Subtract: the same y
3x + 2y = 16
x + 2y = 8
Subtract: 2x = 8, so x = 4
4 + 2y = 8, so y = 2
Maths · Foundation and Higher · AQA · 45 minutes
Simultaneous equations Why is the answer to two equations where their lines cross?
Students find two numbers from two clues, then see on a graph why the answer is where two lines cross, and solve by elimination. Slides to present, a vote that shows who's stuck, practice and homework.
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Move c to 2, 5 and 8. Where do the lines cross? That x and y work in both equations.
They cross at (2, 3)
The idea in one minute
One equation in x and y has many answers: every point on its line. Two equations together usually have one answer, the point on both lines, where they cross. Elimination finds it without drawing: add or take away the equations so that one letter goes, solve for the other, then put it back.
The lesson goes picture first. Students find two numbers from two clues, see every point on one line work, then watch a second line pick out the one point that works in both. Only then do they eliminate: subtract, add, and multiply to make a letter match. The core is linear pairs, for Foundation and Higher; the stretch meets a line and a curve.
By the end, students can
- explain why the solution is where the two lines cross
- solve a pair of equations by adding or subtracting them
- multiply an equation to make one letter match first
- check a solution in both equations
Key words
- simultaneous
- eliminate
- substitute
- solution
- point of intersection
Before this lesson
Students should solve a linear equation such as 2x = 8 and know that every point on a line fits its equation. Elimination is taught here, from worked examples.
Taught in Straight-line graphs.
The lesson, step by step
30 minutes in total. Each step says what changes for this length.
3 min 1 · Hook · students think
Guess my numbers
Students find two numbers from two clues before anyone explains.
For this length: Show the two clues as equations at once; skip the guessing round
Say “I am thinking of two numbers. They add to 5. The second is double the first, take away 1. Find them. Don't show anyone yet.”
Ask- How many pairs add to 5?
- Which of those pairs fits the second clue too?
Watch for Stopping at the first clue, such as 1 and 4: ask whether 4 is double 1, take 1.
For the teacher
Give two minutes of silent work, then ask how they found it. Most try pairs that add to 5 until one fits the second clue.
Don't explain yet. The question is the point: is there a quicker way than trying pairs, and how do we know there's only one answer?
Most will try pairs that add to 5. Gather the ways they found 2 and 3: it is the next step's picture.
Shorter: Show the two clues as equations at once; skip the guessing round (−2 min).
8 min 2 · Explain · you explain
Two lines, one point; then elimination
Every point on one line works; only the crossing works in both. Then eliminate.
For this length: Show the crossing without moving c; one elimination example
Say “Each clue is a line of answers. The answer to both is on both lines.”
Ask- x + y = 5: give me three answers. Where are they on the graph?
- If I move the first line up, where is the answer now?
Watch for Subtracting when one equation has −2y and the other +2y: the letter goes when they are added.
Multiply first: make y match
2x + y = 7, times 2: 4x + 2y = 14
3x − 2y = 7
Add: 7x = 21, so x = 3
6 + y = 7, so y = 1
For the teacher
First slide: x + y = 5 has endless answers, every point on its line: (0, 5), (1, 4), (2, 3), (4, 1). Second slide: the second clue is y = 2x − 1, another line. Only (2, 3) is on both, so x = 2, y = 3 is the only answer to both. Move c: the crossing moves, and so does the answer.
Elimination finds the crossing without drawing. When the y-terms are the same, subtract the equations and y goes. When they have opposite signs, add. When they don't match, multiply a whole equation first, every term.
When both equations say what y is (y = … and y = …), set the two right-hand sides equal and solve. Always find the second letter and check the pair in both equations.
Vote before moving c (Ask the class): where will the lines cross when c is 8?
Shorter: Show the crossing without moving c; one elimination example (−4 min).
5 min 3 · Your turn · students write
Worked, then yours
Each worked example beside a twin that students solve themselves.
For this length: Do the first two pairs only
Say “Watch me do one. Then you do its twin, which is almost the same.”
Ask- Add or subtract? How do you know before you start?
Watch for Stopping after x: the answer is a pair.
Add: opposite signs
x + y = 7
x − y = 1
Add: 2x = 8, so x = 4
4 + y = 7, so y = 3
Then theirs
x + y = 9
x − y = 3
Subtract: the same sign
2x + y = 10
x + y = 7
Subtract: x = 3
3 + y = 7, so y = 4
Then theirs
3x + y = 14
x + y = 6
Multiply first
x + 2y = 7
3x − y = 7, times 2: 6x − 2y = 14
Add: 7x = 21, so x = 3
3 + 2y = 7, so y = 2
Then theirs
2x + 3y = 12
x − y = 1
For the teacher
Work the first example aloud, writing each line. Then students solve its twin while you watch who decides add or subtract first.
Do the same for each pair. The third pair catches students who multiply only one term.
Open the answers slide once most have finished their twins.
