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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.

For the following system to be consistent, 3x + 5y - 7z = 4, -9x + 11y + kz = -1, -6x + 3y - 3z = -2, we must have k ≠ ?

Learn how to determine the value of k that ensures consistency for a system of linear equations. The determinant of the coefficient matrix plays a crucial role, and in this problem, we derive k ≠ 13 for consistency. Suitable for Grades 10-12.

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