Solve (1/x-y+2/x^2-xy)+x+2/5x

$$( \frac { 1 } { x - y } + \frac { 2 } { x ^ { 2 } - x y } ) + \frac { x + 2 } { 5 x }$$

Evaluate

$$\frac{\left(x+2\right)\left(x-y+5\right)}{5x\left(x-y\right)}$$

Short Solution Steps

Factor x^{2}-xy.

$$\frac{1}{x-y}+\frac{2}{x\left(x-y\right)}+\frac{x+2}{5x}$$

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

$$\frac{x}{x\left(x-y\right)}+\frac{2}{x\left(x-y\right)}+\frac{x+2}{5x}$$

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

$$\frac{x+2}{x\left(x-y\right)}+\frac{x+2}{5x}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x\left(x-y\right) and 5x is 5x\left(x-y\right). Multiply \frac{x+2}{x\left(x-y\right)} times \frac{5}{5}. Multiply \frac{x+2}{5x} times \frac{x-y}{x-y}.

$$\frac{5\left(x+2\right)}{5x\left(x-y\right)}+\frac{\left(x+2\right)\left(x-y\right)}{5x\left(x-y\right)}$$

Since \frac{5\left(x+2\right)}{5x\left(x-y\right)} and \frac{\left(x+2\right)\left(x-y\right)}{5x\left(x-y\right)} have the same denominator, add them by adding their numerators.

$$\frac{5\left(x+2\right)+\left(x+2\right)\left(x-y\right)}{5x\left(x-y\right)}$$

Do the multiplications in 5\left(x+2\right)+\left(x+2\right)\left(x-y\right).

$$\frac{5x+10+x^{2}-xy+2x-2y}{5x\left(x-y\right)}$$

Combine like terms in 5x+10+x^{2}-xy+2x-2y.

$$\frac{7x+10+x^{2}-xy-2y}{5x\left(x-y\right)}$$

Expand 5x\left(x-y\right).

$$\frac{7x+10+x^{2}-xy-2y}{5x^{2}-5xy}$$

Expand

$$\frac{-x^{2}+xy-7x+2y-10}{5x\left(y-x\right)}$$

Short Solution Steps

Factor x^{2}-xy.

$$\frac{1}{x-y}+\frac{2}{x\left(x-y\right)}+\frac{x+2}{5x}$$

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

$$\frac{x}{x\left(x-y\right)}+\frac{2}{x\left(x-y\right)}+\frac{x+2}{5x}$$

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

$$\frac{x+2}{x\left(x-y\right)}+\frac{x+2}{5x}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x\left(x-y\right) and 5x is 5x\left(x-y\right). Multiply \frac{x+2}{x\left(x-y\right)} times \frac{5}{5}. Multiply \frac{x+2}{5x} times \frac{x-y}{x-y}.

$$\frac{5\left(x+2\right)}{5x\left(x-y\right)}+\frac{\left(x+2\right)\left(x-y\right)}{5x\left(x-y\right)}$$

Since \frac{5\left(x+2\right)}{5x\left(x-y\right)} and \frac{\left(x+2\right)\left(x-y\right)}{5x\left(x-y\right)} have the same denominator, add them by adding their numerators.

$$\frac{5\left(x+2\right)+\left(x+2\right)\left(x-y\right)}{5x\left(x-y\right)}$$

Do the multiplications in 5\left(x+2\right)+\left(x+2\right)\left(x-y\right).

$$\frac{5x+10+x^{2}-xy+2x-2y}{5x\left(x-y\right)}$$

Combine like terms in 5x+10+x^{2}-xy+2x-2y.

$$\frac{7x+10+x^{2}-xy-2y}{5x\left(x-y\right)}$$

Expand 5x\left(x-y\right).

$$\frac{7x+10+x^{2}-xy-2y}{5x^{2}-5xy}$$