How many solutions? The graph of a system

Three possibilities

• A system of two linear equations has one of three outcomes, and each has a clear picture:

• One solution — the lines have different gradients, so they cross exactly once.
• No solution — the lines are parallel (same gradient, different intercept), so they never meet.
• Infinitely many solutions — the two equations describe the same line, so every point on it works.

Spotting the case from the algebra

• You do not need the graph — the elimination step tells you which case you are in:
  • If you reach a normal value like $x = 2$, there is one solution.
  • If both variables vanish and you get a false statement like $0 = 5$, there is no solution.
  • If both variables vanish and you get a true statement like $0 = 0$, there are infinitely many solutions.

Why it matters in context

• In a real problem, no solution means the conditions contradict each other (they cannot all be met).
• Infinitely many means one condition repeats another, so the problem is under-determined — you need more information to pin down a single answer.