Algebra Examples

$\frac{x + 2}{x} + \frac{x - 6}{3 x} = \frac{2 x + 9}{5 x}$

Step 1

Subtract $\frac{2 x + 9}{5 x}$ from both sides of the equation.

$\frac{x + 2}{x} + \frac{x - 6}{3 x} - \frac{2 x + 9}{5 x} = 0$

Step 2

Simplify $\frac{x + 2}{x} + \frac{x - 6}{3 x} - \frac{2 x + 9}{5 x}$ .

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Step 2.1

Find the common denominator.

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Step 2.1.1

Multiply $\frac{x + 2}{x}$ by $\frac{15}{15}$ .

$\frac{x + 2}{x} \cdot \frac{15}{15} + \frac{x - 6}{3 x} - \frac{2 x + 9}{5 x} = 0$

Step 2.1.2

Multiply $\frac{x + 2}{x}$ by $\frac{15}{15}$ .

$\frac{(x + 2) \cdot 15}{x \cdot 15} + \frac{x - 6}{3 x} - \frac{2 x + 9}{5 x} = 0$

Step 2.1.3

Multiply $\frac{x - 6}{3 x}$ by $\frac{5}{5}$ .

$\frac{(x + 2) \cdot 15}{x \cdot 15} + \frac{x - 6}{3 x} \cdot \frac{5}{5} - \frac{2 x + 9}{5 x} = 0$

Step 2.1.4

Multiply $\frac{x - 6}{3 x}$ by $\frac{5}{5}$ .

$\frac{(x + 2) \cdot 15}{x \cdot 15} + \frac{(x - 6) \cdot 5}{3 x \cdot 5} - \frac{2 x + 9}{5 x} = 0$

Step 2.1.5

Multiply $\frac{2 x + 9}{5 x}$ by $\frac{3}{3}$ .

$\frac{(x + 2) \cdot 15}{x \cdot 15} + \frac{(x - 6) \cdot 5}{3 x \cdot 5} - (\frac{2 x + 9}{5 x} \cdot \frac{3}{3}) = 0$

Step 2.1.6

Multiply $\frac{2 x + 9}{5 x}$ by $\frac{3}{3}$ .

$\frac{(x + 2) \cdot 15}{x \cdot 15} + \frac{(x - 6) \cdot 5}{3 x \cdot 5} - \frac{(2 x + 9) \cdot 3}{5 x \cdot 3} = 0$

Step 2.1.7

Reorder the factors of $3 x \cdot 5$ .

$\frac{(x + 2) \cdot 15}{x \cdot 15} + \frac{(x - 6) \cdot 5}{3 \cdot 5 x} - \frac{(2 x + 9) \cdot 3}{5 x \cdot 3} = 0$

Step 2.1.8

Multiply $3$ by $5$ .

$\frac{(x + 2) \cdot 15}{x \cdot 15} + \frac{(x - 6) \cdot 5}{15 x} - \frac{(2 x + 9) \cdot 3}{5 x \cdot 3} = 0$

Step 2.1.9

Reorder the factors of $5 x \cdot 3$ .

$\frac{(x + 2) \cdot 15}{x \cdot 15} + \frac{(x - 6) \cdot 5}{15 x} - \frac{(2 x + 9) \cdot 3}{3 \cdot 5 x} = 0$

Step 2.1.10

Multiply $3$ by $5$ .

$\frac{(x + 2) \cdot 15}{x \cdot 15} + \frac{(x - 6) \cdot 5}{15 x} - \frac{(2 x + 9) \cdot 3}{15 x} = 0$

Step 2.2

Combine the numerators over the common denominator.

$\frac{(x + 2) \cdot 15 + (x - 6) \cdot 5 - (2 x + 9) \cdot 3}{15 x} = 0$

Step 2.3

Simplify each term.

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Step 2.3.1

Apply the distributive property.

$\frac{x \cdot 15 + 2 \cdot 15 + (x - 6) \cdot 5 - (2 x + 9) \cdot 3}{15 x} = 0$

Step 2.3.2

Move $15$ to the left of $x$ .

$\frac{15 \cdot x + 2 \cdot 15 + (x - 6) \cdot 5 - (2 x + 9) \cdot 3}{15 x} = 0$

Step 2.3.3

Multiply $2$ by $15$ .

$\frac{15 \cdot x + 30 + (x - 6) \cdot 5 - (2 x + 9) \cdot 3}{15 x} = 0$

Step 2.3.4

Apply the distributive property.

$\frac{15 x + 30 + x \cdot 5 - 6 \cdot 5 - (2 x + 9) \cdot 3}{15 x} = 0$

Step 2.3.5

Move $5$ to the left of $x$ .

$\frac{15 x + 30 + 5 \cdot x - 6 \cdot 5 - (2 x + 9) \cdot 3}{15 x} = 0$

Step 2.3.6

Multiply $- 6$ by $5$ .

$\frac{15 x + 30 + 5 \cdot x - 30 - (2 x + 9) \cdot 3}{15 x} = 0$

Step 2.3.7

Apply the distributive property.

$\frac{15 x + 30 + 5 x - 30 + (- (2 x) - 1 \cdot 9) \cdot 3}{15 x} = 0$

Step 2.3.8

Multiply $2$ by $- 1$ .

$\frac{15 x + 30 + 5 x - 30 + (- 2 x - 1 \cdot 9) \cdot 3}{15 x} = 0$

Step 2.3.9

Multiply $- 1$ by $9$ .

$\frac{15 x + 30 + 5 x - 30 + (- 2 x - 9) \cdot 3}{15 x} = 0$

Step 2.3.10

Apply the distributive property.

$\frac{15 x + 30 + 5 x - 30 - 2 x \cdot 3 - 9 \cdot 3}{15 x} = 0$

Step 2.3.11

Multiply $3$ by $- 2$ .

$\frac{15 x + 30 + 5 x - 30 - 6 x - 9 \cdot 3}{15 x} = 0$

Step 2.3.12

Multiply $- 9$ by $3$ .

$\frac{15 x + 30 + 5 x - 30 - 6 x - 27}{15 x} = 0$

Step 2.4

Combine the opposite terms in $15 x + 30 + 5 x - 30 - 6 x - 27$ .

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Step 2.4.1

Subtract $30$ from $30$ .

$\frac{15 x + 5 x + 0 - 6 x - 27}{15 x} = 0$

Step 2.4.2

Add $15 x + 5 x$ and $0$ .

$\frac{15 x + 5 x - 6 x - 27}{15 x} = 0$

Step 2.5

Add $15 x$ and $5 x$ .

$\frac{20 x - 6 x - 27}{15 x} = 0$

Step 2.6

Subtract $6 x$ from $20 x$ .

$\frac{14 x - 27}{15 x} = 0$

Step 3

Set the numerator equal to zero.

$14 x - 27 = 0$

Step 4

Solve the equation for $x$ .

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Step 4.1

Add $27$ to both sides of the equation.

$14 x = 27$

Step 4.2

Divide each term in $14 x = 27$ by $14$ and simplify.

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Step 4.2.1

Divide each term in $14 x = 27$ by $14$ .

$\frac{14 x}{14} = \frac{27}{14}$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Cancel the common factor of $14$ .

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Step 4.2.2.1.1

Cancel the common factor.

$\frac{14 x}{14} = \frac{27}{14}$

Step 4.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{27}{14}$