Algebra Examples

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

Step 1

Simplify terms.

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Step 1.1

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

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Step 1.1.1

Factor $2$ out of $2 x$ .

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

Step 1.1.2

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

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

Step 1.1.3

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

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

Step 1.2

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

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

Step 1.3

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

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

Step 1.4

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

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

Step 1.5

Simplify the expression.

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Step 1.5.1

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

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

Step 1.5.2

Reorder terms.

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

Step 1.6

Combine the numerators over the common denominator.

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

Step 2

Simplify the numerator.

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Step 2.1

Multiply $- 1$ by $2$ .

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

Step 2.2

Apply the distributive property.

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

Step 2.3

Multiply $- 1$ by $- 2$ .

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

Step 2.4

Add $4 y$ and $2 y$ .

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

Step 2.5

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

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Step 2.5.1

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

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

Step 2.5.2

Factor $2$ out of $6 y$ .

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

Step 2.5.3

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.

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Step 3.1

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

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Step 3.1.1

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

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

Step 3.1.2

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

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

Step 3.1.3

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

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

Step 3.1.4

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

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

Step 3.1.5

Cancel the common factor.

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

Step 3.1.6

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

$2 \cdot (- 1)$

Step 3.2

Multiply $2$ by $- 1$ .

$- 2$