Algebra Examples

$- 16 = - \frac{15}{16} (\frac{4}{5} q + \frac{32}{3})$

Step 1

Rewrite the equation as $- \frac{15}{16} \cdot (\frac{4}{5} q + \frac{32}{3}) = - 16$ .

$- \frac{15}{16} \cdot (\frac{4}{5} q + \frac{32}{3}) = - 16$

Step 2

Multiply both sides of the equation by $- \frac{16}{15}$ .

$- \frac{16}{15} (- \frac{15}{16} \cdot (\frac{4}{5} q + \frac{32}{3})) = - \frac{16}{15} \cdot - 16$

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 $- \frac{16}{15} (- \frac{15}{16} \cdot (\frac{4}{5} q + \frac{32}{3}))$ .

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Step 3.1.1.1

Simplify terms.

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Step 3.1.1.1.1

Combine $\frac{4}{5}$ and $q$ .

$- \frac{16}{15} (- \frac{15}{16} \cdot (\frac{4 q}{5} + \frac{32}{3})) = - \frac{16}{15} \cdot - 16$

Step 3.1.1.1.2

Apply the distributive property.

$- \frac{16}{15} (- \frac{15}{16} \cdot \frac{4 q}{5} - \frac{15}{16} \cdot \frac{32}{3}) = - \frac{16}{15} \cdot - 16$

Step 3.1.1.1.3

Cancel the common factor of $5$ .

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Step 3.1.1.1.3.1

Move the leading negative in $- \frac{15}{16}$ into the numerator.

$- \frac{16}{15} (\frac{- 15}{16} \cdot \frac{4 q}{5} - \frac{15}{16} \cdot \frac{32}{3}) = - \frac{16}{15} \cdot - 16$

Step 3.1.1.1.3.2

Factor $5$ out of $- 15$ .

$- \frac{16}{15} (\frac{5 (- 3)}{16} \cdot \frac{4 q}{5} - \frac{15}{16} \cdot \frac{32}{3}) = - \frac{16}{15} \cdot - 16$

Step 3.1.1.1.3.3

Cancel the common factor.

$- \frac{16}{15} (\frac{5 \cdot - 3}{16} \cdot \frac{4 q}{5} - \frac{15}{16} \cdot \frac{32}{3}) = - \frac{16}{15} \cdot - 16$

Step 3.1.1.1.3.4

Rewrite the expression.

$- \frac{16}{15} (\frac{- 3}{16} (4 q) - \frac{15}{16} \cdot \frac{32}{3}) = - \frac{16}{15} \cdot - 16$

Step 3.1.1.1.4

Cancel the common factor of $4$ .

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Step 3.1.1.1.4.1

Factor $4$ out of $16$ .

$- \frac{16}{15} (\frac{- 3}{4 (4)} (4 q) - \frac{15}{16} \cdot \frac{32}{3}) = - \frac{16}{15} \cdot - 16$

Step 3.1.1.1.4.2

Factor $4$ out of $4 q$ .

$- \frac{16}{15} (\frac{- 3}{4 (4)} (4 (q)) - \frac{15}{16} \cdot \frac{32}{3}) = - \frac{16}{15} \cdot - 16$

Step 3.1.1.1.4.3

Cancel the common factor.

$- \frac{16}{15} (\frac{- 3}{4 \cdot 4} (4 q) - \frac{15}{16} \cdot \frac{32}{3}) = - \frac{16}{15} \cdot - 16$

Step 3.1.1.1.4.4

Rewrite the expression.

$- \frac{16}{15} (\frac{- 3}{4} q - \frac{15}{16} \cdot \frac{32}{3}) = - \frac{16}{15} \cdot - 16$

Step 3.1.1.1.5

Combine $\frac{- 3}{4}$ and $q$ .

$- \frac{16}{15} (\frac{- 3 q}{4} - \frac{15}{16} \cdot \frac{32}{3}) = - \frac{16}{15} \cdot - 16$

Step 3.1.1.1.6

Cancel the common factor of $3$ .

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Step 3.1.1.1.6.1

Move the leading negative in $- \frac{15}{16}$ into the numerator.

$- \frac{16}{15} (\frac{- 3 q}{4} + \frac{- 15}{16} \cdot \frac{32}{3}) = - \frac{16}{15} \cdot - 16$

Step 3.1.1.1.6.2

Factor $3$ out of $- 15$ .

$- \frac{16}{15} (\frac{- 3 q}{4} + \frac{3 (- 5)}{16} \cdot \frac{32}{3}) = - \frac{16}{15} \cdot - 16$

Step 3.1.1.1.6.3

Cancel the common factor.

$- \frac{16}{15} (\frac{- 3 q}{4} + \frac{3 \cdot - 5}{16} \cdot \frac{32}{3}) = - \frac{16}{15} \cdot - 16$

Step 3.1.1.1.6.4

Rewrite the expression.

