Solve x/x+y-x-1/x+2y-xy/x^2+3xy+2y^2

$$\frac { x } { x + y } - \frac { x - 1 } { x + 2 y } - \frac { x y } { x ^ { 2 } + 3 x y + 2 y ^ { 2 } }$$

Evaluate

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

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x+y and x+2y is \left(x+y\right)\left(x+2y\right). Multiply \frac{x}{x+y} times \frac{x+2y}{x+2y}. Multiply \frac{x-1}{x+2y} times \frac{x+y}{x+y}.

$$\frac{x\left(x+2y\right)}{\left(x+y\right)\left(x+2y\right)}-\frac{\left(x-1\right)\left(x+y\right)}{\left(x+y\right)\left(x+2y\right)}-\frac{xy}{x^{2}+3xy+2y^{2}}$$

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

$$\frac{x\left(x+2y\right)-\left(x-1\right)\left(x+y\right)}{\left(x+y\right)\left(x+2y\right)}-\frac{xy}{x^{2}+3xy+2y^{2}}$$

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

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

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

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

Factor x^{2}+3xy+2y^{2}.

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

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

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

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

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

Cancel out x+y in both numerator and denominator.

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

Expand

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

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x+y and x+2y is \left(x+y\right)\left(x+2y\right). Multiply \frac{x}{x+y} times \frac{x+2y}{x+2y}. Multiply \frac{x-1}{x+2y} times \frac{x+y}{x+y}.

$$\frac{x\left(x+2y\right)}{\left(x+y\right)\left(x+2y\right)}-\frac{\left(x-1\right)\left(x+y\right)}{\left(x+y\right)\left(x+2y\right)}-\frac{xy}{x^{2}+3xy+2y^{2}}$$

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

$$\frac{x\left(x+2y\right)-\left(x-1\right)\left(x+y\right)}{\left(x+y\right)\left(x+2y\right)}-\frac{xy}{x^{2}+3xy+2y^{2}}$$

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

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

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

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

Factor x^{2}+3xy+2y^{2}.

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

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

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

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

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

Cancel out x+y in both numerator and denominator.

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