The system of linear equations $x+y+z=6$, $x+2y+3z=10,x+2y+az=b$ has no solution when

$[A\mid B]=\left[\begin{array}{lllc}1& 1& 1& 6\\ 1& 2& 3& 10\\ 1& 2& a& b\end{array}\right]$

$\text{ [Applying }{R}_2\to R_2-R_1,R_3\to R_3-R_1\text{ ] }$

$[A\mid B]=\left[\begin{array}{cccc}1& 1& 1& 6\\ 0& 1& 2& 4\\ 0& 1& a-1& b-6\end{array}\right]$[Applying $R_3\to R_3-R_1$ ]

$=\left[\begin{array}{cccc}1& 1& 1& 6\\ 0& 1& 2& 4\\ 0& 0& a-3& b-10\end{array}\right]$

It has no solution for $a=3,b\ne 10$