Algebra Examples

$- \frac{16}{9} = - \frac{4}{3} (\frac{5}{3} + n)$

Step 1

Rewrite the equation as $- \frac{4}{3} \cdot (\frac{5}{3} + n) = - \frac{16}{9}$ .

$- \frac{4}{3} \cdot (\frac{5}{3} + n) = - \frac{16}{9}$

Step 2

Multiply both sides of the equation by $- \frac{3}{4}$ .

$- \frac{3}{4} (- \frac{4}{3} \cdot (\frac{5}{3} + n)) = - \frac{3}{4} (- \frac{16}{9})$

Step 3

Simplify both sides of the equation.

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Step 3.1

Simplify the left side.

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Step 3.1.1

Simplify $- \frac{3}{4} (- \frac{4}{3} \cdot (\frac{5}{3} + n))$ .

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Step 3.1.1.1

Apply the distributive property.

$- \frac{3}{4} (- \frac{4}{3} \cdot \frac{5}{3} - \frac{4}{3} n) = - \frac{3}{4} (- \frac{16}{9})$

Step 3.1.1.2

Multiply $- \frac{4}{3} \cdot \frac{5}{3}$ .

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Step 3.1.1.2.1

Multiply $\frac{5}{3}$ by $\frac{4}{3}$ .

$- \frac{3}{4} (- \frac{5 \cdot 4}{3 \cdot 3} - \frac{4}{3} n) = - \frac{3}{4} (- \frac{16}{9})$

Step 3.1.1.2.2

Multiply $5$ by $4$ .

$- \frac{3}{4} (- \frac{20}{3 \cdot 3} - \frac{4}{3} n) = - \frac{3}{4} (- \frac{16}{9})$

Step 3.1.1.2.3

Multiply $3$ by $3$ .

$- \frac{3}{4} (- \frac{20}{9} - \frac{4}{3} n) = - \frac{3}{4} (- \frac{16}{9})$

Step 3.1.1.3

Combine $n$ and $\frac{4}{3}$ .

$- \frac{3}{4} (- \frac{20}{9} - \frac{n \cdot 4}{3}) = - \frac{3}{4} (- \frac{16}{9})$

Step 3.1.1.4

Move $4$ to the left of $n$ .

$- \frac{3}{4} (- \frac{20}{9} - \frac{4 n}{3}) = - \frac{3}{4} (- \frac{16}{9})$

Step 3.1.1.5

Simplify terms.

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Step 3.1.1.5.1

Apply the distributive property.

$- \frac{3}{4} (- \frac{20}{9}) - \frac{3}{4} (- \frac{4 n}{3}) = - \frac{3}{4} (- \frac{16}{9})$

Step 3.1.1.5.2

Cancel the common factor of $3$ .

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Step 3.1.1.5.2.1

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

$\frac{- 3}{4} (- \frac{20}{9}) - \frac{3}{4} (- \frac{4 n}{3}) = - \frac{3}{4} (- \frac{16}{9})$

Step 3.1.1.5.2.2

Move the leading negative in $- \frac{20}{9}$ into the numerator.

$\frac{- 3}{4} \cdot \frac{- 20}{9} - \frac{3}{4} (- \frac{4 n}{3}) = - \frac{3}{4} (- \frac{16}{9})$

Step 3.1.1.5.2.3

Factor $3$ out of $- 3$ .

$\frac{3 (- 1)}{4} \cdot \frac{- 20}{9} - \frac{3}{4} (- \frac{4 n}{3}) = - \frac{3}{4} (- \frac{16}{9})$

Step 3.1.1.5.2.4

Factor $3$ out of $9$ .

$\frac{3 \cdot - 1}{4} \cdot \frac{- 20}{3 \cdot 3} - \frac{3}{4} (- \frac{4 n}{3}) = - \frac{3}{4} (- \frac{16}{9})$

Step 3.1.1.5.2.5

Cancel the common factor.

$\frac{3 \cdot - 1}{4} \cdot \frac{- 20}{3 \cdot 3} - \frac{3}{4} (- \frac{4 n}{3}) = - \frac{3}{4} (- \frac{16}{9})$

Step 3.1.1.5.2.6

Rewrite the expression.

$\frac{- 1}{4} \cdot \frac{- 20}{3} - \frac{3}{4} (- \frac{4 n}{3}) = - \frac{3}{4} (- \frac{16}{9})$

Step 3.1.1.5.3

Cancel the common factor of $4$ .

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Step 3.1.1.5.3.1

Factor $4$ out of $- 20$ .

$\frac{- 1}{4} \cdot \frac{4 (- 5)}{3} - \frac{3}{4} (- \frac{4 n}{3}) = - \frac{3}{4} (- \frac{16}{9})$

Step 3.1.1.5.3.2

Cancel the common factor.

$\frac{- 1}{4} \cdot \frac{4 \cdot - 5}{3} - \frac{3}{4} (- \frac{4 n}{3}) = - \frac{3}{4} (- \frac{16}{9})$

Step 3.1.1.5.3.3

Rewrite the expression.

$- \frac{- 5}{3} - \frac{3}{4} (- \frac{4 n}{3}) = - \frac{3}{4} (- \frac{16}{9})$

Step 3.1.1.5.4

Cancel the common factor of $3$ .

