How do you solve x-y=1 ; x+y=3. explain

You isolate one variable. Let's say that x will be that variable in the first equation:
x-y=1
 +y  +y
x=y+1
So now you have the equation for x.  All you have to do is substitute it into the other:
x=y+1
(y+1)+y=3
Combine like terms and solve:
2y+1=3
    -1  -1
2y=2
2y/2=2/2
y=1
Now that you have the value of one, substitute it into either of the equations to find x.
If you put it in the first:
x-1=1
 +1  +1
x=2
If you put it into the other, you would get the same answer.
Therefore, y=1 and x=2

X - y = 1
x + y = 3

2x = 4
__ _
2 2
x = 2
x + y = 3
2 + y = 3
-2 -2

y = 1
(x, y) = (2, 1)