What are the solutions to a system of equations involving x, y, and z?

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Discussion Overview

The discussion revolves around finding all pairs (x,y) in the real numbers that satisfy a system of equations involving fourth-degree polynomials and a quadratic equation. The scope includes mathematical reasoning and problem-solving approaches.

Discussion Character

  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant presents the equations to solve: \(\frac{x^4-16}{8x}=\frac{y^4-1}{y}\) and \(x^2-2xy+y^2=8\).
  • Another participant suggests a more precise notation for the pairs, indicating that the solutions should be expressed in terms of \((x,y) \in \mathbb{R}^2\) or \(x, y \in \mathbb{R}\).
  • Concerns are raised about the complexity of the equations, with one participant speculating that there may not be a simple solution.
  • One participant mentions that Mathematica finds 8 complex solutions, indicating a potential disconnect with the original request for real solutions.
  • A participant expresses their lack of familiarity with complex numbers and requests a complete solution suitable for their understanding.
  • Another participant suggests that proving the existence of exactly 8 solutions may require additional argumentation beyond mere enumeration.
  • A different approach is proposed by rewriting the equations, which some participants believe may simplify the problem.
  • One participant notes that substituting \(x=2z\) leads to a new equation but highlights issues with continuity in the function involved.

Areas of Agreement / Disagreement

Participants express differing views on the nature of the solutions, with some focusing on real solutions while others mention complex solutions. The discussion remains unresolved regarding the exact nature and number of solutions.

Contextual Notes

There are limitations related to the assumptions made about the continuity of functions involved and the nature of the solutions (real vs. complex). The complexity of the equations may also restrict straightforward approaches to finding solutions.

phucghe
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Find all pairs [tex](x,y) \in R[/tex] such that :
[tex]\frac{x^4-16}{8x}=\frac{y^4-1}{y}[/tex] and [tex]x^2-2xy+y^2=8[/tex]
 
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phucghe said:
Find all pairs [tex](x,y) \in R[/tex] such that :
[tex]\frac{x^4-16}{8x}=\frac{y^4-1}{y}[/tex] and [tex]x^2-2xy+y^2=8[/tex]

1) You should write "pairs [tex](x,y) \in R^2[/tex]" or "[tex]x, y \in R[/tex]".

2) What form do you need the answer in? Looking at those 4th powers and mixed terms, I'm guessing that there might not be a simple or intuitive solution for this.

3) What work have you done on the problem so far?
 
Mathematica finds 8 complex solutions.
 
I'am a elemantary pupil so I don't know about complex number
Could anyone give me a complete solution.
 
CRGreathouse said:
Mathematica finds 8 complex solutions.

Mathematica > my rough analysis of the problem.

If there's only 8 solutions, you can probably find them all by trial and error. To prove that there are exactly 8 solutions, and you have accounted for them all probably requires you to do a little arguing.
 
phucghe said:
Find all pairs [tex](x,y) \in R[/tex] such that :
[tex]\frac{x^4-16}{8x}=\frac{y^4-1}{y}[/tex] and [tex]x^2-2xy+y^2=8[/tex]
Rewrite your equations as:
[tex]\frac{(x-2)(x+2)(x^{2}+4)}{8x}=\frac{(y-1)(y+1)(y^{2}+1)}{y},(x-y)^{2}=8[/tex]
this ought to help a bit.
 
thanks but it seems to be not necessary for this problem.
Though by putting x=2z I had : [tex]\frac{z^4-1}{z}=\frac{y^4-1}{y}[/tex],cossidering the function [tex]f(x)=x^3-\frac{1}{x}[/tex] and its monotonousness ,there are still some troubles for example f(x) is not continuous at x=0
 

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