Find one, then all
3x + 5y = 22: try y = 0, 1, 2… y = 2 gives 3x = 12, x = 4
Step 5 across, 3 down: (9, −1), (14, −4)…
and back: (−1, 5), (−6, 8)…
All: x = 4 + 5t, y = 2 − 3t for any whole number t
Whole-number solutions of ax + by = c Which whole numbers solve 3x + 5y = c, and how can you be sure you have found them all?
Maths olympiad · 45 minutes
Students find which totals 3p and 5p stamps can make, see the whole-number solutions as points on a line, discover that they step along by (5, −3), and prove why. Then: when there are none, and how to count the positive ones. Slides, a vote that shows who's stuck, problems up to olympiad level, and short proofs to mark.
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Stamps cost 3p and 5p. Move c, the total. The marked points are the whole-number solutions of 3x + 5y = c. Which totals can you make with x and y both 0 or more, and which can't you make?
Whole-number points: (−1, 4), (4, 1), (9, −2)
The idea in one minute
An equation such as 3x + 5y = 22 is a line, and its whole-number solutions are the grid points on it: (4, 2), (9, −1), (−1, 5) and so on. They come at equal steps of 5 across and 3 down, because if two solutions differ by (d, e), then 3d + 5e = 0, so 5 divides 3d, and as 3 and 5 share no factor, 5 divides d. When a and b share a factor that does not divide c, there are no whole-number solutions at all.
The lesson goes problem first. Students try to make every total from 1p to 20p with 3p and 5p stamps, then watch the line 3x + 5y = c move with its grid points marked. They list solutions, spot the step, and only then prove why it must be so. From there: equations with no solutions, counting the positive solutions with an inequality, and the largest total that can't be made, as olympiad-style problems with written arguments the teacher marks.
By the end, students can
- find one whole-number solution of ax + by = c, then all of them
- explain why the solutions step by (b, −a) when a and b share no factor
- show that an equation has no whole-number solutions using a common factor
- count the solutions with x and y positive, and solve stamp and coin problems
Key words
- whole-number (integer) solution
- linear Diophantine equation
- lattice point
- highest common factor (gcd)
- general solution
Before this lesson
Students should substitute into an equation, know factors and the highest common factor, and read points on a straight-line graph.
Taught in Counting routes on a grid.
The lesson, step by step
45 minutes in total. The stretch is optional.
5 min 1 · Hook · students think
Stamps of 3p and 5p
Students find which totals 3p and 5p stamps can make before anyone explains.
Say“You have as many 3p and 5p stamps as you like. Which totals from 1p to 20p can you make exactly? Which can't you make? Is there a total after which you can make everything?”
Ask- How did you make 17p? Is there another way?
- You can make 8p, 9p and 10p. Why does that mean you can make 11p, 12p and 13p?
Watch forStudents who stop at 'I can't make 7p' with no reason: ask how they know they tried every way.
For the teacher
Give three minutes with a table: total, and how to make it. Most find 3, 5, 6, 8, 9, 10 quickly and struggle to be sure about 7.
Don't explain yet. Ask how many ways there are to make 30p (three: ten 3p; five 3p and three 5p; six 5p) and whether the ways have a pattern.
Collect the totals that can't be made (1, 2, 4 and 7) without confirming. Keep the question 'how do you know?' for the end of the lesson: it is the stretch.
Shorter: Ask for 1p to 12p only, and the ways to make 30p (−2 min).
12 min 2 · Explain · you explain
Discover it, then name it
The solutions as grid points on a line, the step between them and why, no solutions, counting.
Say“Every whole-number solution of 3x + 5y = 22 is a grid point on its line. Find three, and look at how you get from one to the next.”
Ask- From (4, 2), what is the next solution? And the one before?
- Could two solutions be closer than 5 apart in x? Why not?
- 6x + 9y = 20: try a few. What do you notice about the left side?
Watch forMoving both x and y up: to keep 3x + 5y fixed, x goes up 5 while y goes down 3.