Shorter: Do the first two pairs only (−3 min).
9 min 4 · Who's stuck? · students vote
Vote, then practise
One question shows who's stuck; then practice in levels, and a stretch.
For this length: After the vote, questions 1–4 only
Say “Choose an answer on your own. Nobody talks until everyone has chosen.”
Ask- How can you check your pair without me?
Watch for Subtracting when the signs say add.
Show your working on paper and check each pair in both equations. The stretch is optional.
The question that shows who's stuck
- Solve 3x + 2y = 16 and x − 2y = 0.
In your Classroom, as they answer (an example) - A10
- B7
- C4
- D4
Each wrong answer is a different mistake. While they answer, your Classroom shows who chose which, so you know in seconds whom to help with what.
- x + y = 8 and x − y = 2, so x = ___ and y = ___
- 3x + y = 11 and x + y = 5, so x = ___ and y = ___
- y = 2x + 1 and y = 7 − x, so x = ___ and y = ___
- 2x + 3y = 13 and 2x + y = 7, so x = ___ and y = ___
- The lines of two equations cross at (3, −1). What does that tell you?
- y = 2x + 1 and y = 2x + 4. How many solutions does the pair have?
- 3 pens and 2 rulers cost £3.10. 1 pen and 2 rulers cost £1.70. A pen costs £___
- Stretch: y = x² and y = x + 2 meet at the smaller x = ___ and the larger x = ___
Support · For question 1: add the two equations, and y goes. Then put x back into x + y = 8 to find y. Check the pair in x − y = 2 too. When both equations say what y is (y = … and y = …), set the two right-hand sides equal.
For the teacher
Everyone chooses before anyone explains. If they subtracted, ask whether +2y and −2y go by adding or by subtracting. If they slipped a sign, ask them to check the pair in x − 2y = 0.
Questions 1–2 build fluency; 3–4 change the look; 5–6 ask for reasons; 7 is a problem in words. The stretch meets a line and a curve.
Students show their working on paper or the board; the quiz keeps their final answers.
If many subtracted, put +2y and −2y side by side and ask which way they cancel, before they practise.
Shorter: After the vote, questions 1–4 only (−6 min).
Longer: The stretch, then: how many times can a line meet a curve? (+10 min).
5 min 5 · Exit · students show
Solve one, show it on a graph
One pair solved and one sentence show who links the answer to the crossing.
Say “Solve the pair in any way you like. Then say how a graph would show your answer.”
Watch for Giving x only: the answer is a pair.
Show your working; one sentence for the second part.
- Solve x + y = 6 and y = x + 2. Then say in one sentence how a graph of the two lines would show your answer.
For the teacher
Solve x + y = 6 and y = x + 2 in any way: elimination, substitution or a sketch.
Then answer the exit question in one sentence.
The sentences show who links the answer to where the lines cross: start the next lesson with a sketch if few do. Set the homework.
Longer: Sketch both lines and mark the crossing (+5 min).
The answers, their slides and the mistake behind each wrong answer come with your copy. The worked examples above are open to everyone.
One-to-one: ask the student to say before each step whether they will add or subtract, and why; let them check every answer in both equations.
Homework: solve and check
Solve each pair. For questions 1, 2, 3 and 5, write whether you add or subtract, and why. Check every pair in both equations. For question 6, sketch both lines and mark where they cross.
- x + y = 10 and x − y = 2
- 2x + y = 9 and x + y = 5
- 3x + 2y = 12 and x − 2y = 4
- y = 3x − 2 and y = x + 4
- Two adult tickets and one child ticket cost £26. One adult ticket and one child ticket cost £16. Find the price of each ticket.
- Sketch y = x + 1 and y = 5 − x. Read off where they cross, then check it in both equations.
Print the worksheet with working space and homework. Teacher answers come with your Library copy.
Teaching simultaneous equations: common questions
How do you explain simultaneous equations visually?
Each equation is a line: every point on it is an answer to that equation. The answer to both is the point on both lines, where they cross. Show one line with its many answers, add the second, and move it: the crossing, and the answer, move with it.
When do you add and when do you subtract in elimination?
Look at the letter you want to remove. If it has the same sign in both equations (+2y and +2y), subtract. If the signs are opposite (+2y and −2y), add. If the numbers differ, multiply a whole equation first, every term, so that they match.
Is this simultaneous equations lesson suitable for Foundation GCSE?
Yes. The core solves linear pairs by elimination and reads the answer from a graph, for students who can solve a linear equation. The stretch, a line meeting the curve y = x², is Higher.
Can I use the lesson without creating an account?
Yes. Read the lesson, print the worksheet and open its slides without signing in; nothing you draw there is kept. A free teacher account keeps the whole lesson in your Library to change and teach, with the quizzes, the answers and the answer slides.
Curriculum reference: AQA GCSE Mathematics 8300: A19 (two simultaneous equations, solved algebraically and by graph). The line-and-curve stretch is Higher.
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