y^{2}+4xy-y^{2}}{xy\left(x+y\right)}$$

Combine like terms in 2x^{2}+y^{2}+4xy-y^{2}.

$$\frac{2x^{2}+4xy}{xy\left(x+y\right)}$$

Factor the expressions that are not already factored in \frac{2x^{2}+4xy}{xy\left(x+y\right)}.

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

Cancel out x in both numerator and denominator.

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

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

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

Use the distributive property to multiply 2 by x+2y.

$$\frac{2x+4y}{xy+y^{2}}$$

Expand

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

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, add them by adding 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.

$$\frac{2x^{2}+y^{2}+4xy}{xy\left(x+y\right)}-\frac{y}{x^{2}+xy}$$

Factor x^{2}+xy.

$$\frac{2x^{2}+y^{2}+4xy}{xy\left(x+y\right)}-\frac{y}{x\left(x+y\right)}$$

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

$$\frac{2x^{2}+y^{2}+4xy}{xy\left(x+y\right)}-\frac{yy}{xy\left(x+y\right)}$$

Since \frac{2x^{2}+y^{2}+4xy}{xy\left(x+y\right)} and \frac{yy}{xy\left(x+y\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{2x^{2}+y^{2}+4xy-yy}{xy\left(x+y\right)}$$

Do the multiplications in 2x^{2}+y^{2}+4xy-yy.

$$\frac{2x^{2}+y^{2}+4xy-y^{2}}{xy\left(x+y\right)}$$

Combine like terms in 2x^{2}+y^{2}+4xy-y^{2}.

$$\frac{2x^{2}+4xy}{xy\left(x+y\right)}$$

Factor the expressions that are not already factored in \frac{2x^{2}+4xy}{xy\left(x+y\right)}.

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

Cancel out x in both numerator and denominator.

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

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

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

Use the distributive property to multiply 2 by x+2y.

$$\frac{2x+4y}{xy+y^{2}}$$