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IB Maths AI SL · Unit 1: Number and Algebra

IB Maths AI SL Systems of Equations Questions

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 the way an IB examiner expects — then try the practice questions and check your working against the mark scheme.

Practise Systems of Equations questions → AI SL formula booklet

What's examined in AI SL systems of equations

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

Systems of Equations worked examples

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

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.

Solution

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 $= \mathbf{\text{€}5}$.

Examiner tip: Define your variables clearly if the question doesn't, and remember to include units ($\text{€}$, 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: $x + 2y - z = 4$, $2x - y + 3z = 13$, $-x + 3y + 2z = 15$.

Solution

1. Recognise a $3 \times 3$ linear system.

2. Open the Equation app $\to$ Simultaneous $\to$ 3 Unknowns.

3. Enter the augmented matrix: $\begin{pmatrix} 1 & 2 & -1 & | & 4 \\ 2 & -1 & 3 & | & 13 \\ -1 & 3 & 2 & | & 15 \end{pmatrix}$.

4. Execute the solver.

5. Read the values: $x = 2$, $y = 3$, $z = 4$.

6. State: $\mathbf{x = 2,\ y = 3,\ z = 4}$.

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 $y = ax^2 + bx + c$ 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

1. Substitute $(0, 5)$: $5 = a(0)^2 + b(0) + c \implies c = 5$.

2. Substitute $(2, 13)$: $13 = 4a + 2b + 5 \implies 4a + 2b = 8$.

3. Substitute $(-1, 4)$: $4 = a - b + 5 \implies a - b = -1$.

4. Reduce to a $2 \times 2$ system on the GDC (equations 2 and 3).

5. Solve: $a = 1$, $b = 2$.

6. State: $\mathbf{a = 1,\ b = 2,\ c = 5}$.

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

Solution

1. Recognise that at intersection $y$-values agree.

2. Equate: $2x - 3 = -0.5x + 7$.

3. Collect the $x$-terms: $2.5x - 3 = 7$.

4. Collect the constants: $2.5x = 10$.

5. Solve: $x = 4$.

6. Substitute back: $y = 2(4) - 3 = 5$. Intersection $= \mathbf{(4,\ 5)}$.

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 $\text{€}50$ 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.

Solution

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: $h = \mathbf{6\text{ hours}}$.

Examiner tip: In a cost model, the fixed fee is the $y$-intercept ($c$) and the hourly rate is the gradient ($m$) of $y = mx + c$.

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.

Question 1 · easy · 5 marks · Paper 1

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

  1. Write down the augmented matrix or the system of equations as it would be entered into your GDC.

  2. 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 \(\text{€}15\) for an adult, \(\text{€}10\) for a child, and \(\text{€}12\) for a student. A total of \(600\) tickets were sold. The total amount of money from ticket sales was \(\text{€}7816\). There were twice as many adult tickets sold as child tickets [4]. 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 \(z\).

  1. Write down three linear equations that express the information given above.

  2. Using your GDC, find the exact number of adult, child, and student tickets sold.

Attempt it and see the mark scheme →

Question 3 · hard · 6 marks · Paper 1

A polynomial function is given by \(f(x) = ax^3 + bx^2 + cx\). It is known that the graph of the function passes through the points \((1, 2)\), \((2, 6)\), and \((-1, -6)\).

  1. Set up a system of three linear equations in terms of \(a\), \(b\), and \(c\).

  2. Using your GDC, solve the system to find the values of \(a\), \(b\), and \(c\).

  3. Hence, find the value of \(f(3)\).

Attempt it and see the mark scheme →

All 19 systems of equations questions with mark schemes →

FAQ

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

There are 19 exam-style systems of equations questions in the AI SL question bank (Paper 1: 9 · Paper 2: 10), graded 3 easy, 6 medium, 5 hard, 3 very hard, 2 starter. 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: 10. 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.

More AI SL Unit 1 topics

Systems of Equations in other IB Maths courses

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