UPSC 2022 Maths Optional Paper 2 Q5b — Step-by-Step Solution

10 marks · Section B

Gaussian Elimination · Numerical Analysis · Read the full method →

Question

Solve by Gauss elimination: $2x+2y+4z=18$ , $x+3y+2z=13$ , $3x+y+3z=14$ .

Standard Gauss elimination with normalization (divide by leading coefficient) at each step.

\left(\begin{array}{ccc|c}2 & 2 & 4 & 18\\ 1 & 3 & 2 & 13\\ 3 & 1 & 3 & 14\end{array}\right).

$R_1\to R_1/2$ :

\left(\begin{array}{ccc|c}1 & 1 & 2 & 9\\ 1 & 3 & 2 & 13\\ 3 & 1 & 3 & 14\end{array}\right).

$R_2\to R_2-R_1$ , $R_3\to R_3-3R_1$ :

\left(\begin{array}{ccc|c}1 & 1 & 2 & 9\\ 0 & 2 & 0 & 4\\ 0 & -2 & -3 & -13\end{array}\right).

$R_2\to R_2/2$ :

\left(\begin{array}{ccc|c}1 & 1 & 2 & 9\\ 0 & 1 & 0 & 2\\ 0 & -2 & -3 & -13\end{array}\right).

$R_3\to R_3+2R_2$ :

\left(\begin{array}{ccc|c}1 & 1 & 2 & 9\\ 0 & 1 & 0 & 2\\ 0 & 0 & -3 & -9\end{array}\right).

$R_3\to R_3/(-3)$ :

\left(\begin{array}{ccc|c}1 & 1 & 2 & 9\\ 0 & 1 & 0 & 2\\ 0 & 0 & 1 & 3\end{array}\right).

$z=3$ . From row 2: $y=2$ . From row 1: $x+y+2z=9\Rightarrow x=9-2-6=1$ .

\boxed{\;x=1,\;y=2,\;z=3.\;}