I cannot replicate this behavior.
In[398]:= Solve[{44826600000*a + 406601000000*b + 134691918588*c +
656526*f + 512206908*e + 170700546*d - 7642032.5 == 0,
3757520000000*b + 406601000000*a + 1233170000000*c + 5925820*f +
4678814294*e + 1544327542*d - 69097900 == 0,
406601000000*c + 134692000000*a + 1233170000000*b + 1967580*f +
1544327592*e + 512206908*d - 22936700 == 0,
170700546*a + 1544327592*b + 512206908*c + 2550*f + 1967580*e +
656526*d - 29327.5 == 0,
512206908*a + 4678814150*b + 1544327592*c + 7634*f + 5925818*e +
1967580*d - 87833 == 0,
10*f + 656526*a + 5925820*b + 1967580*c + 7634*e + 2550*d - 113.5 ==
0}, {a, b, c, d, e, f}]
During evaluation of In[398]:= RowReduce::luc: Result for RowReduce of badly conditioned matrix {{656526.,5.12206908*10^8,1.70700546*10^8,1.34691918588*10^11,4.06601*10^11,4.48266*10^10,-7.6420325*10^6},{5.92582*10^6,4.678814294*10^9,1.544327542*10^9,1.23317*10^12,3.75752*10^12,4.06601*10^11,-6.90979*10^7},{<<1>>},<<1>>,{7634.,5.925818*10^6,<<4>>,-87833.},{10.,7634.,2550.,1.96758*10^6,5.92582*10^6,656526.,-113.5}} may contain significant numerical errors.
Out[398]= {{a -> 1.90207400851, b -> 0.00137897080916,
c -> -0.413684977324, d -> -649.780295374, e -> 101.12375112,
f -> 44210.0251526}}