TITLE: How to solve this linear equations?
QUESTION [5 upvotes]: I am dealing with the following example:
$$
\pmatrix{
    -1 & -2& -3 \\
     3 & 2 & 1 \\
1& 1 & 1 \\
    2 & 4& 1 
    }* 
\pmatrix{
   x_1\\
x_2 \\
x_3 \\
    }= \pmatrix{
   -6\\
6 \\
3 \\
7 \\ 
 }$$
I know I should solve that matrix with gauß' elimination, but my problem is that there are only $x_1 $ to $x_3 $, so I do not know how to interprete this example?

REPLY [4 votes]: As @Clayton noted do the G.E., as you always do for any matrix. Here you have $$A:=\left( \begin{array}
    ~-1 & -2& -3&~~ -6 \\
     3 & 2 & 1&~~~~~~~   6 \\
1& 1 & 1&~~~~~~~   3 \\
    2 & 4& 1&~~~~~~~   7 \\ 
    \end{array} \right)$$ which by doing some process became:  $$\sim\left( \begin{array}
        11 & 1& 1&~~~~~~~ 3 \\
     0 & 1 & 2&~~~~~~~   3 \\
0& 0 & -5&~~~  -5 \\
    0 & 0& 0&~~~~~~~   0 \\ 
    \end{array} \right)$$ This means that the forth equation in the original system was dependent to other 3 equations. It is clear now that your solution $(x_1,x_2,x_3)$ is unique and can be found easily.