Science:Math Exam Resources/Courses/MATH152/April 2022/Question B1 (b)/Solution 1

We have the following reduced augmented matrix:

$$\left[{\begin{array}{cccc|c}1&0&0&1&5\\0&0&0&0&0\\0&1&0&0&1\\0&0&1&2&4\\0&0&0&0&H-9\end{array}}\right],$$

which corresponds to the system of equations

$${\begin{cases}n_{A}+n_{D}&=5\\n_{B}&=1\\n_{C}+2n_{D}&=4\end{cases}}.$$

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 $n_{D}$, so let $t=n_{D}$. Then we have

$$n_{A}=5-t{\text{ and }}n_{C}=4-2t.$$

In parametric form,

$$\left[{\begin{array}{c}n_{A}\\n_{B}\\n_{C}\\n_{D}\end{array}}\right]=\left[{\begin{array}{c}5\\1\\4\\0\end{array}}\right]+t\left[{\begin{array}{c}-1\\0\\-2\\1\end{array}}\right].$$