Gaussian Elimination

Gaussian elimination is used to solve linear systems of simultaneous equations by reducing the associated matrix of coefficients to upper triangular form, substitute values of variables obtained from the bottom row into the row next from bottom, then these two values into the next from bottom row and so on.
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.
\[x_1+2+2 \times 3=9 \rightarrow x_1 =1\]
.

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