Why can't a system of 2 linear equations have precisely 2 solutions

Question

Why can't a system of two linear equations have precisely two solutions?

Answer 1

I think a better answer to this question is Two equations representing two lines may intersect at

1 point, or no points, or there may be, in fact, only one distinct line, in which case there are infinitely many points.  This is the exact & complete answer.

 

That is, the two equations might be dependent (infinitely many points), or parallel lines (no points), or intersecting (lines have different y-intercepts but same slope).

Answer 2

For a system of linear equations, each equation represents a line.  From Euclidean geometry, two lines either intersect in only one point or are parallel so they do not intersect.  It is for this reason that the linear systems has either one or two solutions, depending if the lines intersect or not.