The line (x-3)/1=(y-4)/2=(z-5)/2 cuts the plane x + y + z = 17 at

The coordinates of any point on the given line are given by 

$\frac{x-3}{1}=\frac{y-4}{2}=\frac{z-5}{2}=λ$

Suppose the givem line cuts the given plane at (λ+3,2λ+4, 2λ + 5).

This lies on the plane.

$ ∴ λ+3+ 2λ+4+2λ+5 = 17 ⇒ λ = 1 $

Hence, the coordinates of the required point are (4, 6, 7).