Try these tasks while you’re waiting
Task from Don Steward
Task from Don Steward
What happens if you add the two equations?��What happens if you subtract them?
Can you make up more questions like this?
Simultaneous Equations in Depth���Jo Morgan�@mathsjem
Curriculum and sequencing
Examples and explanation
Assessment
Misconceptions
Tasks and resources
Enrichment, history, narrative
Methods
Vocabulary
What does it mean for a teacher to ‘know’ a topic in depth?
The perfect time to deliver a ‘topics in depth’ session to your team is just before you all teach the topic.
Curriculum Content
KS5
KS4
KS3
Methods for Linear Simultaneous Questions
What are the three commonly used algebraic methods for solving linear simultaneous equations?
Elimination by addition or subtraction | Substitution | Equating / Comparison |
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Rearrange | | |
Sub into | | |
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Scale and rearrange | | |
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Equate | | |
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Elimination | Substitution | Equating |
Heavily taught Used when hard to rearrange | Follows on to solving quadratic simultaneous equations | Useful in coordinate geometry |
Choice of method typically depends on how the equations are presented
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Euclid’s First Axiom ‘Things which are equal to the same thing are also equal to one another’
Choose a suitable method to solve each pair of simultaneous equations
Comparison (equate)
Comparison (equate)
Comparison (equate)
Substitution
Substitution
Substitution
Substitution
Elimination
Elimination
Elimination
Durrell’s Algebra 1914
Artificial Intelligence (ChatGPT) 2026
CHALLENGE
Can you explain why your chosen method is the most efficient? Solve two of the equations to check your choices.
Note: equating by setting 15y = 15y
Graphical methods are sometimes required
Methods I’m not talking about today…
Bar models
(visual elimination)
Vedic maths (see my book!)
Matrices
Algebra discs
Difficulties with elimination
All algebraic methods requires strong prerequisite skills: algebraic manipulation, substitution, changing the subject, working with negatives, solving equations
Elementary Algebra
(Hamblin Smith, 1871)
Surprised?
In this book, when solving simultaneous equations by elimination, the instruction is always to subtract.
If necessary, coefficients (including signs) are made to match before subtracting.
Two ‘no decision required’ alternatives to “always subtract”
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x -2
“Truth Set”
“An overweight girl…”
Skemp, 1966
Writing out the subtraction instead of doing it mentally?
Atomisation
All add
All subtract: everything is positive
All subtract: involves negatives
Mix of add and subtract
Scale up one equation then add or subtract
Scale up both equations then add or subtract
Rearrange first
From Corbett Maths (L2 Further Maths)
My Y11 Further Maths students found these difficult – perhaps they had not seen simultaneous equations presented in a non-standard format before.
Equating
Substitution
Graphical (without time spent plotting)
“solving simultaneous equations usually involves an awareness of invariance:��that you can transform a linear relationship using a multiplier without changing the (full) set of points that fit the rule”��Don Steward
Prerequisite skill: algebraic fluency
Interwoven Maths
Equivalent Equations
Interwoven Maths
variationtheory.com
Interwoven Maths
The multi-step nature of the topic means it lends itself well to ‘backwards fading’ and techniques like ‘the tick trick’
Dale Chapman
Starting Points Maths
Dr Austin
Ashton Coward
Resources and Deeper Thinking
MathsPad
A zoo has several ostriches and several giraffes.
In total they have 30 eyes and 44 legs.
How many ostriches and how many giraffes are in the zoo?
Example of interweaving:
Mean from a frequency table (Don Steward)
Boss Maths
Thinking about what it means to have no solutions or infinite solutions
Underground Maths
How do you know when two simultaneous equations have an infinite number of solutions? Why do small variations produce quite different solutions?
Don Steward
Generalisations
Don Steward
Curriculum and sequencing
Is this topic allocated sufficient time?
Examples and explanation
Assessment
Misconceptions
Tasks and resources
Enrichment, history, narrative
Methods
Vocabulary
What does it mean for a teacher to ‘know’ a topic in depth?
Thanks for listening!
Twitter: mathsjem
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Circle: HoDSpace
Blog: resourceaholic.com
Book: A Compendium of Mathematical Methods