At a glance. One lesson of about 60 minutes. It assumes elimination and substitution from Year 9, and equal gradients for parallel lines. Students decide the number of solutions from the coefficients, find a parameter that fixes the case, and model a break-even.
Calculator: GDC optional. The lesson is by hand. The optional section shows GDC users the graphs and the intersect tool, and warns that lines with nearly equal gradients can look parallel.
What students will learn
Know
- Two lines meet once, never or everywhere.
- ax + by = c and px + qy = r meet once exactly when aq − bp is not 0.
- Equivalence transformations: add the same to both sides, multiply by a non-zero number, substitute.
Understand
- The coefficients decide the number of solutions before anything is solved.
- 0 = 0 and 0 = 5 are different outcomes of elimination.
- A step that can multiply by zero is not an equivalence.
Do
- Decide the number of solutions from the coefficients.
- Find a parameter that gives no solution or infinitely many.
- Model a break-even with two linear costs and interpret it.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: elimination, substitution and coefficient tests (questions 1 to 8).
- Criterion B. Investigating patterns: the first figure. Students record a, b, c and the number of solutions for several settings and state the rule that links them.
- Criterion C. Communicating: naming each equivalence transformation and ending with a sentence about the number of solutions (worked examples, faded example).
- Criterion D. Applying mathematics in real-life contexts: plans with fees and rates in the hook and the second figure, and the interpretation of a break-even (question 5).
The one thing to do with the figure
Make students hunt for the boundary. Keep a and b where the lines cross once, and ask students to change c: nothing changes about how many solutions there are. Then ask for a change to a or b that makes the lines parallel. Students find a ratio, and can say it before anyone writes it.
In the second figure, ask: “Which plan would you choose for 10 gigabytes? For 60?” The break-even is where the choice changes.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 5 min | Hook: the different ways two lines can meet. Collect ideas. |
| 12 min | First figure; key idea; the warning card. |
| 8 min | Second figure; worked examples 1 to 3; pairs finish the faded example. |
| 5 min | Question 1, the check. If many choose B, return to the key idea. |
| 20 min | Your turn, Levels 1 to 3, at each student's own pace. |
| 5 min | Something to think about, then the exit ticket, question 8. |
The answers
| Warm-up | x = 5 and y = 2; every sign changes, on both sides; parallel. |
|---|---|
| Hook | Once, never or everywhere. Never: for example 4x + 6y = 5 or 2x + 3y = 10; the same line: for example 4x + 6y = 12. |
| Second figure | Fee 120: Plan A costs 150 + 3 × 30 = 240 at 30 gigabytes, and 120 + 4 × 30 = 240. |
| Worked examples | (2, 3); no solution; k = 4. |
| Faded example | k = 6, no solution. |
| 1. Which system has no solution | C. A and D each have one solution; B is the same line. |
| 2. Elimination | 3. 19x = 57. |
| 3. The same line | 20. Twice 10. |
| 4. A parameter for no solution | 10. 4 ÷ 6 = 10 ÷ 15. |
| 5. Break-even with a parameter | 105. 230 − 125. |
| 6. Two unknown coefficients, one line | 9. a = 6 and b = 3. |
| 7. A triangle between lines | 36. ½ × 12 × 6. |
| 8. A parameter for the same line | A. B drops the sign of the multiplier; C takes parallel for the same line; D reads opposite signs as different gradients. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A, B or D in question 1 | Mixed up one solution, the same line and none. | Ask: what is the gradient of each line? What does doubling one equation give? |
| 2 or 57 in question 2 | Gave y, or stopped at 19x = 57. | Write the multiplied equations in full, add them, then divide by 19. |
| 14 or 10 in question 3 | Added to the right side, or left it unchanged. | Multiply every term of the equation, on both sides. |
| 22.5 or 9 in question 4 | Inverted the ratio, or copied a right-hand side. | Write the ratio of the x coefficients equal to the ratio of the y coefficients. |
| 255 or 55 in question 5 | Added the gap in the rates, or ignored Plan A's gigabyte cost. | Work out what Plan A costs at 25 gigabytes first, then match it with B. |
| 6, 3 or 15 in question 6 | Gave one coefficient, or reused the 9. | Find the multiplier from the right sides first. |
| 72 or 24 in question 7 | Left out the half, or used the wrong height. | Draw the three points and mark the base along the y-axis. |
| B, C or D in question 8 | Dropped the sign of the multiplier, took equal gradients to mean the same line, or read opposite signs as different gradients. | Ask: what number times −x + 2y gives 3x − 6y? Apply it to k as well. |
Questions worth asking
- How can you tell from the coefficients alone that two lines are parallel? Listen for: the coefficients of x and y are in the same ratio.
