>>13911028In terms of rationals - the equation forms a line, so you just find two arbitrary solutions and get all other as scalar multiples of the difference between them as vectors. Same goes in terms of any field
In terms of integers:
Note that 3 and 10 are co-prime, which means that the equation 3x+10y=1 has a solution and 3x+10y is also a linear function - scalar multiplying or adding pairs of numbers will give output that is the sum/scalar multiple of the product. So the algorithm to get all the integer solutions goes as follows:
1. Find a solution to 3x+10y=1 by the euclidean algorithm
2. Multiply the pair (x,y) by 101.
3. Find all the integer solutions of 3x+10y=0, which is trivial.
4. Note that the difference between any integer solution of 3x+10y=101 is an integer solution of 3x+10y=0, which are already found
Done.