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Step 3.1.1.5.4.1

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

$- \frac{- 5}{3} + \frac{- 3}{4} (- \frac{4 n}{3}) = - \frac{3}{4} (- \frac{16}{9})$

Step 3.1.1.5.4.2

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

$- \frac{- 5}{3} + \frac{- 3}{4} \cdot \frac{- 4 n}{3} = - \frac{3}{4} (- \frac{16}{9})$

Step 3.1.1.5.4.3

Factor $3$ out of $- 3$ .

$- \frac{- 5}{3} + \frac{3 (- 1)}{4} \cdot \frac{- 4 n}{3} = - \frac{3}{4} (- \frac{16}{9})$

Step 3.1.1.5.4.4

Cancel the common factor.

$- \frac{- 5}{3} + \frac{3 \cdot - 1}{4} \cdot \frac{- 4 n}{3} = - \frac{3}{4} (- \frac{16}{9})$

Step 3.1.1.5.4.5

Rewrite the expression.

$- \frac{- 5}{3} + \frac{- 1}{4} (- 4 n) = - \frac{3}{4} (- \frac{16}{9})$

Step 3.1.1.5.5

Cancel the common factor of $4$ .

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Step 3.1.1.5.5.1

Factor $4$ out of $- 4 n$ .

$- \frac{- 5}{3} + \frac{- 1}{4} (4 (- n)) = - \frac{3}{4} (- \frac{16}{9})$

Step 3.1.1.5.5.2

Cancel the common factor.

$- \frac{- 5}{3} + \frac{- 1}{4} (4 (- n)) = - \frac{3}{4} (- \frac{16}{9})$

Step 3.1.1.5.5.3

Rewrite the expression.

$- \frac{- 5}{3} - - n = - \frac{3}{4} (- \frac{16}{9})$

Step 3.1.1.5.6

Multiply.

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Step 3.1.1.5.6.1

Multiply $- 1$ by $- 1$ .

$- \frac{- 5}{3} + 1 n = - \frac{3}{4} (- \frac{16}{9})$

Step 3.1.1.5.6.2

Multiply $n$ by $1$ .

$- \frac{- 5}{3} + n = - \frac{3}{4} (- \frac{16}{9})$

Step 3.1.1.6

Simplify each term.

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Step 3.1.1.6.1

Move the negative in front of the fraction.

$- - \frac{5}{3} + n = - \frac{3}{4} (- \frac{16}{9})$

Step 3.1.1.6.2

Multiply $- - \frac{5}{3}$ .

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Step 3.1.1.6.2.1

Multiply $- 1$ by $- 1$ .

$1 (\frac{5}{3}) + n = - \frac{3}{4} (- \frac{16}{9})$

Step 3.1.1.6.2.2

Multiply $\frac{5}{3}$ by $1$ .

$\frac{5}{3} + n = - \frac{3}{4} (- \frac{16}{9})$

Step 3.2

Simplify the right side.

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Step 3.2.1

Simplify $- \frac{3}{4} (- \frac{16}{9})$ .

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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 $- \frac{3}{4}$ into the numerator.

$\frac{5}{3} + n = \frac{- 3}{4} (- \frac{16}{9})$

Step 3.2.1.1.2

Move the leading negative in $- \frac{16}{9}$ into the numerator.

$\frac{5}{3} + n = \frac{- 3}{4} \cdot \frac{- 16}{9}$

Step 3.2.1.1.3

Factor $3$ out of $- 3$ .

$\frac{5}{3} + n = \frac{3 (- 1)}{4} \cdot \frac{- 16}{9}$

Step 3.2.1.1.4

Factor $3$ out of $9$ .

$\frac{5}{3} + n = \frac{3 \cdot - 1}{4} \cdot \frac{- 16}{3 \cdot 3}$

Step 3.2.1.1.5

Cancel the common factor.

$\frac{5}{3} + n = \frac{3 \cdot - 1}{4} \cdot \frac{- 16}{3 \cdot 3}$

Step 3.2.1.1.6

Rewrite the expression.

$\frac{5}{3} + n = \frac{- 1}{4} \cdot \frac{- 16}{3}$

Step 3.2.1.2

Cancel the common factor of $4$ .

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Step 3.2.1.2.1

Factor $4$ out of $- 16$ .

$\frac{5}{3} + n = \frac{- 1}{4} \cdot \frac{4 (- 4)}{3}$

Step 3.2.1.2.2

Cancel the common factor.

$\frac{5}{3} + n = \frac{- 1}{4} \cdot \frac{4 \cdot - 4}{3}$

Step 3.2.1.2.3

Rewrite the expression.

$\frac{5}{3} + n = - \frac{- 4}{3}$

Step 3.2.1.3

Move the negative in front of the fraction.

$\frac{5}{3} + n = - - \frac{4}{3}$

Step 3.2.1.4

Multiply $- - \frac{4}{3}$ .

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Step 3.2.1.4.1

Multiply $- 1$ by $- 1$ .

$\frac{5}{3} + n = 1 (\frac{4}{3})$

Step 3.2.1.4.2

Multiply $\frac{4}{3}$ by $1$ .

$\frac{5}{3} + n = \frac{4}{3}$

Step 4

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

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Step 4.1

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

$n = \frac{4}{3} - \frac{5}{3}$

Step 4.2

Combine the numerators over the common denominator.

$n = \frac{4 - 5}{3}$

Step 4.3

Subtract $5$ from $4$ .

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

Step 4.4

Move the negative in front of the fraction.

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

Step 5

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$n = - 0.\overline{3}$