How do I solve a system of equations through Gaussian elimination using Mathematica?

Question

I have absolutely no experience using Mathematica or similar packages, so please bear in mind with me. I am an IB student that has gotten themselves a copy of Mathematica for the purpose of simply performing Gaussian elimination, a method that I am largely unfamiliar with (as it is not part of our syllabus).

I have the above system of equations that I would like to solve, and I have the following numerical values:

I would like to ask, how can I apply Gaussian elimination in Mathematica to obtain the following numbers, through solving system S?

Again, I want to mention that I know little about Gaussian elimination and even less about Mathematica, which is the reason I came to this exchange. Any help would be really greatly appreciated, thanks in advance!

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{
 2 (x2 - x1) x + 2 (y2 - y1) y + 2 (z2 - z1) z == r1^2 - x1^2 - r2^2 + x2^2,
 2 (x3 - x1) x + 2 (y3 - y1) y + 2 (z3 - z1) z == r1^2 - x1^2 - r3^2 + x3^2,
 2 (x4 - x1) x + 2 (y4 - y1) y + 2 (z4 - z1) z == r1^2 - x1^2 - r4^2 + x4^2,
 x1 == 2088202.299, x3 == 35606984.591,
 y1 == -11757191.37, y3 == 94447027.237,
 z1 == 25391471.881, z3 == 9101378.572,
 x2 == 11092568.240, x4 == 3966929.048,
 y2 == -14198201.090, y4 == 7362851.831,
 z2 == 21471165.950, z4 == 26388447.172,
 r1 == 23204698.51, r2 == 21585835.37,
 r3 == 31364260.01, r4 == 24966798.73
}

Answer 1

If you really want to use Gaussian elimination (HW requirement) then you can do the following:

Clear["Global`*"]
eqs = {2 (x2 - x1) x + 2 (y2 - y1) y + 2 (z2 - z1) z == 
    r1^2 - x1^2 - r2^2 + x2^2, 
   2 (x3 - x1) x + 2 (y3 - y1) y + 2 (z3 - z1) z == 
    r1^2 - x1^2 - r3^2 + x3^2, 
   2 (x4 - x1) x + 2 (y4 - y1) y + 2 (z4 - z1) z == 
    r1^2 - x1^2 - r4^2 + x4^2};
par = {x1 -> 2088202.299, x3 -> 35606984.591, y1 -> -11757191.37, 
   y3 -> 94447027.237, z1 -> 25391471.881, z3 -> 9101378.572, 
   x2 -> 11092568.240, x4 -> 3966929.048, y2 -> -14198201.090, 
   y4 -> 7362851.831, z2 -> 21471165.950, z4 -> 26388447.172, 
   r1 -> 23204698.51, r2 -> 21585835.37, r3 -> 31364260.01, 
   r4 -> 24966798.73};
eqs = eqs /. par
vars = {x, y, z};
{b, A} = Normal@CoefficientArrays[eqs, vars];
Row[{"we need to solve ", MatrixForm@A . MatrixForm@vars == MatrixForm[-b]}]

Now you can do Gaussian elimination using displayRREF which gives

displayRREF[A, -b]

To verify, you can use NSolve as suggested, or use

 LinearSolve[A, -b] // MatrixForm

Answer 2

Typing in your equations, then using NSolve:

eqs = {
   2 (x2 - x1) x + 2 (y2 - y1) y + 2 (z2 - z1) z == r1^2 - x1^2 - r2^2 + x2^2,
   2 (x3 - x1) x + 2 (y3 - y1) y + 2 (z3 - z1) z == r1^2 - x1^2 - r3^2 + x3^2,
   2 (x4 - x1) x + 2 (y4 - y1) y + 2 (z4 - z1) z == r1^2 - x1^2 - r4^2 + x4^2,
   x1 == 2088202.299,   x3 == 35606984.591,   y1 == -11757191.37,
   y3 == 94447027.237,   z1 == 25391471.881,   z3 == 9101378.572,
   x2 == 11092568.240,   x4 == 3966929.048,   y2 == -14198201.090,
   y4 == 7362851.831,   z2 == 21471165.950,   z4 == 26388447.172,
   r1 == 23204698.51,   r2 == 21585835.37,   r3 == 31364260.01,
   r4 == 24966798.73
};

NSolve[eqs]

(* Out: {
 {r1 -> 2.32047*10^7, r2 -> 2.15858*10^7, r3 -> 3.13643*10^7, 
  r4 -> 2.49668*10^7, x1 -> 2.0882*10^6, x2 -> 1.10926*10^7, 
  x3 -> 3.5607*10^7, y1 -> -1.17572*10^7, y3 -> 9.4447*10^7, 
  x4 -> 3.96693*10^6, y4 -> 7.36285*10^6, z1 -> 2.53915*10^7, 
  z4 -> 2.63884*10^7, z3 -> 9.10138*10^6, y2 -> -1.41982*10^7, 
  z2 -> 2.14712*10^7,
  x -> -2.99306*10^6, y -> 1880.12, z -> -3.12609*10^7}
} *)

You will notice that the results for $x$, $y$, $z$ are similar, but not identical, to those you proposed.