Algebra Examples

$\frac{y^{2} - 49}{{(y - 7)}^{2}} \div \frac{5 y + 35}{y^{2} - 7 y}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{y^{2} - 49}{{(y - 7)}^{2}} \cdot \frac{y^{2} - 7 y}{5 y + 35}$

Step 2

Simplify the numerator.

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Step 2.1

Rewrite $49$ as $7^{2}$ .

$\frac{y^{2} - 7^{2}}{{(y - 7)}^{2}} \cdot \frac{y^{2} - 7 y}{5 y + 35}$

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 $b = 7$ .

$\frac{(y + 7) (y - 7)}{{(y - 7)}^{2}} \cdot \frac{y^{2} - 7 y}{5 y + 35}$

Step 3

Simplify terms.

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Step 3.1

Cancel the common factor of $y - 7$ and ${(y - 7)}^{2}$ .

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Step 3.1.1

Factor $y - 7$ out of $(y + 7) (y - 7)$ .

$\frac{(y - 7) (y + 7)}{{(y - 7)}^{2}} \cdot \frac{y^{2} - 7 y}{5 y + 35}$

Step 3.1.2

Cancel the common factors.

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Step 3.1.2.1

Factor $y - 7$ out of ${(y - 7)}^{2}$ .

$\frac{(y - 7) (y + 7)}{(y - 7) (y - 7)} \cdot \frac{y^{2} - 7 y}{5 y + 35}$

Step 3.1.2.2

Cancel the common factor.

$\frac{(y - 7) (y + 7)}{(y - 7) (y - 7)} \cdot \frac{y^{2} - 7 y}{5 y + 35}$

Step 3.1.2.3

Rewrite the expression.

$\frac{y + 7}{y - 7} \cdot \frac{y^{2} - 7 y}{5 y + 35}$

Step 3.2

Factor $y$ out of $y^{2} - 7 y$ .

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Step 3.2.1

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

$\frac{y + 7}{y - 7} \cdot \frac{y \cdot y - 7 y}{5 y + 35}$

Step 3.2.2

Factor $y$ out of $- 7 y$ .

$\frac{y + 7}{y - 7} \cdot \frac{y \cdot y + y \cdot - 7}{5 y + 35}$

Step 3.2.3

Factor $y$ out of $y \cdot y + y \cdot - 7$ .

$\frac{y + 7}{y - 7} \cdot \frac{y (y - 7)}{5 y + 35}$

Step 3.3

Factor $5$ out of $5 y + 35$ .

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Step 3.3.1

Factor $5$ out of $5 y$ .

$\frac{y + 7}{y - 7} \cdot \frac{y (y - 7)}{5 (y) + 35}$

Step 3.3.2

Factor $5$ out of $35$ .

$\frac{y + 7}{y - 7} \cdot \frac{y (y - 7)}{5 y + 5 \cdot 7}$

Step 3.3.3

Factor $5$ out of $5 y + 5 \cdot 7$ .

$\frac{y + 7}{y - 7} \cdot \frac{y (y - 7)}{5 (y + 7)}$

Step 3.4

Cancel the common factor of $y + 7$ .

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Step 3.4.1

Factor $y + 7$ out of $5 (y + 7)$ .

$\frac{y + 7}{y - 7} \cdot \frac{y (y - 7)}{(y + 7) \cdot 5}$

Step 3.4.2

Cancel the common factor.

$\frac{y + 7}{y - 7} \cdot \frac{y (y - 7)}{(y + 7) \cdot 5}$

Step 3.4.3

Rewrite the expression.

$\frac{1}{y - 7} \cdot \frac{y (y - 7)}{5}$

Step 3.5

Cancel the common factor of $y - 7$ .

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Step 3.5.1

Factor $y - 7$ out of $y (y - 7)$ .

$\frac{1}{y - 7} \cdot \frac{(y - 7) y}{5}$

Step 3.5.2

Cancel the common factor.

$\frac{1}{y - 7} \cdot \frac{(y - 7) y}{5}$

Step 3.5.3

Rewrite the expression.

$\frac{y}{5}$