IB Maths: Systems of Linear Equations FAQ

Set up equations from the context first, then solve using your GDC's simultaneous equation solver, substitution, or row reduction. Always state the solution clearly (e.g. x = 2, y = -1, z = 3) and check it satisfies every original equation — examiners award the final mark for a verified, clearly stated answer.

On a TI-84 or Casio, enter the coefficients as a 3x4 augmented matrix, then use the equation-solver or rref (row-reduce) function to read off x, y and z directly. Practise this on your own calculator well before the exam — syntax differs between models.

Substitution and elimination work well for two equations and two unknowns and are expected as hand-written working on Paper 1. Matrix methods (row reduction or inverse matrices) scale better to three unknowns and are the standard approach when a GDC is allowed on Paper 2.

Row-reduce the augmented matrix: a unique solution gives three pivot rows with no contradictions; no solution appears as a row like 0 = 5 (a contradiction); infinite solutions appear as a row of all zeros, meaning one equation depends on the others. IB questions often ask you to find the value of a parameter that causes each case.

Set up the augmented matrix with k left as a variable, row-reduce as far as possible, then find the value(s) of k that make a row zero or contradictory. This tests the same three-case logic (unique, none, infinite) but requires algebraic manipulation instead of pure numbers.

Systems of linear equations with two unknowns appear in both AA and AI at SL and HL. Three-unknown systems solved via matrices and row reduction are HL content only, in both AA and AI, under the Number and Algebra topic. According to the IB, the current Mathematics guides had first teaching in 2019 and first exams in 2021, still current for 2025 assessment.

Yes — HL students need matrix representation of systems, including writing a system as Ax = b, calculating the determinant to check for a unique solution, and using row reduction or an inverse matrix to solve it. This links directly to the Matrices sub-topic in the HL-only content.

The IB's approach to assessment objectives rewards applying maths to unfamiliar contexts, so systems of equations questions are usually dressed up as cost, mixture, or motion problems rather than given as bare equations. The maths is identical — the skill examiners test is translating words into equations accurately.

In my experience marking mocks, the biggest losses are arithmetic slips in hand-elimination, forgetting to state units or context in the final answer, and rounding mid-calculation instead of carrying exact values. A close second is misreading the question and setting up the wrong equations from the word problem.

Two-unknown systems are genuinely one of the more accessible topics on the SL papers — mechanical once you know the method. Three-unknown systems with a parameter, tested mainly at HL, are harder and regularly appear as a Paper 2 or Paper 3 discriminating question worth 6-8 marks.

A standalone two-unknown system is usually a short 3-5 mark question, often within a longer Paper 1 or Paper 2 question. Three-unknown, parameter-based systems at HL can be worth 6-8 marks and sometimes anchor a full Paper 3 investigation question.

Paper 1 (no GDC) expects clean, integer-friendly systems solved by substitution or elimination with full working shown for method marks. Paper 2 and Paper 3 allow a GDC, so messier numbers and three-unknown systems appear, and you're expected to use matrix functions rather than hand-eliminate.

Your child needs repeated exposure to both clean textbook-style systems and messier, worded exam-style ones with a parameter. On RevisionPrep, the Maths AA/AI Topical Worksheets group Number and Algebra questions by exact sub-topic, and the Mock Papers let students time themselves under real Paper 1-3 conditions.

A-Level Maths (Pure content) covers two-unknown simultaneous equations at a similar depth to IB SL, but doesn't require the matrix-based, three-unknown, parameter-dependent systems that sit in IB HL's Number and Algebra topic. Students moving from A-Level to IB HL Maths typically need extra practice specifically on matrices and row reduction.