Two equations, two unknowns
Use elimination (make one variable match, then add or subtract) or substitution (make one variable the subject and put it into the other equation).
Linear + quadratic: substitute
Rearrange the linear equation and substitute into the quadratic to get a single quadratic in one variable; solve it, then find the matching values of the other variable.
Pair each x with its own y
A linear + quadratic pair usually has two solution points; each goes with its own — keep the pairs together and quote them as points.
Drawn from real examiner reports.
Squaring the linear equation
Squaring to get is wrong — , not . Substitute the linear equation into the quadratic instead.
Reported on the November 2024 Higher Paper 2H (Q22).
Denominator 2 not 2a in the formula
Writing the quadratic-formula denominator as instead of when the leading coefficient is not 1. The whole numerator is divided by .
Reported on the November 2024 Higher Paper 2H (Q22).
Only finding one variable
Solving for (or ) and stopping, without back-substituting to find its partner. Every solution point needs both coordinates.
Coefficients not matched to eliminate
Elimination only works when a variable has equal or equal-and-opposite coefficients. If not, multiply one whole equation up first — otherwise adding or subtracting removes nothing.
(general exam technique)
Sign slip subtracting equations
When subtracting one equation from another, subtract every term, including the constants: gives , not .
(general exam technique)
Check both pairs in both equations
Each pair must satisfy both original equations — substitute back for a fast, complete check.
Choose the easier method
Use substitution when one variable already has coefficient 1; use elimination when coefficients line up for adding or subtracting. Either is fine if shown clearly.
Quote answers as points
For a line-meets-curve question, give the full pairs, e.g. and , not a loose list of numbers — the pairing carries marks.
Elimination: make one variable match, then add or subtract to remove it.
Substitution: make one variable the subject and put it into the other equation. Useful when a variable already has coefficient 1.
Two methods for linear simultaneous equations?
Solve the simultaneous equations and .
