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.
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.
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.