Solve 2x/x-y-3y/x+y

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

Evaluate

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

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x-y and x+y is \left(x+y\right)\left(x-y\right). Multiply \frac{2x}{x-y} times \frac{x+y}{x+y}. Multiply \frac{3y}{x+y} times \frac{x-y}{x-y}.

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

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

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

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

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

Combine like terms in 2x^{2}+2xy-3yx+3y^{2}.

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

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

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

Differentiate w.r.t. x

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