If the system of simultaneous linear equations  $x+y+z=\lambda$,  $5x-y+\mu .z=10$ and $2x+3y-z=6$has unique solution, then 

The System of simultaneous linear equations has unique solution, then $\Delta \ne 0$  

$\left|\begin{array}{ccc}1& 1& 1\\ 5& -1& \mu \\ 2& 3& -1\end{array}\right|\ne 0$

$23-\mu \ne 0\Rightarrow \mu \ne 23$

and  $\lambda$ can be any real number