Linear Systems (AI SL)
A linear system is a set of equations, each of degree 1, that must all be true at once - two equations for two unknowns, three for three. IB AI SL doesn't require any hand method: you're expected to enter the equations directly into your GDC's simultaneous-equation solver and read off the answer. This page walks through that process with worked examples and the errors that most often cost marks. It's part of the broader Systems & Polynomial Equations topic.
37 questions on this sub-topic.
Solving with technology
Covered under IB syllabus reference SL1.8. There's nothing to substitute into by hand: your job is to translate the problem into correctly-formed equations and let the GDC solve the system. See the GDC guide for calculator-specific steps.
2-variable system
\(a_1x+b_1y=d_1\)
\(a_2x+b_2y=d_2\)
Not in the formula booklet - enter both equations into the 2-variable solver on your GDC.
3-variable system
\(a_1x+b_1y+c_1z=d_1\)
\(a_2x+b_2y+c_2z=d_2\)
\(a_3x+b_3y+c_3z=d_3\)
Needs three independent equations. IB exam systems always have a unique solution.
Need the full syllabus wording and formula-booklet reference table? See Systems & Polynomial Equations.
Worked examples
Solve the system \(x+y+z=6,\ x+2y+3z=14,\ x+4y+9z=36.\)
(a) State \(x.\)
(b) State \(y.\)
(c) State \(z.\)
Worked solution
Enter all three equations into the 3-variable solver. M1
\(x=1.\) A1
\(y=2.\) A1 \(z=3.\) A1
Three pens, two pencils and one rubber cost $8. One pen, three pencils and two rubbers cost $7. Two pens, one pencil and three rubbers cost $7.
Find the cost of each item.
(a) State the cost of a pen.
(b) State the cost of a pencil.
(c) State the cost of a rubber.
Worked solution
Let pen \(=p\), pencil \(=q\), rubber \(=r\): \(3p+2q+r=8,\ p+3q+2r=7,\ 2p+q+3r=7.\) M1 A1
Solving with technology: M1 \(p=$1.56,\ q=$1.22,\ r=$0.89.\) A1 A1
Use technology to solve \(2x+3y=12\) and \(x-y=1.\)
Worked solution
Enter the two equations into the simultaneous-equation solver. M1
\(x=3.\) A1
\(y=2.\) A1
Common mistakes
- Missing an equation. Three unknowns need three independent equations - forgetting to translate one of the given conditions into an equation leaves the system unsolvable or gives infinitely many answers.
- Muddling the variable order. In a word problem like the pen-pencil-rubber system, it's easy to enter a coefficient under the wrong variable. Label \(p\), \(q\), \(r\) explicitly before typing anything into the GDC.
- Treating a "no solution" or "infinite solutions" result as a mistake. If the GDC reports either, that can be the genuinely correct mathematical answer (parallel or identical equations) rather than a sign you mistyped something - check the original equations before assuming an error.
Ready to practise properly?
38 linear-system questions, marked instantly like the real exam.
Quick answers
How do I solve a system of linear equations for IB AI SL?
Enter each equation into your GDC's simultaneous equation solver (2 or 3 variables). No algebraic method is required in the exam - technology is expected to do the solving.
How many equations do I need for 3 unknowns?
Three independent equations, one for each unknown. IB exam questions with 3 variables always give exactly three genuinely different conditions, so a unique solution exists.