整数方程: 

10x+6y=12

The point here is not to solve this specific equation but rather to generate an algorithm to handle the more general problem:

ax+by=c  eq(1), all variables are integers.

first extract common denominator between a,b and c to the form

a1x+b1y=c1. eq(2) 

c1 is prime to both a1 and b1.

Now, if a1 and b1 have common denominator, then there is no solution.

So we look at 

a1x+b1y=1 eq(3).