To simplify the expression \( 9\left(k + \frac{1}{3}\right) + 10\left(\frac{1}{2}k - 1\right) \) using the properties of operations, we can follow these steps:
- Distribute the 9 and the 10 into their respective expressions:
\[ 9(k + \frac{1}{3}) = 9k + 9 \cdot \frac{1}{3} = 9k + 3 \]
\[ 10\left(\frac{1}{2}k - 1\right) = 10 \cdot \frac{1}{2}k - 10 \cdot 1 = 5k - 10 \]
- Combine the results:
\[ 9k + 3 + 5k - 10 \]
- Combine like terms:
- Combine the \(k\) terms: \(9k + 5k = 14k\)
- Combine the constant terms: \(3 - 10 = -7\)
Putting it all together, we have:
\[ 14k - 7 \]
Thus, the result of adding \( 9(k + \frac{1}{3}) + 10(\frac{1}{2}k - 1) \) is
\[ \boxed{14k - 7} \]