Rearranging 5 equations

Question

I am trying to rearrange 5 equations in terms of 5 variables. I have the equations for y1, y2, y3, y4 and y5 in terms of x1, x2, x3, x4 and x5, but I want to solve for x1, x2, x3, x4 and x5 in terms of y1, y2, y3, y4 and y5. I tried the following:

Solve[{y1 == x4*((x2*x3 - x5^2)/(x1*x2*x3 - (x5^2)*(x1 + x2 + x3) - 2*x5^3)), 
       y2 == x4*((x1*x3 - x5^2)/(x1*x2*x3 - (x5^2)*(x1 + x2 + x3) - 2*x5^3)), 
       y3 == x4*((x1*x2 - x5^2)/(x1*x2*x3 - (x5^2)*(x1 + x2 + x3) - 2*x5^3)), 
       y4 == x4*(1/(2*x5 + x2 + x3)),
       y5 == x4*(1/(2*x5 + x1 + x3))}, {x1, x2, x3, x4, x5}]

and the output

{}

I looked at this post and thought that maybe I should be using Reduce, so I tried the following:

Reduce[{y1 == x4*((x2*x3 - x5^2)/(x1*x2*x3 - (x5^2)*(x1 + x2 + x3) - 2*x5^3)), 
        y2 == x4*((x1*x3 - x5^2)/(x1*x2*x3 - (x5^2)*(x1 + x2 + x3) - 2*x5^3)), 
        y3 == x4*((x1*x2 - x5^2)/(x1*x2*x3 - (x5^2)*(x1 + x2 + x3) - 2*x5^3)), 
        y4 == x4*(1/(2*x5 + x2 + x3)), 
        y5 == x4*(1/(2*x5 + x1 + x3))}, {x1, x2, x3, x4, x5}]

as well as this:

Reduce[{y1 == x4*((x2*x3 - x5^2)/(x1*x2*x3 - (x5^2)*(x1 + x2 + x3) - 2*x5^3)), 
        y2 == x4*((x1*x3 - x5^2)/(x1*x2*x3 - (x5^2)*(x1 + x2 + x3) - 2*x5^3)), 
        y3 == x4*((x1*x2 - x5^2)/(x1*x2*x3 - (x5^2)*(x1 + x2 + x3) - 2*x5^3)), 
        y4 == x4*(1/(2*x5 + x2 + x3)),
        y5 == x4*(1/(2*x5 + x1 + x3)), 
        y1 > 0, y2 > 0, y3 > 0, y4 > 0, y5 > 0}, {x1, x2, x3, x4, x5}]

and Mathematica just gets stuck calculating for a long time and doesn't display anything. Is this the correct way to solve this problem? Any suggestions?

Answer 1

As @DanielLichtblau noted in the comments, Solve correctly returns an empty set because there are no solutions to the system. But if you want to rearrange the equations of the system in terms of x1,x2,x3,x4, and x5, you can do something like the following:

Module[{res,
    vars = {x1, x2, x3, x4, x5},
    eqn1 = y1 == x4*((x2*x3 - x5^2)/(x1*x2*x3 - (x5^2)*(x1 + x2 + x3) - 2*x5^3)),
    eqn2 = y2 == x4*((x1*x3 - x5^2)/(x1*x2*x3 - (x5^2)*(x1 + x2 + x3) - 2*x5^3)),
    eqn3 = y3 == x4*((x1*x2 - x5^2)/(x1*x2*x3 - (x5^2)*(x1 + x2 + x3) - 2*x5^3)),
    eqn4 = y4 == x4*(1/(2*x5 + x2 + x3)),
    eqn5 = y5 == x4*(1/(2*x5 + x1 + x3))},
  res = Table[Simplify@Flatten@Solve[j, {i}], {i, vars}, {j, {eqn1, eqn2, eqn3, eqn4, eqn5}}];
  res = vars /. Flatten@res[[All, -2 ;; -1]];
  Table[(Thread[vars == #] & /@ res)[[i, i]], {i, Length[res]}]]

(*{{x1 == -x3 - 2 x5 + x4/y5},
   {x2 == -x3 - 2 x5 + x4/y4},
   {x3 == -x2 - 2 x5 + x4/y4},
   {x4 == (x2 + x3 + 2 x5) y4},
   {x5 == (x4 - (x2 + x3) y4)/(2 y4)}}*)