IB Maths AA HL · Unit 1: Number and Algebra
IB Maths AA HL Systems of Equations Questions
Exam-style IB Maths AA HL systems of equations questions with worked solutions. Start with the 3 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.
- 23 questions
- Paper 1: 17
- Paper 2: 6
- 5 easy
- 7 medium
- 8 hard
- 3 starter
- 3 worked examples
Practise Systems of Equations questions →
AA HL formula booklet
What's examined in AA HL systems of equations
The question bank covers these systems of equations question types (number of questions in brackets):
- Consistency of linear systems (15)
- Solve linear systems (5)
- Geometric meaning of solutions (3)
Systems of Equations worked examples
Worked example 1: Solving 3x3 systems via technology · easy
Solve the following system of linear equations using technology: $$\begin{aligned}2x - y + z &= 3 \\ x + y + z &= 6 \\ x - y + 2z &= 5\end{aligned}$$
1. Identify the coefficients of the variables $x, y, z$ to form an augmented matrix or directly input into the GDC.
2. Set up the formal matrix equation: $\begin{bmatrix} 2 & -1 & 1 \\ 1 & 1 & 1 \\ 1 & -1 & 2 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 3 \\ 6 \\ 5 \end{bmatrix}$.
3. Access the simultaneous equation solver on your graphic display calculator (CG50: MENU A (Equation) -> F1 (Simultaneous) -> 3 Unknowns).
4. Input the values exactly as they appear in the system, ensuring the constant terms are correctly placed on the right-hand side of the equals sign.
5. Execute the solver function to calculate the values.
6. State the final exact coordinates of the point of intersection: $\mathbf{x = 1, y = 2, z = 3}$.
Examiner tip: In Paper 2, no manual algebraic row reduction is required for standard 3x3 intersections; students simply need to state the system, specify they are using GDC, and correctly interpret the calculator output to claim full marks.
Worked example 2: Elimination to find geometric intersections · medium
Consider the planes $\Pi_1: x + y + z = 6$, $\Pi_2: 2x - y + 3z = 9$, and $\Pi_3: -x + 2y - z = 2$. By using manual algebraic elimination, determine the exact coordinates of their unique point of intersection.
1. Label the equations: (1) $x + y + z = 6$, (2) $2x - y + 3z = 9$, (3) $-x + 2y - z = 2$.
2. Add equation (1) and equation (3) to efficiently eliminate both $x$ and $z$ at the same time: $(x - x) + (y + 2y) + (z - z) = 6 + 2$.
3. Solve the resulting equation to immediately find $y$: $3y = 8 \implies y = \frac{8}{3}$.
4. Eliminate $x$ using equations (1) and (2). Take Eq(2) $- 2 \times$ Eq(1): $(2x - 2x) + (-y - 2y) + (3z - 2z) = 9 - 12$.
5. Simplify to find an equation purely in $y$ and $z$: $-3y + z = -3$.
6. Substitute $y = \frac{8}{3}$ into the new equation to find $z$: $-3\left(\frac{8}{3}\right) + z = -3 \implies z = 5$. Finally, use Eq(1) to find $x$: $x + \frac{8}{3} + 5 = 6 \implies x = -\frac{5}{3}$. The point is $\mathbf{\left(-\frac{5}{3}, \frac{8}{3}, 5\right)}$.
Examiner tip: Strategic choices when eliminating variables (such as spotting that adding equations 1 and 3 immediately cancels two variables simultaneously) can save vast amounts of time compared to blind, rigid substitution.
Worked example 3: Investigating inconsistent systems with parameters · hard
A system of equations is given by $x + y + 2z = 4$, $2x + 3y + 5z = 7$, and $3x + 4y + kz = 12$. Determine the value of the constant $k$ for which this system has no solutions.
1. Label the equations: (1) $x + y + 2z = 4$, (2) $2x + 3y + 5z = 7$, (3) $3x + 4y + kz = 12$.
2. Eliminate $x$ from equation (2) by calculating Eq(2) $- 2 \times$ Eq(1): $(2x - 2x) + (3y - 2y) + (5z - 4z) = 7 - 8$, yielding $y + z = -1$.
3. Eliminate $x$ from equation (3) by calculating Eq(3) $- 3 \times$ Eq(1): $(3x - 3x) + (4y - 3y) + (kz - 6z) = 12 - 12$.
4. Simplify to yield a second equation in $y$ and $z$: $y + (k-6)z = 0$.
5. Subtract the result of step 2 from the result of step 4 to eliminate $y$: $(y - y) + ((k-6)z - z) = 0 - (-1)$. This condenses to $(k-7)z = 1$.
6. Analyze the final statement. For the system to have no solutions (geometrically, planes forming a triangular prism), we require an impossible statement such as $0z = 1$. This occurs when the coefficient of $z$ is zero, thus $\mathbf{k = 7}$.
Examiner tip: When finding conditions for infinitely many solutions or no solutions, systematically eliminate variables row-by-row to reach a final equation of the form $az = b$. If $a=0$ and $b \neq 0$, there are no solutions.
Try these IB Maths AA HL 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 · 2 marks · Paper 1
Three planes, \(\Pi_1\), \(\Pi_2\), and \(\Pi_3\), are defined by a system of linear equations. When attempting to solve the system algebraically, a student arrives at the false statement \(0 = 12\).
State two possible geometric interpretations of how these three planes intersect in 3D space.
Question 2 · medium · 6 marks · Paper 1
Solve the following system of equations algebraically:
\[\begin{aligned}
x + y + z &= 6 \\
2x - y + 3z &= 9 \\
3x + 2y - z &= 4\end{aligned}\]
Question 3 · hard · 8 marks · Paper 1
Consider the system of equations with real parameter \(k\):
\[\begin{aligned}
x + y + 2z &= 4 \\
2x + 3y + 5z &= 7 \\
3x + 4y + kz &= 12\end{aligned}\]
Use row reduction (elimination) to show that the system can be reduced to an equation of the form \((k-7)z = c\), finding the value of \(c\). [6 marks]
Hence, determine the value of \(k\) that makes the system inconsistent. [2 marks]
All 23 systems of equations questions with mark schemes →
FAQ
How many IB Maths AA HL systems of equations questions are there?
There are 23 exam-style systems of equations questions in the AA HL question bank (Paper 1: 17 · Paper 2: 6), graded 5 easy, 7 medium, 8 hard, 3 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: 17 · Paper 2: 6. Paper 1 is non-calculator, so practise the exact-value algebra as well as the GDC methods.
Where can I get the mark schemes?
Open the AA HL 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.
Systems of Equations in other IB Maths courses
← All IB Maths AA HL topics