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I'm coming up blank on Wikipedia and other sources, though this seems elementary. I'd like to know what techniques or processes are used to find all (integer) solutions to an equation such as $3x+2y = 380$ using linear algebra.

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If its integers you want, it's called a "Diophantine equation", a linear one in this case. Do you want all solutions (in which case there are an infinite number if there are any) or only positive solutions? –  marty cohen Jul 30 '12 at 0:12
    
Only positive solutions. What should I know about Diophantine equations? –  mirai Jul 30 '12 at 0:13
    
You should know that there are general techniques for solving linear diophantine equations in any number of variables, and that these techniques can be found in introductory Number Theory textbooks (and, no doubt, on the web, as well). –  Gerry Myerson Jul 30 '12 at 0:30
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up vote 3 down vote accepted

I can't imagine using linear algebra to find the integer solutions to your equation, when it's so simple to note that $x$ must be even, say, $x=2z$, so the equation becomes $3z+y=190$, and the solution is $z$ is arbitrary, $y=190-3z$.

EDIT: For linear diophantine equations in several variables, a good starting place is the Wikipedia piece on Bezout's identity, http://en.wikipedia.org/wiki/Bezout%27s_identity

MORE EDIT: There's a nice discussion by Gilbert and Pathria of systems of linear diophantine equations here; now linear algebra comes into it.

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True, though I'd like to know what techniques to use with arbitrary coefficients and an arbitrary number of variables. –  mirai Jul 30 '12 at 0:14
    
There's an exposition by Leon Bernstein at fq.math.ca/Scanned/6-3/bernstein.pdf –  Gerry Myerson Jul 30 '12 at 0:38
    
I just came across the Gilbert PDF as well. Thanks for the Bernstein link. –  mirai Jul 30 '12 at 1:26
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3x+2y=380=(3-2)380

=>3(x+380)=-2(y+380)

Clearly, 2|(x+380) =>x+380=2a for some integer a.

=>x=2a-380=2(a-190)=2b where b=a-190.

Putting x=2b, 3(2b)+2y=380 =>3b+y=190=>y=190-3b.

So, (x,y)=(2b,190-3b) where b is some integer.

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