Solving Diophantine Equation: 97x+35y=13

  • MHB
  • Thread starter Petrus
  • Start date
In summary: Finally, I did find my mistake...! I somehow did have -97k in my paper... and confused myself..! Can you confirm that this is correct...?Yes, that is correct.
  • #1
Petrus
702
0
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,
\(\displaystyle x=169-35K, \ y= -468-97K\)
I get the 'homogen soloution' that \(\displaystyle x=-35k, \ y=97k\)
'Solve equation \(\displaystyle 97x+35y=13\)'

Regards,
 
Last edited:
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  • #2
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,
\(\displaystyle x=169-35K, \ y= -468-97K\)
I get the 'homogen soloution' that \(\displaystyle x=-35k, \ y=97k\)
'Solve equation \(\displaystyle 97x+35y=13\)'

Regards,

Erm... what is your question?
 
  • #3
Solve equation \(\displaystyle 97x+35y=13\)
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,
 
  • #4
Petrus said:
Solve equation \(\displaystyle 97x+35y=13\)
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?)
 
  • #5
I like Serena said:
...
(Btw, isn't this university-level stuff instead of pre-algebra?)

Good call...moved to the Number Theory sub-forum.
 
  • #6
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
 
  • #7
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?
 
  • #8
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?
\(\displaystyle x=169-35K, \ y= -468-97K\)
cause that website says the answer shall be
http://staff.www.ltu.se/~larserik/applmath/chap10/eq409.gif
 
  • #9
Petrus said:
so this answer is correct?
\(\displaystyle x=169-35K, \ y= -468-97K\)
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 \(\displaystyle x=x_0 , y=t_0\) (particulate solution) that is what we find we euclidmens algorithmen.
*Find all soloution to homogeneous equation \(\displaystyle ax'+by'=0\) and that is the one i do it wrong then...
 
  • #11
Petrus said:
*Find all soloution to homogeneous equation \(\displaystyle ax'+by'=0\) 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 \(\displaystyle x'=-kb, \ y'= ka\)
 
  • #13
Petrus said:
that is the part I strugle, my book say \(\displaystyle x'=-kb, \ y'= ka\)

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?
\(\displaystyle x' = -35k , \ y'= 97k\)
 
  • #15
Petrus said:
\(\displaystyle x' = -35k , \ y'= 97k\)

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

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?
\(\displaystyle 97(-35k)+35*97k\)
my book says \(\displaystyle x=x_0+x' , y=y_0+y'\)
and our \(\displaystyle x_0=169\) and \(\displaystyle y_0=-468\)
 
  • #19
Petrus said:
\(\displaystyle 97(-35k)+35*97k\)

Do you see that this is zero?

my book says \(\displaystyle x=x_0+x' , y=y_0+y'\)
and our \(\displaystyle x_0=169\) and \(\displaystyle y_0=-468\)

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

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

Erm.

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

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

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

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

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

See the difference?
Finally, I did find my mistake...! I somehow did have \(\displaystyle -97k\) 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 \(\displaystyle -97k\) 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! ;)
 

FAQ: Solving Diophantine Equation: 97x+35y=13

How do you solve a Diophantine equation?

Solving a Diophantine equation involves finding integer solutions for the given equation. This can be done using various techniques such as substitution, elimination, or by using modular arithmetic.

What is the significance of the coefficients in a Diophantine equation?

The coefficients in a Diophantine equation represent the values of the variables and their corresponding powers. They are crucial in determining the possible solutions and the complexity of solving the equation.

Can a Diophantine equation have multiple solutions?

Yes, a Diophantine equation can have multiple solutions. In fact, infinite solutions are possible for some equations. However, for equations with more than two variables, finding all the solutions can be challenging.

How do you approach a Diophantine equation with large coefficients?

In cases where the coefficients are large, it is helpful to use a combination of techniques such as substitution, elimination, and modular arithmetic. It may also be beneficial to use a computer program or calculator to handle the complex calculations.

Are there any real-world applications of Diophantine equations?

Yes, Diophantine equations have various applications in fields such as number theory, cryptography, and computer science. They are also used in designing secure communication systems and in the study of rational points on elliptic curves.

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