At a glance. One lesson of about 60 minutes. The hook starts with two bills that differ by one packet. The figure shows the four cases: opposite terms, matching terms, and one or both equations scaled. The side-by-side check against substitution happens in the what-if and in lesson 7.
Calculator: no calculator. Nothing here needs one. Every multiplier is 1 to 5 and every solution is a whole number.
What students will learn
Know
- Equations can be added or subtracted, term by term and total by total.
- Multiply every term of an equation to make one unknown match.
- Add when the matching terms have opposite signs; subtract when they have the same sign.
Understand
- Elimination keeps the balance: the same is done to both sides.
- A multiplier changes how the equation is written, not the line it describes.
- Substitution and elimination reach the same point.
Do
- Eliminate an unknown by adding or subtracting, after scaling if needed.
- Find the second unknown and check in both equations.
- Solve a context problem with two unknowns.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: scaling, adding and subtracting equations (questions 1 to 8).
- Criterion B. Investigating patterns: use the figure to find, for each system, every multiplier pair and operation that eliminates an unknown.
- Criterion C. Communicating: show the scaled equations, the operation, and the check.
- Criterion D. Applying mathematics in real-life contexts: the shop in the hook and the market in question 5.
The one thing to do with the figure
Ask which unknown to eliminate before sliding. “For 2x + 5y = 18 and 3x + 2y = 16, what should the multipliers be to match the x terms?” Students say 3 and 2; let them test it.
Then ask for a different pair that eliminates y (2 and 5), and ask which is shorter.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 10 min | Hook; students predict, then use the figure for the four systems. |
| 10 min | Key idea; worked examples on the shop and on scaling. |
| 10 min | Faded example in pairs; compare the first line. |
| 5 min | Question 1, the check. |
| 15 min | Your turn, Levels 1 to 3, at each student's own pace. |
| 5 min | Exit ticket, question 8, on paper or on screen. |
The answers
| Warm-up | 7; 8x + 12y = 28; x = 3. |
|---|---|
| The shop | A jar is 3 rupees and a packet is 5 rupees. |
| Worked example 2 | (4, 3). |
| Worked example 3 | (2, 3). |
| Faded example | (4, 3). |
| 1. Add or subtract | A. B doubles the y terms; C makes them unequal; D misses that +3y and −3y cancel. |
| 2. Add to eliminate | 8. 2x = 16. |
| 3. Subtract to eliminate | 5. 4y = 20. |
| 4. Scale one equation first | 5. 2x + 4y = 18; 3x + 4y = 23; x = 5. |
| 5. Write and solve from a context | 2. 10r = 50; r = 5; b = 22 − 20. |
| 6. Scale both equations | −3. y = 2; 2x + 6 = 0. |
| 7. Scale both, adding opposites | −3. x = 2; 10 − 2y = 16. |
| 8. First step of elimination | A. B and D do not match the y terms; C matches them but adds. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| B, C or D in question 1 | Did not see that +3y and −3y are opposites, or scaled needlessly. | Write the two y terms one above the other and say what adding them gives. |
| 16, 4 or 12 in question 2 | Stopped early or gave a given total or the other unknown. | Underline the unknown the question asks for. |
| 20, 8.5 or 3 in question 3 | Stopped early, added the totals when subtracting, or gave x. | Write each subtraction in a column: terms and totals. |
| 14, 2 or −5 in question 4 | Scaled only the left side, gave y, or reversed the order for the totals. | Draw an arrow from the multiplier to every term including the total. |
| 50, 5 or 6 in question 5 | Stopped early or gave rice or 3 kg of beans. | Reread the question and label the unknowns r and b. |
| 2, 3 or −6 in question 6 | Gave y, dropped the sign, or stopped before dividing. | Substitute back into one equation and check against the other. |
| 2, 3 or 6 in question 7 | Gave x, dropped the sign, or stopped before dividing. | Underline y and write “y =” as the last line. |
| B, C or D in question 8 | Added when the signs are the same, or did not scale. | Write the scaled equation first, then look at the signs on the matching terms. |
Questions worth asking
- How do you know whether to add or subtract? Listen for “the matching terms have the same sign, so subtract; opposite, so add”.
- Why can we multiply the whole equation? The same is done to both sides, so the balance and the line are unchanged.
- Which unknown is easier to eliminate in 2x + 5y = 18 and 3x + 2y = 16? Either works; compare the multipliers: 3 and 2 against 2 and 5.
- When would you choose substitution? When one equation already says y = … or x = …
Differentiation
A way in for anyone stuck
- Give a four-step card: scale, add or subtract, solve, check.
- Start with systems where the terms already match, then opposites, then one scaled.
- Colour the matching terms before adding or subtracting.
Stretch
- Question 7, then: eliminate x instead of y and compare the working.
- Why does elimination give no solution, or infinitely many, when the lines are parallel or the same? Try x + y = 5 and 2x + 2y = 9.
Students learning in English as an additional language
“Eliminate” means “remove completely”. “Coefficient” is the number in front of the letter. “Opposite” here means the same number with a different sign.
| Word | Plain meaning | Example |
|---|---|---|
| eliminate | remove an unknown by adding or subtracting | adding 3y and −3y |
| coefficient | the number in front of an unknown | in 5y the coefficient is 5 |
| opposite | the same size with the other sign | +3y and −3y |
| multiply through | multiply every term of an equation | 2(x + 2y = 9) is 2x + 4y = 18 |
| back-substitute | put a found value in to get the other unknown | x = 5, so y = 2 |
| check | put the pair into both original equations | 5 + 4 = 9 and 15 + 8 = 23 |
Sentence starters for explaining aloud or in writing:
- I multiply the first equation by … so that the … terms match.
- The signs are the same / different, so I add / subtract.
- I check: in the first equation …, and in the second equation …
Two things not to say
“Just get rid of one of them.” Say how: match, then add or subtract.
“Multiply the top line only.” Every term of the equation is multiplied, total included.
Questions to set
Extra practice in three tiers, including one question from earlier work for spacing. Every answer has been checked.
1Fluency
The idea on its own, with friendly numbers. Set these first.
- Addx + y = 9 and x − y = 1. Find x.
x = 5. - Subtract2x + y = 11 and x + y = 7. Find x.
x = 4.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- Context2 pens and 1 ruler cost 14 pesos; 1 pen and 1 ruler cost 9 pesos. What does a pen cost?
5 pesos: subtract to get one pen. - Earlier workSolve y = x + 1 and x + y = 7 by substitution.
(3, 4).
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningWhy does replacing one of x + y = 5 and x − y = 1 by the sum of both, and keeping the other, not change the solution?
The replaced equation can be recovered by subtracting the kept one from the sum, so both pairs have the same solutions. - Working backwardsWrite two equations with solution (2, 5) where adding eliminates y.
Any correct pair, for example x + y = 7 and x − y = −3.
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
All four sliders work with the keyboard (tab to a slider, then the arrow keys) and by touch. Under the figure, “Show me the steps” gives the same states in words.
No device? Write each equation on a strip of paper and add or subtract the strips term by term.