Algebra Examples

$- q + \frac{4}{3} (q - 2) + 1 = \frac{2}{9} q - 4$

Step 1

Simplify $- q + \frac{4}{3} \cdot (q - 2) + 1$ .

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Step 1.1

Simplify each term.

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Step 1.1.1

Apply the distributive property.

$- q + \frac{4}{3} q + \frac{4}{3} \cdot - 2 + 1 = \frac{2}{9} q - 4$

Step 1.1.2

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

$- q + \frac{4 q}{3} + \frac{4}{3} \cdot - 2 + 1 = \frac{2}{9} q - 4$

Step 1.1.3

Multiply $\frac{4}{3} \cdot - 2$ .

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Step 1.1.3.1

Combine $\frac{4}{3}$ and $- 2$ .

$- q + \frac{4 q}{3} + \frac{4 \cdot - 2}{3} + 1 = \frac{2}{9} q - 4$

Step 1.1.3.2

Multiply $4$ by $- 2$ .

$- q + \frac{4 q}{3} + \frac{- 8}{3} + 1 = \frac{2}{9} q - 4$

Step 1.1.4

Move the negative in front of the fraction.

$- q + \frac{4 q}{3} - \frac{8}{3} + 1 = \frac{2}{9} q - 4$

Step 1.2

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

$- q \cdot \frac{3}{3} + \frac{4 q}{3} - \frac{8}{3} + 1 = \frac{2}{9} q - 4$

Step 1.3

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

$\frac{- q \cdot 3}{3} + \frac{4 q}{3} - \frac{8}{3} + 1 = \frac{2}{9} q - 4$

Step 1.4

Combine the numerators over the common denominator.

$\frac{- q \cdot 3 + 4 q}{3} - \frac{8}{3} + 1 = \frac{2}{9} q - 4$

Step 1.5

Combine the numerators over the common denominator.

$1 + \frac{- q \cdot 3 + 4 q - 8}{3} = \frac{2}{9} q - 4$

Step 1.6

Multiply $3$ by $- 1$ .

$1 + \frac{- 3 q + 4 q - 8}{3} = \frac{2}{9} q - 4$

Step 1.7

Add $- 3 q$ and $4 q$ .

$1 + \frac{q - 8}{3} = \frac{2}{9} q - 4$

Step 1.8

Write $1$ as a fraction with a common denominator.

$\frac{3}{3} + \frac{q - 8}{3} = \frac{2}{9} q - 4$

Step 1.9

Combine the numerators over the common denominator.

$\frac{3 + q - 8}{3} = \frac{2}{9} q - 4$

Step 1.10

Subtract $8$ from $3$ .

$\frac{q - 5}{3} = \frac{2}{9} q - 4$

Step 2

Combine $\frac{2}{9}$ and $q$ .

$\frac{q - 5}{3} = \frac{2 q}{9} - 4$

Step 3

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

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Step 3.1

Subtract $\frac{2 q}{9}$ from both sides of the equation.

$\frac{q - 5}{3} - \frac{2 q}{9} = - 4$

Step 3.2

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

$\frac{q - 5}{3} \cdot \frac{3}{3} - \frac{2 q}{9} = - 4$

Step 3.3

Write each expression with a common denominator of $9$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.3.1

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

$\frac{(q - 5) \cdot 3}{3 \cdot 3} - \frac{2 q}{9} = - 4$

Step 3.3.2

Multiply $3$ by $3$ .

$\frac{(q - 5) \cdot 3}{9} - \frac{2 q}{9} = - 4$

Step 3.4

Combine the numerators over the common denominator.

$\frac{(q - 5) \cdot 3 - 2 q}{9} = - 4$

Step 3.5

Simplify the numerator.

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Step 3.5.1

Apply the distributive property.

$\frac{q \cdot 3 - 5 \cdot 3 - 2 q}{9} = - 4$

Step 3.5.2

Move $3$ to the left of $q$ .

$\frac{3 \cdot q - 5 \cdot 3 - 2 q}{9} = - 4$

Step 3.5.3

Multiply $- 5$ by $3$ .

$\frac{3 \cdot q - 15 - 2 q}{9} = - 4$

Step 3.5.4

Subtract $2 q$ from $3 q$ .

$\frac{q - 15}{9} = - 4$

Step 4

Multiply both sides of the equation by $9$ .

$9 \frac{q - 15}{9} = 9 \cdot - 4$

Step 5

Simplify both sides of the equation.

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Step 5.1

Simplify the left side.

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Step 5.1.1

Cancel the common factor of $9$ .

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Step 5.1.1.1

Cancel the common factor.

$9 \frac{q - 15}{9} = 9 \cdot - 4$

Step 5.1.1.2

Rewrite the expression.

$q - 15 = 9 \cdot - 4$

Step 5.2

Simplify the right side.

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Step 5.2.1

Multiply $9$ by $- 4$ .

$q - 15 = - 36$

Step 6

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

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Step 6.1

Add $15$ to both sides of the equation.

$q = - 36 + 15$

Step 6.2

Add $- 36$ and $15$ .

$q = - 21$