AHL 1.16—Solution of systems of linear equations

You're already comfortable solving systems of two linear equations , for instance:

$\{\begin{array}{l}2x+y=6\\ 5x-y=1\end{array}$

In AA HL, the challenge is extended to systems of three equations in three unknowns, where each equation takes the general form:

$a_{i}x+b_{i}y+c_{i}z=d_i$

Three solving methods are available for such systems:

1. Substitution , express one variable in terms of the others, then substitute
2. Elimination , add or subtract multiples of equations to cancel variables
3. Gaussian Elimination , use matrix row operations to reduce the system systematically

Of these, Gaussian elimination is by far the most powerful and efficient for $3\times 3$ systems.

When solving any system of linear equations, exactly one of three outcomes is possible:

Unique Solution: The system has exactly one set of values $(x,y,z)$ satisfying all equations simultaneously. Geometrically, three planes intersect at a single point.

Infinite Solutions: The system has infinitely many solutions, expressible using one or more free parameters. Geometrically, the planes share a common line or are coincident.

Inconsistent System: 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. $0=5$) → no solution
• Arriving at an always-true statement (e.g. $0=0$) → infinite solutions
• Arriving at specific values for all variables → unique solution