Why the step is (5, −3)
Two solutions: 3x + 5y = 22 and 3p + 5q = 22
Subtract: 3(x − p) = 5(q − y)
5 divides 3(x − p) and shares no factor with 3, so 5 divides x − p
So x moves in steps of 5, and then y moves 3 the other way
Count the positive ones
3x + 5y = 100: one solution is x = 0, y = 20
All: x = 5t, y = 20 − 3t
x ≥ 1 needs t ≥ 1; y ≥ 1 needs 3t ≤ 19, so t ≤ 6
t = 1, 2, 3, 4, 5, 6: six solutions
For the teacher
First slide: the line 3x + 5y = 22 with its grid points. Students find (4, 2) by trying y = 0, 1, 2. Then they find others and say the step: 5 across, 3 down.
Second slide: why. Two solutions subtracted give 3(x − p) = 5(q − y). 5 divides the right side, so it divides the left; 5 shares no factor with 3, so 5 divides x − p. That is a proof: it covers every solution, not just the ones we found. Name it: an equation like this, solved in whole numbers, is a linear Diophantine equation, and x = 4 + 5t, y = 2 − 3t is its general solution.
Third slide: none. 2x + 4y is always even and 5 is odd: the line passes between the grid points. Likewise 6x + 9y is always a multiple of 3, and 20 is not. In general ax + by = c has whole-number solutions exactly when the highest common factor of a and b divides c (the 'if' part, that a solution then exists, is a fact worth knowing; its proof can wait). When they share a factor that does divide c, divide it out first: 4x + 6y = 20 is 2x + 3y = 10, so its solutions step by (3, −2), not (6, −4).
Fourth slide: count. Write the general solution, then turn 'positive' into inequalities on t.
Vote before you move c on the slide (Ask the class): will 3x + 5y = 7 have any marked points with both x and y 0 or more?
Shorter: Find one, the step and why; no solutions and counting as one slide to read (−4 min).
8 min 3 · Your turn · students write
Worked, then yours
Each worked example beside a twin that students solve themselves.
Say“Watch me do one. Then you do its twin. Check every solution by putting it back in.”
Ask- Which way does y go when x goes up? How do you know before you work it out?
Watch forStepping by (3, 5) or (2, 7) the wrong way round: x steps by y's number.
Find one, then all
2x + 7y = 30: y = 0 gives x = 15
All: x = 15 + 7t, y = −2t (or x = 1 + 7t, y = 4 − 2t)
Then theirs
4x + 3y = 25: one solution, then all
Any at all?
8x + 12y = 30
The left side is a multiple of 4; 30 is not
No whole-number solutions
Then theirs
9x + 6y = 51: any solutions? Find one if so
How many positive?
2x + 7y = 30: x = 1 + 7t, y = 4 − 2t
x ≥ 1: t ≥ 0. y ≥ 1: t ≤ 1
Two: (1, 4) and (8, 2)
Then theirs
4x + 3y = 25: how many with x and y positive?
For the teacher
Work the first example aloud, writing each line. Then students solve its twin, and check each solution in the equation.
Do the same for each pair. The third pair uses the first: the general solution, then the inequalities.
Open the answers slide once most have finished their twins. For the second twin, ask for a reason either way: a common factor that settles it, or a solution put back in.
Shorter: Do the first two pairs only (−3 min).
15 min 4 · Who's stuck? · students vote
Vote, then climb
One question shows who's stuck; then problems that climb from fluency to olympiad level.
Say“Choose an answer on your own. Nobody talks until everyone has chosen. Then work up the problems as far as you can; the last ones are olympiad level.”
Ask- Have you checked the first and last solution you counted in the equation?
Watch forSolutions with x or y equal to 0 counted as positive; a slip at one end of the range of t; the word 'positive' forgotten.
Whole numbers may be negative unless the question says positive. Show your working on paper. Questions 5 and 8 need a written explanation.
The question that shows who's stuck
- How many solutions of 3x + 5y = 60 have x and y both positive whole numbers?
In your Classroom, as they answer (an example) - A10
- B7
- C4
- D5
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.
- 3x + 5y = 31 has the solution x = 2, y = 5. The next solution with a bigger x is x = ___, y = ___
- 5x + 2y = 29 has three solutions with x and y both positive. In the one with the smallest y, what is x?
- Which equation has no whole-number solutions?
- Pens cost 7p and rubbers 4p. Sam spends exactly 50p and buys at least one of each. In how many different ways can Sam do it?
