Simplify (x-3y)/(x^3-y^3)-(2x)/(y^3-x^3)

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

Simplify each term.

Since both terms are perfect cubes, factor using the difference of cubes formula, $a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$ where $a = x$ and $b = y$ .

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

Since both terms are perfect cubes, factor using the difference of cubes formula, $a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$ where $a = y$ and $b = x$ .

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

Simplify with factoring out.

Factor $- 1$ out of $y$ .

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

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

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

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

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

Reorder.

Reorder terms.

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

Reorder terms.

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

Reorder terms.

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

To write $\frac{x - 3 y}{(x - y) (x^{2} + y^{2} + x y)}$ as a fraction with a common denominator, multiply by $\frac{- 1}{- 1}$ .

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

Write each expression with a common denominator of $(x - y) (x^{2} + y^{2} + x y) \cdot - 1$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{x - 3 y}{(x - y) (x^{2} + y^{2} + x y)}$ by $\frac{- 1}{- 1}$ .

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

Reorder the factors of $(x - y) (x^{2} + y^{2} + x y) \cdot - 1$ .

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Apply the distributive property.

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

Move $- 1$ to the left of $x$ .

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

Multiply $- 1$ by $- 3$ .

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

Rewrite $- 1 x$ as $- x$ .

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

Subtract $2 x$ from $- x$ .

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

Factor $3$ out of $- 3 x + 3 y$ .

Factor $3$ out of $- 3 x$ .

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

Factor $3$ out of $3 y$ .

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

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

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

Reduce the expression by cancelling the common factors.

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

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

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

Factor $- 1$ out of $y$ .

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

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

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

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Dividing two negative values results in a positive value.

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

Simplify (x-3y)/(x^3-y^3)-(2x)/(y^3-x^3)

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