Algebra Examples

$\frac{y + \frac{y - 3}{3}}{\frac{5}{9} + \frac{2}{3 y}}$

Step 1

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

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Step 1.1

Multiply $\frac{y + \frac{y - 3}{3}}{\frac{5}{9} + \frac{2}{3 y}}$ by $\frac{9 y}{9 y}$ .

$\frac{9 y}{9 y} \cdot \frac{y + \frac{y - 3}{3}}{\frac{5}{9} + \frac{2}{3 y}}$

Step 1.2

Combine.

$\frac{9 y (y + \frac{y - 3}{3})}{9 y (\frac{5}{9} + \frac{2}{3 y})}$

Step 2

Apply the distributive property.

$\frac{9 y \cdot y + 9 y \frac{y - 3}{3}}{9 y \frac{5}{9} + 9 y \frac{2}{3 y}}$

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 $9 y$ .

$\frac{9 y \cdot y + 3 (3 y) \frac{y - 3}{3}}{9 y \frac{5}{9} + 9 y \frac{2}{3 y}}$

Step 3.1.2

Cancel the common factor.

$\frac{9 y \cdot y + 3 (3 y) \frac{y - 3}{3}}{9 y \frac{5}{9} + 9 y \frac{2}{3 y}}$

Step 3.1.3

Rewrite the expression.

$\frac{9 y \cdot y + 3 y (y - 3)}{9 y \frac{5}{9} + 9 y \frac{2}{3 y}}$

Step 3.2

Cancel the common factor of $9$ .

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Step 3.2.1

Factor $9$ out of $9 y$ .

$\frac{9 y \cdot y + 3 y (y - 3)}{9 (y) \frac{5}{9} + 9 y \frac{2}{3 y}}$

Step 3.2.2

Cancel the common factor.

$\frac{9 y \cdot y + 3 y (y - 3)}{9 y \frac{5}{9} + 9 y \frac{2}{3 y}}$

Step 3.2.3

Rewrite the expression.

$\frac{9 y \cdot y + 3 y (y - 3)}{y \cdot 5 + 9 y \frac{2}{3 y}}$

Step 3.3

Cancel the common factor of $3 y$ .

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Step 3.3.1

Factor $3 y$ out of $9 y$ .

$\frac{9 y \cdot y + 3 y (y - 3)}{y \cdot 5 + 3 y (3) \frac{2}{3 y}}$

Step 3.3.2

Cancel the common factor.

$\frac{9 y \cdot y + 3 y (y - 3)}{y \cdot 5 + 3 y \cdot 3 \frac{2}{3 y}}$

Step 3.3.3

Rewrite the expression.

$\frac{9 y \cdot y + 3 y (y - 3)}{y \cdot 5 + 3 \cdot 2}$

Step 3.4

Multiply $3$ by $2$ .

$\frac{9 y \cdot y + 3 y (y - 3)}{y \cdot 5 + 6}$

Step 4

Simplify the numerator.

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Step 4.1

Multiply $y$ by $y$ by adding the exponents.

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Step 4.1.1

Move $y$ .

$\frac{9 (y \cdot y) + 3 y (y - 3)}{y \cdot 5 + 6}$

Step 4.1.2

Multiply $y$ by $y$ .

$\frac{9 y^{2} + 3 y (y - 3)}{y \cdot 5 + 6}$

Step 4.2

Apply the distributive property.

$\frac{9 y^{2} + 3 y \cdot y + 3 y \cdot - 3}{y \cdot 5 + 6}$

Step 4.3

Multiply $y$ by $y$ by adding the exponents.

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Step 4.3.1

Move $y$ .

$\frac{9 y^{2} + 3 (y \cdot y) + 3 y \cdot - 3}{y \cdot 5 + 6}$

Step 4.3.2

Multiply $y$ by $y$ .

$\frac{9 y^{2} + 3 y^{2} + 3 y \cdot - 3}{y \cdot 5 + 6}$

Step 4.4

Multiply $- 3$ by $3$ .

$\frac{9 y^{2} + 3 y^{2} - 9 y}{y \cdot 5 + 6}$

Step 4.5

Add $9 y^{2}$ and $3 y^{2}$ .

$\frac{12 y^{2} - 9 y}{y \cdot 5 + 6}$

Step 4.6

Factor $3 y$ out of $12 y^{2} - 9 y$ .

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Step 4.6.1

Factor $3 y$ out of $12 y^{2}$ .

$\frac{3 y (4 y) - 9 y}{y \cdot 5 + 6}$

Step 4.6.2

Factor $3 y$ out of $- 9 y$ .

$\frac{3 y (4 y) + 3 y (- 3)}{y \cdot 5 + 6}$

Step 4.6.3

Factor $3 y$ out of $3 y (4 y) + 3 y (- 3)$ .

$\frac{3 y (4 y - 3)}{y \cdot 5 + 6}$

Step 5

Move $5$ to the left of $y$ .

$\frac{3 y (4 y - 3)}{5 y + 6}$