solve the system by graphing 
x + y = 3
x + y = -1

The idea is to graph both lines and find the
point where they cross, or intersect.

Get three points on the first line:

x + y = 3

Pick an arbitrary value for x or y.
I'll arbitrarily pick the value 2 
and arbitrarily pick the letter x
to substitute it for:

2 + y = 3
    y = 3 - 2
    y = 1

So (x,y) = (2,1) is one point on the first line

Next I'll arbitrarily pick the value 4 
and arbitrarily pick the letter y
to substitute it for:

x + 4 = 3
    x = 3 - 4
    x = -1

So (x,y) = (-1,4) is another point on the first line

As a check, I'll find a third point

Next I'll arbitrarily pick the value -3 
and arbitrarily pick the letter y
to substitute it for:

x + (-3) = 3
   x - 3 = 3
       x = 3 + 3
       x = 6 

So (x,y) = (6,-3) is another point on the first line

Plot those points



Now I'll draw a smooth dark red straight line through them:

 

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Now I'll do the same for the other equation:

Get three points on the second line:

x + y = -1

Pick an arbitrary value for x or y.
I'll arbitrarily pick the value 3 
and arbitrarily pick the letter x
to substitute it for:

3 + y = -1
    y = -1 - 3
    y = -4

So (x,y) = (3,-4) is one point on the 2nd line

Next I'll arbitrarily pick the value 0 
and arbitrarily pick the letter y
to substitute it for:

x + 0 = -1
    x = -1
    

So (x,y) = (-1,0) is another point on the 2nd line

As a check, I'll find a third point

Next I'll arbitrarily pick the value -2 
and arbitrarily pick the letter x
to substitute it for:

(-2) + y = -1
  -2 + y = -1
       y = -1 + 2
       x = 1 

So (x,y) = (1,-2) is another point on the 2nd line

Plot those points



Now draw a smooth green straight line through them:

 

Oh. my! It looks like the dark red line and the
green line run parallel to each other, so they
will never intersect. That means there is no
solution.  This is called an inconsistent
system of equations.

Edwin