Find the equation of the plane through the intersection of the planes:

Equation of plane passing through the intersection of x+y+z=1 and 2x+3y+4z=5 is
(x+y+z-1)+k(2x+3y+4z-5) =0

i.e (1+2k)x+(1+3k)y+(1+4k)z-1-5k=0......(i)

Also (i) is perpendicular to the plane x-y+z=0.
using the condition of perpendicularity

( 1+2k).1+(1+3k)-1+(1+4k)1=0
this gives k =-1/3
substitute in (i) to get the required equation