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The problem asks us to determine the values of \( k \) such that the given matrix represents an augmented matrix of a consistent linear system. The augmented matrix is:

\[
\begin{pmatrix}
1 & k & -3 \\
-2 & 4 & 6
\end{pmatrix}
\]

This corresponds to the system of equations:

\[
\begin{aligned}
1 \cdot x_1 + k \cdot x_2 &= -3, \\
-2 \cdot x_1 + 4 \cdot x_2 &= 6.
\end{aligned}
\]

To determine the values of \( k \) that make this system consistent, we need to check for conditions where the system has no solutions or infinitely many solutions, which are tied to the matrix's row equivalence.

We can use Gaussian elimination to simplify the system and check for values of \( k \) that result in inconsistency.

Let's proceed with the row reduction process to analyze the consistency of the system.After performing Gaussian elimination, the augmented matrix in reduced row echelon form (RREF) is:

\[
\begin{pmatrix}
1 & 0 & -3 \\
0 & 1 & 0
\end{pmatrix}
\]

This shows that the system of equations is consistent for all values of \( k \). The matrix has no rows that result in an inconsistency (e.g., a row like \( [0 \, 0 \, b] \) where \( b \neq 0 \)).

Thus, **the system is consistent for all values of \( k \)**.

Would you like a detailed explanation or have any questions?

Here are 5 related questions to expand on this topic:
1. What are the conditions for a linear system to be inconsistent?
2. How is Gaussian elimination applied to augmented matrices in general?
3. What is the difference between a consistent and an inconsistent system?
4. How can you determine if a system has infinitely many solutions?
5. How does row reduction help identify the rank of a matrix?

**Tip:** Gaussian elimination is an essential tool for solving systems of equations and determining consistency or inconsistency efficiently.

Determine the values of k such that the matrix is the augmented matrix of a consistent linear system: [ [ 1, k, -3 ], [ -2, 4, 6 ] ]

Find the values of k that make the matrix [ [ 1, k, -3 ], [ -2, 4, 6 ] ] represent a consistent linear system. This problem involves Gaussian elimination, row reduction, and concepts from linear algebra. Suitable for students in Grades 10-12.

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