Linear Algebra Examples

$\frac{n}{2} - \frac{3 n}{4} + \frac{5 n}{6} = 21$

Step 1

Simplify $\frac{n}{2} - \frac{3 n}{4} + \frac{5 n}{6}$ .

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Step 1.1

Find the common denominator.

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Step 1.1.1

Multiply $\frac{n}{2}$ by $\frac{6}{6}$ .

$\frac{n}{2} \cdot \frac{6}{6} - \frac{3 n}{4} + \frac{5 n}{6} = 21$

Step 1.1.2

Multiply $\frac{n}{2}$ by $\frac{6}{6}$ .

$\frac{n \cdot 6}{2 \cdot 6} - \frac{3 n}{4} + \frac{5 n}{6} = 21$

Step 1.1.3

Multiply $\frac{3 n}{4}$ by $\frac{3}{3}$ .

$\frac{n \cdot 6}{2 \cdot 6} - (\frac{3 n}{4} \cdot \frac{3}{3}) + \frac{5 n}{6} = 21$

Step 1.1.4

Multiply $\frac{3 n}{4}$ by $\frac{3}{3}$ .

$\frac{n \cdot 6}{2 \cdot 6} - \frac{3 n \cdot 3}{4 \cdot 3} + \frac{5 n}{6} = 21$

Step 1.1.5

Multiply $\frac{5 n}{6}$ by $\frac{2}{2}$ .

$\frac{n \cdot 6}{2 \cdot 6} - \frac{3 n \cdot 3}{4 \cdot 3} + \frac{5 n}{6} \cdot \frac{2}{2} = 21$

Step 1.1.6

Multiply $\frac{5 n}{6}$ by $\frac{2}{2}$ .

$\frac{n \cdot 6}{2 \cdot 6} - \frac{3 n \cdot 3}{4 \cdot 3} + \frac{5 n \cdot 2}{6 \cdot 2} = 21$

Step 1.1.7

Multiply $2$ by $6$ .

$\frac{n \cdot 6}{12} - \frac{3 n \cdot 3}{4 \cdot 3} + \frac{5 n \cdot 2}{6 \cdot 2} = 21$

Step 1.1.8

Reorder the factors of $4 \cdot 3$ .

$\frac{n \cdot 6}{12} - \frac{3 n \cdot 3}{3 \cdot 4} + \frac{5 n \cdot 2}{6 \cdot 2} = 21$

Step 1.1.9

Multiply $3$ by $4$ .

$\frac{n \cdot 6}{12} - \frac{3 n \cdot 3}{12} + \frac{5 n \cdot 2}{6 \cdot 2} = 21$

Step 1.1.10

Reorder the factors of $6 \cdot 2$ .

$\frac{n \cdot 6}{12} - \frac{3 n \cdot 3}{12} + \frac{5 n \cdot 2}{2 \cdot 6} = 21$

Step 1.1.11

Multiply $2$ by $6$ .

$\frac{n \cdot 6}{12} - \frac{3 n \cdot 3}{12} + \frac{5 n \cdot 2}{12} = 21$

Step 1.2

Combine the numerators over the common denominator.

$\frac{n \cdot 6 - 3 n \cdot 3 + 5 n \cdot 2}{12} = 21$

Step 1.3

Simplify each term.

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Step 1.3.1

Move $6$ to the left of $n$ .

$\frac{6 \cdot n - 3 n \cdot 3 + 5 n \cdot 2}{12} = 21$

Step 1.3.2

Multiply $3$ by $- 3$ .

$\frac{6 n - 9 n + 5 n \cdot 2}{12} = 21$

Step 1.3.3

Multiply $2$ by $5$ .

$\frac{6 n - 9 n + 10 n}{12} = 21$

Step 1.4

Simplify by adding terms.

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Step 1.4.1

Subtract $9 n$ from $6 n$ .

$\frac{- 3 n + 10 n}{12} = 21$

Step 1.4.2

Add $- 3 n$ and $10 n$ .

$\frac{7 n}{12} = 21$

Step 2

Multiply both sides of the equation by $\frac{12}{7}$ .

$\frac{12}{7} \cdot \frac{7 n}{12} = \frac{12}{7} \cdot 21$

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 $\frac{12}{7} \cdot \frac{7 n}{12}$ .

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Step 3.1.1.1

Cancel the common factor of $12$ .

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Step 3.1.1.1.1

Cancel the common factor.

$\frac{12}{7} \cdot \frac{7 n}{12} = \frac{12}{7} \cdot 21$

Step 3.1.1.1.2

Rewrite the expression.

$\frac{1}{7} (7 n) = \frac{12}{7} \cdot 21$

Step 3.1.1.2

Cancel the common factor of $7$ .

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Step 3.1.1.2.1

Factor $7$ out of $7 n$ .

$\frac{1}{7} (7 (n)) = \frac{12}{7} \cdot 21$

Step 3.1.1.2.2

Cancel the common factor.

$\frac{1}{7} (7 n) = \frac{12}{7} \cdot 21$

Step 3.1.1.2.3

Rewrite the expression.

$n = \frac{12}{7} \cdot 21$

Step 3.2

Simplify the right side.

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Step 3.2.1

Simplify $\frac{12}{7} \cdot 21$ .

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Step 3.2.1.1

Cancel the common factor of $7$ .

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Step 3.2.1.1.1

Factor $7$ out of $21$ .

$n = \frac{12}{7} \cdot (7 (3))$

Step 3.2.1.1.2

Cancel the common factor.

$n = \frac{12}{7} \cdot (7 \cdot 3)$

Step 3.2.1.1.3

Rewrite the expression.

$n = 12 \cdot 3$

Step 3.2.1.2

Multiply $12$ by $3$ .

$n = 36$

Step 4

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

${n | n = 36}$

Step 5