The airplane effective speed, when flying into a head wind, is   = 350 mph. 
This speed is the difference  u-v  of the airplane speed in still air and the wind speed:  u - v = 350 mph.


The airplane effective speed, when flying with the wind, is   = 450 mph.
This speed is the sum u+v of the airplane speed in still air and the wind speed: u + v = 450 mph.


To determine the airplane speed in still air  "u"  and the wind speed  "v", you need to solve this system of two equations:

u - v = 350,   (1)
u + v = 450.   (2)


Add the equations (1) and (2). You will get

2u = 350 + 450
2u = 800,

u =  = 400 miles per hour.  It is the airplane speed in still air.

Then from (1) you find  v = u - 400 = 450 - 400 = 50 miles per hour.  It is the wind speed.