Two solutions can be guessed mentally and instantly:

    (a,b) = (2,-1)  and  (a,b) = (-1,2).    (1)



It can be proved mentally and visually.


The curve x^4 + y^4 = 17  is  similar to the circle  x^2 + y^2 = 17.


It is not a circle, of course, but is a similar convex shape.


The line x + y = 1 is the straight line that cuts this curve x^4 + y^4 = 17.


It is clear, that there are two intersection points, and can not be more than 2 intersection points.


Thus, the listed solutions (1) is the full set of real solutions.



If you want to get an algebra solution, then express

    a = 1 - b

from the first equation and substitute into the second equation.


You will get a polynomial equation  P(b) = 17  for  "b"  of degree 4.


Two roots are b = -1 and b = 2.

Hence, two linear divisors are (b+1) and (b-2).


Divide the polynomial  P(b)-17  by the product (b+1)*(b-2).


You will get the quotient as a quadratic polynomial, which has no real roots.


From it, you will conclude that the listed solutions (1) are the only solutions, and there are no other solutions.