Use row reduction by writing the system as an augmented matrix, applying valid row operations until it reaches row-echelon form, and then reading or calculating the unknowns. This is AHL 1.16 content for IB Mathematics: Analysis and Approaches HL.
The Method
For the system
write the coefficients and constants as one matrix:
The permitted elementary row operations are swapping two rows, multiplying a row by a non-zero constant, and adding a multiple of one row to another. Each operation preserves the solution set because it corresponds to an equivalent algebraic step performed on an equation. Therefore, row reduction simplifies the system without changing its solutions.
Apply and :
Swap and , then apply :
The leading non-zero entry in each row is a pivot. Moving pivots progressively right creates row-echelon form and makes the final equation contain only one unknown. A pivot also identifies a variable that is not free.
Use back-substitution:
Therefore, the unique solution is .
Classifying the System
Row reduction can produce three outcomes:
| Outcome | Reduced-matrix evidence |
|---|---|
| Unique solution | A pivot in every variable column |
| No solution | A contradictory row such as |
| Infinitely many solutions | A zero row and at least one free variable; use a parameter |
Check final values by substituting them into the original equations. This catches arithmetic errors during elimination. If the system has free variables, define a parameter explicitly and give the general solution rather than listing one possible solution.
A common misconception is that row operations change the solution. They do not: permitted operations produce equivalent systems.
Exam Technique
For Paper 1, show each row operation and the back-substitution clearly. On Paper 2, matrix or RREF technology may be used, but for solve, record the matrix set-up and enough working to justify the solution. Always classify the result as unique, inconsistent, or dependent.