You can tell if there is no solution if you end up with a false statement after performing elimination. For example:
-x - y = 4
x + y = 6
Add the equations together
-x - y = 4
x + y = 6
0 + 0 = 10
0 = 10.
The line "0 = 10" is false, therefore there is no solution.
If the equations were to be graphed, the resulting lines wouldn't intersect each other. So they would not share any points in common.
