S3 Maths | Simultaneous Linear Equations: substitution, elimination and number of solutions

S3 Maths | Simultaneous Linear Equations: substitution, elimination and number of solutions – tutor prep guide

2026-11-14

Tutor lesson-prep guides · Secondary Mathematics

Students can start a simultaneous pair and then lose the item at the last step: they stop once x is found and never go back for y, or they multiply both equations by the same number so nothing cancels. This lesson turns substitution and elimination into fixed routines and closes with counting solutions.

Objectives — Students can: (1) solve a pair by substitution when one equation already gives x or y on its own; (2) solve a pair of general-form equations by elimination, including cases needing multipliers; (3) use gradients and intercepts to count solutions (one intersection, parallel with none, coincident with infinitely many) and set up two equations from a word problem with two unknowns.

01 Prerequisites (3-minute check)

If any of these fails, fix it before the lesson starts.

Skill	Quick check	Pass mark
Solving a linear equation	Solve 3x plus 5 equals 17.	x equals 4, subtract 5 then divide by 3
Rearranging	From x plus y equals 7, what is x?	x equals 7 minus y, so the sign flips
Gradient of a line	Do y equals 2x plus 1 and y equals 2x minus 3 share a gradient?	Yes, both are 2, so the lines are parallel

02 Lesson flow (45 minutes)

Time	Stage	Tutor does	Student does
0-5 min	One equation is not enough	Write x plus y equals 7 and ask for x, forcing the point that two unknowns need two equations	Explain that a second equation is required
5-15 min	Substitution	With x plus y equals 7 and x minus y equals 1, rearrange to x equals y plus 1, substitute, get y equals 3, then x equals 4	Repeat it themselves and show the back-substitution step
15-27 min	Elimination	Add 3x plus 2y equals 16 and 3x minus 2y equals 8 to get 6x equals 24, then handle 3x plus 2y equals 12 and 5x minus 3y equals 1 with multipliers	Finish one item needing multipliers and check the constants were multiplied too
27-37 min	Counting solutions	Sketch three pairs: crossing, parallel, coincident, matching one solution, none and infinitely many	Classify equations directly into the three cases
37-43 min	Two-unknown word problems	Adult tickets 95 and child tickets 50: two adults with three children cost 340, one adult with two children 195, so set a and c	Define unknowns, form both equations, solve and give units
43-45 min	Exit check	Collect three items: one substitution, one elimination, one counting solutions	Answer and hand in

03 Teaching script (read it out)

Scene 1: Substitution: isolate once, then swap

Substitution is three mechanical moves: rewrite one equation as one unknown equals something (x equals 7 minus y), put that into the other equation wherever x appears, then solve for the remaining unknown. For x plus y equals 7 and x minus y equals 1, the second gives x equals y plus 1, and substituting into the first gives 2y plus 1 equals 7, so y equals 3 and x equals 4. The commonest failure is stopping too early: the answer to a pair is an ordered pair, so both values must be written and checked in the other equation.

Have students solve y equals 2x with x plus y equals 9 and explain why back-substitution is needed, reaching x equals 3 and y equals 6

Scene 2: Elimination: the target is one unknown gone

Elimination works when the coefficients of one unknown are numerically equal: if their signs differ, add; if they match, subtract. Adding 3x plus 2y equals 16 to 3x minus 2y equals 8 cancels the y terms to give 6x equals 24, so x equals 4 and y equals 2. When coefficients differ, scale first: with 3x plus 2y equals 12 and 5x minus 3y equals 1, multiply the first by 3 and the second by 2 to get 9x plus 6y equals 36 and 10x minus 6y equals 2, add to reach 19x equals 38, so x equals 2 and y equals 3. The discipline: use different multipliers and multiply every term, constants included.

Ask which multiplier each equation needs to kill y, then check that the constant term was scaled as well

Scene 3: Counting solutions: equal gradients mean parallel

Each linear equation is a line and the intersection is the solution. Different gradients give one crossing and a unique solution; equal gradients with different intercepts give parallel lines and no solution; equal gradients and equal intercepts give one line and infinitely many solutions. Rearranging to y equals mx plus c allows a direct comparison, so y equals 2x plus 1 and 2y equals 4x plus 2 are the same line once the second is halved. Students often write no solution as x equals 0, which is a point rather than an impossibility.

