When should I substitute and when should I add or subtract the equations?

Substitution is fastest when one equation is already written as x = ... or y = ..., while adding or subtracting the two equations removes a variable at once when its coefficients match or are opposites.

Why check the answer in both equations?

Substituting into only one equation can satisfy that line but fail the other, so put the pair back into both equations and confirm both sides match.

S2 Maths: Simultaneous Linear Equations (Challenge, 10 Questions)

2026-11-08

Tutor Prep Guide · S2 Mathematics

The challenge paper asks students to decide whether a system has one solution, none or infinitely many, to clear fractional coefficients and to build equations from context, all with answers and full reasoning.

Quick answer Answers in order: (4, 3), (6, 6), 11 cm and 6 cm, 6 kg and 4 kg, infinitely many, none, (3, -2), 47, 16 years, (5, 2).

01 Paper details (check answers afterwards)

Item	Detail
Questions	10 questions
Suggested time	about 35 minutes
Difficulty mix	Challenge 10
Chapter	Chapter: S2 Maths: Simultaneous Linear Equations
Level	S2

02 Questions (10 items)

ChallengeSolve 4x + 5y = 31 and 4x – 5y = 1.
ChallengeSolve x/2 + y/3 = 5 and x/3 + y/2 = 5.
ChallengeA rectangle has perimeter 34 cm and its length is 5 cm more than its width. Find both sides.
ChallengeCoffee A costs 60 dollars per kg and coffee B 80 dollars per kg. A 10 kg blend averages 68 dollars per kg. Find each amount.
ChallengeHow many solutions does 2x + 3y = 6 with 4x + 6y = 12 have?
ChallengeHow many solutions does 2x + y = 5 with 2x + y = 7 have?
ChallengeSolve x + 2y = -1 and 3x – y = 11.
ChallengeIn a two-digit number the digits add to 11; swapping them makes the number 27 larger. Find the number.
ChallengeA father is 40 and his son is 12. In how many years will the father be twice as old as the son?
ChallengeSolve 7x + 4y = 43 and 3x + 4y = 23.

03 Answers and one-line notes

#	Answer	One-line note
1	x = 4, y = 3	Adding gives 8x = 32 so x = 4, and 16 + 5y = 31 then gives y = 3.
2	x = 6, y = 6	Multiplying both by 6 gives 3x + 2y = 30 and 2x + 3y = 30; subtracting leaves x – y = 0, so x = y = 6.
3	length 11 cm, width 6 cm	With l + w = 17 from the perimeter and l – w = 5, adding gives l = 11 and w = 6.
4	A 6 kg, B 4 kg	Let A be a kg: 60a + 80(10 – a) = 680, giving 20a = 120, so A is 6 kg and B is 4 kg.
5	infinitely many	The second equation is twice the first, so both describe the same line and every point on it satisfies both equations.
6	none	The left sides are identical but the right sides differ, so no pair can satisfy both; the two lines are parallel.
7	x = 3, y = -2	The second equation gives y = 3x – 11; substituting leaves 7x – 22 = -1, so x = 3 and y = -2.
8	47	With tens digit t and units u: t + u = 11 and u – t = 3, so t = 4 and u = 7, giving 47.
9	16 years	With t years to come: 40 + t = 2(12 + t), so t = 16.
10	x = 5, y = 2	Subtracting removes y and gives 4x = 20, so x = 5 and 15 + 4y = 23 gives y = 2.

04 What to do after marking

After marking, redo only the wrong items and write down which step went wrong (formula, unit, or misread number).
Repeat the same item two days later; move to the next paper only when both attempts are correct.
Parents only need the answer and note columns; there is no need to explain every item.

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Questions and answers are written originally by the TutorZone team and are not taken from any school, publisher or examination authority paper. Every value is checked by computer.