Official solution
3x - 2y + 4z - 7 + \[\lambda(x + 5y - 2z + 9) = 0
(3 + \[\lambda\])x + (5\[\lambda\] - 2)y + (4 - 2\[\lambda\])z + 9\[\lambda\] - 7 = 0
passing through (1, 4, -3)
\[\Rightarrow\] 3 + \[\lambda\] + 20\[\lambda\] - 8 - 12 + 6\[\lambda\] + 9\[\lambda\] - 7 = 0
\[\Rightarrow\] \[\lambda\] = \[{2 \over 3}\]
\[\Rightarrow\] equation of plane is
-11x - 4y - 8z + 3 = 0
\[\Rightarrow\] \[\alpha\] + \[\beta\] + \[\gamma\] = -23