Systems of Equations (AA HL)
A system of linear equations can have exactly one solution, infinitely many, or none at all - and the AA HL syllabus expects you to solve systems of up to three equations in three unknowns both algebraically and with technology, and to recognise which of those three cases you're in. This topic covers elimination, substitution, row reduction, and writing a general solution in parametric form when there are infinitely many.
What the syllabus says
This topic maps onto one point in the official IB Analysis & Approaches syllabus.
| Code | Syllabus content |
|---|---|
| AHL1.16 | Solutions of systems of linear equations, with a maximum of three equations in three unknowns, solved using both algebraic and technological methods (for example row reduction or matrices) - including cases with a unique solution, an infinite number of solutions, or no solution. Finding a general solution for a system with an infinite number of solutions. |
This links to the intersection of lines and planes in the vectors topic (AHL3.18) - a system of three equations is really three planes, and its solution set is where they meet.
Key terms
Five words worth knowing cold before you touch the formulas below - each with a worked example showing exactly what it means.
What is a system of linear equations?
A system of linear equations is a set of two or three equations, each linear in the same unknowns, that must all be satisfied simultaneously. Solving the system means finding every combination of values that makes every equation true at once.
e.g. \(x+y=5\) and \(x-y=1\) together give the unique solution \(x=3,\ y=2.\)
What does a "unique solution" mean?
A unique solution is a single point (or triple of values) that satisfies every equation in the system, and no other point does. Geometrically, for a 3x3 system this means the three planes meet at exactly one point in space.
e.g. \(5x+2y=16\) and \(3x-2y=0\) reduce (adding) to \(8x=16\), giving the unique solution \(x=2,\ y=3.\)
What does "infinitely many solutions" mean?
A system has infinitely many solutions when the equations aren't independent enough to pin down a single point - after elimination, one equation becomes \(0=0\), leaving a free variable. Geometrically, the planes meet along a whole line (or coincide entirely).
e.g. \(x+y+z=1,\ 2x+2y+2z=2,\ 3x+3y+3z=3\) are all the same plane, so any point on it solves the system.
What does "no solution" (inconsistent) mean?
A system is inconsistent when the equations contradict each other - after elimination, one equation reduces to something impossible, like \(0=5\). No combination of values can satisfy every equation at once.
e.g. \(3x+6y=7\) and \(3x+6y=12\) can't both hold, since the left-hand sides are identical but the right-hand sides differ.
What is a parametric general solution?
When a system has infinitely many solutions, the general solution expresses every variable in terms of one free parameter (often \(t\)), describing the whole line of solutions in one formula rather than listing individual points.
e.g. Letting \(z=t\) in a dependent system might give the general solution \((x,y,z)=(3-2t,\ t-1,\ t).\)
Key formulas
This topic is method-driven rather than formula-driven - the table below summarises the main techniques, and the comparison table after it shows how to tell the three solution cases apart.
Method reference
None of these are formula-booklet entries - solving a system is a technique you apply, whether by hand or by GDC.
| Method | Used for | Booklet? |
|---|---|---|
| Elimination (add/subtract multiples of equations) | Removing one variable at a time | Not in booklet - method |
| Substitution | Solving for one variable and substituting into the others | Not in booklet - method |
| Row reduction of the augmented matrix | Systematic elimination for 3x3 systems, by hand or GDC | Not in booklet - method |
| Coefficient determinant \(=0\) | Testing whether a 3x3 system has a unique solution | Not in booklet - not core AA content, but a valid classification test |
Unique, infinite, or no solution
After row-reducing, the pattern left behind tells you exactly which case you're in.
| Case | Row-reduced pattern | Geometric meaning (3 planes) |
|---|---|---|
| Unique solution | Reduces to \(x=\ldots,\ y=\ldots,\ z=\ldots\) | Planes meet at a single point |
| Infinitely many solutions | A row reduces to \(0=0\) (consistent) | Planes meet in a line, or are coincident |
| No solution | A row reduces to \(0=k\) with \(k\neq0\) (inconsistent) | No common point - e.g. the planes form a triangular prism |
Solving systems algebraically
These are the three hand methods examined at AA HL - all valid, and often combined.
Elimination
Add or subtract scaled equations so one variable cancels, reducing the system by one unknown at a time.
Substitution
Solve one equation for one variable, then substitute that expression into the remaining equations.
Row reduction
Write the augmented matrix and use row operations to reach a triangular or fully reduced form.
General solutions in parametric form
When a system is dependent, one variable is left free - the general solution is built around that freedom.
Introduce a parameter
Let the free variable equal a parameter, e.g. \(z=t\), once row reduction shows it can't be pinned down.
Back-substitute
Use the remaining (reduced) equations to express every other variable in terms of \(t\).
Worked examples
Two full exam-style questions, marked exactly like the real thing. Try each one yourself before checking the worked solution.
