let's recall that perpendicular lines have negative reciprocal slopes, so say for a line with a slope of a/b

a perpendicular one will be -b/a.
what does that mean? well, that the lines are perpendicular of course, they meet at a 90° angle, and two straight lines meeting at 90° can only meet once.
since a solution to a system of equations is where the graph meet, and these two meet perpendicularly once, then that means only one solution.