The system is overdetermined (as others have noted). All the same it appears NSolve
can handle it.
eqns = {-1201.1` (1479.5354631799682` + 4 n + 14 x +
94.35073779795688` z) +
42.82` (61.948` x + 63.546` x y + 14.007` (n + 5 x + 4 z) +
15.999` (1233.94621931664` + m + n + 9 x +
45.67536889897844` z) +
12.011` (1479.5354631799682` + 4 n + 14 x +
94.35073779795688` z) +
1.008` (2467.89243863328` + 2 m + 3 n + x (25 - y) +
184.70147559591376` z)) ==
0, -100.8` (2467.89243863328` + 2 m + 3 n + x (25 - y) +
184.70147559591376` z) +
7.385` (61.948` x + 63.546` x y + 14.007` (n + 5 x + 4 z) +
15.999` (1233.94621931664` + m + n + 9 x +
45.67536889897844` z) +
12.011` (1479.5354631799682` + 4 n + 14 x +
94.35073779795688` z) +
1.008` (2467.89243863328` + 2 m + 3 n + x (25 - y) +
184.70147559591376` z)) ==
0, -1599.9` (1233.94621931664` + m + n + 9 x +
45.67536889897844` z) +
44.685` (61.948` x + 63.546` x y + 14.007` (n + 5 x + 4 z) +
15.999` (1233.94621931664` + m + n + 9 x +
45.67536889897844` z) +
12.011` (1479.5354631799682` + 4 n + 14 x +
94.35073779795688` z) +
1.008` (2467.89243863328` + 2 m + 3 n + x (25 - y) +
184.70147559591376` z)) ==
0, -1400.7` (n + 5 x + 4 z) +
2.7` (61.948` x + 63.546` x y + 14.007` (n + 5 x + 4 z) +
15.999` (1233.94621931664` + m + n + 9 x +
45.67536889897844` z) +
12.011` (1479.5354631799682` + 4 n + 14 x +
94.35073779795688` z) +
1.008` (2467.89243863328` + 2 m + 3 n + x (25 - y) +
184.70147559591376` z)) ==
0, -6194.8` x +
0.96` (61.948` x + 63.546` x y + 14.007` (n + 5 x + 4 z) +
15.999` (1233.94621931664` + m + n + 9 x +
45.67536889897844` z) +
12.011` (1479.5354631799682` + 4 n + 14 x +
94.35073779795688` z) +
1.008` (2467.89243863328` + 2 m + 3 n + x (25 - y) +
184.70147559591376` z)) ==
0, -6354.6` x y +
1.45` (61.948` x + 63.546` x y + 14.007` (n + 5 x + 4 z) +
15.999` (1233.94621931664` + m + n + 9 x +
45.67536889897844` z) +
12.011` (1479.5354631799682` + 4 n + 14 x +
94.35073779795688` z) +
1.008` (2467.89243863328` + 2 m + 3 n + x (25 - y) +
184.70147559591376` z)) == 0};
polys = Apply[Subtract, eqns, {1}];
Solve and check residuals.
NSolve[polys, {n, m, x, y, z}]
polys /. approxSol
(* Out[8]= {{n -> 44.6232963489, m -> 518.132047914, x -> 14.9169862073,
y -> 1.47243401106, z -> 16.5848900493}}
Out[9]= {{-9.31322574615*10^-10, 3.49245965481*10^-10,
3.72529029846*10^-9, -1.57160684466*10^-9,
1.30967237055*10^-10, -2.91038304567*10^-11}} *)