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The system of equations you provided is:

\[
\begin{aligned}
-7x + 4y + 3z &= 5 \\
-5x + 3y + kz &= -2 \\
31x - 18y - 3z &= -8
\end{aligned}
\]

We are tasked with finding the condition on \( k \) for the system to be consistent.

### Approach:

To ensure consistency, the determinant of the coefficient matrix must not be zero, meaning the system has a unique solution. This matrix is:

\[
\begin{pmatrix}
-7 & 4 & 3 \\
-5 & 3 & k \\
31 & -18 & -3
\end{pmatrix}
\]

We will calculate the determinant of this matrix, and set the condition that it must not equal zero.

### Step-by-step:

\[
\text{Determinant} =
\begin{vmatrix}
-7 & 4 & 3 \\
-5 & 3 & k \\
31 & -18 & -3
\end{vmatrix}
\]

1. First, expand the determinant along the first row:

\[
\text{Determinant} = -7 \cdot 
\begin{vmatrix}
3 & k \\
-18 & -3
\end{vmatrix}
- 4 \cdot
\begin{vmatrix}
-5 & k \\
31 & -3
\end{vmatrix}
+ 3 \cdot
\begin{vmatrix}
-5 & 3 \\
31 & -18
\end{vmatrix}
\]

2. Compute the 2x2 determinants:
   - For \(\begin{vmatrix} 3 & k \\ -18 & -3 \end{vmatrix}\):
   
   \[
   3(-3) - (-18)(k) = -9 + 18k = 18k - 9
   \]
   
   - For \(\begin{vmatrix} -5 & k \\ 31 & -3 \end{vmatrix}\):
   
   \[
   (-5)(-3) - (k)(31) = 15 - 31k = 15 - 31k
   \]
   
   - For \(\begin{vmatrix} -5 & 3 \\ 31 & -18 \end{vmatrix}\):
   
   \[
   (-5)(-18) - (31)(3) = 90 - 93 = -3
   \]

3. Now substitute these values into the original expansion:

\[
\text{Determinant} = -7(18k - 9) - 4(15 - 31k) + 3(-3)
\]

4. Simplify each term:

\[
= -7(18k - 9) = -126k + 63
\]
\[
= -4(15 - 31k) = -60 + 124k
\]
\[
= 3(-3) = -9
\]

5. Combine all the terms:

\[
\text{Determinant} = (-126k + 63) + (-60 + 124k) - 9
\]

\[
= -126k + 124k + 63 - 60 - 9
\]

\[
= -2k - 6
\]

For the system to be consistent, the determinant should not be zero:

\[
-2k - 6 \neq 0
\]

Solve for \( k \):

\[
-2k \neq 6
\]

\[
k \neq -3
\]

### Conclusion:

The system will be consistent as long as \( k \neq -3 \).

---

Would you like more details or have any other questions?

Here are some follow-up questions to explore:

1. What happens to the system if \( k = -3 \)?
2. How would we find the solutions for the system if it is consistent?
3. Can we solve this system using row reduction techniques?
4. How does the determinant relate to the consistency of systems of equations?
5. What if we had a 4x4 system? How would the determinant affect consistency then?

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

For the following system to be consistent: -7x + 4y + 3z = 5, -5x + 3y + kz = -2, 31x - 18y - 3z = -8, we must have k ≠ ?

Learn how to determine the condition on k for the system of linear equations to be consistent. We explore the determinant of a 3x3 matrix and its role in the consistency of systems of equations. This problem is suitable for advanced high school and college students.

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