Linear Algebra Examples

$\frac{3}{5} = \frac{3}{10} d - 12$

Step 1

Rewrite the equation as $\frac{3}{10} d - 12 = \frac{3}{5}$ .

$\frac{3}{10} d - 12 = \frac{3}{5}$

Step 2

Combine $\frac{3}{10}$ and $d$ .

$\frac{3 d}{10} - 12 = \frac{3}{5}$

Step 3

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

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Step 3.1

Add $12$ to both sides of the equation.

$\frac{3 d}{10} = \frac{3}{5} + 12$

Step 3.2

To write $12$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{3 d}{10} = \frac{3}{5} + 12 \cdot \frac{5}{5}$

Step 3.3

Combine $12$ and $\frac{5}{5}$ .

$\frac{3 d}{10} = \frac{3}{5} + \frac{12 \cdot 5}{5}$

Step 3.4

Combine the numerators over the common denominator.

$\frac{3 d}{10} = \frac{3 + 12 \cdot 5}{5}$

Step 3.5

Simplify the numerator.

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Step 3.5.1

Multiply $12$ by $5$ .

$\frac{3 d}{10} = \frac{3 + 60}{5}$

Step 3.5.2

Add $3$ and $60$ .

$\frac{3 d}{10} = \frac{63}{5}$

Step 4

Multiply both sides of the equation by $\frac{10}{3}$ .

$\frac{10}{3} \cdot \frac{3 d}{10} = \frac{10}{3} \cdot \frac{63}{5}$

Step 5

Simplify both sides of the equation.

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Step 5.1

Simplify the left side.

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Step 5.1.1

Simplify $\frac{10}{3} \cdot \frac{3 d}{10}$ .

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Step 5.1.1.1

Cancel the common factor of $10$ .

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Step 5.1.1.1.1

Cancel the common factor.

$\frac{10}{3} \cdot \frac{3 d}{10} = \frac{10}{3} \cdot \frac{63}{5}$

Step 5.1.1.1.2

Rewrite the expression.

$\frac{1}{3} (3 d) = \frac{10}{3} \cdot \frac{63}{5}$

Step 5.1.1.2

Cancel the common factor of $3$ .

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Step 5.1.1.2.1

Factor $3$ out of $3 d$ .

$\frac{1}{3} (3 (d)) = \frac{10}{3} \cdot \frac{63}{5}$

Step 5.1.1.2.2

Cancel the common factor.

$\frac{1}{3} (3 d) = \frac{10}{3} \cdot \frac{63}{5}$

Step 5.1.1.2.3

Rewrite the expression.

$d = \frac{10}{3} \cdot \frac{63}{5}$

Step 5.2

Simplify the right side.

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Step 5.2.1

Simplify $\frac{10}{3} \cdot \frac{63}{5}$ .

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Step 5.2.1.1

Cancel the common factor of $5$ .

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Step 5.2.1.1.1

Factor $5$ out of $10$ .

$d = \frac{5 (2)}{3} \cdot \frac{63}{5}$

Step 5.2.1.1.2

Cancel the common factor.

$d = \frac{5 \cdot 2}{3} \cdot \frac{63}{5}$

Step 5.2.1.1.3

Rewrite the expression.

$d = \frac{2}{3} \cdot 63$

Step 5.2.1.2

Cancel the common factor of $3$ .

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Step 5.2.1.2.1

Factor $3$ out of $63$ .

$d = \frac{2}{3} \cdot (3 (21))$

Step 5.2.1.2.2

Cancel the common factor.

$d = \frac{2}{3} \cdot (3 \cdot 21)$

Step 5.2.1.2.3

Rewrite the expression.

$d = 2 \cdot 21$

Step 5.2.1.3

Multiply $2$ by $21$ .

$d = 42$

Step 6

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

${d | d = 42}$

Step 7