If the system of linear equations

$\begin{array}{l}2\mathrm{x}+\mathrm{y}-\mathrm{z}=7\\ \mathrm{x}-3\mathrm{y}+2\mathrm{z}=1\\ \mathrm{x}+4\mathrm{y}+\mathrm{\delta z}=\mathrm{k},\text{ where }\mathrm{\delta },\mathrm{k}\in \mathrm{R}\end{array}$

has infinitely many solutions, then $\delta +k$ is equal to :

$\begin{array}{l}\mathrm{The} \mathrm{system} \mathrm{of} \mathrm{linear} \mathrm{equations}\\ 2\mathrm{x}+\mathrm{y}-\mathrm{z}=7,\mathrm{x}-3\mathrm{y}+2\mathrm{z}=1,\mathrm{x}+4\mathrm{y}+\mathrm{\delta z}=\mathrm{k},\\ \mathrm{has} \mathrm{infinitely} \mathrm{many} \mathrm{solutions}.\\ \left|\begin{array}{ccc}2& 1& -1\\ 1& -3& 2\\ 1& 4& \delta \end{array}\right|=0\\ \Rightarrow \delta =-3\\ \left|\begin{array}{ccc}7& 1& -1\\ 1& -3& 2\\ \mathrm{K}& 4& -3\end{array}\right|=0\Rightarrow \mathrm{K}=6\\ \Rightarrow \delta +\mathrm{K}=3\end{array}$