Question

X+Y-Z=4

2X+Z=7

3X-2Y=5

Answer 1

Let we solve the sysyem by using Gauss method:

$\begin{pmatrix} 1 & 1 &-1&=&4\\ 2 & 0 & 1&=&7\\ 3& -2& 0&=&5 \end{pmatrix}$

1) Add 2 equation to the1 equation

$\begin{pmatrix} 3 & 1 &0&=&11\\ 2 & 0 & 1&=&7\\ 3& -2& 0&=&5 \end{pmatrix}$

2) We add 1 equation multiplied by 2 to to the 3 equation:

$\begin{pmatrix} 3 & 1 &0&=&4\\ 2 & 0 & 1&=&7\\ 9& 0& 0&=&27 \end{pmatrix}$

3) Divide 3 eqution by 9

$\begin{pmatrix} 3 & 1 &0&=&11\\ 2 & 0 & 1&=&7\\ 1& 0& 0&=&\frac{27}{9}=3 \end{pmatrix}$

4) Substract from 1 equation 3 equation multiplied by 3 and fron 2 equation 3 equation multiplied by 2

$\begin{pmatrix} 0 & 1 &0&=&11-3\cdot 3=2\\ 0 & 0 & 1&=&7-2\cdot 3=1\\ 1& 0& 0&=&3 \end{pmatrix}$

Matrix to the left is unit now then problem is solved.

Answer:

$X=3\\ Y=2\\ Z=1$