To determine the value of \( k \) for which the system of equations has no solutions, we can represent the system in matrix form as follows:
\[
\begin{bmatrix}
1 & 1 & 3 \\
-4 & 2 & 5 \\
k & 0 & 1
\end{bmatrix}
\begin{bmatrix}
x \\ y \\ z
\end{bmatrix}
=
\begin{bmatrix}
10 \\ 7 \\ 3
\end{bmatrix}
\]
To analyze the conditions under which the system has no solution, we can use the concept of the rank of the coefficient matrix compared to the augmented matrix. Specifically, a system will have no solutions if the rank of the coefficient matrix is less than the rank of the augmented matrix.
First, we set up the augmented matrix:
\[
\left[\begin{array}{ccc|c}
1 & 1 & 3 & 10 \\
-4 & 2 & 5 & 7 \\
k & 0 & 1 & 3
\end{array}\right]
\]
Next, we will perform row operations to bring this matrix to row echelon form.
1. **Row 1** stays the same.
2. **Row 2** can be modified by adding 4 times Row 1 to it:
\[
R_2 = R_2 + 4R_1 \\
\Rightarrow -4 + 4(1) = 0, \\
2 + 4(1) = 6, \\
5 + 4(3) = 17, \\
7 + 4(10) = 47 \\
\Rightarrow R_2 = [0, 6, 17 | 47]
\]
3. Now, we will modify **Row 3**. To eliminate the first term in Row 3, we can do the following. Let's say we perform:
\[
R_3 = R_3 - kR_1
\]
This gives:
\[
R_3 = [k - k, 0 - k, 1 - 3k | 3 - 10k] \\
\Rightarrow R_3 = [0, -k, 1 - 3k | 3 - 10k]
\]
The new augmented matrix looks like this:
\[
\left[\begin{array}{ccc|c}
1 & 1 & 3 & 10 \\
0 & 6 & 17 & 47 \\
0 & -k & 1 - 3k & 3 - 10k
\end{array}\right]
\]
Next, we need to see when the rank of the coefficient matrix differs from the rank of the augmented matrix. For that, the third row has to be a nontrivial linear combination of the first two rows such that it produces a contradiction.
From the second row of the system, we can conclude that, as long as \( k \neq 0 \), Row 3 provides a new pivot. Thus, we must examine what happens when \( k = -3 \):
If \( k = -3 \)
\[
R_3 = [0, 3, 10 | 27] \quad \text{(after substituting \( k = -3 \))}
\]
Now, Row 3 should be checked against the other equations. If we solve with the last row resulting in a contradiction, we find inconsistent results leading to no solutions.
Thus, the value of \( k \) which results in this system having no solutions is:
\[
\boxed{-3}
\]
