Solve x+3y/x^{2-y^{2}}-x+2y/x^2-y^2+2x-3y/x^2-y^2

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

Evaluate

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

Short Solution Steps

Since \frac{x+3y}{x^{2}-y^{2}} and \frac{x+2y}{x^{2}-y^{2}} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in x+3y-\left(x+2y\right).

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

Combine like terms in x+3y-x-2y.

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

Since \frac{y}{x^{2}-y^{2}} and \frac{2x-3y}{x^{2}-y^{2}} have the same denominator, add them by adding their numerators.

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

Combine like terms in y+2x-3y.

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

Factor the expressions that are not already factored in \frac{-2y+2x}{x^{2}-y^{2}}.

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

Cancel out x-y in both numerator and denominator.

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

Expand

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

Short Solution Steps

Since \frac{x+3y}{x^{2}-y^{2}} and \frac{x+2y}{x^{2}-y^{2}} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in x+3y-\left(x+2y\right).

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

Combine like terms in x+3y-x-2y.

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

Since \frac{y}{x^{2}-y^{2}} and \frac{2x-3y}{x^{2}-y^{2}} have the same denominator, add them by adding their numerators.

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

Combine like terms in y+2x-3y.

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

Factor the expressions that are not already factored in \frac{-2y+2x}{x^{2}-y^{2}}.

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

Cancel out x-y in both numerator and denominator.

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