dm92 asks for info on Newton method. Several links out there, this one is OK. http://m.youtube.com/watch?v=ER5B_YBFMJo&feature=related
Successive approximations.
You solve f(x)=0. You give an initial guess x0. Then x1 = x0 - f(x0)/f'(x0). Repeat.
In several dimensions, like this problem with real numbers a x and y, the functions are
f1 = 6-sum abc
f2 = 8-sum2 abc
f3 = 5-sum3 abc
Calculate a 3x3 matrix M of partial derivatives of f123 wrt axy
Guess axy, and repeat axy = axy - M-1 x f123
You can use delta's to approximate the derivatives if f is messy.