Algebra Examples

$\frac{5 y}{y^{2} - 7 y} - \frac{4}{2 y - 14} + \frac{9}{y}$

Step 1

Simplify each term.

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Step 1.1

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

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Step 1.1.1

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

$\frac{5 y}{y \cdot y - 7 y} - \frac{4}{2 y - 14} + \frac{9}{y}$

Step 1.1.2

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

$\frac{5 y}{y \cdot y + y \cdot - 7} - \frac{4}{2 y - 14} + \frac{9}{y}$

Step 1.1.3

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

$\frac{5 y}{y (y - 7)} - \frac{4}{2 y - 14} + \frac{9}{y}$

Step 1.2

Cancel the common factor of $y$ .

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Step 1.2.1

Cancel the common factor.

$\frac{5 y}{y (y - 7)} - \frac{4}{2 y - 14} + \frac{9}{y}$

Step 1.2.2

Rewrite the expression.

$\frac{5}{y - 7} - \frac{4}{2 y - 14} + \frac{9}{y}$

Step 1.3

Cancel the common factor of $4$ and $2 y - 14$ .

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Step 1.3.1

Factor $2$ out of $4$ .

$\frac{5}{y - 7} - \frac{2 \cdot 2}{2 y - 14} + \frac{9}{y}$

Step 1.3.2

Cancel the common factors.

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Step 1.3.2.1

Factor $2$ out of $2 y$ .

$\frac{5}{y - 7} - \frac{2 \cdot 2}{2 (y) - 14} + \frac{9}{y}$

Step 1.3.2.2

Factor $2$ out of $- 14$ .

$\frac{5}{y - 7} - \frac{2 \cdot 2}{2 (y) + 2 (- 7)} + \frac{9}{y}$

Step 1.3.2.3

Factor $2$ out of $2 (y) + 2 (- 7)$ .

$\frac{5}{y - 7} - \frac{2 \cdot 2}{2 (y - 7)} + \frac{9}{y}$

Step 1.3.2.4

Cancel the common factor.

$\frac{5}{y - 7} - \frac{2 \cdot 2}{2 (y - 7)} + \frac{9}{y}$

Step 1.3.2.5

Rewrite the expression.

$\frac{5}{y - 7} - \frac{2}{y - 7} + \frac{9}{y}$

Step 2

Combine fractions.

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Step 2.1

Combine the numerators over the common denominator.

$\frac{5 - 2}{y - 7} + \frac{9}{y}$

Step 2.2

Subtract $2$ from $5$ .

$\frac{3}{y - 7} + \frac{9}{y}$

Step 3

To write $\frac{3}{y - 7}$ as a fraction with a common denominator, multiply by $\frac{y}{y}$ .

$\frac{3}{y - 7} \cdot \frac{y}{y} + \frac{9}{y}$

Step 4

To write $\frac{9}{y}$ as a fraction with a common denominator, multiply by $\frac{y - 7}{y - 7}$ .

$\frac{3}{y - 7} \cdot \frac{y}{y} + \frac{9}{y} \cdot \frac{y - 7}{y - 7}$

Step 5

Write each expression with a common denominator of $(y - 7) y$ , by multiplying each by an appropriate factor of $1$ .

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Step 5.1

Multiply $\frac{3}{y - 7}$ by $\frac{y}{y}$ .

$\frac{3 y}{(y - 7) y} + \frac{9}{y} \cdot \frac{y - 7}{y - 7}$

Step 5.2

Multiply $\frac{9}{y}$ by $\frac{y - 7}{y - 7}$ .

$\frac{3 y}{(y - 7) y} + \frac{9 (y - 7)}{y (y - 7)}$

Step 5.3

Reorder the factors of $(y - 7) y$ .

$\frac{3 y}{y (y - 7)} + \frac{9 (y - 7)}{y (y - 7)}$

Step 6

Combine the numerators over the common denominator.

$\frac{3 y + 9 (y - 7)}{y (y - 7)}$

Step 7

Simplify the numerator.

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Step 7.1

Factor $3$ out of $3 y + 9 (y - 7)$ .

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Step 7.1.1

Factor $3$ out of $3 y$ .

$\frac{3 (y) + 9 (y - 7)}{y (y - 7)}$

Step 7.1.2

Factor $3$ out of $9 (y - 7)$ .

$\frac{3 (y) + 3 (3 (y - 7))}{y (y - 7)}$

Step 7.1.3

Factor $3$ out of $3 (y) + 3 (3 (y - 7))$ .

$\frac{3 (y + 3 (y - 7))}{y (y - 7)}$

Step 7.2

Apply the distributive property.

$\frac{3 (y + 3 y + 3 \cdot - 7)}{y (y - 7)}$

Step 7.3

Multiply $3$ by $- 7$ .

$\frac{3 (y + 3 y - 21)}{y (y - 7)}$

Step 7.4

Add $y$ and $3 y$ .

$\frac{3 (4 y - 21)}{y (y - 7)}$