Linear Algebra Examples

$\frac{1}{3} x - \frac{1}{4} x + \frac{1}{12} x = 3$

Step 1

Simplify $\frac{1}{3} x - \frac{1}{4} x + \frac{1}{12} x$ .

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Step 1.1

Simplify each term.

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Step 1.1.1

Combine $\frac{1}{3}$ and $x$ .

$\frac{x}{3} - \frac{1}{4} x + \frac{1}{12} x = 3$

Step 1.1.2

Combine $x$ and $\frac{1}{4}$ .

$\frac{x}{3} - \frac{x}{4} + \frac{1}{12} x = 3$

Step 1.1.3

Combine $\frac{1}{12}$ and $x$ .

$\frac{x}{3} - \frac{x}{4} + \frac{x}{12} = 3$

Step 1.2

Find the common denominator.

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Step 1.2.1

Multiply $\frac{x}{3}$ by $\frac{4}{4}$ .

$\frac{x}{3} \cdot \frac{4}{4} - \frac{x}{4} + \frac{x}{12} = 3$

Step 1.2.2

Multiply $\frac{x}{3}$ by $\frac{4}{4}$ .

$\frac{x \cdot 4}{3 \cdot 4} - \frac{x}{4} + \frac{x}{12} = 3$

Step 1.2.3

Multiply $\frac{x}{4}$ by $\frac{3}{3}$ .

$\frac{x \cdot 4}{3 \cdot 4} - (\frac{x}{4} \cdot \frac{3}{3}) + \frac{x}{12} = 3$

Step 1.2.4

Multiply $\frac{x}{4}$ by $\frac{3}{3}$ .

$\frac{x \cdot 4}{3 \cdot 4} - \frac{x \cdot 3}{4 \cdot 3} + \frac{x}{12} = 3$

Step 1.2.5

Multiply $3$ by $4$ .

$\frac{x \cdot 4}{12} - \frac{x \cdot 3}{4 \cdot 3} + \frac{x}{12} = 3$

Step 1.2.6

Reorder the factors of $4 \cdot 3$ .

$\frac{x \cdot 4}{12} - \frac{x \cdot 3}{3 \cdot 4} + \frac{x}{12} = 3$

Step 1.2.7

Multiply $3$ by $4$ .

$\frac{x \cdot 4}{12} - \frac{x \cdot 3}{12} + \frac{x}{12} = 3$

Step 1.3

Combine the numerators over the common denominator.

$\frac{x \cdot 4 - x \cdot 3 + x}{12} = 3$

Step 1.4

Simplify each term.

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Step 1.4.1

Move $4$ to the left of $x$ .

$\frac{4 \cdot x - x \cdot 3 + x}{12} = 3$

Step 1.4.2

Multiply $3$ by $- 1$ .

$\frac{4 x - 3 x + x}{12} = 3$

Step 1.5

Simplify terms.

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Step 1.5.1

Subtract $3 x$ from $4 x$ .

$\frac{x + x}{12} = 3$

Step 1.5.2

Add $x$ and $x$ .

$\frac{2 x}{12} = 3$

Step 1.5.3

Cancel the common factor of $2$ and $12$ .

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Step 1.5.3.1

Factor $2$ out of $2 x$ .

$\frac{2 (x)}{12} = 3$

Step 1.5.3.2

Cancel the common factors.

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Step 1.5.3.2.1

Factor $2$ out of $12$ .

$\frac{2 x}{2 \cdot 6} = 3$

Step 1.5.3.2.2

Cancel the common factor.

$\frac{2 x}{2 \cdot 6} = 3$

Step 1.5.3.2.3

Rewrite the expression.

$\frac{x}{6} = 3$

Step 2

Multiply both sides of the equation by $6$ .

$6 \frac{x}{6} = 6 \cdot 3$

Step 3

Simplify both sides of the equation.

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Step 3.1

Simplify the left side.

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Step 3.1.1

Cancel the common factor of $6$ .

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Step 3.1.1.1

Cancel the common factor.

$6 \frac{x}{6} = 6 \cdot 3$

Step 3.1.1.2

Rewrite the expression.

$x = 6 \cdot 3$

Step 3.2

Simplify the right side.

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Step 3.2.1

Multiply $6$ by $3$ .

$x = 18$

Step 4

The domain is the set of all valid $x$ values.

${x | x = 18}$

Step 5