Here is how I would solve such a problem:

15x= 45 (mod 50). Since both 15 and 45 are divisible by 5, that becomes 3x= 9 (mod 50).

That means that 3x= 9+ 50y for some integer y. That is the same as 3x- 50y= 9

Now, 3 divides into 50 16 times with a remainder 2. 2 divides into 3 once with remainder 1.

That is, 3- 2= 1 and 50- 3(16)= 2. Then 3- (50- 3(16))= 3(17)- 50= 1 and, multiplying by 9, 3(153)- 50(9)= 9. That is, x= 153= 3 (mod 50),

Check: 15*3= 45 which is, of course, 45 (mod 15).