Simultaneous equation questions in Additional Math usually have one linear and one non-linear equation, or two equations that can be turned into a quadratic with a substitution. The reports show that students can start the problem and then lose marks in the algebra and in the final steps. This guide collects what Cambridge examiners say students get wrong, based on recent reports.
Summarised in our own words from a reading of the examiner reports.
Mistakes that cost marks
- Multiplying only part of an equation. Forgetting to multiply every term of an equation when preparing to eliminate, or omitting to multiply all terms by x² or by x + 1 when clearing fractions.
- Squaring term by term. One of the most common incorrect methods was to square each term in each equation, which gave an incorrect result.
- Confusing substitutions. Reports say candidates who used x or y as the substitute pair confused themselves later, and that those who used x = x² or y = x² for their substituted unknown did not recover. Choose a new letter with care.
- Not reaching a quadratic. Not rearranging to the form ax² + bx + c = 0. For example, x²(x² − 5) = −4 became x² = −4 and x² − 5 = −4. Another weaker response offered an equation that had not been rearranged, such as 2x²(2x² − 11) = 12.
- Missing solutions. Forgetting the negative square roots in the final step, finding all four roots and then rejecting the negative ones as not possible, or losing a pair of solutions by cancelling y from a two-term quadratic straight away. Sometimes the root from the first factor was ignored.
- Not finding both coordinates. Not finding the y-coordinates and losing the last accuracy marks, and sign slips when writing a negative coordinate (−9 became 9).
- Wrong methods for a cubic. Applying the quadratic formula to a cubic, or solving the derivative of the cubic equal to 0. Reports say a very small number found the quadratic factor.
- Simplifying incorrectly. Cancelling the 2 with the 6 only, to give an answer of 3 ± √8, and performing incorrect operations on fractions.
- Calculator answers. Answers found with a calculator and given as rounded decimals, when the method had to be shown.
- Stating something false to reach the answer. In one question some students obtained the result by stating incorrectly that x + y² = 10 and then using it, which can be seen to give the required result by chance. This was not credited.
- The discriminant approach. Common, but earlier simplification errors led to incorrect critical values.
What examiners advise
- Show the method. Calculators are valuable checking tools, but showing little method, especially when an error has been made, is unlikely to earn full marks.
- Arithmetic errors do not lose method marks when the working is shown: candidates who made an error but showed the method for finding the midpoint and gradient could still earn method marks.
- Check your solutions: expand brackets to check, check that each point satisfies both equations, and check that solutions are feasible.
- Choose an easier substitution where possible, and a letter that does not clash with x or y.
Habits that help
- After eliminating a variable, write the new equation in the form "= 0" before doing anything else.
- When you have found x, substitute it into the linear equation to find y.
- List your final answers as coordinate pairs to avoid mixing up the signs.
Questions students ask
What is the safest way to solve a linear and a non-linear equation together?
Substitute from the linear equation into the non-linear one, rearrange to the form ax² + bx + c = 0, solve, then substitute x into the linear equation to find y. List final answers as coordinate pairs.
Why do I lose marks when clearing fractions?
By multiplying only part of an equation. Multiply every term when preparing to eliminate, and every term by x² or x + 1 when clearing fractions. Squaring term by term is one of the most common incorrect methods.
How do I avoid missing solutions?
Do not forget the negative square roots, and do not reject solutions that are valid. Cancelling y from an equation can lose a pair of solutions. Check that each point satisfies both equations.
Can I use the calculator to solve simultaneous equations?
Use it to check. Answers found on a calculator and given as rounded decimals, with no method shown, are unlikely to earn full marks. Method marks are still earned for shown working even if there is an arithmetic error.
Which letter should I use for a substitution?
One that does not clash with x or y. Candidates who used x or y as the substitute pair confused themselves later.
Practice this topic
Ready to try some? Work through past paper questions on this topic one at a time. Attempt each question fully first, and open the mark scheme only afterwards.