Official solution
Equation of plane P can be assumed as
P : x + 2y + 3z + 1 + \[\lambda\] (x - y - z - 6) = 0
\[\Rightarrow\] P : (1 + \[\lambda\])x + (2 - \[\lambda\])y + (3 - \[\lambda\])z + 1 - 6\[\lambda\] = 0
\[ \Rightarrow {\overrightarrow n _1} = (1 + \lambda )\widehat i + (2 - \lambda )\widehat j + (3 - \lambda )\widehat k\]
\[\therefore\] \[{\overrightarrow n _1}\,.\,{\overrightarrow n _2} = 0\]
\[\Rightarrow\] 2(1 + \[\lambda\]) - (2 - \[\lambda\]) - (3 - \[\lambda\]) = 0
\[\Rightarrow\] 2 + 2\[\lambda\] - 2 + \[\lambda\] - 3 + \[\lambda\] = 0 \[\Rightarrow\] \[\lambda\] = \[{3 \over 4}\]
\[\Rightarrow\] \[P:{{7x} \over 4} + {5 \over 4}y + {{9z} \over 4} - {{14} \over 4} = 0\]
\[\Rightarrow\] 7x + 5y + 9z = 14
(0, 1, 1) lies on P.