Using Euclid's Algorithm to find a solution in the integers?

I have 2 problems. One has been solved and the other I tried, following the example of the 1st, but I am missing a piece of the puzzle.

1) 4x + 6y = 18
Doing Euclid's Algorithm: 6/4 = 1R2; 4/2 = 2R0 thererfore, 2 is the GCD of 4 & 6 and divides 18, so a solution exists.
Now divide everything in original equation by 2 giving 2x + 3y = 9.
Now the part I don't get, but have written down, says to let y=9 then 2x=-18 and x= -9. How was it determined that y=9?


2) The next problem is 21x + 51y = 3
After doing Euclid's Algorithm, I found that 3 is the GCD, and divided the original equation by 3, giving 7x + 5y = 1
Where do I go from here?

1) For the first equation , 2x+3y = 9 note that 3 must divide x
so write x= 3t. Then 6t+3y=9 and y=3-2t. The solution is
x=3t, y=3-2t for all integers t .

2) The second equation reduces to 7x+17y = 3 . To find
the solution reduce this mod 7 to get 0+3y = 3 mod 7 so
that y=1mod 7, and set y= 7t+1. Plug this in and solve for x,
7x+17(7t+1) = 3 and x = - 17t - 2. So all solutions in
integers are given by x= -17t -2 and y=7t+1 for all integer
values of t.