Answer
We know the system has no solution when we have a false statement.
Work Step by Step
Example system: $x+y=2$ and $-x-y=1$
$x+y=2$
$-x-y=1$
$x+y+(-x-y)=2+1$
$0 = 3$
We know this is not a true statement, so we also know there are no solutions to the system.