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To determine the value of \( k \) that makes the system consistent, let's first write down the system of linear equations:

\[
\begin{aligned}
    3x - 4y + 7z &= -3 \\
    -7x + 12y + kz &= -6 \\
    2x - 4y + 3z &= 5
\end{aligned}
\]

The system will be inconsistent if there is no solution, which usually happens if we get contradictory statements from the rows in the augmented matrix. To avoid such contradictions, we need to check the conditions under which the determinant of the coefficient matrix is non-zero. This would ensure the existence of a unique solution and hence consistency of the system.

The coefficient matrix is:

\[
A = \begin{pmatrix}
    3 & -4 & 7 \\
    -7 & 12 & k \\
    2 & -4 & 3
\end{pmatrix}
\]

We need to calculate the determinant of this matrix and ensure it's not zero for consistency.

Let me calculate the determinant of \( A \) now.The determinant of the coefficient matrix \( A \) is:

\[
\text{det}(A) = 4k + 52
\]

For the system to be consistent, the determinant should not be zero. Therefore, we solve:

\[
4k + 52 \neq 0
\]

\[
k \neq -13
\]

Thus, for the system to be consistent, \( k \neq -13 \).

Would you like more details or have any questions? Here are some related questions for further practice:

1. How do you solve a system of equations using Cramer's rule?
2. What does it mean for a system of equations to be inconsistent?
3. How does the determinant of a matrix relate to the invertibility of the matrix?
4. What methods can you use to solve a system of linear equations if the determinant is zero?
5. How can you express the solution of a system of linear equations geometrically?

**Tip:** When dealing with systems of linear equations, checking the determinant of the coefficient matrix can quickly help determine whether a unique solution exists (i.e., if the system is consistent and has a solution).

For the following system to be consistent, 3x - 4y + 7z = -3, -7x + 12y + kz = -6, 2x - 4y + 3z = 5, we must have k ≠ ?

Explore how to solve a system of linear equations by determining the value of k that ensures consistency. This includes calculating the determinant of a 3x3 matrix. Suitable for undergraduate-level students.

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