IB Maths AA HLTopic 1 — Number & AlgebraPaper 1 & 2HL only~9 min read
Systems of Linear Equations
Up until now, you’ve mostly worked with one equation in one unknown — solve for x and you’re done. But real problems usually involve several unknowns at once: how much do tickets cost at three different price levels, how many of each ingredient go into a recipe, what currents flow through the branches of an electrical circuit. To pin all those unknowns down, you need several equations linking them together — a system of linear equations. The HL syllabus covers two-variable systems (two equations, two unknowns) and three-variable systems (three equations, three unknowns). This first note introduces the language, the geometric picture, and how to solve one quickly using your GDC. The next two notes cover row reduction (the algebraic technique) and what happens when there isn’t a single, clean answer.
📘 What you need to know
A linear equation is one where every variable appears to the first power only — no x2, no xy, no √x, no 1/x.
A system of linear equations is two or more linear equations in the same unknowns, all required to hold simultaneously.
To solve a system uniquely, you generally need at least as many equations as unknowns. With n unknowns, expect at least n equations.
2×2 system standard form:a1x + b1y = c1, a2x + b2y = c2.
3×3 system standard form:aix + biy + ciz = di for i = 1, 2, 3.
Geometrically: each linear equation is a straight line in 2D or a plane in 3D. The solution is where they all intersect.
Your GDC has a built-in solver for systems of linear equations — usually under “simultaneous equations” in the algebra menu.
For word problems, introduce variables for the unknowns first, then translate each piece of information into one linear equation.
What is a linear equation?
A linear equation in the variables x, y, … is one where each variable appears to the power 1 and there are no products of different variables. Constants in front are fine; nonlinear functions like squaring, square-rooting, or multiplying two unknowns together are not.
Linear ✓
2x + 3y = 5 x + y + z = 7
4a − 2b = 9
first powers only, no products
Nonlinear ✗
x2 + y = 4 xy = 6
√x + y = 3
squares, products, and roots disallowed
Linear equations are special because their graphs are straight lines (in 2D) or flat planes (in 3D). That straightness is exactly what makes them solvable using elimination, substitution, or matrix methods — techniques that wouldn’t work on curved equations.
Standard form for 2×2 and 3×3 systems
The IB exam will ask you to handle either a 2×2 system (two equations, two unknowns) or a 3×3 system (three equations, three unknowns). They’re written using the same template:
2×2 system standard forma1x + b1y = c1 a2x + b2y = c2
Whenever an equation isn’t already in this form, rearrange it before solving — push all the variable terms to the left and the constant to the right.
For example, “x + y equals z minus 4″ looks like words, but rearranging gives x + y − z = −4. Now it’s in standard form, ready for the solver. Always rearrange before typing into your GDC.
What systems of linear equations look like geometrically
Here’s the picture that makes systems make sense. Each linear equation is a straight object on its own:
Each equation is a line (2D) or a plane (3D) — solutions are where they cross
2×2 system → 2D
two lines in the plane
If they cross at exactly one point, that point’s (x, y) is the unique solution.
3×3 system → 3D
three planes in space
If all three meet at exactly one point, that point’s (x, y, z) is the unique solution.
🤔 Why “lines” and “planes” specifically?
Because the equations are linear. The graph of ax + by = c is always a straight line (you’ve seen this since pre-IB). And the graph of ax + by + cz = d is the natural extension — a flat plane in three dimensions. Geometric “straightness” is the defining property of linear equations.
What if they don’t cross? If the lines are parallel (in 2D) or the planes don’t meet (in 3D), there’s no solution. If they overlap exactly, there are infinitely many. We’ll explore these special cases in the third note of this section.
Forming a system from a word problem
Most exam questions don’t hand you the equations directly — they describe a situation and expect you to translate it. The skill is twofold: introduce variables for the unknowns, then extract one equation per piece of information.
🧭 Recipe — building a system from a word problem
Identify the unknowns in the problem — usually quantities, prices, ages, or rates.
Assign letters (typically x, y, z) and write a one-line definition for each: “Let x = price of …”.