Give three pairs and ask only for the number of solutions, with the reason stated by comparing m and c

04 Question types and worked steps

Type	Example	Steps	Answer
Substitution	Solve x plus y equals 7 and x minus y equals 1.	(1) Rearrange to x equals y plus 1 (2) Substitute into the first (3) Solve y equals 3 (4) Back-substitute x equals 4 (5) Check in the first equation	x equals 4, y equals 3
Elimination	Solve 3x plus 2y equals 16 and 3x minus 2y equals 8.	(1) Add to cancel y (2) 6x equals 24 (3) x equals 4 (4) Back-substitute for y equals 2	x equals 4, y equals 2
Needing multipliers	Solve 3x plus 2y equals 12 and 5x minus 3y equals 1.	(1) Multiply the first by 3 and the second by 2 (2) Add to get 19x equals 38 (3) x equals 2 (4) y equals 3	x equals 2, y equals 3
Counting solutions	How many solutions do y equals 2x plus 1 and y equals 2x minus 3 have?	(1) Compare gradients, both 2 (2) Compare intercepts 1 and minus 3, they differ (3) Parallel (4) No solution	No solution, the lines are parallel

Sanity check (teach this): Three checks. Back-substitute first: the pair must satisfy both original equations, and checking one only verifies half. Multipliers second: whatever factor is used must be applied to the constant as well, and missing the constant is the commonest arithmetic slip. Count third: two lines meet at most once, so two distinct answer pairs usually mean the equations were really the same line or a division went wrong.

05 Independent practice (3 questions)

Question	Answer	What to watch
Solve y equals 2x and x plus y equals 9.	x equals 3, y equals 6	Whether the answer was checked by substitution
Solve 3x plus 2y equals 12 and 5x minus 3y equals 1.	x equals 2, y equals 3	Whether different multipliers were used and constants scaled
Two adults and three children cost 340, one adult and two children cost 195. Find the adult price.	Adult 95, child 50	Whether two unknowns and two equations were set up

06 Common mistakes and what to say

Mistake	Student writes	Say this
Stopping at x	x equals 4, so the answer is 4.	A pair needs an ordered pair. Back-substitute to find y, then write x equals 4, y equals 2 and check in the other equation.
Multiplying only the left side	Multiplying the first by 3 gives 9x plus 6y equals 12.	The whole equation is multiplied, so the right side 12 becomes 36. Scaling is done to both sides of the equals sign.
Using the same multiplier on both equations	To cancel y, multiply both equations by 3.	Multiplying both by the same number keeps the coefficient ratio, so nothing ever cancels. The coefficients must end up equal or opposite.
Writing no solution as x equals 0	The lines are parallel, so x equals 0.	Parallel lines never meet, so no pair of numbers works and the answer is no solution. x equals 0 is a specific point.

07 Homework (paper, section and question numbers)

Linear equation word problems (Q1-Q4) — Single-unknown word problems first, then add the second unknown
Equations of straight lines (Q3-Q6) — Gradient and intercept decide the number of solutions, same tool
Secondary Mathematics index (pick two equation papers, timed) — Twenty minutes timed; every answer needs both x and y

08 Exit check (last 5 minutes)

Question	Answer	Decision
What solves x plus y equals 7 and x minus y equals 1?	x equals 4, y equals 3	x alone does not count
Two lines share a gradient but have different intercepts. How many solutions?	None	None with a parallel explanation is a pass
When scaling for elimination, what happens to the constant?	It is multiplied by the same factor	Missing it means the equality idea is not secure

09 Tutor FAQ

Should both substitution and elimination be taught?

Yes. When one equation already says y equals something, substitution is quickest; when both are in general form, elimination saves time. One method alone leaves students stuck on the other format.

Do solutions need a graph?

Comparing m and c after rearranging is fastest, with a rough sketch as support. Exams are timed, so teach the comparison first and let the sketch confirm it.

How should unknowns be set up in word problems?

Count the unknown quantities first: two means two letters. Then translate each sentence into one equation, so two sentences give two equations, which is the most reliable check.

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A TutorZone lesson-prep guide. Written by the TutorZone team; questions and lesson flow are original and are not taken from any school or publisher material.

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