I got bogged down in the algebra of substitution too but had a fresh look last night.
Tag the LHS of the 4 equations L1,L2,L3,L4 top to bottom.
L4 = (x+y)*L3  (xy)*L2
L3 = (x+y)*L2  (xy)*L1
691 = (x+y)*217 (xy)*79
217 = (x+y)*79  (xy)*23
Solve to get (x+y) = 1, xy = 6.
Then ax^4 + by^4 = (x+y)*L4  (xy)*L3 = 1*691  (6)*217 = 1993.

Posted by xdog
on 20131007 12:21:31 