Solving Diophantine Equation: 97x+35y=13

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Discussion Overview

The discussion revolves around solving the Diophantine equation $$97x + 35y = 13$$. Participants explore methods for finding particular and general solutions, including the application of Euclid's algorithm and the concept of homogeneous solutions. The conversation includes attempts to clarify the steps involved in reaching a solution and verifying correctness.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Some participants present their solutions as $$x = 169 - 35K$$ and $$y = -468 - 97K$$, derived from their application of the method outlined in their textbooks.
  • Others mention that Wolfram Alpha provides a different solution of $$x = 29$$ and $$y = -80$$, suggesting that these are effectively the same due to the nature of Diophantine equations.
  • A participant expresses confusion regarding the method for finding homogeneous solutions and the correct substitution process, leading to a discussion about the signs in their equations.
  • There is a back-and-forth regarding the correct formulation of the general solution, with participants correcting each other’s substitutions and clarifying the relationships between the variables.
  • One participant acknowledges a mistake in their earlier calculations, leading to a confirmation of the correct answer by others.

Areas of Agreement / Disagreement

Participants generally agree on the final form of the solution after some initial confusion and corrections. However, there are competing views on the intermediate steps and methods used to arrive at the solution, indicating that some aspects of the discussion remain unresolved.

Contextual Notes

Some participants express uncertainty about the application of Euclid's algorithm and the nature of the solutions, indicating potential limitations in their understanding of the method. The discussion also reflects varying interpretations of the steps involved in solving the equation.

Who May Find This Useful

This discussion may be useful for individuals studying number theory, particularly those interested in Diophantine equations and the methods for solving them, as well as those seeking clarification on the application of Euclid's algorithm.

Petrus
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Hello MHB,
I got problem with this http://staff.www.ltu.se/~larserik/applmath/chap10/part4.html (It's 'Exempel 1' and yes it's on Swedish), I follow my book method and get,
$$x=169-35K, \ y= -468-97K$$
I get the 'homogen soloution' that $$x=-35k, \ y=97k$$
'Solve equation $$97x+35y=13$$'

Regards,
 
Last edited:
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Petrus said:
Hello MHB,
I got problem with this http://staff.www.ltu.se/~larserik/applmath/chap10/part4.html (It's 'Exemple 1' and yes it's on Swedish), I follow my book method and get,
$$x=169-35K, \ y= -468-97K$$
I get the 'homogen soloution' that $$x=-35k, \ y=97k$$
'Solve equation $$97x+35y=13$$'

Regards,

Erm... what is your question?
 
Solve equation $$97x+35y=13$$
ehmm. I am suposed to find one soultion with gcd, eucliders algoritmen then find all other soloution, cause it got infinity soloution, for x and y. Do it make sense now?

Regards,
 
Petrus said:
Solve equation $$97x+35y=13$$
ehmm. I am suposed to find one soultion with gcd, eucliders algoritmen then find all other soloution, cause it got infinity soloution, for x and y. Do it make sense now?

Regards,

Ah, okay.
So do you know how to apply Euclid's algorithm?
(Btw, isn't this university-level stuff instead of pre-algebra?)
 
I like Serena said:
...
(Btw, isn't this university-level stuff instead of pre-algebra?)

Good call...moved to the Number Theory sub-forum.
 
I like Serena said:
Ah, okay.
So do you know how to apply Euclid's algorithm?
(Btw, isn't this university-level stuff instead of pre-algebra?)
Yes, I get x=169 and y=-468
 
Petrus said:
Yes, I get x=169 and y=-468

That looks about right.
Wolfram says x=29, y=-80, which is effectively the same answer.
(Since you can add a multiple of 35 to x if you also subtract the same multiple of 97 from y.)
So you're all good?
 
I like Serena said:
That looks about right.
Wolfram says x=29, y=-80, which is effectively the same answer.
(Since you can add a multiple of 35 to x if you also subtract the same multiple of 97 from y.)
So you're all good?
so this answer is correct?
$$x=169-35K, \ y= -468-97K$$
cause that website says the answer shall be
http://staff.www.ltu.se/~larserik/applmath/chap10/eq409.gif
 
Petrus said:
so this answer is correct?
$$x=169-35K, \ y= -468-97K$$
cause that website says the answer shall be
http://staff.www.ltu.se/~larserik/applmath/chap10/eq409.gif
The website answer is correct. If you add a multiple of 35 to $x$ then you have to subtract that multiple of 97 from $y$.
 
