Algebra Examples

$\frac{1}{2} x - 7 = \frac{1}{3} (x - 12)$

Step 1

Combine $\frac{1}{2}$ and $x$ .

$\frac{x}{2} - 7 = \frac{1}{3} \cdot (x - 12)$

Step 2

Simplify $\frac{1}{3} \cdot (x - 12)$ .

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Step 2.1

Apply the distributive property.

$\frac{x}{2} - 7 = \frac{1}{3} x + \frac{1}{3} \cdot - 12$

Step 2.2

Combine $\frac{1}{3}$ and $x$ .

$\frac{x}{2} - 7 = \frac{x}{3} + \frac{1}{3} \cdot - 12$

Step 2.3

Cancel the common factor of $3$ .

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Step 2.3.1

Factor $3$ out of $- 12$ .

$\frac{x}{2} - 7 = \frac{x}{3} + \frac{1}{3} \cdot (3 (- 4))$

Step 2.3.2

Cancel the common factor.

$\frac{x}{2} - 7 = \frac{x}{3} + \frac{1}{3} \cdot (3 \cdot - 4)$

Step 2.3.3

Rewrite the expression.

$\frac{x}{2} - 7 = \frac{x}{3} - 4$

Step 3

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

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Step 3.1

Subtract $\frac{x}{3}$ from both sides of the equation.

$\frac{x}{2} - 7 - \frac{x}{3} = - 4$

Step 3.2

To write $\frac{x}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{x}{2} \cdot \frac{3}{3} - \frac{x}{3} - 7 = - 4$

Step 3.3

To write $- \frac{x}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{x}{2} \cdot \frac{3}{3} - \frac{x}{3} \cdot \frac{2}{2} - 7 = - 4$

Step 3.4

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.4.1

Multiply $\frac{x}{2}$ by $\frac{3}{3}$ .

$\frac{x \cdot 3}{2 \cdot 3} - \frac{x}{3} \cdot \frac{2}{2} - 7 = - 4$

Step 3.4.2

Multiply $2$ by $3$ .

$\frac{x \cdot 3}{6} - \frac{x}{3} \cdot \frac{2}{2} - 7 = - 4$

Step 3.4.3

Multiply $\frac{x}{3}$ by $\frac{2}{2}$ .

$\frac{x \cdot 3}{6} - \frac{x \cdot 2}{3 \cdot 2} - 7 = - 4$

Step 3.4.4

Multiply $3$ by $2$ .

$\frac{x \cdot 3}{6} - \frac{x \cdot 2}{6} - 7 = - 4$

Step 3.5

Combine the numerators over the common denominator.

$\frac{x \cdot 3 - x \cdot 2}{6} - 7 = - 4$

Step 3.6

Simplify each term.

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Step 3.6.1

Simplify the numerator.

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Step 3.6.1.1

Factor $x$ out of $x \cdot 3 - x \cdot 2$ .

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Step 3.6.1.1.1

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

$\frac{x (3) - x \cdot 2}{6} - 7 = - 4$

Step 3.6.1.1.2

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

$\frac{x (3) + x (- 1 \cdot 2)}{6} - 7 = - 4$

Step 3.6.1.1.3

Factor $x$ out of $x (3) + x (- 1 \cdot 2)$ .

$\frac{x (3 - 1 \cdot 2)}{6} - 7 = - 4$

Step 3.6.1.2

Multiply $- 1$ by $2$ .

$\frac{x (3 - 2)}{6} - 7 = - 4$

Step 3.6.1.3

Subtract $2$ from $3$ .

$\frac{x \cdot 1}{6} - 7 = - 4$

Step 3.6.2

Multiply $x$ by $1$ .

$\frac{x}{6} - 7 = - 4$

Step 4

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

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Step 4.1

Add $7$ to both sides of the equation.

$\frac{x}{6} = - 4 + 7$

Step 4.2

Add $- 4$ and $7$ .

$\frac{x}{6} = 3$

Step 5

Multiply both sides of the equation by $6$ .

$6 \frac{x}{6} = 6 \cdot 3$

Step 6

Simplify both sides of the equation.

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Step 6.1

Simplify the left side.

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Step 6.1.1

Cancel the common factor of $6$ .

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Step 6.1.1.1

Cancel the common factor.

$6 \frac{x}{6} = 6 \cdot 3$

Step 6.1.1.2

Rewrite the expression.

$x = 6 \cdot 3$

Step 6.2

Simplify the right side.

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Step 6.2.1

Multiply $6$ by $3$ .

$x = 18$