We have the following reduced augmented matrix:
![{\displaystyle \left[{\begin{array}{cccc|c}1&0&0&1&5\\0&0&0&0&0\\0&1&0&0&1\\0&0&1&2&4\\0&0&0&0&H-9\end{array}}\right],}](https://wiki.ubc.ca/api/rest_v1/media/math/render/svg/b4729c98aafa176cd37dec3a25168b0fb119046c)
which corresponds to the system of equations
![{\displaystyle {\begin{cases}n_{A}+n_{D}&=5\\n_{B}&=1\\n_{C}+2n_{D}&=4\end{cases}}.}](https://wiki.ubc.ca/api/rest_v1/media/math/render/svg/f48e68bd2d830626592671a5ede1f7a90a624c11)
We see that there are be multiple solutions (since there are more variables than equations), and that the system can be solved in terms of
, so let
. Then we have
![{\displaystyle n_{A}=5-t{\text{ and }}n_{C}=4-2t.}](https://wiki.ubc.ca/api/rest_v1/media/math/render/svg/4f785675bd801127d5f6fd7d0350d42ca45c8bcb)
In parametric form,
![{\displaystyle \left[{\begin{array}{c}n_{A}\\n_{B}\\n_{C}\\n_{D}\end{array}}\right]=\left[{\begin{array}{c}5\\1\\4\\0\end{array}}\right]+t\left[{\begin{array}{c}-1\\0\\-2\\1\end{array}}\right].}](https://wiki.ubc.ca/api/rest_v1/media/math/render/svg/eb146363f5f7c1c043ad80ab8b95771ffeafe5a5)