In[14]:= eq1 = ((120 - a)/2) + (a/(-I 5)) + ((b - a)/(-I 5)) +
5 ((14.14 + I 14.14)/-I ) == 0 // FullSimplify
Out[14]= (-10.7 + 70.7 I) + (0. + 0.2 I) b == 0.5 a
In[15]:= eq2 = ((b - I 120)/
4) + (b/(I 4)) + ((a - b)/(-I 5)) + (-14.14 - I 14.14)/(I 4) == 0 // FullSimplify
Well, it seems that I get your results:
Out[15]= (-132.325 + 17.675 I) + (1. + 0. I) a - (2.25 + 1.25 I) b == 0
In[16]:= Solve[{eq1, eq2}, {a, b}] // Rationalize
Out[16]= {{a -> -(319233/5785) + (733732 I)/5785, b -> -(84267/2314) + (97717 I)/1157}}