Using exact numbers Solve give you exact solution.Try:
sol = Solve[{130858 (28341 + x) == (21125 + z) (59784 + y),
115387 (6008 + y) == (36596 + z) (82117 + x),
151649 (3611 + z) == (45855 + x) (38993 + y)}, {x, y, z}]
N[sol]
(* {{x -> -197504., y -> -190642., z -> 148038.}, {x -> -197504.,
y -> 109379., z -> -151983.}, {x -> 102517., y -> -190642.,
z -> -151983.}, {x -> -2830.53, y -> 30133.1,
z -> 16000.8}, {x -> 8578.47, y -> 43071.8, z -> 25845.7}} *)
NSolve is a numeric solver.
NSolve[{130858 (28341 + x) == (21125 + z) (59784 + y),
115387 (6008 + y) == (36596 + z) (82117 + x),
151649 (3611 + z) == (45855 + x) (38993 + y)}, {x, y, z}]
(* {{x -> -197504., y -> -190642., z -> 148038.}, {x -> -197504.,
y -> 109379., z -> -151983.}, {x -> 102517., y -> -190642.,
z -> -151983.}, {x -> -2830.53, y -> 30133.1,
z -> 16000.8}, {x -> 8578.47, y -> 43071.8, z -> 25845.7}} *)