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?

You get an empty solution set because there are no solutions. As it happens, anything that might make the equations vanish also makes denominators zero. So you are getting an appropriate result. — Daniel Lichtblau, Sep 18 '14 at 23:26
So the problem is that I didn't specify that the denominators can't be zero? Is there a way to specify this is not the case? I tried: "Solve[{y1 == x4((x2x3 - x5^2)/(x1x2x3 - (x5^2)(x1 + x2 + x3) - 2x5^3)), y2 == x4((x1x3 - x5^2)/(x1x2x3 - (x5^2)(x1 + x2 + x3) - 2x5^3)), y3 == x4((x1x2 - x5^2)/(x1x2x3 - (x5^2)(x1 + x2 + x3) - 2x5^3)), y4 == x4(1/(2x5 + x2 + x3)), y5 == x4(1/(2x5 + x1 + x3)), (x1x2x3 - (x5^2)(x1 + x2 + x3) - 2x5^3) != 0, (2*x5 + x2 + x3) != 0}, {x1, x2, x3, x4, x5}]" but that didn't work. — Inversion, Sep 18 '14 at 23:33
Hi belisarius, I'm aware that y scales with x4, but I'm not sure where you're going with your statement. Could you explain? — Inversion, Sep 18 '14 at 23:49
Actually Solve will automatically enforce that denominators do not vanish. In effect it does what you did explicitly. And it turns out that no solutions exist. — Daniel Lichtblau, Sep 18 '14 at 23:59
What do you mean by no solutions exist? I have 5 independent variables and I just want to rearrange the equations in terms of the other 5 variables, which I thought would be possible with 5 independent equations. — Inversion, Sep 19 '14 at 00:09
@belisarius Oops. I'm new and didn't realize I'm supposed to use the @ symbol to direct my comments. — Inversion, Sep 19 '14 at 20:58

dionys · Answer 1 · 2015-09-09T15:59:10.193

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)}}*)

Rearranging 5 equations

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