$- \frac{16}{15} (\frac{- 3 q}{4} + \frac{- 5}{16} \cdot 32) = - \frac{16}{15} \cdot - 16$

Step 3.1.1.1.7

Cancel the common factor of $16$ .

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Step 3.1.1.1.7.1

Factor $16$ out of $32$ .

$- \frac{16}{15} (\frac{- 3 q}{4} + \frac{- 5}{16} \cdot (16 (2))) = - \frac{16}{15} \cdot - 16$

Step 3.1.1.1.7.2

Cancel the common factor.

$- \frac{16}{15} (\frac{- 3 q}{4} + \frac{- 5}{16} \cdot (16 \cdot 2)) = - \frac{16}{15} \cdot - 16$

Step 3.1.1.1.7.3

Rewrite the expression.

$- \frac{16}{15} (\frac{- 3 q}{4} - 5 \cdot 2) = - \frac{16}{15} \cdot - 16$

Step 3.1.1.1.8

Simplify the expression.

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Step 3.1.1.1.8.1

Multiply $- 5$ by $2$ .

$- \frac{16}{15} (\frac{- 3 q}{4} - 10) = - \frac{16}{15} \cdot - 16$

Step 3.1.1.1.8.2

Move the negative in front of the fraction.

$- \frac{16}{15} (- \frac{3 q}{4} - 10) = - \frac{16}{15} \cdot - 16$

Step 3.1.1.1.9

Apply the distributive property.

$- \frac{16}{15} (- \frac{3 q}{4}) - \frac{16}{15} \cdot - 10 = - \frac{16}{15} \cdot - 16$

Step 3.1.1.1.10

Cancel the common factor of $4$ .

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Step 3.1.1.1.10.1

Move the leading negative in $- \frac{16}{15}$ into the numerator.

$\frac{- 16}{15} (- \frac{3 q}{4}) - \frac{16}{15} \cdot - 10 = - \frac{16}{15} \cdot - 16$

Step 3.1.1.1.10.2

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

$\frac{- 16}{15} \cdot \frac{- 3 q}{4} - \frac{16}{15} \cdot - 10 = - \frac{16}{15} \cdot - 16$

Step 3.1.1.1.10.3

Factor $4$ out of $- 16$ .

$\frac{4 (- 4)}{15} \cdot \frac{- 3 q}{4} - \frac{16}{15} \cdot - 10 = - \frac{16}{15} \cdot - 16$

Step 3.1.1.1.10.4

Cancel the common factor.

$\frac{4 \cdot - 4}{15} \cdot \frac{- 3 q}{4} - \frac{16}{15} \cdot - 10 = - \frac{16}{15} \cdot - 16$

Step 3.1.1.1.10.5

Rewrite the expression.

$\frac{- 4}{15} (- 3 q) - \frac{16}{15} \cdot - 10 = - \frac{16}{15} \cdot - 16$

Step 3.1.1.1.11

Cancel the common factor of $3$ .

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Step 3.1.1.1.11.1

Factor $3$ out of $15$ .

$\frac{- 4}{3 (5)} (- 3 q) - \frac{16}{15} \cdot - 10 = - \frac{16}{15} \cdot - 16$

Step 3.1.1.1.11.2

Factor $3$ out of $- 3 q$ .

$\frac{- 4}{3 (5)} (3 (- q)) - \frac{16}{15} \cdot - 10 = - \frac{16}{15} \cdot - 16$

Step 3.1.1.1.11.3

Cancel the common factor.

$\frac{- 4}{3 \cdot 5} (3 (- q)) - \frac{16}{15} \cdot - 10 = - \frac{16}{15} \cdot - 16$

Step 3.1.1.1.11.4

Rewrite the expression.

$\frac{- 4}{5} (- q) - \frac{16}{15} \cdot - 10 = - \frac{16}{15} \cdot - 16$

Step 3.1.1.1.12

Combine $\frac{- 4}{5}$ and $q$ .

$- \frac{- 4 q}{5} - \frac{16}{15} \cdot - 10 = - \frac{16}{15} \cdot - 16$

Step 3.1.1.1.13

Cancel the common factor of $5$ .

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Step 3.1.1.1.13.1

Move the leading negative in $- \frac{16}{15}$ into the numerator.

$- \frac{- 4 q}{5} + \frac{- 16}{15} \cdot - 10 = - \frac{16}{15} \cdot - 16$

Step 3.1.1.1.13.2

Factor $5$ out of $15$ .

$- \frac{- 4 q}{5} + \frac{- 16}{5 (3)} \cdot - 10 = - \frac{16}{15} \cdot - 16$

Step 3.1.1.1.13.3

Factor $5$ out of $- 10$ .

$- \frac{- 4 q}{5} + \frac{- 16}{5 \cdot 3} \cdot (5 \cdot - 2) = - \frac{16}{15} \cdot - 16$

Step 3.1.1.1.13.4

Cancel the common factor.

