Answer to Question #175574 in Linear Algebra for william

Question

solve the system of three variable linear equations

x + 2y = 1

3x + 2y + 4z = 7

-2x + y - 2z = -1

Answer 1

The augmented matrix of the given system is:

$$\begin{bmatrix} 1 & 2 &0:1\\ 3 & 2&4:7\\ -2&1&-2:-1 \end{bmatrix}$$

Now, $R_3\rightarrow3R_1-R_3;\ \ R_3\rightarrow2R_1+R_3$

$$\begin{bmatrix} 1 & 2 &0:1\\ 0 & 4&-4:4\\ 0&5&-2:1 \end{bmatrix}$$

$$R_2\rightarrow R_2/4;$$

$$\begin{bmatrix} 1 & 2 &0:1\\ 0 & 1&-1:1\\ 0&5&-2:1 \end{bmatrix}$$

$$R_1\rightarrow R_1-2R_2;\ \ R_3\rightarrow R_3-5R_2$$

$$\Rightarrow \begin{bmatrix} 1 & 0 &2:3\\ 0 & 1&-1:1\\ 0&0&3:6 \end{bmatrix}$$

Comparing on third row,

$$3z=6\Rightarrow [z=2]$$

On second row,

$$y-z=-1\Rightarrow y=-1+z\\ \Rightarrow [y=1]$$

On first row,

$$x=3-2z\Rightarrow [x=-1]$$

$$\Rightarrow\begin{bmatrix} x \\ y\\ z \end{bmatrix}=\begin{bmatrix} -1\\ 1\\ 2 \end{bmatrix}$$