Answer to Question #205547 in Linear Algebra for Abel

Question

Solve by gaussian method

X+2Y-3Y=11

3X+2Y+Z=1

2X+Y-5Z=11

Answer 1

Augmented matrix

$$\begin{pmatrix} 1 & 2 & -3 & & 11 \\ 3 & 2 & 1 & & 1 \\ 2 & 1 & -5 & & 11 \\ \end{pmatrix}$$

$R_2=R_2-3R_1$

$$\begin{pmatrix} 1 & 2 & -3 & & 11 \\ 0 & -4 & 10 & & -32 \\ 2 & 1 & -5 & & 11 \\ \end{pmatrix}$$

$R_3=R_3-2R_1$

$$\begin{pmatrix} 1 & 2 & -3 & & 11 \\ 0 & -4 & 10 & & -32 \\ 0 & -3 & 1 & & -11 \\ \end{pmatrix}$$

$R_2=-R_2/4$

$$\begin{pmatrix} 1 & 2 & -3 & & 11 \\ 0 & 1 & -5/2 & & 8 \\ 0 & -3 & 1 & & -11 \\ \end{pmatrix}$$

$R_1=R_1-2R_2$

$$\begin{pmatrix} 1 & 0 & 2 & & -5 \\ 0 & 1 & -5/2 & & 8 \\ 0 & -3 & 1 & & -11 \\ \end{pmatrix}$$

$R_3=R_3+3R_2$

$$\begin{pmatrix} 1 & 0 & 2 & & -5 \\ 0 & 1 & -5/2 & & 8 \\ 0 & 0 & -13/2 & & 13 \\ \end{pmatrix}$$

$R_3=-(2/13)R_3$

$$\begin{pmatrix} 1 & 0 & 2 & & -5 \\ 0 & 1 & -5/2 & & 8 \\ 0 & 0 & 1 & & -2\\ \end{pmatrix}$$

$R_1=R_1-2R_3$

$$\begin{pmatrix} 1 & 0 & 0 & & -1 \\ 0 & 1 & -5/2 & & 8 \\ 0 & 0 & 1 & & -2\\ \end{pmatrix}$$

$R_2=R_2+(5/2)R_3$

$$\begin{pmatrix} 1 & 0 & 0 & & -1 \\ 0 & 1 & 0 & & 3 \\ 0 & 0 & 1 & & -2\\ \end{pmatrix}$$

Then

$$x=-1, y=3, z=-2$$

$(-1, 3, -2)$