If the system of equations $\alpha x+y+z=5,x+2y+3z=4,x+3y+5z=\beta$ has infinitely many solutions, then the ordered pair $(\alpha ,\beta )$ is equal to:

Given system of equations have  infinitely many solutions,

$then \Delta ={\Delta }_x={\Delta }_y={\Delta }_z=0$

$$\begin{gathered} \Delta =\left|\begin{array}{lll}\alpha & 1& 1\\ 1& 2& 3\\ 1& 3& 5\end{array}\right|=0 \\ \Rightarrow \alpha (10-9)-1(5-3)+1(3-2)=0 \\ \Rightarrow \alpha =1 \\ {\Delta }_x=\left|\begin{array}{lll}5& 1& 1\\ 4& 2& 3\\ \beta & 3& 5\end{array}\right|=0 \\ \Rightarrow 5(10-9)-1(20-3\beta )+1(12-2\beta )=0 \\ \Rightarrow 5-20+3\beta +12-2\beta =0 \\ \Rightarrow \beta =3 \end{gathered}$$