10 Find the set of values of \(a\) for which the system of equations $$\begin{aligned} x + 4 y + 12 z & = 5 \\ 2 x + a y + 12 z & = a - 1 \\ 3 x + 12 y + 2 a z & = 10 \end{aligned}$$ has a unique solution. Show that the system does not have any solution in the case \(a = 18\). Given that \(a = 8\), show that the number of solutions is infinite and find the solution for which \(x + y + z = 1\).

10 Find the set of values of $a$ for which the system of equations

$$\begin{aligned}
x + 4 y + 12 z & = 5 \\
2 x + a y + 12 z & = a - 1 \\
3 x + 12 y + 2 a z & = 10
\end{aligned}$$

has a unique solution.

Show that the system does not have any solution in the case $a = 18$.

Given that $a = 8$, show that the number of solutions is infinite and find the solution for which $x + y + z = 1$.

\hfill \mbox{\textit{CAIE FP1 2010 Q10}}