If the system of linear equations

$$ \begin{aligned} & 7 x+11 y+\alpha z=13 \\\\ & 5 x+4 y+7 z=\beta \\\\ & 175 x+194 y+57 z=361 \end{aligned} $$

has infinitely many solutions, then $\alpha+\beta+2$ is equal to :

Given, <br/><br/>$$ \begin{aligned} & 7 x+11 y+\alpha z=13 \\\\ & 5 x+4 y+7 z=\beta \\\\ & 175 x+194 y+57 z=361 \end{aligned} $$ <br/><br/>$$ \text { For infinite solution, }\left|\begin{array}{ccc} 7 & 11 & \alpha \\ 5 & 4 & 7 \\ 175 & 194 & 57 \end{array}\right|=0 $$ <br/><br/>$$ \Rightarrow\left|\begin{array}{ccc} 7 & 11 & \alpha \\ 5 & 4 & 7 \\ 0 & -81 & 57-25 \alpha \end{array}\right|=0 $$ <br/><br/>$$ \begin{aligned} & \Rightarrow 81(49-5 \alpha)+(57-25 \alpha)(-27)=0 \\\\ & \Rightarrow 270 \alpha=-2430 \Rightarrow \alpha=-9 \end{aligned} $$ <br/><br/>And $\Delta_1=0$ <br/><br/>$$ \begin{aligned} & \left|\begin{array}{ccc} 13 & 11 & -9 \\ \beta & 4 & 7 \\ 361 & 194 & 57 \end{array}\right|=0 \\\\ & \Rightarrow \beta=11 \end{aligned} $$ <br/><br/>$\therefore \alpha+\beta+2=4$