Algebra Examples

$\frac{12 - y}{6 y - 36} - \frac{6}{y^{2} - 6 y}$

Step 1

Simplify each term.

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Step 1.1

Factor $6$ out of $6 y - 36$ .

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Step 1.1.1

Factor $6$ out of $6 y$ .

$\frac{12 - y}{6 (y) - 36} - \frac{6}{y^{2} - 6 y}$

Step 1.1.2

Factor $6$ out of $- 36$ .

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

Step 1.1.3

Factor $6$ out of $6 y + 6 \cdot - 6$ .

$\frac{12 - y}{6 (y - 6)} - \frac{6}{y^{2} - 6 y}$

Step 1.2

Factor $y$ out of $y^{2} - 6 y$ .

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Step 1.2.1

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

$\frac{12 - y}{6 (y - 6)} - \frac{6}{y \cdot y - 6 y}$

Step 1.2.2

Factor $y$ out of $- 6 y$ .

$\frac{12 - y}{6 (y - 6)} - \frac{6}{y \cdot y + y \cdot - 6}$

Step 1.2.3

Factor $y$ out of $y \cdot y + y \cdot - 6$ .

$\frac{12 - y}{6 (y - 6)} - \frac{6}{y (y - 6)}$

Step 2

To write $\frac{12 - y}{6 (y - 6)}$ as a fraction with a common denominator, multiply by $\frac{y}{y}$ .

$\frac{12 - y}{6 (y - 6)} \cdot \frac{y}{y} - \frac{6}{y (y - 6)}$

Step 3

To write $- \frac{6}{y (y - 6)}$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$\frac{12 - y}{6 (y - 6)} \cdot \frac{y}{y} - \frac{6}{y (y - 6)} \cdot \frac{6}{6}$

Step 4

Write each expression with a common denominator of $6 (y - 6) y$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

Multiply $\frac{12 - y}{6 (y - 6)}$ by $\frac{y}{y}$ .

$\frac{(12 - y) y}{6 (y - 6) y} - \frac{6}{y (y - 6)} \cdot \frac{6}{6}$

Step 4.2

Multiply $\frac{6}{y (y - 6)}$ by $\frac{6}{6}$ .

$\frac{(12 - y) y}{6 (y - 6) y} - \frac{6 \cdot 6}{y (y - 6) \cdot 6}$

Step 4.3

Reorder the factors of $6 (y - 6) y$ .

$\frac{(12 - y) y}{6 y (y - 6)} - \frac{6 \cdot 6}{y (y - 6) \cdot 6}$

Step 4.4

Reorder the factors of $y (y - 6) \cdot 6$ .

$\frac{(12 - y) y}{6 y (y - 6)} - \frac{6 \cdot 6}{6 y (y - 6)}$

Step 5

Combine the numerators over the common denominator.

$\frac{(12 - y) y - 6 \cdot 6}{6 y (y - 6)}$

Step 6

Simplify the numerator.

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Step 6.1

Apply the distributive property.

$\frac{12 y - y \cdot y - 6 \cdot 6}{6 y (y - 6)}$

Step 6.2

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

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Step 6.2.1

Move $y$ .

$\frac{12 y - (y \cdot y) - 6 \cdot 6}{6 y (y - 6)}$

Step 6.2.2

Multiply $y$ by $y$ .

$\frac{12 y - y^{2} - 6 \cdot 6}{6 y (y - 6)}$

Step 6.3

Multiply $- 6$ by $6$ .

$\frac{12 y - y^{2} - 36}{6 y (y - 6)}$

Step 6.4

Reorder terms.

$\frac{- y^{2} + 12 y - 36}{6 y (y - 6)}$

Step 6.5

Factor by grouping.

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Step 6.5.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = - 1 \cdot - 36 = 36$ and whose sum is $b = 12$ .

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Step 6.5.1.1

Factor $12$ out of $12 y$ .

$\frac{- y^{2} + 12 (y) - 36}{6 y (y - 6)}$

Step 6.5.1.2

Rewrite $12$ as $6$ plus $6$

$\frac{- y^{2} + (6 + 6) y - 36}{6 y (y - 6)}$

Step 6.5.1.3

Apply the distributive property.

$\frac{- y^{2} + 6 y + 6 y - 36}{6 y (y - 6)}$

Step 6.5.2

Factor out the greatest common factor from each group.

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Step 6.5.2.1

Group the first two terms and the last two terms.

$\frac{(- y^{2} + 6 y) + 6 y - 36}{6 y (y - 6)}$

Step 6.5.2.2

Factor out the greatest common factor (GCF) from each group.

$\frac{y (- y + 6) - 6 (- y + 6)}{6 y (y - 6)}$

Step 6.5.3

Factor the polynomial by factoring out the greatest common factor, $- y + 6$ .

$\frac{(- y + 6) (y - 6)}{6 y (y - 6)}$

Step 7

Simplify the numerator.

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Step 7.1

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

$\frac{(- (y) + 6) (y - 6)}{6 y (y - 6)}$

Step 7.2

Rewrite $6$ as $- 1 (- 6)$ .

$\frac{(- (y) - 1 (- 6)) (y - 6)}{6 y (y - 6)}$

Step 7.3

Factor $- 1$ out of $- (y) - 1 (- 6)$ .

$\frac{- (y - 6) (y - 6)}{6 y (y - 6)}$

Step 7.4

Rewrite $- (y - 6)$ as $- 1 (y - 6)$ .

$\frac{- 1 (y - 6) (y - 6)}{6 y (y - 6)}$

Step 7.5

Raise $y - 6$ to the power of $1$ .

$\frac{- 1 ({(y - 6)}^{1} (y - 6))}{6 y (y - 6)}$

Step 7.6

Raise $y - 6$ to the power of $1$ .

$\frac{- 1 ({(y - 6)}^{1} {(y - 6)}^{1})}{6 y (y - 6)}$

Step 7.7

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

$\frac{- 1 {(y - 6)}^{1 + 1}}{6 y (y - 6)}$

Step 7.8

Add $1$ and $1$ .

$\frac{- 1 {(y - 6)}^{2}}{6 y (y - 6)}$

Step 8

Reduce the expression by cancelling the common factors.

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Step 8.1

Cancel the common factor of ${(y - 6)}^{2}$ and $y - 6$ .

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Step 8.1.1

Factor $y - 6$ out of $- 1 {(y - 6)}^{2}$ .

$\frac{(y - 6) (- 1 (y - 6))}{6 y (y - 6)}$

Step 8.1.2

Cancel the common factors.

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Step 8.1.2.1

Factor $y - 6$ out of $6 y (y - 6)$ .

$\frac{(y - 6) (- 1 (y - 6))}{(y - 6) (6 y)}$

Step 8.1.2.2

Cancel the common factor.

$\frac{(y - 6) (- 1 (y - 6))}{(y - 6) (6 y)}$

Step 8.1.2.3

Rewrite the expression.

$\frac{- 1 (y - 6)}{6 y}$

Step 8.2

Move the negative in front of the fraction.

$- \frac{y - 6}{6 y}$