Example, 3x3 system with Gauss

Let us apply Gaussian elimination to a $3\times 3$ system, working on the augmented matrix (coefficients plus constant terms) until it reaches triangular form.

Example — $3\times 3$ system with Gauss

$\begin{cases} x + y + z = 6 \\ 2x - y + 3z = 9 \\ -x + 2y + z = 5\end{cases} \quad\to\quad \begin{pmatrix}\begin{array}{ccc|c} 1 & 1 & 1 & 6 \\ 2 & -1 & 3 & 9 \\ -1 & 2 & 1 & 5\end{array}\end{pmatrix}.$ Step 1: $R_2\to R_2 - 2R_1$ and $R_3\to R_3 + R_1$. $\begin{pmatrix}\begin{array}{ccc|c} 1 & 1 & 1 & 6 \\ 0 & -3 & 1 & -3 \\ 0 & 3 & 2 & 11\end{array}\end{pmatrix}.$ Step 2: $R_3\to R_3 + R_2$. $\begin{pmatrix}\begin{array}{ccc|c} 1 & 1 & 1 & 6 \\ 0 & -3 & 1 & -3 \\ 0 & 0 & 3 & 8\end{array}\end{pmatrix}.$ Back-substitution: $z = 8/3$, $\;-3y + 8/3 = -3 \Rightarrow y = (3+8/3)/3 = 17/9$, $\;x = 6 - 17/9 - 8/3 = (54-17-24)/9 = 13/9$.

Check: $13/9 + 17/9 + 24/9 = 54/9 = 6$ ✓.

Links

Topics: Linear systems
Concepts: Gauss’s method · Triangular system
Methods: Gauss elimination
Skills: Calculate · Solve systems
People: Carl Friedrich Gauss