- Explain why there are no whole numbers x and y with 10x + 4y = 33.
- How many solutions of 5x + 7y = 200 have x and y both positive whole numbers?
- With as many 4p and 7p stamps as you like, what is the largest total you cannot make exactly?
- Stretch: prove that every whole-number total of 8p or more can be made with 3p and 5p stamps.
- Stretch: how many solutions of 3x + 5y = 1000 have x and y both positive whole numbers?
Support · For question 1: from the solution you are given, x goes up 5 and y goes down 3. For the counting questions: find one solution, write all of them with t, then write 'x ≥ 1' and 'y ≥ 1' as inequalities in t. For question 7: list the totals you can make; once you can make four in a row, you can make every total after them.
For the teacher
Everyone chooses before anyone explains. Then ask a student for each answer to list the solutions behind it, and the class to test each one: is it a solution, and are x and y both positive? Zeros, the ends of the range of t and the word 'positive' all show up here.
Questions 1–2 build fluency; 3–4 give the same idea a new look; 5 asks for a written reason; 6–7 are competition style; 8 and 9 are olympiad level.
Students show their working on paper or the board; the quiz keeps their final answers, and the written ones come to you to mark.
If the class is split, ask every student to write out the solutions they counted and tick each one twice: is it in the equation, and are x and y both positive? Long answers (5, 8) are marked by you: one mark for the idea, one for each step with its reason, one for the conclusion. Model answers are in each question's explanation.
Shorter: After the vote, questions 1–5 only (−6 min).
Longer: Question 8 written up in full, then read aloud and improved together (+10 min).
5 min 5 · Exit · students show
All of them, and why
One general solution with a reason shows who knows they have every solution.
Say“Find every whole-number solution, and say in a sentence how you know there are no others.”
Watch forA list of three solutions with 'and so on': the answer is a rule for all of them, with a reason.
One solution, the rule for all of them, and a sentence.
- Find all whole-number solutions of 4x + 7y = 3. How do you know there are no others?
For the teacher
Find all whole-number solutions of 4x + 7y = 3.
A full answer gives one solution, the rule for all of them, and why no others exist.
The reasons show who has the proof and who has only the pattern: start the next lesson with one of each (anonymous). Set the homework.
Longer: Swap answers and mark a partner's reason: is it a proof or a pattern? (+5 min).
One to one: ask the student to find three solutions before you say anything about steps, and to predict the next one; then ask them to explain why it can't be any other step, out loud first, then in writing.
Homework: find them all
Whole numbers may be negative unless the question says positive. Check every solution by putting it back in. Question 6 is a written proof: write it as you would in an olympiad.
- Find one whole-number solution of 5x + 3y = 31, then write down all of them.
- Does 10x + 15y = 42 have any whole-number solutions? Explain.
- How many solutions of 4x + 9y = 100 have x and y both positive?
- Adult tickets cost £7 and child tickets £4. A group spends exactly £100 and buys at least one of each. List every possibility.
- With 5p and 8p stamps, what is the largest total you cannot make? Show that every larger total can be made.
- Prove that if two solutions of 3x + 5y = c are different, their x values differ by at least 5.
Print the worksheet with working space and homework. Teacher answers come with your Library copy.
Teaching whole-number solutions of ax + by = c: common questions
Why do the whole-number solutions of ax + by = c come at equal steps?
Take two solutions and subtract: a times the change in x equals minus b times the change in y. If a and b share no factor, b must divide the change in x, so x moves in steps of b and y in steps of a the other way. If they share a factor d, the steps are b ÷ d and a ÷ d.
When does ax + by = c have no whole-number solutions?
When the highest common factor of a and b does not divide c. Every value of ax + by is a multiple of that factor, so c must be one too. When it is, there are always solutions, and infinitely many: one solution and the step give them all.
What is the stamps (or coins) problem?
With stamps of two values that share no factor, every large enough total can be made, and there is a largest total that can't. For 3p and 5p it is 7p. Students find it by listing totals, and prove it by finding a run of consecutive totals as long as the smaller stamp, then adding that stamp.
Which competitions use problems like these?
Whole-number solutions of equations, coins and stamps come up in junior and intermediate maths challenges, their olympiad rounds and national olympiads. This lesson's problems are written for it; it uses no past papers.
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