Algebra Examples

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

Step 1

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

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Step 1.1

Subtract $\frac{2 x}{3}$ from both sides of the equation.

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

Step 1.2

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

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

Step 1.3

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

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

Step 1.4

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

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Step 1.4.1

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

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

Step 1.4.2

Multiply $2$ by $3$ .

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

Step 1.4.3

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

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

Step 1.4.4

Multiply $3$ by $2$ .

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

Step 1.5

Combine the numerators over the common denominator.

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

Step 1.6

Combine the numerators over the common denominator.

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

Step 1.7

Simplify each term.

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Step 1.7.1

Multiply $3$ by $3$ .

$\frac{9 x - 2 x \cdot 2 + 1}{6} = - \frac{2}{3}$

Step 1.7.2

Multiply $2$ by $- 2$ .

$\frac{9 x - 4 x + 1}{6} = - \frac{2}{3}$

Step 1.8

Subtract $4 x$ from $9 x$ .

$\frac{5 x + 1}{6} = - \frac{2}{3}$

Step 2

Multiply both sides by $6$ .

$\frac{5 x + 1}{6} \cdot 6 = - \frac{2}{3} \cdot 6$

Step 3

Simplify.

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Step 3.1

Simplify the left side.

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Step 3.1.1

Cancel the common factor of $6$ .

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Step 3.1.1.1

Cancel the common factor.

$\frac{5 x + 1}{6} \cdot 6 = - \frac{2}{3} \cdot 6$

Step 3.1.1.2

Rewrite the expression.

$5 x + 1 = - \frac{2}{3} \cdot 6$

Step 3.2

Simplify the right side.

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Step 3.2.1

Simplify $- \frac{2}{3} \cdot 6$ .

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Step 3.2.1.1

Cancel the common factor of $3$ .

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Step 3.2.1.1.1

Move the leading negative in $- \frac{2}{3}$ into the numerator.

$5 x + 1 = \frac{- 2}{3} \cdot 6$

Step 3.2.1.1.2

Factor $3$ out of $6$ .

$5 x + 1 = \frac{- 2}{3} \cdot (3 (2))$

Step 3.2.1.1.3

Cancel the common factor.

$5 x + 1 = \frac{- 2}{3} \cdot (3 \cdot 2)$

Step 3.2.1.1.4

Rewrite the expression.

$5 x + 1 = - 2 \cdot 2$

Step 3.2.1.2

Multiply $- 2$ by $2$ .

$5 x + 1 = - 4$

Step 4

Solve for $x$ .

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Step 4.1

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

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Step 4.1.1

Subtract $1$ from both sides of the equation.

$5 x = - 4 - 1$

Step 4.1.2

Subtract $1$ from $- 4$ .

$5 x = - 5$

Step 4.2

Divide each term in $5 x = - 5$ by $5$ and simplify.

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Step 4.2.1

Divide each term in $5 x = - 5$ by $5$ .

$\frac{5 x}{5} = \frac{- 5}{5}$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Cancel the common factor of $5$ .

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Step 4.2.2.1.1

Cancel the common factor.

$\frac{5 x}{5} = \frac{- 5}{5}$

Step 4.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 5}{5}$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

Divide $- 5$ by $5$ .

$x = - 1$