Pre-Algebra Examples

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

Step 1

Combine $\frac{2}{3}$ and $x$ .

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

Step 2

Combine $\frac{3}{5}$ and $x$ .

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

Step 3

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

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Step 3.1

Subtract $\frac{3 x}{5}$ from both sides of the equation.

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

Step 3.2

To write $\frac{2 x}{3}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

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

Step 3.3

To write $- \frac{3 x}{5}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 3.4

Write each expression with a common denominator of $15$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.4.1

Multiply $\frac{2 x}{3}$ by $\frac{5}{5}$ .

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

Step 3.4.2

Multiply $3$ by $5$ .

$\frac{2 x \cdot 5}{15} - \frac{3 x}{5} \cdot \frac{3}{3} + \frac{1}{2} = - \frac{5}{6}$

Step 3.4.3

Multiply $\frac{3 x}{5}$ by $\frac{3}{3}$ .

$\frac{2 x \cdot 5}{15} - \frac{3 x \cdot 3}{5 \cdot 3} + \frac{1}{2} = - \frac{5}{6}$

Step 3.4.4

Multiply $5$ by $3$ .

$\frac{2 x \cdot 5}{15} - \frac{3 x \cdot 3}{15} + \frac{1}{2} = - \frac{5}{6}$

Step 3.5

Combine the numerators over the common denominator.

$\frac{2 x \cdot 5 - 3 x \cdot 3}{15} + \frac{1}{2} = - \frac{5}{6}$

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 $2 x \cdot 5 - 3 x \cdot 3$ .

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Step 3.6.1.1.1

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

$\frac{x (2 \cdot 5) - 3 x \cdot 3}{15} + \frac{1}{2} = - \frac{5}{6}$

Step 3.6.1.1.2

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

$\frac{x (2 \cdot 5) + x (- 3 \cdot 3)}{15} + \frac{1}{2} = - \frac{5}{6}$

Step 3.6.1.1.3

Factor $x$ out of $x (2 \cdot 5) + x (- 3 \cdot 3)$ .

$\frac{x (2 \cdot 5 - 3 \cdot 3)}{15} + \frac{1}{2} = - \frac{5}{6}$

Step 3.6.1.2

Multiply $2$ by $5$ .

$\frac{x (10 - 3 \cdot 3)}{15} + \frac{1}{2} = - \frac{5}{6}$

Step 3.6.1.3

Multiply $- 3$ by $3$ .

$\frac{x (10 - 9)}{15} + \frac{1}{2} = - \frac{5}{6}$

Step 3.6.1.4

Subtract $9$ from $10$ .

$\frac{x \cdot 1}{15} + \frac{1}{2} = - \frac{5}{6}$

Step 3.6.2

Multiply $x$ by $1$ .

$\frac{x}{15} + \frac{1}{2} = - \frac{5}{6}$

Step 4

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

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Step 4.1

Subtract $\frac{1}{2}$ from both sides of the equation.

$\frac{x}{15} = - \frac{5}{6} - \frac{1}{2}$

Step 4.2

To write $- \frac{1}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{x}{15} = - \frac{5}{6} - \frac{1}{2} \cdot \frac{3}{3}$

Step 4.3

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

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Step 4.3.1

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

$\frac{x}{15} = - \frac{5}{6} - \frac{3}{2 \cdot 3}$

Step 4.3.2

Multiply $2$ by $3$ .

$\frac{x}{15} = - \frac{5}{6} - \frac{3}{6}$

Step 4.4

Combine the numerators over the common denominator.

$\frac{x}{15} = \frac{- 5 - 3}{6}$

Step 4.5

Subtract $3$ from $- 5$ .

$\frac{x}{15} = \frac{- 8}{6}$

Step 4.6

Cancel the common factor of $- 8$ and $6$ .

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Step 4.6.1

Factor $2$ out of $- 8$ .

$\frac{x}{15} = \frac{2 (- 4)}{6}$

Step 4.6.2

Cancel the common factors.

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Step 4.6.2.1

Factor $2$ out of $6$ .

$\frac{x}{15} = \frac{2 \cdot - 4}{2 \cdot 3}$

Step 4.6.2.2

Cancel the common factor.

$\frac{x}{15} = \frac{2 \cdot - 4}{2 \cdot 3}$

Step 4.6.2.3

Rewrite the expression.

$\frac{x}{15} = \frac{- 4}{3}$

Step 4.7

Move the negative in front of the fraction.

$\frac{x}{15} = - \frac{4}{3}$

Step 5

Multiply both sides of the equation by $15$ .

$15 \frac{x}{15} = 15 (- \frac{4}{3})$

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 $15$ .

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Step 6.1.1.1

Cancel the common factor.

$15 \frac{x}{15} = 15 (- \frac{4}{3})$

Step 6.1.1.2

Rewrite the expression.

$x = 15 (- \frac{4}{3})$

Step 6.2

Simplify the right side.

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Step 6.2.1

Simplify $15 (- \frac{4}{3})$ .

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Step 6.2.1.1

Cancel the common factor of $3$ .

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Step 6.2.1.1.1

Move the leading negative in $- \frac{4}{3}$ into the numerator.

$x = 15 (\frac{- 4}{3})$

Step 6.2.1.1.2

Factor $3$ out of $15$ .

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

Step 6.2.1.1.3

Cancel the common factor.

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

Step 6.2.1.1.4

Rewrite the expression.

$x = 5 \cdot - 4$

Step 6.2.1.2

Multiply $5$ by $- 4$ .

$x = - 20$