Simplify (4y)/(x-3y)+(2x-2y)/(3y-x)

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

Step 1

Simplify terms.

Factor $2$ out of $2 x - 2 y$ .

Factor $2$ out of $2 x$ .

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

Factor $2$ out of $- 2 y$ .

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

Factor $2$ out of $2 (x) + 2 (- y)$ .

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

Factor $- 1$ out of $3 y$ .

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

Factor $- 1$ out of $- x$ .

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

Factor $- 1$ out of $- (- 3 y) - (x)$ .

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

Simplify the expression.

Move a negative from the denominator of $\frac{2 (x - y)}{- (- 3 y + x)}$ to the numerator.

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

Reorder terms.

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

Combine the numerators over the common denominator.

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

Step 2

Simplify the numerator.

Multiply $- 1$ by $2$ .

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

Apply the distributive property.

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

Multiply $- 1$ by $- 2$ .

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

Add $4 y$ and $2 y$ .

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

Factor $2$ out of $- 2 x + 6 y$ .

Factor $2$ out of $- 2 x$ .

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

Factor $2$ out of $6 y$ .

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

Factor $2$ out of $2 (- x) + 2 (3 y)$ .

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

Step 3

Reduce the expression by cancelling the common factors.

Cancel the common factor of $- x + 3 y$ and $x - 3 y$ .

Factor $- 1$ out of $- x$ .

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

Factor $- 1$ out of $3 y$ .

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

Factor $- 1$ out of $- (x) - (- 3 y)$ .

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

Rewrite $- (x - 3 y)$ as $- 1 (x - 3 y)$ .

$\frac{2 (- 1 (x - 3 y))}{x - 3 y}$

Cancel the common factor.

$\frac{2 (- 1 (x - 3 y))}{x - 3 y}$

Divide $2 \cdot (- 1)$ by $1$ .

$2 \cdot (- 1)$

Multiply $2$ by $- 1$ .

$- 2$

Simplify (4y)/(x-3y)+(2x-2y)/(3y-x)

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