Practice question · Multiple choice
A system of two linear equations in two unknowns has exactly one solution, no solution, or infinitely many - and never, say, exactly two. Why are those the only possibilities?
Hints
- Draw two straight lines in the plane. Enumerate the ways they can be positioned relative to each other.
- Could two distinct straight lines cross twice?
Show the answer
A. Because two lines meet once, never, or everywhere — not twice.
Why
Each equation is a line, and two lines cross once, run parallel, or coincide, if they share two points they share all of them. So the trichotomy is geometry rather than a fact about the method. It generalises: a linear solution set is empty or an affine subspace, and a subspace with more than one point has infinitely many. Nonlinearity breaks it immediately.
Practise Systems of Linear Equations
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More questions on Systems of Linear Equations
- Sort each manipulation by whether it is a legitimate row operation on a system.
- Three equations in three unknowns can have no solution even when no two of them contradict each other. How?
- Order the stages of solving a linear system by Gaussian elimination.
- Select every system below that is INCONSISTENT, meaning it has no solution.
- A consistent linear system with more unknowns than equations always has infinitely many solutions.