Simplify (2/x-4/y)/(-5/y+3/x)

$\frac{\frac{2}{x} - \frac{4}{y}}{\frac{- 5}{y} + \frac{3}{x}}$

Step 1

Multiply the numerator and denominator of the complex fraction by $x y$ .

Multiply $\frac{\frac{2}{x} - \frac{4}{y}}{\frac{- 5}{y} + \frac{3}{x}}$ by $\frac{x y}{x y}$ .

$\frac{x y}{x y} \cdot \frac{\frac{2}{x} - \frac{4}{y}}{\frac{- 5}{y} + \frac{3}{x}}$

Combine.

$\frac{x y (\frac{2}{x} - \frac{4}{y})}{x y (\frac{- 5}{y} + \frac{3}{x})}$

Step 2

Apply the distributive property.

$\frac{x y \frac{2}{x} + x y (- \frac{4}{y})}{x y \frac{- 5}{y} + x y \frac{3}{x}}$

Step 3

Simplify by cancelling.

Cancel the common factor of $x$ .

Factor $x$ out of $x y$ .

$\frac{x (y) \frac{2}{x} + x y (- \frac{4}{y})}{x y \frac{- 5}{y} + x y \frac{3}{x}}$

Cancel the common factor.

$\frac{x y \frac{2}{x} + x y (- \frac{4}{y})}{x y \frac{- 5}{y} + x y \frac{3}{x}}$

Rewrite the expression.

$\frac{y \cdot 2 + x y (- \frac{4}{y})}{x y \frac{- 5}{y} + x y \frac{3}{x}}$

Cancel the common factor of $y$ .

Move the leading negative in $- \frac{4}{y}$ into the numerator.

$\frac{y \cdot 2 + x y \frac{- 4}{y}}{x y \frac{- 5}{y} + x y \frac{3}{x}}$

Factor $y$ out of $x y$ .

$\frac{y \cdot 2 + y x \frac{- 4}{y}}{x y \frac{- 5}{y} + x y \frac{3}{x}}$

Cancel the common factor.

$\frac{y \cdot 2 + y x \frac{- 4}{y}}{x y \frac{- 5}{y} + x y \frac{3}{x}}$

Rewrite the expression.

$\frac{y \cdot 2 + x \cdot - 4}{x y \frac{- 5}{y} + x y \frac{3}{x}}$

Cancel the common factor of $y$ .

Factor $y$ out of $x y$ .

$\frac{y \cdot 2 + x \cdot - 4}{y x \frac{- 5}{y} + x y \frac{3}{x}}$

Cancel the common factor.

$\frac{y \cdot 2 + x \cdot - 4}{y x \frac{- 5}{y} + x y \frac{3}{x}}$

Rewrite the expression.

$\frac{y \cdot 2 + x \cdot - 4}{x \cdot - 5 + x y \frac{3}{x}}$

Cancel the common factor of $x$ .

Factor $x$ out of $x y$ .

$\frac{y \cdot 2 + x \cdot - 4}{x \cdot - 5 + x (y) \frac{3}{x}}$

Cancel the common factor.

$\frac{y \cdot 2 + x \cdot - 4}{x \cdot - 5 + x y \frac{3}{x}}$

Rewrite the expression.

$\frac{y \cdot 2 + x \cdot - 4}{x \cdot - 5 + y \cdot 3}$

Step 4

Simplify the numerator.

Factor $2$ out of $y \cdot 2 + x \cdot - 4$ .

Factor $2$ out of $y \cdot 2$ .

$\frac{2 \cdot y + x \cdot - 4}{x \cdot - 5 + y \cdot 3}$

Factor $2$ out of $x \cdot - 4$ .

$\frac{2 \cdot y + 2 (x \cdot - 2)}{x \cdot - 5 + y \cdot 3}$

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

$\frac{2 (y + x \cdot - 2)}{x \cdot - 5 + y \cdot 3}$

Move $- 2$ to the left of $x$ .

$\frac{2 (y - 2 x)}{x \cdot - 5 + y \cdot 3}$

Step 5

Simplify the denominator.

Move $- 5$ to the left of $x$ .

$\frac{2 (y - 2 x)}{- 5 \cdot x + y \cdot 3}$

Move $3$ to the left of $y$ .

$\frac{2 (y - 2 x)}{- 5 x + 3 y}$

Step 6

Simplify with factoring out.

Factor $- 1$ out of $- 5 x$ .

$\frac{2 (y - 2 x)}{- (5 x) + 3 y}$

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

$\frac{2 (y - 2 x)}{- (5 x) - (- 3 y)}$

Factor $- 1$ out of $- (5 x) - (- 3 y)$ .

$\frac{2 (y - 2 x)}{- (5 x - 3 y)}$

Rewrite negatives.

Rewrite $- (5 x - 3 y)$ as $- 1 (5 x - 3 y)$ .

$\frac{2 (y - 2 x)}{- 1 (5 x - 3 y)}$

Move the negative in front of the fraction.

$- \frac{2 (y - 2 x)}{5 x - 3 y}$

Simplify (2/x-4/y)/(-5/y+3/x)

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