For planes to intersect on a line

$\Rightarrow$ there should be infinite solution of the given system of equations for infinite solutions

$\Delta=\left|\begin{array}{ccc}{1} & {4} & {-2} \\ {1} & {7} & {-5} \\ {1} & {5} & {\alpha}\end{array}\right|=0 \Rightarrow 3 \alpha+9=0 \Rightarrow \alpha=-3$

$\Delta_{z}=\left|\begin{array}{ccc}{1} & {4} & {1} \\ {1} & {7} & {\beta} \\ {1} & {5} & {5}\end{array}\right|=0 \Rightarrow 13-\beta=0 \Rightarrow \beta=13$

Also for $\alpha=-3$ and $b=13 \Delta_{x}=\Delta_{y}=0$

$\alpha+\beta=-3+13=10$