Solve for \( m \). \( 2(3 m+1)-\frac{1}{2} m=\frac{1}{3}(5+9 m) \)

To solve for \( m \) in the equation \( 2(3 m+1)-\frac{1}{2} m=\frac{1}{3}(5+9 m) \), we begin by simplifying both sides. First, expand the left side: \[ 2(3m + 1) = 6m + 2 \] Thus, the left side becomes: \[ 6m + 2 - \frac{1}{2} m \] To combine the terms involving \( m \): \[ 6m - \frac{1}{2} m = \frac{12m}{2} - \frac{1m}{2} = \frac{11m}{2} \] So, the left side can be rewritten as: \[ \frac{11m}{2} + 2 \] Now, simplifying the right side: \[ \frac{1}{3}(5 + 9m) = \frac{5}{3} + 3m \] Now we set the two simplified expressions equal to each other: \[ \frac{11m}{2} + 2 = \frac{5}{3} + 3m \] Now, to eliminate the fractions, we find a common denominator. The least common multiple of 2 and 3 is 6. Multiply every term by 6: \[ 6(\frac{11m}{2}) + 6(2) = 6(\frac{5}{3}) + 6(3m) \] This simplifies to: \[ 33m + 12 = 10 + 18m \] Now, moving all terms involving \( m \) to one side and constant terms to the other side: \[ 33m - 18m = 10 - 12 \] This simplifies to: \[ 15m = -2 \] Now, solve for \( m \): \[ m = -\frac{2}{15} \] Thus, the solution for \( m \) is: \[ \boxed{-\frac{2}{15}} \]