Example:
Consider the system of equations
\[x_1+x_2+2x_3=9\]
\[2x_1+4x_2-3x_3=1\]
\[3x_1+6x_2-5x_3=0\]
The augmented matrix associated with this system is
\[ \left( \begin{array}{ccc} 1 & 1 & 2 \\ 2 & 4 & -3 \\ 3 & 6 & -5 \end{array} \right| \left| \begin{array}{c} 9 \\ 1 \\ 0 \end{array} \right)\]
Subtract twice the first row from the second and three times the first row from the third..
\[ \left( \begin{array}{ccc} 1 & 1 & 2 \\ 0 & 2 & -7 \\ 0 & 3 & -11 \end{array} \right| \left| \begin{array}{c} 9 \\ -17 \\ -27 \end{array} \right) \]
\[ \left( \begin{array}{ccc} 1 & 1 & 2 \\ 0 & 1 & -7/2 \\ 0 & 3 & -11 \end{array} \right| \left| \begin{array}{c} 9 \\ -17/2 \\ -27 \end{array} \right)\]
Subtract one and a half times the second row from the third.
\[ \left( \begin{array}{ccc} 1 & 1 & 2 \\ 0 & 2 & -7 \\ 0 & 0 & -1/2 &\end{array} \right| \left| \begin{array}{c} 9 \\ -17 \\ -3/2 \end{array} \right)\]
This is equivalent to the system of equations
\[x_1+x_2+2x_3=9\]
\[2x_2-7x_3=-17\]
\[-1/2x_3=-3/2\]
The third equation gives
\[x_3=3\]
Substitute this into the second equation..
\[2x_2-7 \times 3=-17 \rightarrow x_2 =2\]
.Substitute
\[x_2=2, \: x_3 =3\]
into the first.\[x_1+2+2 \times 3=9 \rightarrow x_1 =1\]
.