Automating the solving of system of equations

Question

I checked on the website, and I didn't find a similar topic, so I decided to ask. I solved a couple of maximisation problems, and I obtained solutions for x, y, z and k. What I need to do is to solve them in a system of equations. Now, the peculiarity is that x, y and k have two solutions each, so I have 8 possible systems of equations. What I am supposed to do is to solve all of them, and then select the solutions I need. Is there a way to automate this process? I think writing by hand all the possibilities is cumbersome.

Let's suppose I obtained this output:

{{x -> 3 - y - Sqrt[4 + 7 z + k^2]}, {x -> 3 - y + Sqrt[4 + 7 z + k^2]}}

{{y -> 1/12 (5 + 2 z - Sqrt[12 + 2 k + x^2])}, {y -> 1/12 (5 + 2 z + Sqrt[12 + 2 k + x^2])}}

{{z -> (-2 x + 3 y - 12 k)/(6 k z)}}

{{k -> 1/9 (-2 + 2 x - Sqrt[4 + 2 y^2])}, {k -> 1/9 (-2 + 2 x + Sqrt[4 + 2 y^2])}}

I want to solve the systems:

Solve[x == 3 - y - Sqrt[4 + 7 z + k^2] && y == 1/12 (5 + 2 z - Sqrt[12 + 2 k + x^2]) && z == (-2 x + 3 y - 12 k)/(6 k z) && k == 1/9 (-2 + 2 x - Sqrt[4 + 2 y^2]), {x, y, z, k}]

The only way I know is to manually write one Solve for each system, but it is really cumbersome.

Answer 1

One way

ClearAll["Global`*"]
(*this is the INPUT *)
list = {{{{x -> 3 - y - Sqrt[4 + 7 z + k^2]}, {x -> 
       3 - y + Sqrt[4 + 7 z + k^2]}},
    {{y -> 1/12 (5 + 2 z - Sqrt[12 + 2 k + x^2])}, {y -> 
       1/12 (5 + 2 z + Sqrt[12 + 2 k + x^2])}},
    {{z -> (-2 x + 3 y - 12 k)/(6 k z)}},
    {{k -> 1/9 (-2 + 2 x - Sqrt[4 + 2 y^2])}, {k -> 
       1/9 (-2 + 2 x + Sqrt[4 + 2 y^2])}}}
   };

(*This is the AUTOMATION part *)

list = list[[1, All, 1]];
vars = Union@Cases[list, {Rule[x_, any__]} :> x];
eqs = Cases[list[[All, 1]], Rule[x_, any_] :> (x == any)];
NSolve[eqs, vars, Reals]

Also Solve works

 Solve[eqs, vars, Reals]