How does Mathematica solve this system of nonlinear equations?

Question

During a calculation for a physics lab, I ran into the following six non-linear equations with unknowns $x_1,x_2,x_3,x_4,x_5,x_6 \in \mathbb{R}$

Solve[{(1 + x1*0.405^2/((0.405)^2 - x2) + 
     x3*0.405^2/((0.405)^2 - x4) + x5*0.405^2/((0.405)^2 - x6) == 
    2.337053326), (1 + x1*(0.635)^2/((0.635)^2 - x2) + 
     x3*(0.635)^2/((0.635)^2 - x4) + x5*(0.635)^2/((0.635)^2 - x6) == 
    2.293038467), (1 + x1*(0.670)^2/((0.670)^2 - x2) + 
     x3*(0.670)^2/((0.670)^2 - x4) + x5*(0.670)^2/((0.670)^2 - x6) == 
    2.289560279), (1 + x1*(0.780)^2/((0.780)^2 - x2) + 
     x3*(0.780)^2/((0.780)^2 - x4) + x5*(0.780)^2/((0.780)^2 - x6) == 
    2.277436534), (1 + x1*(0.808)^2/((0.808)^2 - x2) + 
     x3*(0.808)^2/((0.808)^2 - x4) + x5*(0.808)^2/((0.808)^2 - x6) == 
    2.275719732), (1 + x1*(0.850)^2/((0.850)^2 - x2) + 
     x3*(0.850)^2/((0.850)^2 - x4) + x5*(0.850)^2/((0.850)^2 - x6) == 
    2.272264106), (1 + x1*(0.880)^2/((0.880)^2 - x2) + 
     x3*(0.880)^2/((0.880)^2 - x4) + x5*(0.880)^2/((0.880)^2 - x6) == 
    2.268823405), (1 + x1*(0.980)^2/((0.980)^2 - x2) + 
     x3*(0.980)^2/((0.980)^2 - x4) + x5*(0.980)^2/((0.980)^2 - x6) == 
    2.265389225)}, {x1, x2, x3, x4, x5, x6}]

This code does not yield any output by Mathematica, and I cannot figure out wether I'm using the wrong command or wrong syntax or something else...

Assuming that Mathematica knows methods to solving this, how do I tell the program to do so and return the values of my $x_1,\ldots,x_6$ ?

Any help is very much appreciated. ;)

EDIT: By deleting the last two equations and writing it as

Solve[
 1 + x1*0.405^2/((0.405)^2 - x2) + x3*0.405^2/((0.405)^2 - x4) + 
    x5*0.405^2/((0.405)^2 - x6) == 2.337053326
  &&
  1 + x1*(0.635)^2/((0.635)^2 - x2) + x3*(0.635)^2/((0.635)^2 - x4) + 
    x5*(0.635)^2/((0.635)^2 - x6) == 2.293038467
  &&
  1 + x1*(0.670)^2/((0.670)^2 - x2) + x3*(0.670)^2/((0.670)^2 - x4) + 
    x5*(0.670)^2/((0.670)^2 - x6) == 2.289560279
  &&
  1 + x1*(0.780)^2/((0.780)^2 - x2) + x3*(0.780)^2/((0.780)^2 - x4) + 
    x5*(0.780)^2/((0.780)^2 - x6) == 2.277436534
  &&
  1 + x1*(0.808)^2/((0.808)^2 - x2) + x3*(0.808)^2/((0.808)^2 - x4) + 
    x5*(0.808)^2/((0.808)^2 - x6) == 2.275719732
  &&
  1 + x1*(0.850)^2/((0.850)^2 - x2) + x3*(0.850)^2/((0.850)^2 - x4) + 
    x5*(0.850)^2/((0.850)^2 - x6) == 2.272264106
 , {x1, x2, x3, x4, x5, x6}
 ]

the computer returns a result, but one that I do not understand. The following output was generated by Mathematica:

{{x1 -> 1.27242, x2 -> 0.00816205, x3 -> -0.000117262, x4 -> 0.563865,
   x5 -> 0.0192823, x6 -> 1.70608},
 {x1 -> 1.27242, x2 -> 0.00816205, 
  x3 -> 0.0192823, x4 -> 1.70608, x5 -> -0.000117262, 
  x6 -> 0.563865},
 {x1 -> -0.000117262, x2 -> 0.563865, x3 -> 1.27242,
   x4 -> 0.00816205, x5 -> 0.0192823, 
  x6 -> 1.70608},
 {x1 -> -0.000117262, x2 -> 0.563865, 
  x3 -> 0.0192823, x4 -> 1.70608, x5 -> 1.27242, 
  x6 -> 0.00816205},
 {x1 -> 0.0192823, x2 -> 1.70608, x3 -> 1.27242, 
  x4 -> 0.00816205, x5 -> -0.000117262, 
  x6 -> 0.563865},
 {x1 -> 0.0192823, x2 -> 1.70608, 
  x3 -> -0.000117262, x4 -> 0.563865, x5 -> 1.27242, 
  x6 -> 0.00816205}}

Shouldn't this just be one value for each of the variables, instead of six each???

