A system of two linear equations can be graphed as two lines.
The solution of the system of equations is a point that is part of both lines.
As a graph, you see two intersecting lines. The only point on both lines is the intersection of the lines. This is the solution of the system of equations.
Now think of two parallel lines. They never intersect. Parallel lines have the same slope. Since there is no point that is part of both lines, there is no solution to the system of equations.