Simultaneous Equations and Changing the Subject
Translate contexts into two equations, solve them graphically or algebraically and rearrange formulae without breaking equivalence.
How to study National 5 Mathematics
Secure one operational skill at a time, connect it to neighbouring techniques, then apply it in unfamiliar contexts with complete working and a checked conclusion.
Core concepts
Concept 1
Two linear equations can be constructed from text and solved by elimination, substitution or graph intersection.
Exam cue: Define the variables before translating each condition.
Concept 2
A simultaneous solution is an ordered pair satisfying both original equations.
Exam cue: Scale equations so one variable eliminates cleanly, then check both equations.
Concept 3
Changing the subject reverses operations in a valid order and can include a simple square or square root.
Exam cue: Isolate the term containing the new subject before applying a root or square.
Risk pitfalls and guardrails
Constructing two versions of the same equation.
Guardrail: Check signs, brackets, restrictions, units, accuracy and whether the answer needs a reason.
Using inconsistent scaling during elimination.
Guardrail: Check signs, brackets, restrictions, units, accuracy and whether the answer needs a reason.
Taking a square root before isolating the squared expression.
Guardrail: Check signs, brackets, restrictions, units, accuracy and whether the answer needs a reason.
Memory anchors
Simultaneous meaning
The solution satisfies both equations at the same time.
Elimination
Scale if needed, then add or subtract to remove one variable.
Graphical solution
The coordinates of the intersection.
Context construction
Define variables and translate two independent conditions.
Change subject
Undo operations around the required variable in reverse order.
Formula check
Substitute sample values into original and rearranged forms.
Checkpoint rule
Do the check-up only after you can summarize each concept in one sentence and identify one dangerous pitfall from memory.
Knowledge Check (after reading)
Short check-up to confirm understanding of this module.
Check-up Questions
Solve x + y = 9 and x − y = 3.
Solve 2x + y = 11 and x − y = 1.
Answer all questions to submit.
Next step personalized recommendations
Continue learning
Move forward only after this module is stable.
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