Linear Algebra Examples

$\frac{m}{4} + \frac{m}{3} = 14$

Step 1

Simplify $\frac{m}{4} + \frac{m}{3}$ .

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Step 1.1

To write $\frac{m}{4}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{m}{4} \cdot \frac{3}{3} + \frac{m}{3} = 14$

Step 1.2

To write $\frac{m}{3}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{m}{4} \cdot \frac{3}{3} + \frac{m}{3} \cdot \frac{4}{4} = 14$

Step 1.3

Write each expression with a common denominator of $12$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.3.1

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

$\frac{m \cdot 3}{4 \cdot 3} + \frac{m}{3} \cdot \frac{4}{4} = 14$

Step 1.3.2

Multiply $4$ by $3$ .

$\frac{m \cdot 3}{12} + \frac{m}{3} \cdot \frac{4}{4} = 14$

Step 1.3.3

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

$\frac{m \cdot 3}{12} + \frac{m \cdot 4}{3 \cdot 4} = 14$

Step 1.3.4

Multiply $3$ by $4$ .

$\frac{m \cdot 3}{12} + \frac{m \cdot 4}{12} = 14$

Step 1.4

Combine the numerators over the common denominator.

$\frac{m \cdot 3 + m \cdot 4}{12} = 14$

Step 1.5

Simplify the numerator.

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Step 1.5.1

Move $3$ to the left of $m$ .

$\frac{3 \cdot m + m \cdot 4}{12} = 14$

Step 1.5.2

Move $4$ to the left of $m$ .

$\frac{3 m + 4 \cdot m}{12} = 14$

Step 1.5.3

Add $3 m$ and $4 m$ .

$\frac{7 m}{12} = 14$

Step 2

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

$\frac{12}{7} \cdot \frac{7 m}{12} = \frac{12}{7} \cdot 14$

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

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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.

$\frac{12}{7} \cdot \frac{7 m}{12} = \frac{12}{7} \cdot 14$

Step 3.1.1.1.2

Rewrite the expression.

$\frac{1}{7} (7 m) = \frac{12}{7} \cdot 14$

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 $7 m$ .

$\frac{1}{7} (7 (m)) = \frac{12}{7} \cdot 14$

Step 3.1.1.2.2

Cancel the common factor.

$\frac{1}{7} (7 m) = \frac{12}{7} \cdot 14$

Step 3.1.1.2.3

Rewrite the expression.

$m = \frac{12}{7} \cdot 14$

Step 3.2

Simplify the right side.

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Step 3.2.1

Simplify $\frac{12}{7} \cdot 14$ .

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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 $14$ .

$m = \frac{12}{7} \cdot (7 (2))$

Step 3.2.1.1.2

Cancel the common factor.

$m = \frac{12}{7} \cdot (7 \cdot 2)$

Step 3.2.1.1.3

Rewrite the expression.

$m = 12 \cdot 2$

Step 3.2.1.2

Multiply $12$ by $2$ .

$m = 24$

Step 4

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

${m | m = 24}$

Step 5