The determinant of the coefficient matrix is $-39k - 507$. For the system to be consistent, this determinant must not be equal to zero. Therefore, we set:

$-39k - 507 \neq 0$

Solving for $k$:

$k \neq \frac{-507}{-39} = 13$

Thus, for the system to be consistent, $k \neq 13$.

Would you like more details or have any questions? Here are 5 related questions to consider:

1. How do we calculate the determinant of a 3x3 matrix?
2. What does it mean for a system of equations to be inconsistent?
3. Can you explain the role of the determinant in determining the consistency of a system?
4. What happens if the determinant equals zero in a system of linear equations?
5. How does a parameter like $k$ affect the behavior of a system of equations?

Tip: For a system of equations to be consistent, its determinant must not be zero, ensuring a unique or infinite solution set.