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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. ie. 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?

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Do you mean rational or integer solutions? You say rational, but Diophantine equations only admit integer solutions. –  smackcrane Nov 9 '11 at 23:00
    
Thanks for mentioning, its primarily the rational solutions I'm interested in. –  BlueFishi Nov 9 '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 1:53
    
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 1:55
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