Introduction to Systems of Linear Equations
You're already comfortable solving systems of two linear equations — for instance:
In AA HL, the challenge is extended to systems of three equations in three unknowns, where each equation takes the general form:
Three solving methods are available for such systems:
- Substitution — express one variable in terms of the others, then substitute
- Elimination — add or subtract multiples of equations to cancel variables
- Gaussian Elimination — use matrix row operations to reduce the system systematically
Of these, Gaussian elimination is by far the most powerful and efficient for systems.
Substitution and elimination become extremely tedious for three-variable systems. Learn Gaussian elimination well — it is the method examiners expect you to use and show clearly.
Types of Solutions
When solving any system of linear equations, exactly one of three outcomes is possible:
The system has exactly one set of values satisfying all equations simultaneously. Geometrically, three planes intersect at a single point.
The system has infinitely many solutions, expressible using one or more free parameters. Geometrically, the planes share a common line or are coincident.
The system has no solution — the equations contradict each other. Geometrically, at least two planes are parallel, or the planes form a "triangular prism" with no common point.
A quick diagnostic when solving algebraically:
- Arriving at a false statement (e.g. ) → no solution
- Arriving at an always-true statement (e.g. ) → infinite solutions
- Arriving at specific values for all variables → unique solution
Students often misread the diagnostic. Remember: means infinitely many solutions, not "zero equals zero, so everything is zero."