Linear Diophantine Equations

I have a test coming up, and part of this test is using linear diophantine equations. Now I can obtain the gcd, but then I get stuck. I know it’s really simple… but I just cant’ get my head around it. Does anyone have a nice simple explanation and steps to follow?
It would be nice if you could use this example to explain, 1713x + 2000y = 3
cheers.

I know you know how to find the gcd, but let’s just start from the beginning:

1713x + 2000y = 3

Let’s find the gcd of 1713 and 2000:

2000 = 1713 * 1 + 287

1713 = 287 * 5 + 278

287 = 278 * 1 + 9

278 = 9 * 30 + 8

9 = 8 * 1 + 1

8 = 1 * 8 + 0

So 1 is the gcd. Write this as linear combination of 1713 and 2000; you can do this using the above equations in reverse order, step by step:

1 = 9 - 8

1 = 9 - (278 - 9*30) = -278 + 31 * 9

1 = -278 + 31 * (287 - 278) = 31 * 287 - 32 * 278

1= 31 * 287 - 32(1713 - 2875) = -321713 + 191*287

1 = -32 * 1713 + 191(2000 - 1713)

so 1 = 191 * 2000 - 223 * 1713

We want the linear combination to be equal to 3, not 1:

3 = 573 * 2000 - 669 * 1713

This is one of the solutions to the original equation. There are infinitely many more, and they are all of the form:

x = -669 + 2000n/gcd, y = 573 - 1713n/gcd, n an integer.

In this case, gcd = 1, of course, so our general solution is:

x = -669 + 2000n, y = 573 - 1713n