Let u be the airspeed and v be the speed of the wind.

Then the ground-speed flying east (with the wind) is (u+v),
while ground-speed flying west (against the wind) is (u-v).


The condition gives you these two equations:

 = u - v   (1)   (flying west against the wind)  and

 = u + v   (2)   (flying east against the wind).


It is the same as

u - v = 200,           (3)
u + v = 300.           (4)

Add equations (3) and (4) (both sides). You will get

2u = 200 + 300 = 500.   Hence,  u =  = 250 miles per hour is the airspeed.

Then from (4)  u = 300 - 250 = 50 miles per hour is the speed of the wind.