  • #10
Opalg said:
The website answer is correct. If you add a multiple of 35 to $x$ then you have to subtract that multiple of 97 from $y$.
I am kinda confused cause I follow the method from my book.
* find a soloution $$x=x_0 , y=t_0$$ (particulate solution) that is what we find we euclidmens algorithmen.
*Find all soloution to homogeneous equation $$ax'+by'=0$$ and that is the one i do it wrong then...
 
  • #11
Petrus said:
*Find all soloution to homogeneous equation $$ax'+by'=0$$ and that is the one i do it wrong then...

The solution is $x'=kb,\ y'=-ka$, for integers k (note the minus sign).
Substitute to verify...
 
  • #12
I like Serena said:
The solution is $x'=kb,\ y'=-ka$, for integers k (note the minus sign).
Substitute to verify...
that is the part I strugle, my book say $$x'=-kb, \ y'= ka$$
 
  • #13
Petrus said:
that is the part I strugle, my book say $$x'=-kb, \ y'= ka$$

That works as well.
Can you substitute that solution in the equation?
 
  • #14
I like Serena said:
That works as well.
Can you substitute that solution in the equation?
$$x' = -35k , \ y'= 97k$$
 
  • #15
Petrus said:
$$x' = -35k , \ y'= 97k$$

Yes.
Can you substitute that in the equation $ax′+by′=0$?
 
  • #16
I like Serena said:
Yes.
Can you substitute that in the equation $ax′+by′=0$?
$$35(-35k)+97*97k=0$$ ?
 
  • #17
Petrus said:
$$35(-35k)+97*97k=0$$ ?

I kind of lost track of the equation. :)
But the equation was $97x'+35y'=0$.

Can you redo the substitution with this equation?
 
  • #18
I like Serena said:
I kind of lost track of the equation. :)
But the equation was $97x'+35y'=0$.

Can you redo the substitution with this equation?
$$97(-35k)+35*97k$$
my book says $$x=x_0+x' , y=y_0+y'$$
and our $$x_0=169$$ and $$y_0=-468$$
 
  • #19
Petrus said:
$$97(-35k)+35*97k$$

Do you see that this is zero?

my book says $$x=x_0+x' , y=y_0+y'$$
and our $$x_0=169$$ and $$y_0=-468$$

Yes?
 
  • #20
I like Serena said:
Do you see that this is zero?
Yes,
I like Serena said:
Yes?
That means the answer is $$x=169-35K, \ y= -468-97K$$
 
  • #21
Petrus said:
$$97(-35k)+35*97k$$
my book says $$x=x_0+x' , y=y_0+y'$$
and our $$x_0=169$$ and $$y_0=-468$$

Petrus said:
That means the answer is $$x=169-35K, \ y= -468-97K$$

Erm.

You have $x'=-35k,\ y'=97k$ and $$x_0=169$$, $$y_0=-468$$

If I substitute that into $$x=x_0+x' , y=y_0+y'$$, I get
$$\qquad x=169+(-35k) , y=-468+(97k)$$.

See the difference?
 
  • #22
I like Serena said:
Erm.

You have $x'=-35k,\ y'=97k$ and $$x_0=169$$, $$y_0=-468$$

If I substitute that into $$x=x_0+x' , y=y_0+y'$$, I get
$$\qquad x=169+(-35k) , y=-468+(97k)$$.

See the difference?
Finally, I did find my mistake...! I somehow did have $$-97k$$ in my paper... and confused myself..! Can you confirm that this is correct answer?
 
  • #23
Petrus said:
Finally, I did find my mistake...! I somehow did have $$-97k$$ in my paper... and confused myself..! Can you confirm that this is correct answer?

Erm, yes. It is the correct answer.
It's what the website solution said...
 
  • #24
I like Serena said:
Erm, yes. It is the correct answer.
It's what the website solution said...
Hello I like Serena,
I am sorry I just got confused, Thanks for taking your time and now I understand diophantine equation a lot better! Hey look at the bright side, you gained some 'thanks' ;)

Regards,
 
  • #25
Yep. Thanks! ;)
 

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