If the system of equations  $x+y+az=b,2x+5y+2z=6,x+2y+3z=3$  has infinitely many solutions, then  $2a+3b$ is equal to ………….

Since given equations have infinitely many solutions then

$$\begin{gathered} \Delta =\left|\begin{array}{ccc}1& 1& a\\ 2& 5& 2\\ 1& 2& 3\end{array}\right|=0 \\ \Rightarrow 1(15-4)-1(6-2)+a(4-5)=0 \\ \begin{array}{l}=11-4-a=0\\ \Rightarrow a=7\end{array} \\ \begin{array}{l}{\Delta }_3=\left|\begin{array}{ccc}1& 1& b\\ 2& 5& 6\\ 1& 2& 3\end{array}\right|=0\Rightarrow b=3\\ Now 2a+3b=23\end{array} \end{gathered}$$