Simultaneous Linear Equations (AI SL)
Many real situations involve two linear quantities compared at once - two cost plans, two tanks draining and filling, two lines crossing on a graph. Solving them together means finding the single \((x,y)\) pair that satisfies both equations simultaneously. This page covers the two standard solving methods, worked examples, and the mistakes that lose the most marks. It's part of the broader Linear Models topic.
11 questions on this sub-topic.
Setting up and solving
Covered under IB syllabus reference SL2.5, which asks you to model and solve situations built from linear functions \(f(x)=mx+c\). There's no single formula to memorise here - the skill is translating a written problem into two equations, then choosing an efficient way to solve them.
Setting up two equations
Translate each piece of given information into an equation in the same two variables.
Read the problem twice - once for each equation. Keep the variables consistent, e.g. always \(x\) for the first quantity and \(y\) for the second.
Elimination or substitution
Add/subtract to cancel a variable, or rearrange one equation and substitute into the other.
Elimination suits equations with matching or easily-matched coefficients; substitution suits an equation already solved for one variable.
Need the fuller picture on linear modelling, including the GDC's simultaneous-equation solver? See Linear Models.
Worked examples
Solve the simultaneous equations \(4x+y=18\), \(2x-y=6\), giving your solution as a coordinate pair \((x,y)\).
Worked solution
\((4x+y)+(2x-y)=18+6\Rightarrow 6x=24\Rightarrow x=4.\) M1
\(4(4)+y=18\Rightarrow y=2.\) So the solution is \((4,2).\) A1
Two water tanks: Tank A holds \(A(t)=200-5t\) litres and Tank B holds \(B(t)=50+10t\) litres, \(t\) in minutes.
(a) Find when the two tanks hold equal amounts.
(b) Find that common amount.
Worked solution
(a) Equal amounts. Set \(A(t) = B(t)\): \(200 - 5t = 50 + 10t\) M1
\(\Rightarrow 150 = 15t \Rightarrow t = 10 \text{ minutes}.\) A1
(b) Common amount. \(A(10) = 200 - 5(10) = 150 \text{ litres}\quad(\text{check } B(10)=50+100\) M1
\(=150).\) A1
Common mistakes
- Subtracting instead of adding, or vice versa. Before combining two equations, check the sign on the variable you want to cancel - if it's already opposite (like \(+y\) and \(-y\)), add; if it's the same sign, subtract.
- Stopping after finding one variable. Simultaneous equations ask for a pair of values (or a point). Always substitute your first answer back into one of the original equations to find the second.
- Not checking the answer in both original equations. A quick substitution of your \((x,y)\) pair into the equation you didn't use for back-substitution catches arithmetic slips before you move on.
Ready to practise properly?
11 simultaneous-equation questions, marked instantly like the real exam.
Quick answers
What is the difference between elimination and substitution?
Elimination adds or subtracts the two equations to cancel a variable, and works best when the coefficients already match or are easy to match. Substitution rearranges one equation for a variable and plugs it into the other, and works best when one equation is already solved for a variable, such as \(y=mx+c\).
How do I know when a real-world problem needs simultaneous equations?
Whenever two linear quantities are being compared or set equal, such as two cost models meeting or two tanks holding the same amount, write each as its own equation in the same two variables and solve them together.