For the system of equations:

$(\mathrm{a})$ there is only one solution

The given system of equations can be written in matrix form as follows:

$\left[\begin{array}{lll}1 & 2 & 3 \\ 2 & 1 & 3 \\ 5 & 5 & 9\end{array}\right]\left[\begin{array}{l}x \\ y \\ z\end{array}\right]=\left[\begin{array}{l}1 \\ 2 \\ 4\end{array}\right]$

Here,

$A=\left[\begin{array}{lll}1 & 2 & 3 \\ 2 & 1 & 3 \\ 5 & 5 & 9\end{array}\right], X=\left[\begin{array}{l}x \\ y \\ z\end{array}\right]$ and $B=\left[\begin{array}{l}1 \\ 2 \\ 4\end{array}\right]$

Now,

$|A|=\left|\begin{array}{lll}1 & 2 & 3 \\ 2 & 1 & 3 \\ 5 & 5 & 9\end{array}\right|$

$=1(9-15)-2(18-15)+3(10-5)$

$=-6-6+15$

$=3 \neq 0$

$\Rightarrow|A| \neq 0$

So, the given system of equations has a unique solution.