I always found it easier to just move it around. It may take more steps but I know it's correct. So maybe check this and see if it looks easier for you too? You don't have to use it but I always found this one made me understand it rather than remember the rules.
For example.
You multiply your one and get
-12x+28y=104
12x+15y=39
Then you take one and move it around. So let's say we want -12x on the bottom one.
Move Y to the other side.
12x=39-15y (sign changes if you move it to the other side)
Now we need -12x not 12x. So all signs change.
-12x = -39 + 15y
Go back to the original
-12x + 28y = 104
-12x = -39 + 15y
So -12x is same as -39 + 15y. So replace -12x in first one with the stuff after equal in second one.
-39 + 15y +28y = 104
Keep only Y on the left.
15y + 28y = 104 + 39 (sign changes when you move to the other side)
Simplify
43y = 143
Then you'd go through the usual. Get y and put it in to find x.