- What does 0 = 0 tell you? What does 0 = 5 tell you? Listen for: the same line, and no solution.
- Why might a model break down away from the break-even? Listen for: fees or rates change, or other costs are not in the model.
- Can a system of two lines have exactly two solutions? Listen for: no, two lines share at most one point or all of them.
- Which step could give x + 1 = 3 a second solution: multiply by (x − 5), multiply by −3, or add 4? Listen for: multiplying by (x − 5), because both sides become 0 at x = 5.
Differentiation
A way in for anyone stuck
- Give a table with columns for the two equations, the multiplier that makes the x coefficients match, and the result of subtracting.
- Rewrite each equation as y = … so gradients can be compared at a glance.
- Use a graph on squared paper for a pair of lines, then check with elimination.
Stretch
- Find k for which kx + 3y = 6 and 4x + (k + 1)y = 8 have infinitely many solutions.
- The lines x + y = 4 and x − y = 2 meet at one point. Find c so that 2x + y = c passes through it too, and explain why three lines chosen at random almost never meet at one point.
Students learning in English as an additional language
“Parallel” means going in the same direction and never meeting. “Coincide” means lie exactly on top of each other. “Break-even” is the point where two costs are equal. “Eliminate” means remove, here, to remove one letter by adding or subtracting.
| Word | Plain meaning | Example |
|---|---|---|
| system of equations | two or more equations that must be true at the same time | x + y = 9 and x − y = 1 |
| elimination | adding or subtracting equations to remove one letter | adding 2y and −2y gives 0y |
| parallel | lines with the same gradient that never meet | y = 4x + 1 and y = 4x − 7 |
| coincident | the same line, so every point is a solution | x + y = 2 and 3x + 3y = 6 |
| break-even | the amount where two costs are equal | where two phone plans cost the same |
| equivalence transformation | a step that keeps the same solutions | adding 5 to both sides |
Sentence starters for explaining aloud or in writing:
- The gradients are … so the lines …
- Elimination leaves …, which means …
- The break-even is where … so for more than that, … is cheaper.
Two things not to say
“No solution means you made a mistake.” No solution is a genuine result for parallel lines.
“Just multiply across.” Multiplying an equation means every term on both sides.
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.
- SolveSolve x + y = 11 and x − y = 5.
x = 8, y = 3. - DecideHow many solutions do 3x + 6y = 9 and x + 2y = 3 have?
Infinitely many: the first is three times the second.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ParameterFor which k do kx + 2y = 5 and 3x + 6y = 7 have no solution?
k = 1. - Earlier work (lesson 3)Find the gradient of the line through (2, 3) and (6, 11).
2.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningExplain why x + y = 4, 2x + 2y = 8 and 3x + 3y = 13 have no common solution.
The first two are the same line and the third is parallel to it but not the same. - ExtendPlan A costs 100 + 4g and Plan B costs 220 + g. Find the break-even and say which plan to choose above it.
g = 40; above 40 gigabytes B is cheaper.
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
Both figures work with the keyboard (tab to a slider, then the arrow keys) and by touch. “Show me the steps” gives the same states in words.
No device? Squared paper and a ruler are enough to draw the lines. The With a GDC section is optional.