$- \frac{- 4 q}{5} + \frac{- 16}{5 \cdot 3} \cdot (5 \cdot - 2) = - \frac{16}{15} \cdot - 16$

Step 3.1.1.1.13.5

Rewrite the expression.

$- \frac{- 4 q}{5} + \frac{- 16}{3} \cdot - 2 = - \frac{16}{15} \cdot - 16$

Step 3.1.1.1.14

Combine $\frac{- 16}{3}$ and $- 2$ .

$- \frac{- 4 q}{5} + \frac{- 16 \cdot - 2}{3} = - \frac{16}{15} \cdot - 16$

Step 3.1.1.1.15

Multiply $- 16$ by $- 2$ .

$- \frac{- 4 q}{5} + \frac{32}{3} = - \frac{16}{15} \cdot - 16$

Step 3.1.1.2

Simplify each term.

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Step 3.1.1.2.1

Move the negative in front of the fraction.

$- - \frac{4 q}{5} + \frac{32}{3} = - \frac{16}{15} \cdot - 16$

Step 3.1.1.2.2

Multiply $- - \frac{4 q}{5}$ .

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Step 3.1.1.2.2.1

Multiply $- 1$ by $- 1$ .

$1 \frac{4 q}{5} + \frac{32}{3} = - \frac{16}{15} \cdot - 16$

Step 3.1.1.2.2.2

Multiply $\frac{4 q}{5}$ by $1$ .

$\frac{4 q}{5} + \frac{32}{3} = - \frac{16}{15} \cdot - 16$

Step 3.2

Simplify the right side.

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Step 3.2.1

Multiply $- \frac{16}{15} \cdot - 16$ .

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Step 3.2.1.1

Multiply $- 16$ by $- 1$ .

$\frac{4 q}{5} + \frac{32}{3} = 16 (\frac{16}{15})$

Step 3.2.1.2

Combine $16$ and $\frac{16}{15}$ .

$\frac{4 q}{5} + \frac{32}{3} = \frac{16 \cdot 16}{15}$

Step 3.2.1.3

Multiply $16$ by $16$ .

$\frac{4 q}{5} + \frac{32}{3} = \frac{256}{15}$

Step 4

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

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Step 4.1

Subtract $\frac{32}{3}$ from both sides of the equation.

$\frac{4 q}{5} = \frac{256}{15} - \frac{32}{3}$

Step 4.2

To write $- \frac{32}{3}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{4 q}{5} = \frac{256}{15} - \frac{32}{3} \cdot \frac{5}{5}$

Step 4.3

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

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Step 4.3.1

Multiply $\frac{32}{3}$ by $\frac{5}{5}$ .

$\frac{4 q}{5} = \frac{256}{15} - \frac{32 \cdot 5}{3 \cdot 5}$

Step 4.3.2

Multiply $3$ by $5$ .

$\frac{4 q}{5} = \frac{256}{15} - \frac{32 \cdot 5}{15}$

Step 4.4

Combine the numerators over the common denominator.

$\frac{4 q}{5} = \frac{256 - 32 \cdot 5}{15}$

Step 4.5

Simplify the numerator.

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Step 4.5.1

Multiply $- 32$ by $5$ .

$\frac{4 q}{5} = \frac{256 - 160}{15}$

Step 4.5.2

Subtract $160$ from $256$ .

$\frac{4 q}{5} = \frac{96}{15}$

Step 4.6

Cancel the common factor of $96$ and $15$ .

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Step 4.6.1

Factor $3$ out of $96$ .

$\frac{4 q}{5} = \frac{3 (32)}{15}$

Step 4.6.2

Cancel the common factors.

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Step 4.6.2.1

Factor $3$ out of $15$ .

$\frac{4 q}{5} = \frac{3 \cdot 32}{3 \cdot 5}$

Step 4.6.2.2

Cancel the common factor.

$\frac{4 q}{5} = \frac{3 \cdot 32}{3 \cdot 5}$

Step 4.6.2.3

Rewrite the expression.

$\frac{4 q}{5} = \frac{32}{5}$

Step 5

Since the expression on each side of the equation has the same denominator, the numerators must be equal.

$4 q = 32$

Step 6

Divide each term in $4 q = 32$ by $4$ and simplify.

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Step 6.1

Divide each term in $4 q = 32$ by $4$ .

$\frac{4 q}{4} = \frac{32}{4}$

Step 6.2

Simplify the left side.

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Step 6.2.1

Cancel the common factor of $4$ .

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Step 6.2.1.1

Cancel the common factor.

$\frac{4 q}{4} = \frac{32}{4}$

Step 6.2.1.2

Divide $q$ by $1$ .

$q = \frac{32}{4}$

Step 6.3

Simplify the right side.

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Step 6.3.1

Divide $32$ by $4$ .

$q = 8$