We want to solve the inhomogeneous linear system
-
over
(or over ).
Firstly, we eliminate by keeping the first row , replacing the second row by , and replacing the third row by . This yields
-
Now, we can eliminate from the
(new)
third row, with the help of the second row. Because of the fractions, we rather eliminate
(which eliminates also ).
We leave the first and the second row as they are, and we replace the third row by . This yields the system, in a new ordering of the variables,
-
Now we can choose an arbitrary
(free)
value for . The third row determines then uniquely, we must have
-
In the second equation, we can choose arbitrarily, this determines via
The first row determines , namely
Hence, the solution set is
-
A particularly simple solution is obtained by equating the free variables
and
with . This yields the special solution
-
The general solution set can also be written as
-
Here,
-
is a description of the general solution of the corresponding homogeneous linear system.