A cinema sells adult tickets for $\text{€}x$ and child tickets for $\text{€}y$. A family of $2$ adults and $3$ children pays $\text{€}45$. A second family of $3$ adults and $1$ child pays $\text{€}50$. Find the price of one adult ticket and one child ticket.

1. Formulate equation 1: $2x + 3y = 45$. 2. Formulate equation 2: $3x + y = 50$. 3. Open the Simultaneous Equation solver (2 unknowns) on the GDC. 4. Enter the augmented coefficients: $\begin{pmatrix} 2 & 3 & | & 45 \\ 3 & 1 & | & 50 \end{pmatrix}$. 5. Solve : $x = 15$, $y = 5$. 6. State : adult ticket $= \mathbf{\text{€}15}$, child ticket $= \mathbf{\text{€}5}$.

Plumber A charges a fixed call-out fee of $\text{€}50$ plus $\text{€}10$ per hour. Plumber B charges $\text{€}20$ plus $\text{€}15$ per hour. Find the exact number of hours for which both plumbers charge the same total.

1. Model Plumber A: $C_A = 50 + 10h$. 2. Model Plumber B: $C_B = 20 + 15h$. 3. Equate the costs: $50 + 10h = 20 + 15h$. 4. Subtract $10h$: $50 = 20 + 5h$. 5. Subtract $20$: $30 = 5h$. 6. Solve : $h = \mathbf{6\text{ hours}}$.

IB Maths AI SL · Unit 1: Number and Algebra

IB Maths AI SL Systems of Equations Questions

Updated October 2026

Exam-style IB Maths AI SL systems of equations questions with worked solutions. Start with the 5 fully worked examples below — each is solved step by step, showing the working that earns the marks — then try the practice questions and check your working against the mark scheme.

45 questions
Paper 1: 9
Paper 2: 36
10 easy
18 medium
13 hard
4 very hard
5 worked examples

Practise Systems of Equations questions →

🖩 How do I do this on my calculator? Keystrokes for the Casio fx-CG50, TI-84 Plus CE and TI-Nspire.

What's examined in AI SL systems of equations

The question bank covers these systems of equations question types (number of questions in brackets):

Two-variable Linear Systems (15)
Polynomial Equations (15)
Three-variable Linear Systems (11)
Polynomial Coefficients (4)

Systems of Equations worked examples

Worked example 1: Solving a 2×2 system from a word problem · easy

A cinema sells adult tickets for $€ x$ and child tickets for $€ y$ . A family of $2$ adults and $3$ children pays $€ 45$ . A second family of $3$ adults and $1$ child pays $€ 50$ . Find the price of one adult ticket and one child ticket.

Solution

6 steps · 1 to fill in

Examiner tip: Define your variables clearly if the question doesn't, and remember to include units (

€

, m, kg…) in the final answer when the context requires them.

Worked example 2: 3×3 linear system on the GDC · medium

Solve the following system of three linear equations using your GDC: $x + 2 y - z = 4$ , $2 x - y + 3 z = 13$ , $- x + 3 y + 2 z = 15$ .

Solution

6 steps · 1 to fill in

Examiner tip: If a variable is missing from one equation (e.g.

x + z = 5

), enter a coefficient of

0

for it — never leave a matrix cell blank.

Worked example 3: Building a 3×3 system from a curve through three points · hard

The graph of $y = a x^{2} + b x + c$ passes through $(0, 5)$ , $(2, 13)$ and $(- 1, 4)$ . Set up a system of three linear equations and find the exact values of $a$ , $b$ and $c$ .

Solution

6 steps · 1 to fill in

Examiner tip: Whenever a curve passes through the

y

-axis (

x = 0

) the constant term

c

is revealed instantly. That immediately reduces a

3 \times 3

system to a much simpler

2 \times 2

Worked example 4: Intersection of two lines · easy

Two straight lines have equations $y = 2 x - 3$ and $y = - 0.5 x + 7$ . Find the exact coordinates of the point where these two lines intersect.

Solution

6 steps · 1 to fill in

Examiner tip: Verify intersection coordinates fast on the GDC: graph both lines, then use the ISCT tool.

Worked example 5: Comparing two linear cost models · medium

Plumber A charges a fixed call-out fee of $€ 50$ plus $€ 10$ per hour. Plumber B charges $€ 20$ plus $€ 15$ per hour. Find the exact number of hours for which both plumbers charge the same total.

Solution

6 steps

Examiner tip: In a cost model, the fixed fee is the

y

-intercept (

c

) and the hourly rate is the gradient (

m

) of

y = m x + c

Try these IB Maths AI SL systems of equations questions

Three questions from the bank, easiest first. Mark schemes and AI marking of your written working are in the practice area. One of them has its full worked solution here, open to everyone.

