One solution, none, or infinitely many
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Flip to reveal answersHow many solutions can a system of linear equations have?
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Question
How many solutions can a system of linear equations have?
Answer
Exactly one, none, or infinitely many — never a finite number greater than one.
Question
What does 0 = 0 at the end of elimination mean?
Answer
An equation was redundant → infinitely many solutions (planes meet in a line).
Question
What does 0 = (non-zero) mean?
Answer
The equations contradict each other → no solution (inconsistent).
Question
Geometrically, what is a unique solution?
Answer
The three planes meet at a single point.
Question
Geometrically, what is 'infinitely many solutions'?
Answer
The three planes meet along a common line (or coincide).
Question
Geometrically, what is 'no solution'?
Answer
The planes have no point common to all three (e.g. they form a triangular prism).
Question
For a parameter k, how do you find the consistent value?
Answer
Eliminate to two copies of the same left side, then set their right-hand sides equal.
Question
Can a linear system have exactly two solutions?
Answer
No — only 0, 1, or infinitely many.
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Topic 1.16 hub
Systems of equations (HL only)
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