Solve x+2y/x^{2-y^{2}}+3y-x/y^2-x^2-3x-4y/x^2-y^2

$$\frac { x + 2 y } { x ^ { 2 } - y ^ { 2 } } + \frac { 3 y - x } { y ^ { 2 } - x ^ { 2 } } - \frac { 3 x - 4 y } { x ^ { 2 } - y ^ { 2 } }$$

Evaluate

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

Short Solution Steps

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

$$\frac{x+2y}{\left(x+y\right)\left(x-y\right)}+\frac{3y-x}{\left(x+y\right)\left(-x+y\right)}-\frac{3x-4y}{x^{2}-y^{2}}$$

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

$$\frac{-\left(x+2y\right)}{\left(x+y\right)\left(-x+y\right)}+\frac{3y-x}{\left(x+y\right)\left(-x+y\right)}-\frac{3x-4y}{x^{2}-y^{2}}$$

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

$$\frac{-\left(x+2y\right)+3y-x}{\left(x+y\right)\left(-x+y\right)}-\frac{3x-4y}{x^{2}-y^{2}}$$

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

$$\frac{-x-2y+3y-x}{\left(x+y\right)\left(-x+y\right)}-\frac{3x-4y}{x^{2}-y^{2}}$$

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

$$\frac{-2x+y}{\left(x+y\right)\left(-x+y\right)}-\frac{3x-4y}{x^{2}-y^{2}}$$

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

$$\frac{-2x+y}{\left(x+y\right)\left(-x+y\right)}-\frac{3x-4y}{\left(x+y\right)\left(x-y\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x+y\right)\left(-x+y\right) and \left(x+y\right)\left(x-y\right) is \left(x+y\right)\left(-x+y\right). Multiply \frac{3x-4y}{\left(x+y\right)\left(x-y\right)} times \frac{-1}{-1}.

$$\frac{-2x+y}{\left(x+y\right)\left(-x+y\right)}-\frac{-\left(3x-4y\right)}{\left(x+y\right)\left(-x+y\right)}$$

Since \frac{-2x+y}{\left(x+y\right)\left(-x+y\right)} and \frac{-\left(3x-4y\right)}{\left(x+y\right)\left(-x+y\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{-2x+y-\left(-\left(3x-4y\right)\right)}{\left(x+y\right)\left(-x+y\right)}$$

Do the multiplications in -2x+y-\left(-\left(3x-4y\right)\right).

$$\frac{-2x+y+3x-4y}{\left(x+y\right)\left(-x+y\right)}$$

Combine like terms in -2x+y+3x-4y.

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

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

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

Expand

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

Short Solution Steps

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

$$\frac{x+2y}{\left(x+y\right)\left(x-y\right)}+\frac{3y-x}{\left(x+y\right)\left(-x+y\right)}-\frac{3x-4y}{x^{2}-y^{2}}$$

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

$$\frac{-\left(x+2y\right)}{\left(x+y\right)\left(-x+y\right)}+\frac{3y-x}{\left(x+y\right)\left(-x+y\right)}-\frac{3x-4y}{x^{2}-y^{2}}$$

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

$$\frac{-\left(x+2y\right)+3y-x}{\left(x+y\right)\left(-x+y\right)}-\frac{3x-4y}{x^{2}-y^{2}}$$

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

$$\frac{-x-2y+3y-x}{\left(x+y\right)\left(-x+y\right)}-\frac{3x-4y}{x^{2}-y^{2}}$$

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

$$\frac{-2x+y}{\left(x+y\right)\left(-x+y\right)}-\frac{3x-4y}{x^{2}-y^{2}}$$

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

$$\frac{-2x+y}{\left(x+y\right)\left(-x+y\right)}-\frac{3x-4y}{\left(x+y\right)\left(x-y\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x+y\right)\left(-x+y\right) and \left(x+y\right)\left(x-y\right) is \left(x+y\right)\left(-x+y\right). Multiply \frac{3x-4y}{\left(x+y\right)\left(x-y\right)} times \frac{-1}{-1}.

$$\frac{-2x+y}{\left(x+y\right)\left(-x+y\right)}-\frac{-\left(3x-4y\right)}{\left(x+y\right)\left(-x+y\right)}$$

Since \frac{-2x+y}{\left(x+y\right)\left(-x+y\right)} and \frac{-\left(3x-4y\right)}{\left(x+y\right)\left(-x+y\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{-2x+y-\left(-\left(3x-4y\right)\right)}{\left(x+y\right)\left(-x+y\right)}$$

Do the multiplications in -2x+y-\left(-\left(3x-4y\right)\right).

$$\frac{-2x+y+3x-4y}{\left(x+y\right)\left(-x+y\right)}$$

Combine like terms in -2x+y+3x-4y.

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

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

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