Linear Algebra Examples

$\frac{x}{6} - \frac{x}{8} = \frac{5}{12} x$

Step 1

Simplify $\frac{x}{6} - \frac{x}{8}$ .

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Step 1.1

To write $\frac{x}{6}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{x}{6} \cdot \frac{4}{4} - \frac{x}{8} = \frac{5}{12} x$

Step 1.2

To write $- \frac{x}{8}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{x}{6} \cdot \frac{4}{4} - \frac{x}{8} \cdot \frac{3}{3} = \frac{5}{12} x$

Step 1.3

Write each expression with a common denominator of $24$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.3.1

Multiply $\frac{x}{6}$ by $\frac{4}{4}$ .

$\frac{x \cdot 4}{6 \cdot 4} - \frac{x}{8} \cdot \frac{3}{3} = \frac{5}{12} x$

Step 1.3.2

Multiply $6$ by $4$ .

$\frac{x \cdot 4}{24} - \frac{x}{8} \cdot \frac{3}{3} = \frac{5}{12} x$

Step 1.3.3

Multiply $\frac{x}{8}$ by $\frac{3}{3}$ .

$\frac{x \cdot 4}{24} - \frac{x \cdot 3}{8 \cdot 3} = \frac{5}{12} x$

Step 1.3.4

Multiply $8$ by $3$ .

$\frac{x \cdot 4}{24} - \frac{x \cdot 3}{24} = \frac{5}{12} x$

Step 1.4

Combine the numerators over the common denominator.

$\frac{x \cdot 4 - x \cdot 3}{24} = \frac{5}{12} x$

Step 1.5

Simplify the numerator.

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Step 1.5.1

Factor $x$ out of $x \cdot 4 - x \cdot 3$ .

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Step 1.5.1.1

Factor $x$ out of $x \cdot 4$ .

$\frac{x (4) - x \cdot 3}{24} = \frac{5}{12} x$

Step 1.5.1.2

Factor $x$ out of $- x \cdot 3$ .

$\frac{x (4) + x (- 1 \cdot 3)}{24} = \frac{5}{12} x$

Step 1.5.1.3

Factor $x$ out of $x (4) + x (- 1 \cdot 3)$ .

$\frac{x (4 - 1 \cdot 3)}{24} = \frac{5}{12} x$

Step 1.5.2

Multiply $- 1$ by $3$ .

$\frac{x (4 - 3)}{24} = \frac{5}{12} x$

Step 1.5.3

Subtract $3$ from $4$ .

$\frac{x \cdot 1}{24} = \frac{5}{12} x$

Step 1.6

Multiply $x$ by $1$ .

$\frac{x}{24} = \frac{5}{12} x$

Step 2

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

$\frac{x}{24} = \frac{5 x}{12}$

Step 3

Move all terms containing $x$ to the left side of the equation.

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Step 3.1

Subtract $\frac{5 x}{12}$ from both sides of the equation.

$\frac{x}{24} - \frac{5 x}{12} = 0$

Step 3.2

To write $- \frac{5 x}{12}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{x}{24} - \frac{5 x}{12} \cdot \frac{2}{2} = 0$

Step 3.3

Write each expression with a common denominator of $24$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.3.1

Multiply $\frac{5 x}{12}$ by $\frac{2}{2}$ .

$\frac{x}{24} - \frac{5 x \cdot 2}{12 \cdot 2} = 0$

Step 3.3.2

Multiply $12$ by $2$ .

$\frac{x}{24} - \frac{5 x \cdot 2}{24} = 0$

Step 3.4

Combine the numerators over the common denominator.

$\frac{x - 5 x \cdot 2}{24} = 0$

Step 3.5

Simplify the numerator.

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Step 3.5.1

Factor $x$ out of $x - 5 x \cdot 2$ .

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Step 3.5.1.1

Raise $x$ to the power of $1$ .

$\frac{x^{1} - 5 x \cdot 2}{24} = 0$

Step 3.5.1.2

Factor $x$ out of $x^{1}$ .

$\frac{x \cdot 1 - 5 x \cdot 2}{24} = 0$

Step 3.5.1.3

Factor $x$ out of $- 5 x \cdot 2$ .

$\frac{x \cdot 1 + x (- 5 \cdot 2)}{24} = 0$

Step 3.5.1.4

Factor $x$ out of $x \cdot 1 + x (- 5 \cdot 2)$ .

$\frac{x (1 - 5 \cdot 2)}{24} = 0$

Step 3.5.2

Multiply $- 5$ by $2$ .

$\frac{x (1 - 10)}{24} = 0$

Step 3.5.3

Subtract $10$ from $1$ .

$\frac{x \cdot - 9}{24} = 0$

Step 3.6

Cancel the common factor of $- 9$ and $24$ .

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Step 3.6.1

Factor $3$ out of $x \cdot - 9$ .

$\frac{3 (x \cdot - 3)}{24} = 0$

Step 3.6.2

Cancel the common factors.

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Step 3.6.2.1

Factor $3$ out of $24$ .

$\frac{3 (x \cdot - 3)}{3 (8)} = 0$

Step 3.6.2.2

Cancel the common factor.

$\frac{3 (x \cdot - 3)}{3 \cdot 8} = 0$

Step 3.6.2.3

Rewrite the expression.

$\frac{x \cdot - 3}{8} = 0$

Step 3.7

Move $- 3$ to the left of $x$ .

$\frac{- 3 \cdot x}{8} = 0$

Step 3.8

Move the negative in front of the fraction.

$- \frac{3 x}{8} = 0$

Step 4

Set the numerator equal to zero.

$3 x = 0$

Step 5

Divide each term in $3 x = 0$ by $3$ and simplify.

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Step 5.1

Divide each term in $3 x = 0$ by $3$ .

$\frac{3 x}{3} = \frac{0}{3}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $3$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{0}{3}$

Step 5.2.1.2

Divide $x$ by $1$ .

$x = \frac{0}{3}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Divide $0$ by $3$ .

$x = 0$

Step 6

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

${x | x = 0}$

Step 7