Answer to Question #176982 in Linear Algebra for william

Question

Solve the following equations using Gauss elimination and Gauss Jordon.

2x + 7y + z = 14

x + 3y - z = 2

x + 7y + 12z = 45

Answer 1

Solution.

$2x + 7y + z = 14$ ;

$x + 3y - z = 2$ ;

$x + 7y + 12z = 45;$

$\begin{bmatrix} 2 & 7&1&14 \\ 1 & 3&-1&2\\ 1&7&12&45 \end{bmatrix}$

Find the pivot in the 1st column and swap the 2nd and the 1st rows

$\begin{bmatrix} 1& 3&-1&2 \\ 2 & 7&1&14\\ 1&7&12&45 \end{bmatrix}$

Eliminate the 1st column

$\begin{bmatrix} 1& 3&-1&2 \\ 0 & 1&3&10\\ 0&4&13&43 \end{bmatrix}$

Find the pivot in the 2nd column in the 2nd row

$\begin{bmatrix} 1& 3&-1&2 \\ 0 & 1&3&10\\ 0&4&13&43 \end{bmatrix}$

Eliminate the 2nd column

$\begin{bmatrix} 1& 0&-10&-2 8\\ 0 & 1&3&10\\ 0&0&1&3 \end{bmatrix}$

Find the pivot in the 3rd column in the 3rd row

$\begin{bmatrix} 1& 0&-10&-2 8\\ 0 & 1&3&10\\ 0&0&1&3 \end{bmatrix}$

Eliminate the 3rd column

$\begin{bmatrix} 1& 0&0&2 \\ 0 & 1&0&1\\ 0&0&1&3 \end{bmatrix}$

Solution set:

$x = 2; y = 1; z = 3.$

Answer: $x = 2; y = 1; z = 3.$