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To solve the given system of equations and determine the condition for consistency, we need to ensure that the determinant of the coefficient matrix is non-zero. This guarantees that the system has a unique solution, making it consistent.

The system of equations provided is:

\[
6x - 3y - 7z = -4
\]
\[
24x + 5y + kz = 3
\]
\[
9x + 4y - 3z = 5
\]

### Step 1: Coefficient Matrix

The coefficient matrix of the system is:

\[
A = \begin{pmatrix}
6 & -3 & -7 \\
24 & 5 & k \\
9 & 4 & -3
\end{pmatrix}
\]

For the system to be consistent, the determinant of the matrix must not be zero. Let’s compute the determinant of \( A \).

\[
\det(A) = \begin{vmatrix}
6 & -3 & -7 \\
24 & 5 & k \\
9 & 4 & -3
\end{vmatrix}
\]

We can expand this determinant along the first row:

\[
\det(A) = 6 \begin{vmatrix} 5 & k \\ 4 & -3 \end{vmatrix} - (-3) \begin{vmatrix} 24 & k \\ 9 & -3 \end{vmatrix} + (-7) \begin{vmatrix} 24 & 5 \\ 9 & 4 \end{vmatrix}
\]

Now I will compute the determinant for each 2x2 matrix.

Let me calculate that for you:The determinant of the matrix \( A \) is:

\[
\det(A) = -51k - 663
\]

### Step 2: Condition for Consistency

For the system to be consistent, the determinant must not be equal to zero:

\[
-51k - 663 \neq 0
\]

Solving for \( k \), we get:

\[
k \neq -\frac{663}{51} = -13
\]

Thus, the system will be consistent as long as \( k \neq -13 \).

Would you like further clarification on any part of this solution? Here are some related questions for further exploration:

1. How do you solve a system of equations using matrix methods like Cramer's Rule?
2. What does it mean for a system of equations to be consistent or inconsistent?
3. Can you explain how to find the inverse of a matrix when the determinant is non-zero?
4. How do you solve a system of linear equations using Gaussian elimination?
5. What are the applications of determinants in linear algebra?

**Tip**: Always check the determinant of a matrix when determining if a system of linear equations has a unique solution.

For the system of equations: 6x - 3y - 7z = -4, 24x + 5y + kz = 3, 9x + 4y - 3z = 5, determine the condition for consistency and find the value of k.

Explore how to determine the condition for consistency in a 3x3 system of linear equations with a parameter k. The solution involves calculating the determinant of the coefficient matrix and solving for the value of k. Suitable for students in Grades 11-12 and early college.

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