Solve for q 3/2-5/4q+2/3q-5/6=-q

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

Step 1

Simplify $\frac{3}{2} - \frac{5}{4} q + \frac{2}{3} q - \frac{5}{6}$ .

Simplify each term.

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

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

Move $5$ to the left of $q$ .

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

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

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

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

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

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

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

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

Multiply $2$ by $3$ .

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Multiply $3$ by $3$ .

$- \frac{5 q}{4} + \frac{2 q}{3} + \frac{9 - 5}{6} = - q$

Subtract $5$ from $9$ .

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

Cancel the common factor of $4$ and $6$ .

Factor $2$ out of $4$ .

$- \frac{5 q}{4} + \frac{2 q}{3} + \frac{2 (2)}{6} = - q$

Cancel the common factors.

Factor $2$ out of $6$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

To write $- \frac{5 q}{4}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

To write $\frac{2 q}{3}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

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

Write each expression with a common denominator of $12$ , by multiplying each by an appropriate factor of $1$ .

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

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

Multiply $4$ by $3$ .

$- \frac{5 q \cdot 3}{12} + \frac{2 q}{3} \cdot \frac{4}{4} + \frac{2}{3} = - q$

Multiply $\frac{2 q}{3}$ by $\frac{4}{4}$ .

$- \frac{5 q \cdot 3}{12} + \frac{2 q \cdot 4}{3 \cdot 4} + \frac{2}{3} = - q$

Multiply $3$ by $4$ .

$- \frac{5 q \cdot 3}{12} + \frac{2 q \cdot 4}{12} + \frac{2}{3} = - q$

Combine the numerators over the common denominator.

$\frac{- 5 q \cdot 3 + 2 q \cdot 4}{12} + \frac{2}{3} = - q$

Simplify each term.

Simplify the numerator.

Factor $q$ out of $- 5 q \cdot 3 + 2 q \cdot 4$ .

Factor $q$ out of $- 5 q \cdot 3$ .

$\frac{q (- 5 \cdot 3) + 2 q \cdot 4}{12} + \frac{2}{3} = - q$

Factor $q$ out of $2 q \cdot 4$ .

$\frac{q (- 5 \cdot 3) + q (2 \cdot 4)}{12} + \frac{2}{3} = - q$

Factor $q$ out of $q (- 5 \cdot 3) + q (2 \cdot 4)$ .

$\frac{q (- 5 \cdot 3 + 2 \cdot 4)}{12} + \frac{2}{3} = - q$

Multiply $- 5$ by $3$ .

$\frac{q (- 15 + 2 \cdot 4)}{12} + \frac{2}{3} = - q$

Multiply $2$ by $4$ .

$\frac{q (- 15 + 8)}{12} + \frac{2}{3} = - q$

Add $- 15$ and $8$ .

$\frac{q \cdot - 7}{12} + \frac{2}{3} = - q$

Move $- 7$ to the left of $q$ .

$\frac{- 7 \cdot q}{12} + \frac{2}{3} = - q$

Move the negative in front of the fraction.

$- \frac{7 q}{12} + \frac{2}{3} = - q$

Step 2

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

Add $q$ to both sides of the equation.

$- \frac{7 q}{12} + \frac{2}{3} + q = 0$

To write $q$ as a fraction with a common denominator, multiply by $\frac{12}{12}$ .

$- \frac{7 q}{12} + q \cdot \frac{12}{12} + \frac{2}{3} = 0$

Combine $q$ and $\frac{12}{12}$ .

$- \frac{7 q}{12} + \frac{q \cdot 12}{12} + \frac{2}{3} = 0$

Combine the numerators over the common denominator.

$\frac{- 7 q + q \cdot 12}{12} + \frac{2}{3} = 0$

Add $- 7 q$ and $q \cdot 12$ .

Reorder $q$ and $12$ .

$\frac{- 7 q + 12 \cdot q}{12} + \frac{2}{3} = 0$

Add $- 7 q$ and $12 \cdot q$ .

$\frac{5 q}{12} + \frac{2}{3} = 0$

Step 3

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

$\frac{5 q}{12} = - \frac{2}{3}$

Step 4

Multiply both sides of the equation by $\frac{12}{5}$ .

$\frac{12}{5} \cdot \frac{5 q}{12} = \frac{12}{5} (- \frac{2}{3})$

Step 5

Simplify both sides of the equation.

Simplify the left side.

Simplify $\frac{12}{5} \cdot \frac{5 q}{12}$ .

Cancel the common factor of $12$ .

Cancel the common factor.

$\frac{12}{5} \cdot \frac{5 q}{12} = \frac{12}{5} (- \frac{2}{3})$

Rewrite the expression.

$\frac{1}{5} (5 q) = \frac{12}{5} (- \frac{2}{3})$

Cancel the common factor of $5$ .

Factor $5$ out of $5 q$ .

$\frac{1}{5} (5 (q)) = \frac{12}{5} (- \frac{2}{3})$

Cancel the common factor.

$\frac{1}{5} (5 q) = \frac{12}{5} (- \frac{2}{3})$

Rewrite the expression.

$q = \frac{12}{5} (- \frac{2}{3})$

Simplify the right side.

Simplify $\frac{12}{5} (- \frac{2}{3})$ .

Cancel the common factor of $3$ .

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

$q = \frac{12}{5} \cdot \frac{- 2}{3}$

Factor $3$ out of $12$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Combine $\frac{4}{5}$ and $- 2$ .

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

Simplify the expression.

Multiply $4$ by $- 2$ .

$q = \frac{- 8}{5}$

Move the negative in front of the fraction.

$q = - \frac{8}{5}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$q = - \frac{8}{5}$

Decimal Form:

$q = - 1.6$

Mixed Number Form:

$q = - 1 \frac{3}{5}$

Solve for q 3/2-5/4q+2/3q-5/6=-q

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