What are the solutions to these diaphontine equations?

  • Thread starter Sam_
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In summary, the given equations are simultaneous equations and the first one does not depend on y. The second equation can be solved by multiplying the first equation by x^2 and solving for y. These equations are considered Diophantine equations, which are solved in terms of positive integers. There are two obvious solutions, but they may not be integers.
  • #1
Sam_
15
0
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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  • #2
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.
 
  • #3
In what sense is this "diophantine equations"? A Diophantine equation is a single equation to be solved in terms of (positive) integers.
 
  • #4
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:
  • #5
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).
 

FAQ: What are the solutions to these diaphontine equations?

What are Diophantine equations?

Diophantine equations are algebraic equations in which only integer solutions are considered. This means that the variables in the equation can only take on integer values, and no other types of numbers (such as fractions or decimals) are allowed.

Who is Diophantus?

Diophantus was a famous Greek mathematician who lived in the 3rd century AD. He is often known as the "father of algebra" for his work on solving Diophantine equations and other algebraic problems.

What are some examples of Diophantine equations?

Some examples of Diophantine equations include:
- 2x + 3y = 7
- x^2 + y^2 = 25
- 4x + 9y = 36
- x^3 + y^3 = 28

How are Diophantine equations solved?

There are various methods for solving Diophantine equations, including:
- Trial and error
- Using algebraic manipulations
- Applying modular arithmetic
- Utilizing advanced techniques such as continued fractions
- Using computer algorithms

What are the real-world applications of Diophantine equations?

Diophantine equations have many applications in fields such as cryptography, number theory, and computer science. They can also be used to model and solve real-world problems, such as in economics, physics, and engineering.

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