Basic Math Examples

$\frac{\frac{y + 5}{y} - \frac{2}{y - 1}}{\frac{y}{y - 1} + \frac{1}{y}}$

Step 1

Multiply the numerator and denominator of the fraction by $y (y - 1)$ .

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Step 1.1

Multiply $\frac{\frac{y + 5}{y} - \frac{2}{y - 1}}{\frac{y}{y - 1} + \frac{1}{y}}$ by $\frac{y (y - 1)}{y (y - 1)}$ .

$\frac{y (y - 1)}{y (y - 1)} \cdot \frac{\frac{y + 5}{y} - \frac{2}{y - 1}}{\frac{y}{y - 1} + \frac{1}{y}}$

Step 1.2

Combine.

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

Step 2

Apply the distributive property.

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

Step 3

Simplify by cancelling.

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Step 3.1

Cancel the common factor of $y$ .

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Step 3.1.1

Cancel the common factor.

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

Step 3.1.2

Rewrite the expression.

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

Step 3.2

Cancel the common factor of $y - 1$ .

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Step 3.2.1

Move the leading negative in $- \frac{2}{y - 1}$ into the numerator.

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

Step 3.2.2

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

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

Step 3.2.3

Cancel the common factor.

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

Step 3.2.4

Rewrite the expression.

$\frac{(y - 1) (y + 5) + y \cdot - 2}{y (y - 1) \frac{y}{y - 1} + y (y - 1) \frac{1}{y}}$

Step 3.3

Cancel the common factor of $y - 1$ .

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Step 3.3.1

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

$\frac{(y - 1) (y + 5) + y \cdot - 2}{(y - 1) y \frac{y}{y - 1} + y (y - 1) \frac{1}{y}}$

Step 3.3.2

Cancel the common factor.

$\frac{(y - 1) (y + 5) + y \cdot - 2}{(y - 1) y \frac{y}{y - 1} + y (y - 1) \frac{1}{y}}$

Step 3.3.3

Rewrite the expression.

$\frac{(y - 1) (y + 5) + y \cdot - 2}{y \cdot y + y (y - 1) \frac{1}{y}}$

Step 3.4

Raise $y$ to the power of $1$ .

$\frac{(y - 1) (y + 5) + y \cdot - 2}{y^{1} y + y (y - 1) \frac{1}{y}}$

Step 3.5

Raise $y$ to the power of $1$ .

$\frac{(y - 1) (y + 5) + y \cdot - 2}{y^{1} y^{1} + y (y - 1) \frac{1}{y}}$

Step 3.6

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{(y - 1) (y + 5) + y \cdot - 2}{y^{1 + 1} + y (y - 1) \frac{1}{y}}$

Step 3.7

Add $1$ and $1$ .

$\frac{(y - 1) (y + 5) + y \cdot - 2}{y^{2} + y (y - 1) \frac{1}{y}}$

Step 3.8

Cancel the common factor of $y$ .

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Step 3.8.1

Cancel the common factor.

$\frac{(y - 1) (y + 5) + y \cdot - 2}{y^{2} + y (y - 1) \frac{1}{y}}$

Step 3.8.2

Rewrite the expression.

$\frac{(y - 1) (y + 5) + y \cdot - 2}{y^{2} + y - 1}$

Step 4

Simplify the numerator.

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Step 4.1

Expand $(y - 1) (y + 5)$ using the FOIL Method.

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Step 4.1.1

Apply the distributive property.

$\frac{y (y + 5) - 1 (y + 5) + y \cdot - 2}{y^{2} + y - 1}$

Step 4.1.2

Apply the distributive property.

$\frac{y \cdot y + y \cdot 5 - 1 (y + 5) + y \cdot - 2}{y^{2} + y - 1}$

Step 4.1.3

Apply the distributive property.

$\frac{y \cdot y + y \cdot 5 - 1 y - 1 \cdot 5 + y \cdot - 2}{y^{2} + y - 1}$

Step 4.2

Simplify and combine like terms.

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Step 4.2.1

Simplify each term.

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Step 4.2.1.1

Multiply $y$ by $y$ .

$\frac{y^{2} + y \cdot 5 - 1 y - 1 \cdot 5 + y \cdot - 2}{y^{2} + y - 1}$

Step 4.2.1.2

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

$\frac{y^{2} + 5 \cdot y - 1 y - 1 \cdot 5 + y \cdot - 2}{y^{2} + y - 1}$

Step 4.2.1.3

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

$\frac{y^{2} + 5 y - y - 1 \cdot 5 + y \cdot - 2}{y^{2} + y - 1}$

Step 4.2.1.4

Multiply $- 1$ by $5$ .

$\frac{y^{2} + 5 y - y - 5 + y \cdot - 2}{y^{2} + y - 1}$

Step 4.2.2

Subtract $y$ from $5 y$ .

$\frac{y^{2} + 4 y - 5 + y \cdot - 2}{y^{2} + y - 1}$

Step 4.3

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

$\frac{y^{2} + 4 y - 5 - 2 \cdot y}{y^{2} + y - 1}$

Step 4.4

Subtract $2 y$ from $4 y$ .

$\frac{y^{2} + 2 y - 5}{y^{2} + y - 1}$