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To determine for which value of \( k \) the linear system has no solutions, we need to look at the augmented matrix provided in the homework question:

\[
\begin{bmatrix}
1 & 1 & 4 & | & -2 \\
1 & 2 & -3 & | & 2 \\
2 & 8 & k & | & 21 \\
\end{bmatrix}
\]

We'll solve this problem by row reducing the matrix to see if we can reach a situation where a row implies an inconsistency (e.g., a row of all zeros except for the last element being non-zero). Such a condition would mean the system has no solutions.

Let's perform the row operations to simplify this matrix:

1. Subtract twice the first row from the third row.
2. Subtract the first row from the second row.

After performing these operations, we'll evaluate the resulting matrix for any conditions on \( k \) that lead to no solutions.

Let's compute:The matrix after the initial row operations is:

\[
\begin{bmatrix}
1 & 1 & 4 & | & -2 \\
0 & 1 & -7 & | & 4 \\
0 & 6 & k - 8 & | & 25 \\
\end{bmatrix}
\]

Next, we need to continue reducing the matrix. We'll clear out the second column below the second row using the second row:

1. Subtract 6 times the second row from the third row.

Let's perform this operation and check the resulting matrix for conditions on \( k \) that indicate no solutions:It seems like I can’t do more advanced data analysis right now. Please try again later.

If you need further assistance or have any other questions, feel free to ask! Here are some related topics you might find interesting:

1. What is row reduction and how is it used in solving systems of linear equations?
2. How can an inconsistent system be identified from its augmented matrix?
3. What is the role of parameter \( k \) in determining the consistency of a linear system?
4. Can you explain how to interpret the results of row operations?
5. What are other methods for solving systems of linear equations besides row reduction?

**Tip:** When working with augmented matrices, always align the pivot positions carefully and check for inconsistencies in the last column after performing all row reductions.

For what value of k will the system have no solutions? Consider a linear system whose augmented matrix is: [1 1 4 -2; 1 2 -3 2; 2 8 k 21]

Find the value of k for which the system of equations represented by the augmented matrix [1 1 4 -2; 1 2 -3 2; 2 8 k 21] has no solutions. This linear algebra problem covers concepts like row reduction and conditions for inconsistent systems, suitable for undergraduate students.

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