Solve x+y/2x-2y+x^2-y^2/6x-6y

$$\frac{ x+y }{ 2x-2y } + \frac{ { x }^{ 2 } - { y }^{ 2 } }{ 6x-6y }$$

Evaluate

$$-\frac{\left(-x+y-3\right)\left(x+y\right)}{6\left(x-y\right)}$$

Short Solution Steps

Factor the expressions that are not already factored in \frac{x^{2}-y^{2}}{6x-6y}.

$$\frac{x+y}{2x-2y}+\frac{\left(x+y\right)\left(x-y\right)}{6\left(x-y\right)}$$

Cancel out x-y in both numerator and denominator.

$$\frac{x+y}{2x-2y}+\frac{x+y}{6}$$

Factor 2x-2y.

$$\frac{x+y}{2\left(x-y\right)}+\frac{x+y}{6}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2\left(x-y\right) and 6 is 6\left(-x+y\right). Multiply \frac{x+y}{2\left(x-y\right)} times \frac{-3}{-3}. Multiply \frac{x+y}{6} times \frac{-x+y}{-x+y}.

$$\frac{-3\left(x+y\right)}{6\left(-x+y\right)}+\frac{\left(x+y\right)\left(-x+y\right)}{6\left(-x+y\right)}$$

Since \frac{-3\left(x+y\right)}{6\left(-x+y\right)} and \frac{\left(x+y\right)\left(-x+y\right)}{6\left(-x+y\right)} have the same denominator, add them by adding their numerators.

$$\frac{-3\left(x+y\right)+\left(x+y\right)\left(-x+y\right)}{6\left(-x+y\right)}$$

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

$$\frac{-3x-3y-x^{2}+xy-yx+y^{2}}{6\left(-x+y\right)}$$

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

$$\frac{-3x-3y-x^{2}+y^{2}}{6\left(-x+y\right)}$$

Expand 6\left(-x+y\right).

$$\frac{-3x-3y-x^{2}+y^{2}}{-6x+6y}$$

Expand

$$\frac{-x^{2}-3x+y^{2}-3y}{6\left(y-x\right)}$$

Short Solution Steps

Factor the expressions that are not already factored in \frac{x^{2}-y^{2}}{6x-6y}.

$$\frac{x+y}{2x-2y}+\frac{\left(x+y\right)\left(x-y\right)}{6\left(x-y\right)}$$

Cancel out x-y in both numerator and denominator.

$$\frac{x+y}{2x-2y}+\frac{x+y}{6}$$

Factor 2x-2y.

$$\frac{x+y}{2\left(x-y\right)}+\frac{x+y}{6}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2\left(x-y\right) and 6 is 6\left(-x+y\right). Multiply \frac{x+y}{2\left(x-y\right)} times \frac{-3}{-3}. Multiply \frac{x+y}{6} times \frac{-x+y}{-x+y}.

$$\frac{-3\left(x+y\right)}{6\left(-x+y\right)}+\frac{\left(x+y\right)\left(-x+y\right)}{6\left(-x+y\right)}$$

Since \frac{-3\left(x+y\right)}{6\left(-x+y\right)} and \frac{\left(x+y\right)\left(-x+y\right)}{6\left(-x+y\right)} have the same denominator, add them by adding their numerators.

$$\frac{-3\left(x+y\right)+\left(x+y\right)\left(-x+y\right)}{6\left(-x+y\right)}$$

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

$$\frac{-3x-3y-x^{2}+xy-yx+y^{2}}{6\left(-x+y\right)}$$

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

$$\frac{-3x-3y-x^{2}+y^{2}}{6\left(-x+y\right)}$$

Expand 6\left(-x+y\right).

$$\frac{-3x-3y-x^{2}+y^{2}}{-6x+6y}$$