Solve for q 1/3(3/4q+5)=3q-3/2(2q-1)

$\frac{1}{3} (\frac{3}{4} q + 5) = 3 q - \frac{3}{2} (2 q - 1)$

Step 1

Since $q$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$3 q - \frac{3}{2} \cdot (2 q - 1) = \frac{1}{3} \cdot (\frac{3}{4} q + 5)$

Step 2

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

Simplify each term.

Apply the distributive property.

$3 q - \frac{3}{2} (2 q) - \frac{3}{2} \cdot - 1 = \frac{1}{3} \cdot (\frac{3}{4} q + 5)$

Cancel the common factor of $2$ .

Move the leading negative in $- \frac{3}{2}$ into the numerator.

$3 q + \frac{- 3}{2} (2 q) - \frac{3}{2} \cdot - 1 = \frac{1}{3} \cdot (\frac{3}{4} q + 5)$

Factor $2$ out of $2 q$ .

$3 q + \frac{- 3}{2} (2 (q)) - \frac{3}{2} \cdot - 1 = \frac{1}{3} \cdot (\frac{3}{4} q + 5)$

Cancel the common factor.

$3 q + \frac{- 3}{2} (2 q) - \frac{3}{2} \cdot - 1 = \frac{1}{3} \cdot (\frac{3}{4} q + 5)$

Rewrite the expression.

$3 q - 3 q - \frac{3}{2} \cdot - 1 = \frac{1}{3} \cdot (\frac{3}{4} q + 5)$

Multiply $- \frac{3}{2} \cdot - 1$ .

Multiply $- 1$ by $- 1$ .

$3 q - 3 q + 1 (\frac{3}{2}) = \frac{1}{3} \cdot (\frac{3}{4} q + 5)$

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

$3 q - 3 q + \frac{3}{2} = \frac{1}{3} \cdot (\frac{3}{4} q + 5)$

Simplify by adding terms.

Subtract $3 q$ from $3 q$ .

$0 + \frac{3}{2} = \frac{1}{3} \cdot (\frac{3}{4} q + 5)$

Add $0$ and $\frac{3}{2}$ .

$\frac{3}{2} = \frac{1}{3} \cdot (\frac{3}{4} q + 5)$

Step 3

Simplify $\frac{1}{3} \cdot (\frac{3}{4} q + 5)$ .

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

$\frac{3}{2} = \frac{1}{3} \cdot (\frac{3 q}{4} + 5)$

Apply the distributive property.

$\frac{3}{2} = \frac{1}{3} \cdot \frac{3 q}{4} + \frac{1}{3} \cdot 5$

Cancel the common factor of $3$ .

Factor $3$ out of $3 q$ .

$\frac{3}{2} = \frac{1}{3} \cdot \frac{3 (q)}{4} + \frac{1}{3} \cdot 5$

Cancel the common factor.

$\frac{3}{2} = \frac{1}{3} \cdot \frac{3 q}{4} + \frac{1}{3} \cdot 5$

Rewrite the expression.

$\frac{3}{2} = \frac{q}{4} + \frac{1}{3} \cdot 5$

Combine $\frac{1}{3}$ and $5$ .

$\frac{3}{2} = \frac{q}{4} + \frac{5}{3}$

Step 4

Since $q$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$\frac{q}{4} + \frac{5}{3} = \frac{3}{2}$

Step 5

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

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

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

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

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

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

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

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

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

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

Multiply $2$ by $3$ .

$\frac{q}{4} = \frac{3 \cdot 3}{6} - \frac{5}{3} \cdot \frac{2}{2}$

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

$\frac{q}{4} = \frac{3 \cdot 3}{6} - \frac{5 \cdot 2}{3 \cdot 2}$

Multiply $3$ by $2$ .

$\frac{q}{4} = \frac{3 \cdot 3}{6} - \frac{5 \cdot 2}{6}$

Combine the numerators over the common denominator.

$\frac{q}{4} = \frac{3 \cdot 3 - 5 \cdot 2}{6}$

Simplify the numerator.

Multiply $3$ by $3$ .

$\frac{q}{4} = \frac{9 - 5 \cdot 2}{6}$

Multiply $- 5$ by $2$ .

$\frac{q}{4} = \frac{9 - 10}{6}$

Subtract $10$ from $9$ .

$\frac{q}{4} = \frac{- 1}{6}$

Move the negative in front of the fraction.

$\frac{q}{4} = - \frac{1}{6}$

Step 6

Multiply both sides of the equation by $4$ .

$4 \frac{q}{4} = 4 (- \frac{1}{6})$

Step 7

Simplify both sides of the equation.

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$4 \frac{q}{4} = 4 (- \frac{1}{6})$

Rewrite the expression.

$q = 4 (- \frac{1}{6})$

Simplify the right side.

Simplify $4 (- \frac{1}{6})$ .

Cancel the common factor of $2$ .

Move the leading negative in $- \frac{1}{6}$ into the numerator.

$q = 4 (\frac{- 1}{6})$

Factor $2$ out of $4$ .

$q = 2 (2) \frac{- 1}{6}$

Factor $2$ out of $6$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Combine $2$ and $\frac{- 1}{3}$ .

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

Simplify the expression.

Multiply $2$ by $- 1$ .

$q = \frac{- 2}{3}$

Move the negative in front of the fraction.

$q = - \frac{2}{3}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$q = - \frac{2}{3}$

Decimal Form:

$q = - 0.\overline{6}$

Solve for q 1/3(3/4q+5)=3q-3/2(2q-1)

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