Finite Math Examples

$\frac{\frac{x^{2} - 9}{27 x}}{\frac{3 - x}{3 x y}}$

Step 1

Multiply the numerator by the reciprocal of the denominator.

$\frac{x^{2} - 9}{27 x} \cdot \frac{3 x y}{3 - x}$

Step 2

Simplify terms.

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Step 2.1

Cancel the common factor of $3 x$ .

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Step 2.1.1

Factor $3 x$ out of $27 x$ .

$\frac{x^{2} - 9}{3 x (9)} \cdot \frac{3 x y}{3 - x}$

Step 2.1.2

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

$\frac{x^{2} - 9}{3 x (9)} \cdot \frac{3 x (y)}{3 - x}$

Step 2.1.3

Cancel the common factor.

$\frac{x^{2} - 9}{3 x \cdot 9} \cdot \frac{3 x y}{3 - x}$

Step 2.1.4

Rewrite the expression.

$\frac{x^{2} - 9}{9} \cdot \frac{y}{3 - x}$

Step 2.2

Multiply $\frac{x^{2} - 9}{9}$ by $\frac{y}{3 - x}$ .

$\frac{(x^{2} - 9) y}{9 (3 - x)}$

Step 3

Simplify the numerator.

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Step 3.1

Rewrite $9$ as $3^{2}$ .

$\frac{(x^{2} - 3^{2}) y}{9 (3 - x)}$

Step 3.2

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

$\frac{(x + 3) (x - 3) y}{9 (3 - x)}$

Step 4

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

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Step 4.1

Factor $- 1$ out of $x$ .

$\frac{(x + 3) (- 1 (- x) - 3) y}{9 (3 - x)}$

Step 4.2

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

$\frac{(x + 3) (- 1 (- x) - 1 (3)) y}{9 (3 - x)}$

Step 4.3

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

$\frac{(x + 3) (- 1 (- x + 3)) y}{9 (3 - x)}$

Step 4.4

Reorder terms.

$\frac{(x + 3) (- 1 (- x + 3)) y}{9 (- x + 3)}$

Step 4.5

Cancel the common factor.

$\frac{(x + 3) (- 1 (- x + 3)) y}{9 (- x + 3)}$

Step 4.6

Rewrite the expression.

$\frac{((x + 3) \cdot (- 1)) y}{9}$

Step 5

Factor out negative.

$\frac{- (x + 3) y}{9}$

Step 6

Move the negative in front of the fraction.

$- \frac{(x + 3) y}{9}$

Step 7

Reorder factors in $- \frac{(x + 3) y}{9}$ .

$- \frac{y (x + 3)}{9}$