Translate each piece of information into one linear equation. Watch for words like “twice”, “more than”, “less than”, “total”.
Rearrange every equation into standard form ax + by + cz = d before solving.
Solve the system (algebraically or with your GDC), then interpret the answer in context — write a sentence using the original quantities.
Watch for “hidden” equations in word problems. A phrase like “the total of A and B is twice C” gives the equation A + B = 2C, which you’d rearrange to A + B − 2C = 0. Numbers don’t need to appear on both sides — sometimes the “constant” is just zero.
Solving with your GDC
The fastest method for non-trivial systems on Paper 2 is your GDC. Most calculators have a built-in simultaneous equations solver, usually under the algebra or equation menu.
🧭 Recipe — solving with the GDC
Open the simultaneous equations / system solver menu.
Choose the number of equations (2 or 3).
Make sure each equation is in standard form first: ax + by = c or ax + by + cz = d.
Type in each coefficient in the matrix grid (use 0 for any missing variable).
Press solve and read off the values of x, y, (and z).
Paper 1 vs Paper 2: on Paper 1 (no calculator), use elimination, substitution, or row reduction by hand. On Paper 2 (calculator allowed), the GDC is your fastest tool. If the question explicitly says “use an algebraic method”, you must show working — even on Paper 2.
Worked examples
WE 1
Identify which equations are linear
State whether each of the following is a linear equation in x, y, z:
(a) 4x − 7y + 2z = 12 (b) x2 + y = 6 (c) 3xy + z = 5 (d) 8 − 5x = 2y (e) x + 1/y = 4
(a) 4x − 7y + 2z = 12all variables to power 1, no products → LINEAR ✓(b) x² + y = 6x² is a square → NONLINEAR ✗(c) 3xy + z = 5xy is a product of two variables → NONLINEAR ✗(d) 8 − 5x = 2yrearrange: −5x − 2y = −8 (or 5x + 2y = 8) → LINEAR ✓(e) x + 1/y = 41/y is y to the power −1, not 1 → NONLINEAR ✗linear: (a) and (d). Nonlinear: (b), (c), (e).“linear” means strictly first-power and no products — even reciprocals or square roots break the rule
WE 2
Set up and solve a 2×2 system
The sum of two numbers is 23, and three times the larger number minus the smaller equals 41. Write a system of linear equations and solve it.
Step 1: Define variablesLet x = the larger number, y = the smaller numberStep 2: Translate the two pieces of information“sum is 23”: x + y = 23“3 times larger minus smaller is 41”: 3x − y = 41Step 3: Solve (add the equations to eliminate y)(x + y) + (3x − y) = 23 + 414x = 64 → x = 16y = 23 − 16 = 7larger number = 16, smaller number = 7check: 16 + 7 = 23 ✓ and 3(16) − 7 = 48 − 7 = 41 ✓
WE 3
Solve a 3×3 system using your GDC
Solve the system using your GDC:
2x + y − z = 1 x − y + 3z = 8
3x + 2y + z = 10
Step 1: Confirm each equation is in standard form ax + by + cz = d ✓all three already in correct formStep 2: Enter coefficients into the GDC’s simultaneous equation solverrow 1: [2, 1, −1 | 1]row 2: [1, −1, 3 | 8]row 3: [3, 2, 1 | 10]Step 3: Read off the GDC outputx = 1, y = 2, z = 3verify by substitution: 2(1) + 2 − 3 = 1 ✓ 1 − 2 + 9 = 8 ✓ 3 + 4 + 3 = 10 ✓ — always do at least one check after reading a GDC answer
WE 4
Word problem — music store guitar picks
A music store sells three types of guitar picks: thin, medium, and thick.
• A pack of 5 thin, 4 medium, and 3 thick picks costs $22.
• A pack of 2 thin, 3 medium, and 5 thick picks costs $23.
• A medium pick costs $1 more than a thin pick.
Find the price of each type of pick.
