Use the properties of operations to add 9 ( k + 1/3 ) + 10 ( 1/2k - 1 )

To add the expression \( 9(k + \frac{1}{3}) + 10(\frac{1}{2}k - 1) \), we will use the distributive property and then combine like terms. Here's the step-by-step breakdown:

1. Distribute the 9 and the 10:

   \[ 9(k + \frac{1}{3}) = 9k + 9 \cdot \frac{1}{3} = 9k + 3 \]

   \[ 10(\frac{1}{2}k - 1) = 10 \cdot \frac{1}{2}k - 10 \cdot 1 = 5k - 10 \]

2. Combine the results:

   \[ 9k + 3 + 5k - 10 \]

3. Combine like terms:

   • Combine the \( k \) terms: \[ 9k + 5k = 14k \]

   • Combine the constant terms: \[ 3 - 10 = -7 \]

4. Put it all together:

   \[ 14k - 7 \]

So, the final result is:

\[ \boxed{14k - 7} \]