What are the solutions to these diaphontine equations?

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The discussion focuses on solving a system of simultaneous equations, specifically 1/(x^2) + x^2 - 7 = 0 and 1/x + x - y = 0. The first equation is addressed first by multiplying through by x^2 to simplify it, allowing for the substitution of x into the second equation to find y. The term "Diophantine equations" is questioned, as these equations are typically expected to have integer solutions, but the solutions derived do not satisfy the first equation when evaluated. Ultimately, it is concluded that while there may be two solutions for y, neither fits the criteria for integer solutions in the context of the first equation. The discussion emphasizes the need to verify solutions against both equations to determine their validity.
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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