Algebra Examples

$\frac{2}{3} \cdot (\frac{1}{2} x + 12) = \frac{1}{2} \cdot (\frac{1}{3} x + 14) - 3$

Step 1

Simplify $\frac{2}{3} \cdot (\frac{1}{2} x + 12)$ .

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Step 1.1

Rewrite.

$0 + 0 + \frac{2}{3} \cdot (\frac{1}{2} x + 12) = \frac{1}{2} \cdot (\frac{1}{3} x + 14) - 3$

Step 1.2

Simplify by adding zeros.

$\frac{2}{3} \cdot (\frac{1}{2} x + 12) = \frac{1}{2} \cdot (\frac{1}{3} x + 14) - 3$

Step 1.3

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

$\frac{2}{3} \cdot (\frac{x}{2} + 12) = \frac{1}{2} \cdot (\frac{1}{3} x + 14) - 3$

Step 1.4

Apply the distributive property.

$\frac{2}{3} \cdot \frac{x}{2} + \frac{2}{3} \cdot 12 = \frac{1}{2} \cdot (\frac{1}{3} x + 14) - 3$

Step 1.5

Cancel the common factor of $2$ .

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Step 1.5.1

Cancel the common factor.

$\frac{2}{3} \cdot \frac{x}{2} + \frac{2}{3} \cdot 12 = \frac{1}{2} \cdot (\frac{1}{3} x + 14) - 3$

Step 1.5.2

Rewrite the expression.

$\frac{1}{3} x + \frac{2}{3} \cdot 12 = \frac{1}{2} \cdot (\frac{1}{3} x + 14) - 3$

Step 1.6

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

$\frac{x}{3} + \frac{2}{3} \cdot 12 = \frac{1}{2} \cdot (\frac{1}{3} x + 14) - 3$

Step 1.7

Cancel the common factor of $3$ .

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Step 1.7.1

Factor $3$ out of $12$ .

$\frac{x}{3} + \frac{2}{3} \cdot (3 (4)) = \frac{1}{2} \cdot (\frac{1}{3} x + 14) - 3$

Step 1.7.2

Cancel the common factor.

$\frac{x}{3} + \frac{2}{3} \cdot (3 \cdot 4) = \frac{1}{2} \cdot (\frac{1}{3} x + 14) - 3$

Step 1.7.3

Rewrite the expression.

$\frac{x}{3} + 2 \cdot 4 = \frac{1}{2} \cdot (\frac{1}{3} x + 14) - 3$

Step 1.8

Multiply $2$ by $4$ .

$\frac{x}{3} + 8 = \frac{1}{2} \cdot (\frac{1}{3} x + 14) - 3$

Step 2

Simplify $\frac{1}{2} \cdot (\frac{1}{3} x + 14) - 3$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

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

$\frac{x}{3} + 8 = \frac{1}{2} \cdot (\frac{x}{3} + 14) - 3$

Step 2.1.2

Apply the distributive property.

$\frac{x}{3} + 8 = \frac{1}{2} \cdot \frac{x}{3} + \frac{1}{2} \cdot 14 - 3$

Step 2.1.3

Multiply $\frac{1}{2} \cdot \frac{x}{3}$ .

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Step 2.1.3.1

Multiply $\frac{1}{2}$ by $\frac{x}{3}$ .

$\frac{x}{3} + 8 = \frac{x}{2 \cdot 3} + \frac{1}{2} \cdot 14 - 3$

Step 2.1.3.2

Multiply $2$ by $3$ .

$\frac{x}{3} + 8 = \frac{x}{6} + \frac{1}{2} \cdot 14 - 3$

Step 2.1.4

Cancel the common factor of $2$ .

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Step 2.1.4.1

Factor $2$ out of $14$ .

$\frac{x}{3} + 8 = \frac{x}{6} + \frac{1}{2} \cdot (2 (7)) - 3$

Step 2.1.4.2

Cancel the common factor.

$\frac{x}{3} + 8 = \frac{x}{6} + \frac{1}{2} \cdot (2 \cdot 7) - 3$

Step 2.1.4.3

Rewrite the expression.

$\frac{x}{3} + 8 = \frac{x}{6} + 7 - 3$

Step 2.2

Subtract $3$ from $7$ .

$\frac{x}{3} + 8 = \frac{x}{6} + 4$

Step 3

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

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Step 3.1

Subtract $\frac{x}{6}$ from both sides of the equation.

$\frac{x}{3} + 8 - \frac{x}{6} = 4$

Step 3.2

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

$\frac{x}{3} \cdot \frac{2}{2} - \frac{x}{6} + 8 = 4$

Step 3.3

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

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Step 3.3.1

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

$\frac{x \cdot 2}{3 \cdot 2} - \frac{x}{6} + 8 = 4$

Step 3.3.2

Multiply $3$ by $2$ .

$\frac{x \cdot 2}{6} - \frac{x}{6} + 8 = 4$

Step 3.4

Combine the numerators over the common denominator.

$\frac{x \cdot 2 - x}{6} + 8 = 4$

Step 3.5

Subtract $x$ from $x \cdot 2$ .

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Step 3.5.1

Reorder $x$ and $2$ .

$\frac{2 \cdot x - x}{6} + 8 = 4$

Step 3.5.2

Subtract $x$ from $2 \cdot x$ .

$\frac{x}{6} + 8 = 4$

Step 4

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

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Step 4.1

Subtract $8$ from both sides of the equation.

$\frac{x}{6} = 4 - 8$

Step 4.2

Subtract $8$ from $4$ .

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

Step 5

Multiply both sides of the equation by $6$ .

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

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.

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

Step 6.1.1.2

Rewrite the expression.

$x = 6 \cdot - 4$

Step 6.2

Simplify the right side.

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Step 6.2.1

Multiply $6$ by $- 4$ .

$x = - 24$