Solving a 3x3 system with Gauss

Problem

Solve using the Gauss method: $\begin{cases} x+2y-z=3 \\ 2x-y+z=1 \\ x+y+z=4\end{cases}$

Solution

Write the augmented matrix and bring it to triangular form: $\begin{pmatrix}\begin{array}{ccc|c} 1 & 2 & -1 & 3 \\ 2 & -1 & 1 & 1 \\ 1 & 1 & 1 & 4\end{array}\end{pmatrix}.$ Step 1: $R_2\to R_2 - 2R_1$ and $R_3\to R_3 - R_1$. $\begin{pmatrix}\begin{array}{ccc|c} 1 & 2 & -1 & 3 \\ 0 & -5 & 3 & -5 \\ 0 & -1 & 2 & 1\end{array}\end{pmatrix}.$ Step 2: $R_3\to R_3 - \tfrac{1}{5}R_2$. $\begin{pmatrix}\begin{array}{ccc|c} 1 & 2 & -1 & 3 \\ 0 & -5 & 3 & -5 \\ 0 & 0 & \tfrac{7}{5} & 2\end{array}\end{pmatrix}.$ Back-substitution: $\tfrac{7}{5}z = 2 \Rightarrow z = \tfrac{10}{7}$; $\;-5y + 3z = -5 \Rightarrow y = \tfrac{13}{7}$; $\;x = 3 - 2y + z = \tfrac{5}{7}$. $\boxed{x = \tfrac{5}{7}, \quad y = \tfrac{13}{7}, \quad z = \tfrac{10}{7}}$ Check in the third equation: $\tfrac{5}{7}+\tfrac{13}{7}+\tfrac{10}{7} = \tfrac{28}{7} = 4$ ✓.

Links

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