A system has no solution if no values satisfy every equation, one solution if exactly one set of values satisfies them, and infinitely many solutions if an unlimited number of sets satisfy them. This is AHL 1.16 content in IB Mathematics: Analysis and Approaches HL.
How the Classification Works
Solve the equations simultaneously using elimination, substitution, or row reduction. The final equations reveal whether the system is consistent (at least one solution) or inconsistent (no solution).
| Classification | Algebraic result after elimination | Geometric meaning |
|---|---|---|
| No solution | A contradiction such as | Lines are parallel and distinct; three-dimensional planes have no common point |
| One solution | Each variable has one determined value | Lines intersect once; planes have exactly one common point |
| Infinitely many solutions | An identity such as , with at least one free variable | Equations represent the same line or plane, or planes share a line |
For example, consider
Adding the equations gives , so and . Therefore, the system has one solution: .
By contrast,
produces after eliminating the variables. This contradiction means the system has no solution.
Finally,
produces . The second equation is a multiple of the first, so one variable is free. Writing gives , producing infinitely many solutions .
A common misconception is that means “the solution is zero.” It actually shows that one equation adds no new restriction; a general solution must be given using a parameter.
Exam Technique
On AA HL Paper 1, show the algebraic elimination or row-reduction steps. On Paper 2, technology may be used, but interpret the result clearly. Examiners expect the classification and, for infinitely many solutions, a parameterized general solution.