We have to solve the system of linear equations
-
We eliminate with the help of the first equation the variable
. The resulting system is
(
,
)
-
We eliminate from
with
the variable
, this yields
(
)
-
We can now solve the system, because
determines the other variables. Set
.
Then
.
Then
-
![{\displaystyle {}z=-4x-5w=-4\cdot 82-5(-61)=-328+305=-23\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a21578aaf044cac84cf9e6acc9172ea366be6a58)
Finally we get
-
![{\displaystyle {}y=-2x-5z-2w=-2(82)-5(-23)-2(-61)=-164+115+122=73\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2befd160c26cabe2d1c3c7097799ecb699920597)
The solution set, which is the kernel, is therefore
-