Let u = the speed of the plane at no wind, in miles per hour.

    v = speed of the wind.


Then the speed "with the wind" is (u+v) mph,

while the speed "against the wind" is (u-v) mph.


Then your equations ("the speed equations") are

 = u - v,   (speed = ;   =  hours = 1 hour 10 minutes)

 = u + v.   (                           =  hours = 50 minutes).


Simplify the equations

u - v =  = 240,     (1)
u + v =  = 336.     (2)


Add the equation (1) and (2) to get

2u = 240 + 336 = 576  ====>  u =  = 288.

Then from  (2),  v = 336 - 288 = 48.


Answer.  The aircraft speed at no wind is 288 mph.  Speed of the wind is 48 mph.