Linear Algebra Examples

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

Step 1

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

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

Step 2

Multiply both sides of the equation by $- 3$ .

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

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

Simplify $- 3 (- \frac{y}{3})$ .

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Step 3.1.1.1

Cancel the common factor of $3$ .

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Step 3.1.1.1.1

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

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

Step 3.1.1.1.2

Factor $3$ out of $- 3$ .

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

Step 3.1.1.1.3

Cancel the common factor.

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

Step 3.1.1.1.4

Rewrite the expression.

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

Step 3.1.1.2

Multiply.

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Step 3.1.1.2.1

Multiply $- 1$ by $- 1$ .

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

Step 3.1.1.2.2

Multiply $y$ by $1$ .

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

Step 3.2

Simplify the right side.

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Step 3.2.1

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

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Step 3.2.1.1

Apply the distributive property.

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

Step 3.2.1.2

Cancel the common factor of $3$ .

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Step 3.2.1.2.1

Factor $3$ out of $- 3$ .

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

Step 3.2.1.2.2

Cancel the common factor.

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

Step 3.2.1.2.3

Rewrite the expression.

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

Step 3.2.1.3

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

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Step 3.2.1.3.1

Multiply $- 1$ by $- 3$ .

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

Step 3.2.1.3.2

Combine $3$ and $\frac{x}{2}$ .

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

Step 4

Reorder $- 1$ and $\frac{3 x}{2}$ .

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

Step 5

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 6