dependent system
Imagine two people each describing the same path, one saying go north-east at this pace and the other saying the same thing in different words. Their statements look distinct but carry identical content. A dependent system is like that: its equations, though perhaps dressed up differently, secretly describe the very same line, so every point on that line solves the whole system.
Because the equations coincide, a dependent system has infinitely many solutions rather than a single point. One equation adds no new information beyond the other — it is a rescaled, rearranged copy. For example, x + y = 3 and 2x + 2y = 6 are the same line; the second is just the first doubled.
When you solve a dependent system algebraically, the variables drop out and you reach a statement that is always true, such as 0 = 0. That tautology is the tell-tale sign of dependence. The solution is then described not as one pair but as a relationship — every (x, y) with x + y = 3, which you can write parametrically.
Eliminating from x + y = 3 and 2x + 2y = 6 gives 0 = 0, always true. Every point on x + y = 3 works — e.g. (0, 3), (1, 2), (3, 0) — infinitely many solutions.
Same line in disguise — endless solutions.
A dependent system is still consistent — it has solutions, just infinitely many. Dependence is about the equations duplicating information, not about lacking a solution.