Solve 2x-2y/y*3y^2/x^2-y^2+6x/x+y

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

Evaluate

$$6$$

Short Solution Steps

Multiply \frac{2x-2y}{y} times \frac{3y^{2}}{x^{2}-y^{2}} by multiplying numerator times numerator and denominator times denominator.

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

Cancel out y in both numerator and denominator.

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

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

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

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 x+y is \left(x+y\right)\left(x-y\right). Multiply \frac{6x}{x+y} times \frac{x-y}{x-y}.

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

Since \frac{3y\left(2x-2y\right)}{\left(x+y\right)\left(x-y\right)} and \frac{6x\left(x-y\right)}{\left(x+y\right)\left(x-y\right)} have the same denominator, add them by adding their numerators.

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

Do the multiplications in 3y\left(2x-2y\right)+6x\left(x-y\right).

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

Combine like terms in 6yx-6y^{2}+6x^{2}-6xy.

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

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

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

Extract the negative sign in -x-y. Extract the negative sign in -x+y.

$$\frac{-\left(-1\right)\times 6\left(x+y\right)\left(x-y\right)}{\left(x+y\right)\left(x-y\right)}$$

Cancel out \left(x+y\right)\left(x-y\right) in both numerator and denominator.

$$-\left(-1\right)\times 6$$

Multiply -1 and -1 to get 1.

$$6$$

Factor

$$2\times 3$$