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
-
![{\displaystyle {}y={\frac {8}{13}}+{\frac {5}{13}}v\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e7993e98dc35a8133482d006fdf258e1c5f6fde9)
In the second equation, we can choose
arbitrarily, this determines
via
![{\displaystyle {}{\begin{aligned}z&=-{\frac {1}{3}}{\left(-{\frac {7}{2}}-u-{\frac {7}{2}}v+{\frac {23}{2}}{\left({\frac {8}{13}}+{\frac {5}{13}}v\right)}\right)}\\&=-{\frac {1}{3}}{\left(-{\frac {7}{2}}-u-{\frac {7}{2}}v+{\frac {92}{13}}+{\frac {115}{26}}v\right)}\\&=-{\frac {1}{3}}{\left({\frac {93}{26}}-u+{\frac {12}{13}}v\right)}\\&=-{\frac {31}{26}}+{\frac {1}{3}}u-{\frac {4}{13}}v.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ac91090c7bd9b3d5b75b030ed5101303764607d1)
The first row determines
, namely
![{\displaystyle {}{\begin{aligned}x&={\frac {1}{2}}{\left(3-2z-5y+v\right)}\\&={\frac {1}{2}}{\left(3-2{\left(-{\frac {31}{26}}+{\frac {1}{3}}u-{\frac {4}{13}}v\right)}-5{\left({\frac {8}{13}}+{\frac {5}{13}}v\right)}+v\right)}\\&={\frac {1}{2}}{\left({\frac {30}{13}}-{\frac {2}{3}}u-{\frac {4}{13}}v\right)}\\&={\frac {15}{13}}-{\frac {1}{3}}u-{\frac {2}{13}}v.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7772c5581d13bfaf4ca35246a3acc1fd5ee29e9d)
Hence, the solution set is
-
A particularly simple solution is obtained by equating the free variables
and
with
. This yields the special solution
-
![{\displaystyle {}(x,y,z,u,v)=\left({\frac {15}{13}},\,{\frac {8}{13}},\,-{\frac {31}{26}},\,0,\,0\right)\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cfe4e78d9e11347582dd2b7d08d1eda1e2e2d65d)
The general solution set can also be written as
-
Here,
-
is a description of the general solution of the corresponding homogeneous linear system.