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I'm currently doing a project and for part of it I've been looking at rational solutions to linear eqautions in two vaiables, i.e. $ax+by=c$. I'd like to add a bit about what we can use these types of equations and their solutions for but can't find anything on the internet. Surely there's more to them than just finding $x$ and $y$. Are there any interesting uses for these types of equations?

sTertooy
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BlueFishi
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    Do you mean rational or integer solutions? You say rational, but Diophantine equations only admit integer solutions. – smackcrane Nov 09 '11 at 23:00
  • Thanks for mentioning, its primarily the rational solutions I'm interested in. – BlueFishi Nov 09 '11 at 23:10
  • See this question or this link. See rational solutions admit integers solutions too. And to find rational solutions its easy, just isolate one of the variables.

    $y=\frac{c-ax}{b} \quad (b \ne 0)$, so the ordered solution pair is

    $$(x,\frac{c-ax}{b}) \quad (b \ne 0)$$

     – GarouDan Nov 11 '11 at 01:53
  • If $b=0$, $y$ can be anything and $x=\frac{c}{a} \quad (a \ne 0)$. If $a$ and $b$ are $0$, so $x$ and $y$ can be anything. – GarouDan Nov 11 '11 at 01:55

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