Substitute the linear into the quadratic
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.
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).
Pair the answers correctly
A linear + quadratic pair usually has two solution points; each goes with its own — keep the pairs together.
Drawn from real examiner reports.
Only finding one variable
Solving for (or ) and stopping, without back-substituting to find its partner.
Quadratic-formula denominator error
Writing the denominator as 2 instead of when the leading coefficient is not 1.
A commonly reported examiner note records this on a recent higher-tier paper.
Trying to "square" the linear equation
Squaring to get is wrong (the right side of is not ); substitute instead.
A commonly reported examiner note records this on a recent higher-tier paper.
Adding when you should subtract
In elimination, add the equations only when the matched terms have opposite signs (e.g. and ); subtract when they have the same sign (e.g. and ). Getting this backwards changes the result entirely.
Scaling only one equation
To eliminate a variable the coefficients must actually match. Multiplying just one equation, or scaling both by the wrong factors, leaves no matching pair — scale each equation so one variable has the same size coefficient before adding or subtracting.
Check each pair in both equations
Each pair must satisfy both original equations — a fast, complete check.
Show your elimination or substitution steps
On a "show clear algebraic working" pair, the marks are for the shown elimination or substitution, not the final numbers. A correct answer with no working scores nothing, so set out the matched coefficients and the combined equation.
Define a letter for each unknown
For a worded problem with two unknown quantities, assign a letter to each (e.g. let a pen cost ), then turn every sentence into one equation. You need as many equations as unknowns before you can solve.
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 .
