a system of two lines
Put two straight-line equations side by side and ask: what do they have in common? Geometrically you are asking how two lines on the plane are arranged, and there are only three possibilities — and seeing which is which is the same as counting the solutions the two equations share.
Two lines can intersect at a single point (one shared solution), be parallel and never meet (no shared solution), or be coincident — the very same line written two ways (infinitely many shared solutions, every point of the line). You can tell which case you are in from the slopes and intercepts: different slopes means the lines cross at one point; equal slopes but different y-intercepts means parallel and no solution; equal slopes and equal intercepts means coincident, the same line. Algebraically, solving the system by substitution or elimination either pins down one (x, y), or collapses to a false statement like 0 = 5 (parallel, no solution), or to an always-true statement like 0 = 0 (coincident, infinitely many).
This trichotomy — exactly one, none, or infinitely many — is the simplest face of linear systems, the gateway to solving larger systems and to linear algebra. A practical translation: a system that has at least one solution is called consistent (intersecting or coincident), and one with no solution is inconsistent (parallel). The honest subtlety is the coincident case: two equations that look different, like x + y = 2 and 2x + 2y = 4, can secretly be the same line — always check whether one equation is just a multiple of the other before declaring 'two lines'.
Consider x + y = 5 and x - y = 1. Adding eliminates y: 2x = 6, x = 3, then y = 2, a single intersection (3, 2) — the lines cross. But x + y = 5 and x + y = 9 have equal slopes and different constants; subtracting gives 0 = 4, impossible, so they are parallel with no solution.
One solution means crossing, an impossible equation means parallel, an identity means the same line.
Equal slope alone does not mean parallel-and-separate: if the intercepts also match, the two equations are the same line (coincident), with infinitely many solutions. Check the intercepts, not just the slopes.