Linear Algebra Examples

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

Step 1

Simplify each term.

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Step 1.1

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

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

Step 1.2

Combine $y$ and $\frac{1}{6}$ .

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

Step 2

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

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

Step 3

Multiply both sides of the equation by $- 6$ .

$- 6 (- \frac{y}{6}) = - 6 (- \frac{2}{3} - \frac{x}{3})$

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

Simplify $- 6 (- \frac{y}{6})$ .

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Step 4.1.1.1

Cancel the common factor of $6$ .

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Step 4.1.1.1.1

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

$- 6 \frac{- y}{6} = - 6 (- \frac{2}{3} - \frac{x}{3})$

Step 4.1.1.1.2

Factor $6$ out of $- 6$ .

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

Step 4.1.1.1.3

Cancel the common factor.

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

Step 4.1.1.1.4

Rewrite the expression.

$- - y = - 6 (- \frac{2}{3} - \frac{x}{3})$

Step 4.1.1.2

Multiply.

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Step 4.1.1.2.1

Multiply $- 1$ by $- 1$ .

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

Step 4.1.1.2.2

Multiply $y$ by $1$ .

$y = - 6 (- \frac{2}{3} - \frac{x}{3})$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $- 6 (- \frac{2}{3} - \frac{x}{3})$ .

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Step 4.2.1.1

Apply the distributive property.

$y = - 6 (- \frac{2}{3}) - 6 (- \frac{x}{3})$

Step 4.2.1.2

Cancel the common factor of $3$ .

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Step 4.2.1.2.1

Move the leading negative in $- \frac{2}{3}$ into the numerator.

$y = - 6 (\frac{- 2}{3}) - 6 (- \frac{x}{3})$

Step 4.2.1.2.2

Factor $3$ out of $- 6$ .

$y = 3 (- 2) \frac{- 2}{3} - 6 (- \frac{x}{3})$

Step 4.2.1.2.3

Cancel the common factor.

$y = 3 \cdot - 2 \frac{- 2}{3} - 6 (- \frac{x}{3})$

Step 4.2.1.2.4

Rewrite the expression.

$y = - 2 \cdot - 2 - 6 (- \frac{x}{3})$

Step 4.2.1.3

Multiply $- 2$ by $- 2$ .

$y = 4 - 6 (- \frac{x}{3})$

Step 4.2.1.4

Cancel the common factor of $3$ .

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Step 4.2.1.4.1

Move the leading negative in $- \frac{x}{3}$ into the numerator.

$y = 4 - 6 \frac{- x}{3}$

Step 4.2.1.4.2

Factor $3$ out of $- 6$ .

$y = 4 + 3 (- 2) \frac{- x}{3}$

Step 4.2.1.4.3

Cancel the common factor.

$y = 4 + 3 \cdot - 2 \frac{- x}{3}$

Step 4.2.1.4.4

Rewrite the expression.

$y = 4 - 2 (- x)$

Step 4.2.1.5

Multiply $- 1$ by $- 2$ .

$y = 4 + 2 x$

Step 5

Reorder $4$ and $2 x$ .

$y = 2 x + 4$

Step 6

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${x | x \in ℝ}$

Step 7