C. Algebra · Linear and Non-Linear RelationsComparing Two Linear Relations: The Point of Intersection
C3.3 C3.1 A1 A2About 75 minutes
Learning goals- I can compare two linear relations of the form y = ax + b using graphs and tables.
- I can find the point of intersection of two lines graphically and algebraically.
- I can explain what the point of intersection means in a real situation, and which option is better before and after it.
- I can recognize when two lines have no point of intersection or are the same line.
Words to know
point of intersection
The point where two lines cross. Its coordinates satisfy both equations.
break-even point
In a cost comparison, the point where two options cost the same.
linear system
Two (or more) linear equations considered together. Solving the system means finding the point(s) that satisfy all of them.
Which ski pass?
A local ski hill offers two choices:
- Plan A: pay $20 for each visit. C = 20v.
- Plan B: buy a $90 season card, then pay $8 per visit. C = 8v + 90.
If you only go once or twice, Plan A is clearly cheaper. If you go every weekend, Plan B probably wins. But where exactly is the switch? That is what the point of intersection tells you. In Grade 8 you found when two linear patterns were equal; now you will do it with graphs and with algebra, and interpret the answer carefully.
Method 1: Graph both lines
Graph both relations on the same axes. The point where they cross has the same v and the same C for both plans: that is where the plans cost the same.
From the graph, the lines cross at (7.5, 150).
Method 2: Use algebra
At the point of intersection both C-values are equal, so set the two expressions equal and solve:
20v = 8v + 90
12v = 90 (subtract 8v from both sides)
v = 7.5 (divide both sides by 12)
Substitute into either equation: C = 20(7.5) = 150. Check in the other: 8(7.5) + 90 = 60 + 90 = 150 ✓.
The algebra gives the exact point, even when it is hard to read from a graph.
Method 3: Interpret in context
The point (7.5, 150) says: at 7.5 visits, both plans would cost $150. But you cannot make half a visit! So compare whole numbers on each side:
- 7 visits: Plan A costs $140, Plan B costs $146. A is cheaper.
- 8 visits: Plan A costs $160, Plan B costs $154. B is cheaper.
Conclusion: choose Plan A for 7 or fewer visits and Plan B for 8 or more. Always finish with a sentence that answers the real question.
Worked exampleExample 1: Catching up
Liam starts walking from home at 4 km/h. At the same moment his sister Maya leaves a café 3 km behind him on the same trail and bikes at 10 km/h in the same direction. When and where does Maya catch up?
- Measure distance from the café. Liam: d = 4t + 3 (he starts 3 km ahead). Maya: d = 10t.
- Set equal: 10t = 4t + 3. Subtract 4t: 6t = 3. Divide by 6: t = 0.5.
- d = 10(0.5) = 5. Check: 4(0.5) + 3 = 5 ✓.
- Before t = 0.5, Liam is ahead; after it, Maya is ahead.
Maya catches up after 0.5 hours (30 minutes), 5 km from the café.
Maya's line (solid) is steeper because she is faster, so it catches up with Liam's line (dashed), which starts 3 km higher. They meet at (0.5, 5).Worked exampleExample 2: One filling, one draining
Tank A holds 500 L and drains at 20 L/min. Tank B holds 100 L and fills at 30 L/min. When do they hold the same amount?
- A: V = −20t + 500. B: V = 30t + 100.
- −20t + 500 = 30t + 100. Add 20t: 500 = 50t + 100. Subtract 100: 400 = 50t. So t = 8.
- V = 30(8) + 100 = 340. Check: −20(8) + 500 = 340 ✓.
After 8 minutes, both tanks hold 340 L.
Worked exampleExample 3: Compare with a table
Two phone plans: Plan X costs $30 plus $5 per GB of data. Plan Y costs $45 plus $2 per GB. Compare them with a table and find the break-even point.
- Table for 0 to 6 GB. X: 30, 35, 40, 45, 50, 55, 60. Y: 45, 47, 49, 51, 53, 55, 57.
- The costs are equal at 5 GB ($55).
- Below 5 GB, X is cheaper; above 5 GB, Y is cheaper.
- Algebra check: 30 + 5g = 45 + 2g gives 3g = 15, g = 5 ✓.
Break-even at 5 GB, $55. Choose X for less than 5 GB of data and Y for more.
Three possibilities
Two lines in the same plane can:
- cross at exactly one point (different slopes),
- never meet (same slope, different y-intercepts: parallel lines), or
- be the same line (same slope and same y-intercept), so every point is shared.
For example, y = 2x + 1 and y = 2x − 3 never meet: setting 2x + 1 = 2x − 3 gives 1 = −3, which is impossible. In context: if two plans have the same rate but different fixed fees, one is always cheaper.
Just for funWhy did the two lines get along so well?
They saw eye to eye at exactly one point.
Hands-on activityWhich plan is better for your family?
You need: two real price options for something your family uses (for example, a monthly transit pass versus paying per ride, a pool or rink membership versus drop-in fees, or two phone plans; ask a grown-up or look online), graph paper, a pencil.
- Write each option as an equation C = (rate)n + (fixed fee).
- Make a table for a sensible range of n.
- Graph both lines on one grid and estimate the point of intersection.
- Solve algebraically to find the exact point.
- Write a recommendation: "If we use it ___ times or fewer, choose ___. Otherwise choose ___." Then ask: how many times does your family really use it?
What to look for: a clear break-even number, rounded sensibly to whole uses, and a recommendation that fits your family's real usage. This is how people in many careers (accountants, planners, buyers) compare options.
