$\Rightarrow \delta=-3$

And $\left|\begin{array}{ccr}7 & 1 & -1 \\ 1 & -3 & 2 \\ K & 4 & -3\end{array}\right|=0 \Rightarrow K =6$

$\Rightarrow \delta+ K =3$

Alternate

$2 x + y - z =7$             $\dots(1)$

$x-3 y+2 z=1$             $\dots(2)$

$x +4 y +\delta z = k$            $\dots(3)$

Equation $(2) + (3)$

We get $2 x+y+(2+\delta) z=1+K$           $\dots(4)$

For infinitely solution

Form equation $(1)$ and $(4)$

$2+\delta=-1 \Rightarrow \delta=-3$

$1+ k =7 \Rightarrow k =6$

$\delta+ k =3$