# Automating the solving of system of equations

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.

• Is there a way to automate this process? yes there is. But it will be easier to post the Mathematica code itself you used. – Nasser Apr 18 at 13:51
• Thanks for the answer. Unluckily, the code is related to my thesis, so I am a bit worried about posting it online. Maybe I can write a script resembling my situation? – Plastic Man Apr 18 at 14:02
• Yes, you can make a MWE (small example) that illustrates the problem you want automated. People here like to work with code example which makes it easier to help you and answer the question. – Nasser Apr 18 at 14:04
• {{x -> 3 - y - Sqrt[4 + 7 z + k^2]}{x -> 3 - y + Sqrt[4 + 7 z + k^2]}} Is there supposed to be a comma between these 2 lists? i.e. like this {{x -> 3 - y - Sqrt[4 + 7 z + k^2]},{x -> 3 - y + Sqrt[4 + 7 z + k^2]}} – Nasser Apr 18 at 15:11
• Yep, sorry, I ma have deleted it by mistake. They are lists, so x, y and k have two possible solutions. I am going to immediately edit the post. – Plastic Man Apr 18 at 15:17

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]
`
• Thanks a lot! Your code is wonderful. Of course it is not working with my functions, but I have learnt a lot the same. Also, thanks for the patience and the support! – Plastic Man Apr 18 at 15:41