Your turn
1Find the point of intersection of y = 3x + 2 and y = 5x − 4.
3x + 2 = 5x − 4 gives 6 = 2x, so x = 3 and y = 3(3) + 2 = 11. (3, 11). Check: 5(3) − 4 = 11 ✓.
2Find the point of intersection of y = −x + 6 and y = 2x.
−x + 6 = 2x gives 6 = 3x, x = 2, y = 4. (2, 4).
3Rink A costs C = 8h; Rink B costs C = 3h + 10. When do they cost the same? Which is cheaper for 3 hours?
8h = 3h + 10 gives 5h = 10, h = 2: both cost $16 at 2 hours. For 3 hours: A costs $24, B costs $19, so B is cheaper.
4Plan X: C = 5g + 30. Plan Y: C = 2g + 45. Which plan is cheaper for 3 GB?
X: 5(3) + 30 = $45. Y: 2(3) + 45 = $51. Plan X is cheaper (3 GB is below the 5 GB break-even point).
5Do y = 2x + 1 and y = 2x − 3 intersect? Explain.
No. They have the same slope (2) and different y-intercepts, so they are parallel.
6Verify that (4, 2) is the point of intersection of y = x − 2 and y = −0.5x + 4.
4 − 2 = 2 ✓ and −0.5(4) + 4 = −2 + 4 = 2 ✓. It satisfies both equations.
7Two lines showing the cost of two bowling options cross at (6, 30), where x is the number of games and y is the total cost in dollars. What does this point mean?
For 6 games, both options cost the same total, $30.
More practice: warm-up
1What is special about the point of intersection of two lines?
Its coordinates make both equations true.
2Is (1, 5) on both y = 2x + 3 and y = 5x?
2(1) + 3 = 5 ✓ and 5(1) = 5 ✓. Yes.
Subtract x: 4 = x + 1, so x = 3.
4Can two different parallel lines intersect?
5Where do y = 4x and y = 12 meet?
More practice: on your own
1Find the point of intersection of y = 0.5x + 7 and y = 2x + 1.
0.5x + 7 = 2x + 1 gives 6 = 1.5x, so x = 4 and y = 0.5(4) + 7 = 9. (4, 9). Check: 2(4) + 1 = 9 ✓.
2Find the point of intersection of y = −2x + 10 and y = 3x − 5.
−2x + 10 = 3x − 5 gives 15 = 5x, x = 3, y = 4. (3, 4).
3A brother starts a race 120 m ahead of his sister. He runs at 4 m/s; she runs at 6 m/s. When and where does she catch him?
Sister: d = 6t. Brother: d = 4t + 120. 6t = 4t + 120 gives 2t = 120, t = 60 s, at d = 360 m from her start.
4Candle A is 30 cm tall and burns 2 cm per hour. Candle B is 20 cm tall and burns 1 cm per hour. When are they the same height? Which is taller after 12 hours?
30 − 2t = 20 − t gives t = 10 hours, when both are 10 cm. After 12 h: A is 30 − 24 = 6 cm and B is 20 − 12 = 8 cm, so B is taller.
5Car rental for one day: Company P charges $40 plus $0.20 per km; Company Q charges $60 plus $0.10 per km. Find the break-even distance. Which company is better for a 150 km trip?
40 + 0.2d = 60 + 0.1d gives 0.1d = 20, d = 200 km (both $80). For 150 km: P costs $70 and Q costs $75, so choose P.
Challenge
1Do the lines y = x + 1, y = −x + 7 and y = 2x − 2 all pass through one point?
First two: x + 1 = −x + 7, so x = 3, y = 4. Third line at x = 3: 2(3) − 2 = 4 ✓. Yes, all three pass through (3, 4).
2Hall A charges $500 plus $25 per guest. Hall B charges $35 per guest. For how many guests is Hall A cheaper?
Equal when 500 + 25g = 35g, so 10g = 500 and g = 50. For more guests, A's smaller rate wins: Hall A is cheaper for 51 or more guests. (Check 51: A = $1775, B = $1785.)
3Find k so that y = kx + 1 and y = 3x − 5 intersect at the point where x = 2.
On y = 3x − 5, x = 2 gives y = 1, so the point is (2, 1). Then 1 = 2k + 1, so k = 0: the first line is the horizontal line y = 1.
Quiz
1. Where do y = 2x + 3 and y = 4x − 1 intersect?
2. Plan A: C = 20v. Plan B: C = 8v + 90. Which is cheaper for 10 visits?
3. Two lines have the same slope and different y-intercepts. How many points of intersection do they have?
4. In a cost comparison, what does the point of intersection represent?
5. Tank A: V = −20t + 500. Tank B: V = 30t + 100. When are they equal?
6. Break-even happens at 7.5 visits. What should you recommend?
Big idea: Expectation C3.3 asks students to compare two linear relations graphically and algebraically and interpret the point of intersection in context. Graphs show the big picture (which option is cheaper on each side); algebra gives the exact point. This prepares students for solving linear systems in Grade 10.
Common difficulties:
- Stopping at the numbers. The answer to "which plan?" is a sentence, not just (7.5, 150).
- Ignoring whole-number contexts. Half a visit is impossible, so test the whole numbers on either side.
- Algebra slips when the variable is on both sides. Solving these equations is taught in the algebra unit on expressions and equations; encourage your child to check by substituting into both equations.
How to help: The family-plan activity is real decision making. Let your child present their recommendation and explain it using the graph. This connects to strand A (making connections to real life and careers).