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In order for the system of equation above to be a consistent system, k must be equal to what?

To determine the value of \( k \) that makes the system of equations consistent, we need to ensure that the three planes represented by these equations intersect at least at one common point.

Let's denote the system of equations as:
\[
\begin{cases}
x - 4y + 7z = -1 \\
-3x + 17y - 16z = -12 \\
-4x + 21y - 23z = k
\end{cases}
\]

We will solve this by using methods like Gaussian elimination or analyzing the determinant of the coefficient matrix. First, we will express the system in matrix form as \( AX = B \).

### Step 1: Matrix Representation

The coefficient matrix \( A \) and the augmented matrix \( [A|B] \) are:
\[
A = \begin{bmatrix} 
1 & -4 & 7 \\ 
-3 & 17 & -16 \\ 
-4 & 21 & -23 
\end{bmatrix}, \quad
B = \begin{bmatrix} -1 \\ -12 \\ k \end{bmatrix}
\]

### Step 2: Condition for Consistency

The system is consistent if and only if the determinant of the coefficient matrix \( A \) is non-zero or if \( k \) is chosen such that the rank of the augmented matrix \( [A|B] \) is equal to the rank of \( A \). Let's calculate the determinant of \( A \):

\[
\text{det}(A) = \begin{vmatrix} 
1 & -4 & 7 \\ 
-3 & 17 & -16 \\ 
-4 & 21 & -23 
\end{vmatrix}
\]

I'll perform this calculation to find \( \text{det}(A) \).The determinant of matrix \( A \) is \( 0 \). This indicates that the coefficient matrix is singular, meaning the system could be either inconsistent or dependent.

### Step 3: Analyzing Consistency
Since \(\text{det}(A) = 0\), the system is consistent if the third equation is a linear combination of the first two. We need to ensure that the third row of the augmented matrix \( [A|B] \) can be written as a linear combination of the first two rows, including the value of \( k \).

Let's find the relationship between the rows to determine \( k \).The row-reduced echelon form (RREF) of the augmented matrix is:

\[
\begin{bmatrix}
1 & 0 & 11 & 0 \\
0 & 1 & 1 & 0 \\
0 & 0 & 0 & 1
\end{bmatrix}
\]

This form reveals that, for the system to be consistent, the right-hand side of the third row should be zero. Therefore, the value of \( k \) must satisfy:

\[
k = 0
\]

So, the system is consistent when \( k = 0 \).

Would you like more details on any of these steps, or do you have any further questions?

### Related Questions
1. What happens if \( k \) is not zero?
2. Can you describe the geometric interpretation of a consistent system of linear equations?
3. How would the solution set change if \( k \) is different?
4. Can you find the general solution for the consistent system?
5. What are the implications of having a singular coefficient matrix?

**Tip**: When the determinant of a matrix is zero, always check for dependencies between the rows to establish consistency conditions.

In order for the system of equations x - 4y + 7z = -1, -3x + 17y - 16z = -12, -4x + 21y - 23z = k to be a consistent system, k must be equal to what?

Learn how to determine the value of k for a system of linear equations to be consistent. This problem involves key concepts in linear algebra, including determinants, rank, and the consistency of systems. Suitable for students in Grades 10-12.

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