Can this system of equations be solved in real numbers?

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Discussion Overview

The discussion revolves around the solvability of a specific system of equations in real numbers, presented in a mathematical context. Participants explore potential solutions and methods for addressing the equations.

Discussion Character

  • Mathematical reasoning, Exploratory

Main Points Raised

  • One participant presents the system of equations to be solved: $a(b+c-a^3)=b(c+a-b^3)=c(a+b-c^3)=1$.
  • Another participant requests clarification on how the solution was derived.
  • A third participant suggests a solution approach by inspection, reformulating the equations to: $ab + ac - a^4 = 1$, $ab + bc - b^4 = 1$, and $ac + bc - c^4 = 1$.
  • A later reply addresses the previous participant directly, indicating ongoing engagement in the discussion.

Areas of Agreement / Disagreement

The discussion does not present a consensus on the solvability of the system, and multiple approaches and interpretations are being explored without resolution.

Contextual Notes

Participants have not provided detailed assumptions or definitions regarding the variables or the nature of the solutions sought, leaving some aspects of the discussion open to interpretation.

anemone
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Solve in real numbers the system below:

$a(b+c-a^3)=b(c+a-b^3)=c(a+b-c^3)=1$
 
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a=1, b=1, c=1
a=-1, b=-1, c=-1
 
Wilmer said:
a=1, b=1, c=1
a=-1, b=-1, c=-1

Would you mind sharing how you found the solution? :D
 
Lazily, by inspection:
ab + ac - a^4 = 1
ab + bc - b^4 = 1
ac + bc - c^4 = 1
 
Wilmer said:
a=1, b=1, c=1
a=-1, b=-1, c=-1

Hi Wilmer,

Your answer (without the working, hehehe...) is correct, but the question remains on how we are going to prove those are the only solutions.(Nod)
 

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