Pre-Algebra Examples

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

Step 1

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

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Step 1.1

Rewrite.

$0 + 0 + \frac{2}{3} \cdot (1 + n) = - \frac{1}{2} n$

Step 1.2

Simplify by adding zeros.

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

Step 1.3

Apply the distributive property.

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

Step 1.4

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

$\frac{2}{3} + \frac{2}{3} n = - \frac{1}{2} n$

Step 1.5

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

$\frac{2}{3} + \frac{2 n}{3} = - \frac{1}{2} n$

Step 2

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

$\frac{2}{3} + \frac{2 n}{3} = - \frac{n}{2}$

Step 3

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

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Step 3.1

Add $\frac{n}{2}$ to both sides of the equation.

$\frac{2}{3} + \frac{2 n}{3} + \frac{n}{2} = 0$

Step 3.2

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

$\frac{2}{3} + \frac{2 n}{3} \cdot \frac{2}{2} + \frac{n}{2} = 0$

Step 3.3

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

$\frac{2}{3} + \frac{2 n}{3} \cdot \frac{2}{2} + \frac{n}{2} \cdot \frac{3}{3} = 0$

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 $\frac{2 n}{3}$ by $\frac{2}{2}$ .

$\frac{2}{3} + \frac{2 n \cdot 2}{3 \cdot 2} + \frac{n}{2} \cdot \frac{3}{3} = 0$

Step 3.4.2

Multiply $3$ by $2$ .

$\frac{2}{3} + \frac{2 n \cdot 2}{6} + \frac{n}{2} \cdot \frac{3}{3} = 0$

Step 3.4.3

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

$\frac{2}{3} + \frac{2 n \cdot 2}{6} + \frac{n \cdot 3}{2 \cdot 3} = 0$

Step 3.4.4

Multiply $2$ by $3$ .

$\frac{2}{3} + \frac{2 n \cdot 2}{6} + \frac{n \cdot 3}{6} = 0$

Step 3.5

Combine the numerators over the common denominator.

$\frac{2}{3} + \frac{2 n \cdot 2 + n \cdot 3}{6} = 0$

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 $n$ out of $2 n \cdot 2 + n \cdot 3$ .

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Step 3.6.1.1.1

Factor $n$ out of $2 n \cdot 2$ .

$\frac{2}{3} + \frac{n (2 \cdot 2) + n \cdot 3}{6} = 0$

Step 3.6.1.1.2

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

$\frac{2}{3} + \frac{n (2 \cdot 2) + n (3)}{6} = 0$

Step 3.6.1.1.3

Factor $n$ out of $n (2 \cdot 2) + n (3)$ .

$\frac{2}{3} + \frac{n (2 \cdot 2 + 3)}{6} = 0$

Step 3.6.1.2

Multiply $2$ by $2$ .

$\frac{2}{3} + \frac{n (4 + 3)}{6} = 0$

Step 3.6.1.3

Add $4$ and $3$ .

$\frac{2}{3} + \frac{n \cdot 7}{6} = 0$

Step 3.6.2

Move $7$ to the left of $n$ .

$\frac{2}{3} + \frac{7 n}{6} = 0$

Step 4

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

$\frac{7 n}{6} = - \frac{2}{3}$

Step 5

Multiply both sides of the equation by $\frac{6}{7}$ .

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

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

Simplify $\frac{6}{7} \cdot \frac{7 n}{6}$ .

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Step 6.1.1.1

Cancel the common factor of $6$ .

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Step 6.1.1.1.1

Cancel the common factor.

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

Step 6.1.1.1.2

Rewrite the expression.

$\frac{1}{7} (7 n) = \frac{6}{7} (- \frac{2}{3})$

Step 6.1.1.2

Cancel the common factor of $7$ .

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Step 6.1.1.2.1

Factor $7$ out of $7 n$ .

$\frac{1}{7} (7 (n)) = \frac{6}{7} (- \frac{2}{3})$

Step 6.1.1.2.2

Cancel the common factor.

$\frac{1}{7} (7 n) = \frac{6}{7} (- \frac{2}{3})$

Step 6.1.1.2.3

Rewrite the expression.

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

Step 6.2

Simplify the right side.

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Step 6.2.1

Simplify $\frac{6}{7} (- \frac{2}{3})$ .

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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 $- \frac{2}{3}$ into the numerator.

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

Step 6.2.1.1.2

Factor $3$ out of $6$ .

$n = \frac{3 (2)}{7} \cdot \frac{- 2}{3}$

Step 6.2.1.1.3

Cancel the common factor.

$n = \frac{3 \cdot 2}{7} \cdot \frac{- 2}{3}$

Step 6.2.1.1.4

Rewrite the expression.

$n = \frac{2}{7} \cdot - 2$

Step 6.2.1.2

Combine $\frac{2}{7}$ and $- 2$ .

$n = \frac{2 \cdot - 2}{7}$

Step 6.2.1.3

Simplify the expression.

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Step 6.2.1.3.1

Multiply $2$ by $- 2$ .

$n = \frac{- 4}{7}$

Step 6.2.1.3.2

Move the negative in front of the fraction.

$n = - \frac{4}{7}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$n = - \frac{4}{7}$

Decimal Form:

$n = - 0.\overline{571428}$