Linear Algebra Examples

$\frac{1}{2} x + \frac{1}{3} = \frac{1}{6}$

Step 1

Combine $\frac{1}{2}$ and $x$ .

$\frac{x}{2} + \frac{1}{3} = \frac{1}{6}$

Step 2

Move all terms not containing $x$ to the right side of the equation.

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Step 2.1

Subtract $\frac{1}{3}$ from both sides of the equation.

$\frac{x}{2} = \frac{1}{6} - \frac{1}{3}$

Step 2.2

To write $- \frac{1}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{x}{2} = \frac{1}{6} - \frac{1}{3} \cdot \frac{2}{2}$

Step 2.3

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.3.1

Multiply $\frac{1}{3}$ by $\frac{2}{2}$ .

$\frac{x}{2} = \frac{1}{6} - \frac{2}{3 \cdot 2}$

Step 2.3.2

Multiply $3$ by $2$ .

$\frac{x}{2} = \frac{1}{6} - \frac{2}{6}$

Step 2.4

Combine the numerators over the common denominator.

$\frac{x}{2} = \frac{1 - 2}{6}$

Step 2.5

Subtract $2$ from $1$ .

$\frac{x}{2} = \frac{- 1}{6}$

Step 2.6

Move the negative in front of the fraction.

$\frac{x}{2} = - \frac{1}{6}$

Step 3

Multiply both sides of the equation by $2$ .

$2 \frac{x}{2} = 2 (- \frac{1}{6})$

Step 4

Simplify both sides of the equation.

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Step 4.1

Simplify the left side.

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Step 4.1.1

Cancel the common factor of $2$ .

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Step 4.1.1.1

Cancel the common factor.

$2 \frac{x}{2} = 2 (- \frac{1}{6})$

Step 4.1.1.2

Rewrite the expression.

$x = 2 (- \frac{1}{6})$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $2 (- \frac{1}{6})$ .

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Step 4.2.1.1

Cancel the common factor of $2$ .

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Step 4.2.1.1.1

Move the leading negative in $- \frac{1}{6}$ into the numerator.

$x = 2 (\frac{- 1}{6})$

Step 4.2.1.1.2

Factor $2$ out of $6$ .

$x = 2 \frac{- 1}{2 (3)}$

Step 4.2.1.1.3

Cancel the common factor.

$x = 2 \frac{- 1}{2 \cdot 3}$

Step 4.2.1.1.4

Rewrite the expression.

$x = \frac{- 1}{3}$

Step 4.2.1.2

Move the negative in front of the fraction.

$x = - \frac{1}{3}$

Step 5

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

${x | x = - \frac{1}{3}}$

Step 6