Step 1: Define variablesLet t = price of thin pickLet m = price of medium pickLet k = price of thick pickStep 2: Translate each statement into an equation5t + 4m + 3k = 222t + 3m + 5k = 23“medium $1 more than thin”: m = t + 1, rearrange: −t + m = 1Step 3: Solve the 3×3 system on the GDCenter coefficients [5,4,3|22], [2,3,5|23], [−1,1,0|1]GDC output: t = 1, m = 2, k = 3Step 4: Interpret in contextthin = $1, medium = $2, thick = $3always answer in the context’s language: “thin costs $1” — not just “x = 1”
WE 5
Word problem — finding the hidden equation
Three friends Alia, Beth, and Carla raised funds for a charity. Together they raised £180. Alia raised twice as much as Beth. Carla raised £20 less than Alia. How much did each friend raise?
Step 1: Define variablesLet a = Alia’s, b = Beth’s, c = Carla’s amount in £Step 2: Translate each piece of info“together £180”: a + b + c = 180“Alia twice Beth”: a = 2b → a − 2b = 0“Carla £20 less than Alia”: c = a − 20 → −a + c = −20Step 3: Solve on GDC (or by substitution)substitute a = 2b and c = a − 20 = 2b − 20 into first equation2b + b + (2b − 20) = 1805b − 20 = 1805b = 200 → b = 40a = 2(40) = 80, c = 80 − 20 = 60Alia raised £80, Beth raised £40, Carla raised £60check: 80 + 40 + 60 = 180 ✓ Alia (80) = 2·Beth (40) ✓ Carla (60) = Alia (80) − 20 ✓
WE 6
Geometric interpretation
The system x + 2y = 5
3x − y = 8
has a unique solution. Describe what this means geometrically, and find the solution.
Step 1: Geometric interpretationeach equation is a STRAIGHT LINE in the xy-plane“unique solution” means the two lines cross at exactly ONE pointStep 2: Solve algebraically (multiply 2nd by 2 and add)x + 2y = 5 …①6x − 2y = 16 …② (after multiplying by 2)add ① and ②: 7x = 21 → x = 3substitute: 3 + 2y = 5 → y = 1unique solution: (x, y) = (3, 1) — the lines cross at the point (3, 1)always frame the answer geometrically when the question asks for “interpretation” — coordinates of an intersection point, not just a list of x and y values
💡 Top tips
Rearrange every equation into standard form before solving — coefficients on the left, constant on the right.
Use 0 as a coefficient for any missing variable. The equation x + z = 4 becomes 1x + 0y + 1z = 4 when entered into the GDC.
For word problems, write your variable definitions out: “Let x = price of …”. Then the equations follow more cleanly.
Watch for hidden equations in phrases like “twice as many”, “5 less than”, “in total”. Each one becomes a linear equation.
Always interpret your final answer back in the words of the problem. Don’t just write “x = 3” — write “thin pick costs $3”.
Use the GDC on calculator papers, but show working when an algebraic method is requested.
Sketch the lines for 2×2 systems on a calculator graph to verify the intersection point — a quick sanity check.
⚠ Common mistakes
Treating an equation with x2, xy, or 1/x as linear. These are all nonlinear and don’t fit the standard methods.
Forgetting to rearrange before entering into the GDC. The solver needs every equation in ax + by + … = d form.
Sign errors when rearranging. “8 − 5x = 2y” becomes 5x + 2y = 8 (or −5x − 2y = −8) — be careful which side things move to.
Skipping the “let” definitions in a word problem. Examiners want to see what each letter stands for before the equations appear.
Answering “x = 3, y = 7″ instead of “$3 per pen, $7 per pencil”. The IB wants a contextual answer for word problems.
Confusing “twice as much” with “twice more”. “Twice as much as B” means 2B; “twice more than B” arguably means 3B — but the IB usually means 2B, so go with the literal multiplication.
Trying to solve a 3×3 system with only 2 equations. You need at least n equations for n unknowns — fewer means infinitely many solutions.
Setting up systems of linear equations is a foundational skill — it shows up not just in the IB exam but in physics, economics, engineering, and chemistry. The next note shows you a more systematic technique called row reduction, which lets you solve any system algebraically without needing a GDC. After that, you’ll see what happens when a system has no solutions or infinitely many solutions — situations that come up surprisingly often, both in exam questions and in real applications.
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