Answer 1

One approach is to convert this to a minimisation problem. If we assume that the errors are in the values on the right hand side of each expression, and that these errors are independent and follow a Gaussian distribution the appropriate way to do it is, I think, as follows:

eqns = {(1 + x1*0.405^2/((0.405)^2 - x2) + 
      x3*0.405^2/((0.405)^2 - x4) + x5*0.405^2/((0.405)^2 - x6) == 
     2.337053326), (1 + x1*(0.635)^2/((0.635)^2 - x2) + 
      x3*(0.635)^2/((0.635)^2 - x4) + x5*(0.635)^2/((0.635)^2 - x6) ==
      2.293038467), (1 + x1*(0.670)^2/((0.670)^2 - x2) + 
      x3*(0.670)^2/((0.670)^2 - x4) + x5*(0.670)^2/((0.670)^2 - x6) ==
      2.289560279), (1 + x1*(0.780)^2/((0.780)^2 - x2) + 
      x3*(0.780)^2/((0.780)^2 - x4) + x5*(0.780)^2/((0.780)^2 - x6) ==
      2.277436534), (1 + x1*(0.808)^2/((0.808)^2 - x2) + 
      x3*(0.808)^2/((0.808)^2 - x4) + x5*(0.808)^2/((0.808)^2 - x6) ==
      2.275719732), (1 + x1*(0.850)^2/((0.850)^2 - x2) + 
      x3*(0.850)^2/((0.850)^2 - x4) + x5*(0.850)^2/((0.850)^2 - x6) ==
      2.272264106), (1 + x1*(0.880)^2/((0.880)^2 - x2) + 
      x3*(0.880)^2/((0.880)^2 - x4) + x5*(0.880)^2/((0.880)^2 - x6) ==
      2.268823405), (1 + x1*(0.980)^2/((0.980)^2 - x2) + 
      x3*(0.980)^2/((0.980)^2 - x4) + x5*(0.980)^2/((0.980)^2 - x6) ==
      2.265389225)};

vars = Union[Cases[eqns, _Symbol, ∞]];

The objective function is given by

obj = Total[eqns /. a_ == b_ -> (a - b)^2]
(* (-1.33705 + (0.164025 x1)/(0.164025 - x2) + (0.164025 x3)/(
   0.164025 - x4) + (0.164025 x5)/(0.164025 - x6))^2 + (-1.29304 + (
   0.403225 x1)/(0.403225 - x2) + (0.403225 x3)/(0.403225 - x4) + (
   0.403225 x5)/(0.403225 - x6))^2 + (-1.28956 + (0.4489 x1)/(
   0.4489 - x2) + (0.4489 x3)/(0.4489 - x4) + (0.4489 x5)/(
   0.4489 - x6))^2 + (-1.27744 + (0.6084 x1)/(0.6084 - x2) + (
   0.6084 x3)/(0.6084 - x4) + (0.6084 x5)/(
   0.6084 - x6))^2 + (-1.27572 + (0.652864 x1)/(0.652864 - x2) + (
   0.652864 x3)/(0.652864 - x4) + (0.652864 x5)/(
   0.652864 - x6))^2 + (-1.27226 + (0.7225 x1)/(0.7225 - x2) + (
   0.7225 x3)/(0.7225 - x4) + (0.7225 x5)/(
   0.7225 - x6))^2 + (-1.26882 + (0.7744 x1)/(0.7744 - x2) + (
   0.7744 x3)/(0.7744 - x4) + (0.7744 x5)/(
   0.7744 - x6))^2 + (-1.26539 + (0.9604 x1)/(0.9604 - x2) + (
   0.9604 x3)/(0.9604 - x4) + (0.9604 x5)/(0.9604 - x6))^2 *)

Minimize seeks a global minimum of this (but I haven't checked whether it succeeds in finding it)

soln = Minimize[obj, vars]
(* {3.30637*10^-6, {x1 -> 1.36007, x2 -> -0.0064115, 
  x3 -> -0.484192, x4 -> -0.0807497, x5 -> 0.361731, 
  x6 -> -0.00424008}} *)

We can see how close the solution is to the original right hand side values

(List @@@ eqns) /. Last[soln]
(* {{2.33707, 2.33705}, {2.29335, 2.29304}, {2.2889, 
  2.28956}, {2.27766, 2.27744}, {2.27535, 2.27572}, {2.27221, 
  2.27226}, {2.2702, 2.26882}, {2.26456, 2.26539}} *)

This looks like quite a good match.