To solve the equation
\[ \frac{2(x - 7)}{3} - 6 = 3, \]
let's follow these steps:
- Add 6 to both sides to eliminate the -6:
\[ \frac{2(x - 7)}{3} = 3 + 6 \]
\[ \frac{2(x - 7)}{3} = 9. \]
- Multiply both sides by 3 to eliminate the fraction:
\[ 2(x - 7) = 9 \times 3 \]
\[ 2(x - 7) = 27. \]
- Divide both sides by 2:
\[ x - 7 = \frac{27}{2}. \]
- Add 7 to both sides to solve for \(x\):
\[ x = \frac{27}{2} + 7. \]
- Convert 7 to a fraction with a common denominator of 2:
\[ 7 = \frac{14}{2}. \]
Thus:
\[ x = \frac{27}{2} + \frac{14}{2} = \frac{27 + 14}{2} = \frac{41}{2}. \]
Therefore, the solution is:
\[ x = \frac{41}{2}. \]