By addition method, it is the same thing as elimination, which is what it is officially called.
Elimination:
So we can mulitply the bottom equation by 2 so that we can then cancle out the "x's" later on
(2x+3y=19)*2
4x+6y=38
Now we have the two equations
4x-y=3 , 4x+6y=38
We can then subtract these two equations to cancel out the "x"
4x-y-(4x+6y)=3-38
-y-6y=-35
-5y=-35
5y=35
y=7
So now we have figured out our y value is 7, and we can substitute it into our equation to get our "x" value now.
4x-7=3
4x=10
x=5/2 or 2.5