Algebra Examples

$\frac{\frac{y}{3} + 1}{\frac{y^{2}}{9} - 1}$

Step 1

Multiply the numerator and denominator of the fraction by $9$ .

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Step 1.1

Multiply $\frac{\frac{y}{3} + 1}{\frac{y^{2}}{9} - 1}$ by $\frac{9}{9}$ .

$\frac{9}{9} \cdot \frac{\frac{y}{3} + 1}{\frac{y^{2}}{9} - 1}$

Step 1.2

Combine.

$\frac{9 (\frac{y}{3} + 1)}{9 (\frac{y^{2}}{9} - 1)}$

Step 2

Apply the distributive property.

$\frac{9 \frac{y}{3} + 9 \cdot 1}{9 \frac{y^{2}}{9} + 9 \cdot - 1}$

Step 3

Simplify by cancelling.

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Step 3.1

Cancel the common factor of $3$ .

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Step 3.1.1

Factor $3$ out of $9$ .

$\frac{3 (3) \frac{y}{3} + 9 \cdot 1}{9 \frac{y^{2}}{9} + 9 \cdot - 1}$

Step 3.1.2

Cancel the common factor.

$\frac{3 \cdot 3 \frac{y}{3} + 9 \cdot 1}{9 \frac{y^{2}}{9} + 9 \cdot - 1}$

Step 3.1.3

Rewrite the expression.

$\frac{3 y + 9 \cdot 1}{9 \frac{y^{2}}{9} + 9 \cdot - 1}$

Step 3.2

Cancel the common factor of $9$ .

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Step 3.2.1

Cancel the common factor.

$\frac{3 y + 9 \cdot 1}{9 \frac{y^{2}}{9} + 9 \cdot - 1}$

Step 3.2.2

Rewrite the expression.

$\frac{3 y + 9 \cdot 1}{y^{2} + 9 \cdot - 1}$

Step 4

Simplify the numerator.

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Step 4.1

Factor $3$ out of $3 y + 9 \cdot 1$ .

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Step 4.1.1

Factor $3$ out of $3 y$ .

$\frac{3 (y) + 9 \cdot 1}{y^{2} + 9 \cdot - 1}$

Step 4.1.2

Factor $3$ out of $9 \cdot 1$ .

$\frac{3 (y) + 3 (3 \cdot 1)}{y^{2} + 9 \cdot - 1}$

Step 4.1.3

Factor $3$ out of $3 (y) + 3 (3 \cdot 1)$ .

$\frac{3 (y + 3 \cdot 1)}{y^{2} + 9 \cdot - 1}$

Step 4.2

Multiply $3$ by $1$ .

$\frac{3 (y + 3)}{y^{2} + 9 \cdot - 1}$

Step 5

Simplify the denominator.

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Step 5.1

Multiply $9$ by $- 1$ .

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

Step 5.2

Rewrite $y^{2} - 9$ in a factored form.

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Step 5.2.1

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

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

Step 5.2.2

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

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

Step 6

Cancel the common factor of $y + 3$ .

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Step 6.1

Cancel the common factor.

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

Step 6.2

Rewrite the expression.

$\frac{3}{y - 3}$