Question 1 · easy · 5 marks · Paper 1

Consider the following system of three linear equations: $\begin{aligned} a + b + c & = 8 \\ 8 a + 4 b + 2 c & = 4 \\ 27 a + 9 b + 3 c & = 0 \end{aligned}$

Write down the augmented matrix or the system of equations as it would be entered into your GDC.
Using your GDC, find the exact values of $a$ , $b$ , and $c$ .

Attempt it and see the mark scheme →

Question 2 · medium · 6 marks · Paper 1

The ticket prices for a concert are $€ 15$ for an adult, $€ 10$ for a child, and $€ 12$ for a student. A total of $600$ tickets were sold. The total amount of money from ticket sales was $€ 7816$ . There were twice as many adult tickets sold as child tickets. Let the number of adult tickets sold be $x$ , the number of child tickets sold be $y$ , and the number of student tickets sold be $z$ .

Write down three linear equations that express the information given above.
Using your GDC, find the exact number of adult, child, and student tickets sold.

Full worked solution Free, no sign-in

Total number of tickets: $x + y + z = 600$ A1
Total money: $15 x + 10 y + 12 z = 7816$ A1
Twice as many adults as children: $x = 2 y ⟹ x - 2 y + 0 z = 0$ A1
Enter the system into the GDC simultaneous equation solver (3 unknowns): $(\begin{matrix} 1 & 1 & 1 & 600 \\ 15 & 10 & 12 & 7816 \\ 1 & - 2 & 0 & 0 \end{matrix})$ . (M1)
Solve: $x = 308$ , $y = 154$ , $z = 138$ . A1A1
308 adult, 154 child and 138 student tickets.
Note: award one accuracy mark for two correct values and both for all three.

Every question here has a full worked solution. See them all →

Question 3 · hard · 6 marks · Paper 1

A polynomial function is given by $f (x) = a x^{3} + b x^{2} + c x$ . It is known that the graph of the function passes through the points $(1, 2)$ , $(2, 6)$ , and $(- 1, - 6)$ .

Set up a system of three linear equations in terms of $a$ , $b$ , and $c$ .
Using your GDC, solve the system to find the values of $a$ , $b$ , and $c$ .
Hence, find the value of $f (3)$ .

Attempt it and see the mark scheme →

Free account — sign in to download Topic PDF: 7 questions with worked answers · 4 pages · 150 KB

All 45 systems of equations questions with mark schemes →

Save your progress and get 5 more questions on every topic, checked instantly. Free account, no card. Join free

Revise systems of equations for the exam

FAQ

How many IB Maths AI SL systems of equations questions are there?

There are 45 exam-style systems of equations questions in the AI SL question bank (Paper 1: 9 · Paper 2: 36), graded 10 easy, 18 medium, 13 hard, 4 very hard. Every question has a full IB-style mark scheme (M, A and R marks).

Is systems of equations on Paper 1 or Paper 2?

Both. In the bank, Paper 1: 9 · Paper 2: 36. Practise with your GDC — AI papers expect calculator methods throughout.

Where can I get the mark schemes?

Open the AI SL Unit 1 practice page: every question has a step-by-step IB-style mark scheme, and you can photograph your working for instant AI marking. The worked examples on this page are free.

What else is free here

Save your answers and track your progressA free account remembers the questions you have done on every topic and brings you back to this page.

Join free

Already have an account? Sign in

Teaching this topic?

Free ready-to-teach lessons (slides and teacher notes): Solving 2×2 systems on the GDC · Solving polynomial equations with technology.

Systems of Equations in other IB Maths courses

IB Maths AA HL Systems of Equations questions

Keep revising AI SL: AI SL revision notes · IB Maths AI SL past papers · AI SL common exam mistakes · AI SL syllabus checklist · AI SL number and algebra knowledge organiser

← All IB Maths AI SL topics

IB Maths AI SL Systems of Equations Questions

Try it first

What's examined in AI SL systems of equations

Systems of Equations worked examples

Worked example 1: Solving a 2×2 system from a word problem · easy

Solution

Worked example 2: 3×3 linear system on the GDC · medium

Solution

Worked example 3: Building a 3×3 system from a curve through three points · hard

Solution

Worked example 4: Intersection of two lines · easy

Solution

Worked example 5: Comparing two linear cost models · medium

Solution

Try these IB Maths AI SL systems of equations questions

Question 1 · easy · 5 marks · Paper 1

Question 2 · medium · 6 marks · Paper 1

Full worked solution Free, no sign-in

Question 3 · hard · 6 marks · Paper 1

Revise systems of equations for the exam

FAQ

What else is free here

Teaching this topic?

More AI SL Unit 1 topics

Systems of Equations in other IB Maths courses