The effective speed with the wind is

     = 320  miles per hour.


It is the sum of the speed of the airplane "u" at no wind plus the wind speed "v".


So, your first equation is 

    u + v = 320.    (1)


The effective speed against the wind is

     = 256  miles per hour.


It is the difference of the speed of the airplane "u" at no wind and the wind speed "v".


So, your second equation is 

    u - v = 256.    (2)


Thus you have this system of two equations in two unknowns

    u + v = 320,    (1)
    u - v = 256.    (2)


To solve it, add the equations.  You will get

   2u = 320 + 256

   2u = 576  ====>  u = 576/2 = 288.


Then from eq(1)  v = 320 - 288 = 32.


Answer.  The plane speed at no wind is  288 miles per hour.

         The wind speed is  32 miles per hour.