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

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

Evaluate

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

Short Solution Steps

Factor x^{2}-y^{2}.

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

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

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

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

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

Combine like terms in x+y+2x-y.

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

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

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

Expand

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

Short Solution Steps

Factor x^{2}-y^{2}.

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

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

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

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

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

Combine like terms in x+y+2x-y.

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

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

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