Algebra Examples

$\frac{6 b + 18}{b^{2}} + \frac{1}{b} = \frac{3}{b}$

Step 1

Subtract $\frac{3}{b}$ from both sides of the equation.

$\frac{6 b + 18}{b^{2}} + \frac{1}{b} - \frac{3}{b} = 0$

Step 2

Simplify $\frac{6 b + 18}{b^{2}} + \frac{1}{b} - \frac{3}{b}$ .

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Step 2.1

Combine the numerators over the common denominator.

$\frac{6 b + 18}{b^{2}} + \frac{1 - 3}{b} = 0$

Step 2.2

Subtract $3$ from $1$ .

$\frac{6 b + 18}{b^{2}} + \frac{- 2}{b} = 0$

Step 2.3

Simplify each term.

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Step 2.3.1

Factor $6$ out of $6 b + 18$ .

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Step 2.3.1.1

Factor $6$ out of $6 b$ .

$\frac{6 (b) + 18}{b^{2}} + \frac{- 2}{b} = 0$

Step 2.3.1.2

Factor $6$ out of $18$ .

$\frac{6 b + 6 \cdot 3}{b^{2}} + \frac{- 2}{b} = 0$

Step 2.3.1.3

Factor $6$ out of $6 b + 6 \cdot 3$ .

$\frac{6 (b + 3)}{b^{2}} + \frac{- 2}{b} = 0$

Step 2.3.2

Move the negative in front of the fraction.

$\frac{6 (b + 3)}{b^{2}} - \frac{2}{b} = 0$

Step 2.4

To write $- \frac{2}{b}$ as a fraction with a common denominator, multiply by $\frac{b}{b}$ .

$\frac{6 (b + 3)}{b^{2}} - \frac{2}{b} \cdot \frac{b}{b} = 0$

Step 2.5

Write each expression with a common denominator of $b^{2}$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.5.1

Multiply $\frac{2}{b}$ by $\frac{b}{b}$ .

$\frac{6 (b + 3)}{b^{2}} - \frac{2 b}{b \cdot b} = 0$

Step 2.5.2

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

$\frac{6 (b + 3)}{b^{2}} - \frac{2 b}{b^{1} b} = 0$

Step 2.5.3

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

$\frac{6 (b + 3)}{b^{2}} - \frac{2 b}{b^{1} b^{1}} = 0$

Step 2.5.4

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

$\frac{6 (b + 3)}{b^{2}} - \frac{2 b}{b^{1 + 1}} = 0$

Step 2.5.5

Add $1$ and $1$ .

$\frac{6 (b + 3)}{b^{2}} - \frac{2 b}{b^{2}} = 0$

Step 2.6

Combine the numerators over the common denominator.

$\frac{6 (b + 3) - 2 b}{b^{2}} = 0$

Step 2.7

Simplify the numerator.

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Step 2.7.1

Factor $2$ out of $6 (b + 3) - 2 b$ .

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Step 2.7.1.1

Factor $2$ out of $6 (b + 3)$ .

$\frac{2 (3 (b + 3)) - 2 b}{b^{2}} = 0$

Step 2.7.1.2

Factor $2$ out of $- 2 b$ .

$\frac{2 (3 (b + 3)) + 2 (- b)}{b^{2}} = 0$

Step 2.7.1.3

Factor $2$ out of $2 (3 (b + 3)) + 2 (- b)$ .

$\frac{2 (3 (b + 3) - b)}{b^{2}} = 0$

Step 2.7.2

Apply the distributive property.

$\frac{2 (3 b + 3 \cdot 3 - b)}{b^{2}} = 0$

Step 2.7.3

Multiply $3$ by $3$ .

$\frac{2 (3 b + 9 - b)}{b^{2}} = 0$

Step 2.7.4

Subtract $b$ from $3 b$ .

$\frac{2 (2 b + 9)}{b^{2}} = 0$

Step 3

Set the numerator equal to zero.

$2 (2 b + 9) = 0$

Step 4

Solve the equation for $b$ .

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Step 4.1

Divide each term in $2 (2 b + 9) = 0$ by $2$ and simplify.

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Step 4.1.1

Divide each term in $2 (2 b + 9) = 0$ by $2$ .

$\frac{2 (2 b + 9)}{2} = \frac{0}{2}$

Step 4.1.2

Simplify the left side.

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Step 4.1.2.1

Cancel the common factor of $2$ .

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Step 4.1.2.1.1

Cancel the common factor.

$\frac{2 (2 b + 9)}{2} = \frac{0}{2}$

Step 4.1.2.1.2

Divide $2 b + 9$ by $1$ .

$2 b + 9 = \frac{0}{2}$

Step 4.1.3

Simplify the right side.

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Step 4.1.3.1

Divide $0$ by $2$ .

$2 b + 9 = 0$

Step 4.2

Subtract $9$ from both sides of the equation.

$2 b = - 9$

Step 4.3

Divide each term in $2 b = - 9$ by $2$ and simplify.

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Step 4.3.1

Divide each term in $2 b = - 9$ by $2$ .

$\frac{2 b}{2} = \frac{- 9}{2}$

Step 4.3.2

Simplify the left side.

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Step 4.3.2.1

Cancel the common factor of $2$ .

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Step 4.3.2.1.1

Cancel the common factor.

$\frac{2 b}{2} = \frac{- 9}{2}$

Step 4.3.2.1.2

Divide $b$ by $1$ .

$b = \frac{- 9}{2}$

Step 4.3.3

Simplify the right side.

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Step 4.3.3.1

Move the negative in front of the fraction.

$b = - \frac{9}{2}$