Warm-up
Three quick ones from earlier work. Try each in your head or on paper before you open the answer.
- What is y when x = 4 in y = 3x + 2?
Answer
14, because 3 × 4 + 2 = 14. - A line has the equation y = 5x + 1. What does the 5 tell you?
Answer
The gradient: y rises by 5 each time x rises by 1. - Solve 7x = 3x + 20.
Answer
x = 5, because 4x = 20.
Words used in this lesson
- system of equations
- two or more equations with the same unknowns. For example: y = 2x + 1 and y = 3x − 2.
- solution
- values that make every equation true. For example: (3, 7) solves both equations above.
- point of intersection
- the point where two lines meet. For example: the lines cross at (3, 7).
- parallel
- lines that never meet. For example: y = 3x + 6 and y = 3x + 20.
- rate
- how much y changes for each step in x. For example: 6 taka an hour.
- coordinates
- the pair (x, y) that names a point. For example: (5, 30) is 5 across and 30 up.
Two stalls, one price
Near a bus station in Dhaka two stalls charge for phone charging. Stall A takes a 10 taka card fee and then 4 taka an hour. Stall B has no fee and charges 6 taka an hour. For a short charge, stall B is cheaper. For a long one, stall A is. In between, there is a number of hours where both cost the same.
Predict first: do the two prices become equal after 2 hours, 5 hours or 8 hours? You can test your choice with a table before the figure.
Stall A: y = 4x + 10. Stall B: y = 6x. Find the hours where both cost the same.
- Hours 0 to 4: A costs 10, 14, 18, 22, 26. B costs 0, 6, 12, 18, 24. B is cheaper at every step.
- Hour 5: A costs 4 × 5 + 10 = 30 and B costs 6 × 5 = 30. The costs meet.
- Hour 6: A costs 34 and B costs 36, so A is now the cheaper one.
- The lines cross at the point (5, 30): 5 hours, 30 taka.
Now see the same thing on a graph. Move the sliders for A and B, and watch the crossing.
A: y = 3x + 6 and B: y = 5x cross at (3, 15). At x = 3 both give y = 15.
Show me the steps
- A: y = 3x + 6 and B: y = 5x cross at (3, 15). Check: 3 × 3 + 6 = 15 and 5 × 3 = 15.
- A: y = 2x + 12 and B: y = 4x cross at (6, 24). Check: 2 × 6 + 12 = 24 and 4 × 6 = 24.
- A: y = 3x + 6 and B: y = 3x + 20 both rise 3 for each step, so they never meet.
- A: y = x + 5 and B: y = 6x + 10 meet at x = −1, which is to the left of the grid.
A system of linear equations is two equations in the same two unknowns. Its solution is the pair (x, y) that makes both true at once. On a graph that is the point where the two lines meet, because every point on a line is a solution of its equation.
A table finds a crossing that lands on whole numbers. A graph shows the crossing and tells you if there is none. For an exact answer when the crossing is not on a grid corner, set the two right-hand sides equal and solve, as in lesson 1; lessons 5 and 6 turn that into two full methods.
A common mistake: giving only one number. The solution of a system is a pair: the point (3, 15) means x = 3 and y = 15. Check a pair in both equations: if it fails either one, it is not on both lines.
Your go. The first row is given. Line A is y = x + 12 and line B is y = 4x. Find where they meet.
- x = 0: A gives 12 and B gives 0.
- Fill in x = 1, 2, 3, 4 yourself and watch the gap, then check.
Check the answer
A gives 13, 14, 15, 16 and B gives 4, 8, 12, 16. They meet at (4, 16): 4 + 12 = 16 and 4 × 4 = 16. x = 4, y = 16.
Check before you go on
1. Two lines cross at the point (4, 18). Which statement is true?
Your turn
Start at Level 1 and go as far as you can. Type negative numbers with a minus sign, for example -7. A wrong answer gets a reason, not just a cross: read it, then try again.
Level 1 · Getting started
2. The lines y = 2x + 3 and y = x + 7 cross at x = 4. What is y at the crossing?
3. Plan A costs y = 8x + 20 pesos and plan B costs y = 12x pesos, where x is the number of months. The lines cross at (5, 60). What does that tell you?
Level 2 · Building
4. Line A gives 6, 9, 12, 15, 18 for x = 0, 1, 2, 3, 4. Line B gives 18, 15, 12, 9, 6 for the same x values. At which value of x do A and B give the same y?
5. At x = 2, line A is at y = 14 and line B is at y = 8. Line A rises 3 for each step in x and line B rises 6. At what value of x do the lines meet?
Level 3 · Stretch
6. A school stall costs y = 15x + 40 (taka, for x items) and takes in y = 25x. Each item adds 25 to the income and 15 to the cost. How many items must it sell before income matches cost?
7. Line A passes through (0, 2) and (4, 10). Line B passes through (0, 14) and (4, 2). What is the x-coordinate of the point where they cross?
Something to think about
Could two different straight lines cross at two points? Use what you saw on the figure to say why or why not.
One way to think about it
No. If two lines share two points they are the same line, because exactly one straight line passes through two points. So two different lines meet once or never; the next lesson adds the case where they are the same line.
Exit ticket
8. Which point is the solution of y = 2x + 1 and y = 3x − 2?