What are the solutions to these diaphontine equations?

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Discussion Overview

The discussion revolves around solving a pair of simultaneous equations, with a focus on whether they can be classified as Diophantine equations and the nature of their solutions, particularly in the context of integer values.

Discussion Character

  • Technical explanation, Debate/contested, Mathematical reasoning

Main Points Raised

  • One participant presents two simultaneous equations and seeks to find the value of y.
  • Another participant suggests solving the first equation independently before substituting the result into the second equation to find y.
  • A third participant questions the classification of the equations as Diophantine, noting that Diophantine equations typically involve a single equation solved for integers.
  • Some participants assert that if the equations are to be solved in integers, there are only a limited number of solutions for the second equation, which do not satisfy the first equation.
  • One participant emphasizes that the solutions to the system should be obvious and that the integer nature of the solutions can be determined after solving the system.

Areas of Agreement / Disagreement

Participants express differing views on the classification of the equations as Diophantine and the existence of integer solutions. There is no consensus on the nature of the solutions or the classification of the equations.

Contextual Notes

There are unresolved assumptions regarding the definitions of Diophantine equations and the conditions under which the solutions are sought (e.g., integer solutions versus real solutions).

Sam_
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So help me with this:

this are simultanious equations.

1/(x^2) + x^2 -7 = 0
&&
1/x + x -y = 0

What is y?
 
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The first one doesn't depend on y, so tackle that one first.
Multiply the first by x^2 and solve it, then plug the result into the second and solve for y.
 
In what sense is this "diophantine equations"? A Diophantine equation is a single equation to be solved in terms of (positive) integers.
 
In fact, if these are expected to be solved in integers, there is only one (and plain sight obvious) solution for the second equation.

P.S.: A solution which does not check on the first equation, anyway.
P.P.S.: Oh, sorry, two obvious solutions. I was forgetting there is such thing as negative numbers. Doh! Neither will fit the first equation.
 
Last edited:
Dodo said:
In fact, if these are expected to be solved in integers, there is only one (and plain sight obvious) solution for the second equation.

It is plain sight obvious how to solve this system and get the two solutions to it, after which it will be even more plain sight obvious whether they are integers (if not, clearly there are no integer solutions).
 

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