Solve \(5x+2y=16,\qquad 3x-2y=0.\)
(a) State \(x.\)
(b) State \(y.\)
(c) Verify your solution by substituting into the second equation.
Worked solution
(a) Step 1 - Add to eliminate \(y\): \(8x=16\Rightarrow x\) M1
\(=2.\) A1
(b) Step 2 - Back-substitute: \(3(2)-2y=0\Rightarrow y=3.\) A1
(c) Check: \(3(2)-2(3)=6-6=0.\) R1
Consider \(x+2y=4\) and \(3x+ky=7.\)
(a) Find the value of \(k\) for which the system has no unique solution.
(b) For that value of \(k\), determine whether the system is inconsistent or has infinitely many solutions.
Worked solution
(a) No unique solution when the gradients match: \(\dfrac{3}{1}=\dfrac{k}{2}\Rightarrow k\) M1
\(=6.\) A1
(b) With \(k=6:\ 3x+6y=7\) vs \(3(x+2y)=12;\) these give \(7\) and \(12,\) inconsistent - no solutions. M1 A1
Common mistakes
The four slip-ups that account for most of the marks lost on this topic - worth reading before you start practising, not just after you get one wrong.
- Assuming a zero row automatically means infinitely many solutions. A row of zero coefficients only gives infinitely many solutions if the corresponding constant is also zero - if it isn't, the system is inconsistent instead.
- Sign errors when subtracting equations. Subtracting one equation from another to eliminate a variable is a common source of arithmetic slips - double-check every term, including the constants.
- Stopping after finding one variable. Finding \(x\) is only part of the job - always back-substitute to find every remaining unknown, and state all of them.
- Introducing a parameter but only checking two of the three equations. A parametric general solution must satisfy all three original equations, not just the two you used to derive it.
Using your GDC
Every step below is a real button sequence, not a vague "use your calculator" hint - covering the TI-84 Plus, TI-Nspire, and Casio fx-9860/fx-CG50. Pick your model to filter down to just the steps that apply to you.
Solve two or three simultaneous equations (linear systems) without elimination by hand.
- Write each equation in the form \(ax+by\ (+cz)=d.\)
- APPS → PlySmlt2 → Simultaneous Eqn Solver; set the number of equations/unknowns and enter the coefficients.TI-84
- menu → Algebra → Solve System of Equations, or use linSolve.Nspire
- Main menu → Equation → Simultaneous, set the number of unknowns, enter the coefficients, SOLVE.Casio
Tip: No solution or infinitely many? The calculator will flag it - that means the lines/planes are parallel or coincident.
Faster and safer than algebra for messy equations - useful when classifying a system with a parameter, e.g. solving \(\det(A)=0\) for the value of the parameter that makes the system singular.
- Graph \(f(x)\) first so you can see how many solutions exist and roughly where they are.
- Rearrange so everything is on one side: \(f(x)=0\) - or graph both sides as separate functions and find intersections.
- MATH → Solver: enter the expression, type a starting guess close to one root, press ALPHA + ENTER. Move the guess to near a different root and repeat for each solution.TI-84
- Type nSolve(f(x)=0, x, guess) - include a guess or interval e.g. nSolve(f(x)=0, x, 2) or nSolve(f(x)=0, x, {1,5}) to target a specific root.Nspire
- Run-Matrix → SolveN(f(x), x) returns all real roots at once; or use the Equation app for a visual approach.Casio
- Always verify each solution by substituting back into the original equation.
Tip: The solver finds ONE root near your starting guess - change the guess to find others. The graph shows you how many to expect.
See the full GDC guide for more calculator models and topics.
Ready to practise properly?
Systems of equations questions, marked instantly like the real exam.
Quick answers
The questions students on this topic ask most often.
How do I know if a 3x3 system has a unique solution?
Row-reduce it. If every variable ends up isolated with a single numerical value, the solution is unique. If a full row of coefficients reduces to zero while the constant also becomes zero, there are infinitely many solutions; if the coefficients reduce to zero but the constant doesn't, the system is inconsistent.
What does it mean geometrically when three planes have no common solution?
Each equation represents a plane in three dimensions. No common solution usually means the three planes form a triangular prism - each pair of planes meets in a line, but the three lines are parallel and never all cross at one point.
How do I find a general solution when there are infinitely many?
Reduce the system until one variable is free. Set that variable equal to a parameter (commonly \(t\)), then back-substitute through the other equations to express every variable in terms of \(t\). The result is a line (one parameter) or a plane (two parameters) of solutions.
Can I use my GDC to solve simultaneous equations?
Yes, on Paper 2 - most calculators have a simultaneous equation solver or row-reduction (rref) function that solves a system of up to three equations directly. On Paper 1 you need elimination, substitution or row reduction by hand. See the GDC guide for model-specific instructions.
Sub-topics
Systems of Equations broken down into its individual skills, each with its own focused page.
Related topics
More Number & Algebra topics from the same AA HL syllabus unit, in case you want to keep going.