Finite Math Examples

$\frac{12 y + 3}{y^{2} - 16} \cdot \frac{4 - y}{4 y^{2} + y}$

Step 1

Factor $3$ out of $12 y + 3$ .

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Step 1.1

Factor $3$ out of $12 y$ .

$\frac{3 (4 y) + 3}{y^{2} - 16} \cdot \frac{4 - y}{4 y^{2} + y}$

Step 1.2

Factor $3$ out of $3$ .

$\frac{3 (4 y) + 3 (1)}{y^{2} - 16} \cdot \frac{4 - y}{4 y^{2} + y}$

Step 1.3

Factor $3$ out of $3 (4 y) + 3 (1)$ .

$\frac{3 (4 y + 1)}{y^{2} - 16} \cdot \frac{4 - y}{4 y^{2} + y}$

Step 2

Simplify the denominator.

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Step 2.1

Rewrite $16$ as $4^{2}$ .

$\frac{3 (4 y + 1)}{y^{2} - 4^{2}} \cdot \frac{4 - y}{4 y^{2} + y}$

Step 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 $b = 4$ .

$\frac{3 (4 y + 1)}{(y + 4) (y - 4)} \cdot \frac{4 - y}{4 y^{2} + y}$

Step 3

Simplify terms.

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Step 3.1

Factor $y$ out of $4 y^{2} + y$ .

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Step 3.1.1

Factor $y$ out of $4 y^{2}$ .

$\frac{3 (4 y + 1)}{(y + 4) (y - 4)} \cdot \frac{4 - y}{y (4 y) + y}$

Step 3.1.2

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

$\frac{3 (4 y + 1)}{(y + 4) (y - 4)} \cdot \frac{4 - y}{y (4 y) + y^{1}}$

Step 3.1.3

Factor $y$ out of $y^{1}$ .

$\frac{3 (4 y + 1)}{(y + 4) (y - 4)} \cdot \frac{4 - y}{y (4 y) + y \cdot 1}$

Step 3.1.4

Factor $y$ out of $y (4 y) + y \cdot 1$ .

$\frac{3 (4 y + 1)}{(y + 4) (y - 4)} \cdot \frac{4 - y}{y (4 y + 1)}$

Step 3.2

Cancel the common factor of $4 y + 1$ .

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Step 3.2.1

Factor $4 y + 1$ out of $3 (4 y + 1)$ .

$\frac{(4 y + 1) \cdot 3}{(y + 4) (y - 4)} \cdot \frac{4 - y}{y (4 y + 1)}$

Step 3.2.2

Factor $4 y + 1$ out of $y (4 y + 1)$ .

$\frac{(4 y + 1) \cdot 3}{(y + 4) (y - 4)} \cdot \frac{4 - y}{(4 y + 1) y}$

Step 3.2.3

Cancel the common factor.

$\frac{(4 y + 1) \cdot 3}{(y + 4) (y - 4)} \cdot \frac{4 - y}{(4 y + 1) y}$

Step 3.2.4

Rewrite the expression.

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

Step 3.3

Multiply $\frac{3}{(y + 4) (y - 4)}$ by $\frac{4 - y}{y}$ .

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

Step 3.4

Cancel the common factor of $4 - y$ and $y - 4$ .

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Step 3.4.1

Rewrite $4$ as $- 1 (- 4)$ .

$\frac{3 (- 1 (- 4) - y)}{(y + 4) (y - 4) y}$

Step 3.4.2

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

$\frac{3 (- 1 (- 4) - (y))}{(y + 4) (y - 4) y}$

Step 3.4.3

Factor $- 1$ out of $- 1 (- 4) - (y)$ .

$\frac{3 (- 1 (- 4 + y))}{(y + 4) (y - 4) y}$

Step 3.4.4

Reorder terms.

$\frac{3 (- 1 (y - 4))}{(y + 4) (y - 4) y}$

Step 3.4.5

Cancel the common factor.

$\frac{3 (- 1 (y - 4))}{(y + 4) (y - 4) y}$

Step 3.4.6

Rewrite the expression.

$\frac{3 \cdot (- 1)}{(y + 4) y}$

Step 4

Multiply $3$ by $- 1$ .

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

Step 5

Simplify the expression.

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Step 5.1

Move the negative in front of the fraction.

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

Step 5.2

Reorder factors in $- \frac{3}{(y + 4) y}$ .

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