Linear Algebra Examples

$\frac{2}{3} \cdot (F - G) = D$

Step 1

Simplify $\frac{2}{3} \cdot (F - G)$ .

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Step 1.1

Apply the distributive property.

$\frac{2}{3} F + \frac{2}{3} (- G) = D$

Step 1.2

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

$\frac{2 F}{3} + \frac{2}{3} (- G) = D$

Step 1.3

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

$\frac{2 F}{3} - \frac{2 G}{3} = D$

Step 2

Add $\frac{2 G}{3}$ to both sides of the equation.

$\frac{2 F}{3} = D + \frac{2 G}{3}$

Step 3

Multiply both sides of the equation by $\frac{3}{2}$ .

$\frac{3}{2} \cdot \frac{2 F}{3} = \frac{3}{2} (D + \frac{2 G}{3})$

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 $\frac{3}{2} \cdot \frac{2 F}{3}$ .

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Step 4.1.1.1

Cancel the common factor of $3$ .

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Step 4.1.1.1.1

Cancel the common factor.

$\frac{3}{2} \cdot \frac{2 F}{3} = \frac{3}{2} (D + \frac{2 G}{3})$

Step 4.1.1.1.2

Rewrite the expression.

$\frac{1}{2} (2 F) = \frac{3}{2} (D + \frac{2 G}{3})$

Step 4.1.1.2

Cancel the common factor of $2$ .

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Step 4.1.1.2.1

Factor $2$ out of $2 F$ .

$\frac{1}{2} (2 (F)) = \frac{3}{2} (D + \frac{2 G}{3})$

Step 4.1.1.2.2

Cancel the common factor.

$\frac{1}{2} (2 F) = \frac{3}{2} (D + \frac{2 G}{3})$

Step 4.1.1.2.3

Rewrite the expression.

$F = \frac{3}{2} (D + \frac{2 G}{3})$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $\frac{3}{2} (D + \frac{2 G}{3})$ .

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Step 4.2.1.1

Apply the distributive property.

$F = \frac{3}{2} D + \frac{3}{2} \cdot \frac{2 G}{3}$

Step 4.2.1.2

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

$F = \frac{3 D}{2} + \frac{3}{2} \cdot \frac{2 G}{3}$

Step 4.2.1.3

Cancel the common factor of $3$ .

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Step 4.2.1.3.1

Cancel the common factor.

$F = \frac{3 D}{2} + \frac{3}{2} \cdot \frac{2 G}{3}$

Step 4.2.1.3.2

Rewrite the expression.

$F = \frac{3 D}{2} + \frac{1}{2} (2 G)$

Step 4.2.1.4

Cancel the common factor of $2$ .

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Step 4.2.1.4.1

Factor $2$ out of $2 G$ .

$F = \frac{3 D}{2} + \frac{1}{2} (2 (G))$

Step 4.2.1.4.2

Cancel the common factor.

$F = \frac{3 D}{2} + \frac{1}{2} (2 G)$

Step 4.2.1.4.3

Rewrite the expression.

$F = \frac{3 D}{2} + G$

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:

${x | x \in ℝ}$