Solving a Linear Diophantine Equation(see description for examples)

Question

Let me start off by clarifying that(before you guys dismiss me), this is not a homework problem and I\'m not a university student. :)

EDIT Thanks to

Answer 1

Adding on to Klas' very accurate answer:

Hilbert's 10th problem asked if an algorithm existed for determining whether an arbitrary Diophantine equation has a solution. Such an algorithm does exist for the solution of first-order Diophantine equations. However, the impossibility of obtaining a general solution was proven by Yuri Matiyasevich in 1970

taken from: Wolfram MathWorld