Final Answer:
The augmented matrix reduces to an inconsistent form, indicating that the system has no solution. The correct option is a) No solution.
Step-by-step explanation:
The given system of equations can be represented as an augmented matrix:
\[ \begin{bmatrix}
1 & -3 & 1 & \mid & 21 \\
-3 & -10 & -6 & \mid & -80 \\
-1 & 4 & -6 & \mid & -46 \\
\end{bmatrix} \]
Applying Gaussian elimination to row reduce the matrix, we obtain:
\[ \begin{bmatrix}
1 & -3 & 1 & \mid & 21 \\
0 & -1 & -3 & \mid & -17 \\
0 & 1 & -5 & \mid & -25 \\
\end{bmatrix} \]
Continuing the process:
\[ \begin{bmatrix}
1 & -3 & 1 & \mid & 21 \\
0 & -1 & -3 & \mid & -17 \\
0 & 0 & -8 & \mid & -8 \\
\end{bmatrix} \]
The last row implies \(0x + 0y - 8z = -8\), which simplifies to \(0 = -8\). This contradiction indicates an inconsistency in the system, meaning there is no set of values for \(x\), \(y\), and \(z\) that satisfies all three equations simultaneously. Therefore, the system has no solution, and the correct answer is (a) No solution.