Pre-Algebra Examples

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

Step 1

Multiply the numerator and denominator of the fraction by $(y - 2) y$ .

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Step 1.1

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

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

Step 1.2

Combine.

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

Step 2

Apply the distributive property.

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

Step 3

Simplify by cancelling.

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Step 3.1

Cancel the common factor of $y - 2$ .

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Step 3.1.1

Factor $y - 2$ out of $(y - 2) y$ .

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

Step 3.1.2

Cancel the common factor.

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

Step 3.1.3

Rewrite the expression.

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

Step 3.2

Cancel the common factor of $y$ .

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Step 3.2.1

Move the leading negative in $- \frac{1}{y}$ into the numerator.

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

Step 3.2.2

Factor $y$ out of $(y - 2) y$ .

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

Step 3.2.3

Cancel the common factor.

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

Step 3.2.4

Rewrite the expression.

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

Step 3.3

Cancel the common factor of $y$ .

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Step 3.3.1

Factor $y$ out of $(y - 2) y$ .

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

Step 3.3.2

Cancel the common factor.

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

Step 3.3.3

Rewrite the expression.

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

Step 3.4

Cancel the common factor of $y - 2$ .

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Step 3.4.1

Factor $y - 2$ out of $(y - 2) y$ .

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

Step 3.4.2

Cancel the common factor.

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

Step 3.4.3

Rewrite the expression.

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

Step 4

Simplify the numerator.

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Step 4.1

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

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

Step 4.2

Apply the distributive property.

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

Step 4.3

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

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

Step 4.4

Multiply $- 2$ by $- 1$ .

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

Step 4.5

Rewrite $- 1 y$ as $- y$ .

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

Step 4.6

Subtract $y$ from $3 y$ .

$\frac{2 y + 2}{y - 2 + y}$

Step 4.7

Factor $2$ out of $2 y + 2$ .

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Step 4.7.1

Factor $2$ out of $2 y$ .

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

Step 4.7.2

Factor $2$ out of $2$ .

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

Step 4.7.3

Factor $2$ out of $2 (y) + 2 (1)$ .

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

Step 5

Simplify the denominator.

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Step 5.1

Add $y$ and $y$ .

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

Step 5.2

Factor $2$ out of $2 y - 2$ .

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Step 5.2.1

Factor $2$ out of $2 y$ .

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

Step 5.2.2

Factor $2$ out of $- 2$ .

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

Step 5.2.3

Factor $2$ out of $2 (y) + 2 (- 1)$ .

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

Step 6

Cancel the common factor of $2$ .

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Step 6.1

Cancel the common factor.

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

Step 6.2

Rewrite the expression.

$\frac{y + 1}{y - 1}$