Preview
IB DP Mathematics: Applications and Interpretation | SL & HL | Number and Algebra — Topic 1.8
📋 Learning Intentions
- Solve systems of two and three linear equations in two and three unknowns using the GDC (matrix method or equation solver).
- Find the roots of polynomial equations of degree 2, 3, and higher using the GDC graphing and solver functions.
- Interpret the solutions of a system of equations in a real-world context, including identifying when a system has no solution or infinitely many solutions.
- Write a system of equations from a word problem and set it up correctly for GDC entry.
- Determine whether a polynomial equation has real or non-real roots by examining the graph on the GDC.
- Justify the reasonableness of GDC solutions by checking answers in the original equations.
Real-World Introduction
A logistics company needs to allocate three types of delivery vehicles — bicycles, vans, and trucks — across three routes. Each route has a fixed total capacity requirement, a fuel budget, and a time constraint. The constraints form three simultaneous equations in three unknowns. Without technology, solving this by hand would take several minutes of careful algebra; with a GDC, the answer appears in seconds.
Similarly, an engineer modelling the deflection of a bridge beam under load produces a cubic polynomial equation. The roots of that polynomial correspond to the positions along the beam where deflection is zero — critical safety information. Finding those roots by factoring is often impossible; the GDC graph and solver make it routine.
🌍 Why This Matters Systems of linear equations appear in supply-chain optimisation, network flow analysis, and economic modelling. Polynomial equations arise in structural engineering, trajectory modelling, and financial break-even analysis. The GDC is the professional tool that makes these calculations accessible and fast — exactly as required in IB Math AI.
Core Concepts & Definitions
When two or more conditions must hold simultaneously, each condition becomes one equation. The set of all equations that must be satisfied together is called a system. The goal is to find values of the unknowns that satisfy every equation at once.
📖 Key Definition — System of Linear Equations A system of linear equations is a collection of two or more equations, each of degree 1, that share the same set of unknowns. A solution is an assignment of values to the unknowns that satisfies every equation in the system simultaneously.
A system of \(n\) linear equations in \(n\) unknowns can be written in matrix form as \(A\mathbf{x} = \mathbf{b}\), where \(A\) is the coefficient matrix, \(\mathbf{x}\) is the column vector of unknowns, and \(\mathbf{b}\) is the column vector of constants. The GDC solves this directly — no manual row reduction required.
2×2 Systems — Two Equations, Two Unknowns
A 2×2 system has the general form: