SL 1.8—Use of technology to solve systems of linear equations and polynomial equations

Solving systems of equations by hand — substitution, elimination, row reduction — works well for simple cases. But as the number of variables grows or polynomial degrees increase, the algebra becomes tedious and error-prone. This is exactly where technology steps in.

In IB Mathematics AI, you are expected to use graphing calculators and software tools (such as GeoGebra or Desmos) to solve:

• Systems of linear equations (up to 3 variables)
• Polynomial equations of any reasonable degree

The skill being assessed is not the manual calculation — it's knowing how to set up the problem, use the tool correctly, and interpret the output in context.

For a system with $n$ variables, you generally need $n$ equations to find a unique solution. However, systems can have:

• One unique solution — lines/planes intersect at exactly one point
• Infinitely many solutions — equations are dependent (lines/planes overlap)
• No solution — equations are inconsistent (parallel lines/planes that never meet)

Technology approaches for linear systems:

1. Graphical method — plot each equation and find the intersection point(s)
2. Matrix method (rref) — enter the augmented matrix and apply reduced row echelon form

Recognising non-unique solutions from rref output:

When you apply rref, not every system gives a clean unique solution. Learn to recognise these two special cases:

• A row of the form $\left[\begin{array}{ccccc}0& 0& 0& |& 0\end{array}\right]$ means the equations are dependent — the system has infinitely many solutions (one equation is redundant).
• A row of the form $\left[\begin{array}{ccccc}0& 0& 0& |& k\end{array}\right]$ where $k\ne 0$ means the system is inconsistent — there is no solution (the equations contradict each other).