To simplify the expression \(9(k+\frac{1}{3}) + 10(\frac{1}{2}k - 1)\), we will distribute and combine like terms.
First, distribute \(9\) and \(10\):
\[ 9(k + \frac{1}{3}) = 9k + 3 \]
\[ 10(\frac{1}{2}k - 1) = 5k - 10 \]
Now, we combine these results:
\[ 9k + 3 + 5k - 10 \]
Combine like terms:
\[ (9k + 5k) + (3 - 10) = 14k - 7 \]
Thus, the final simplified expression is:
\[ \boxed{14k - 7} \]