Q7. Solve using Gaussian elimination:
\(\begin{cases}-3x_1-x_2+x_3=-11 \\ 2x_1+4x_2-5x_3=12 \\ x_1-2x_2+3x_3=1\end{cases}\)
\(x_1 = 3\), \(x_2 = 4\), \(x_3 = 2\)
Form the augmented matrix, use row operations to get upper-triangular, then back-substitute.
Start: \( \begin{bmatrix} -3 & -1 & 1 & -11 \\ 2 & 4 & -5 & 12 \\ 1 & -2 & 3 & 1 \end{bmatrix} \)
After row reduction to RREF, we get: \( \begin{bmatrix} 1 & 0 & 0 & 3 \\ 0 & 1 & 0 & 4 \\ 0 & 0 & 1 & 2 \end{bmatrix} \)
This gives the solution \(x_1 = 3\), \(x_2 = 4\), \(x_3 = 2\).