Warm-up
Three quick ones from earlier work. Try each in your head or on paper before you open the answer.
- (Lesson 3) State the gradient and the y-intercept of y = 3x − 4.
Answer
Gradient 3; y-intercept −4, the point (0, −4). - (Lesson 2) Find the gradient of the line through (1, 3) and (5, 11).
Answer
2: the rise is 8 and the run is 4. - Work out 7 − 2 × (−3).
Answer
13: multiply first to get −6, then 7 − (−6) = 13.
Words used in this lesson
- point-slope form
- an equation written from a point and a gradient (also called point-gradient form). For example: y − 3 = 2(x − 1).
- substitute
- put a number in place of a letter. For example: put x = 2 in y = 4x + c.
- gradient-intercept form
- the form y = mx + c. For example: y = 2x + 1.
- expand
- multiply out a bracket. For example: 2(x − 1) = 2x − 2.
- equivalent
- different in form but the same line. For example: y − 3 = 2(x − 1) and y = 2x + 1.
- check
- test an answer with the other point. For example: put the second point back in.
What will the battery show?
A solar lamp's battery reads 30% after 1 hour of charging and 50% after 3 hours, and it charges at a steady rate. You want a rule that works for any time, not a table of readings. Two readings are enough, because they fix a straight line. The question is how to turn two points into an equation.
Predict first: how many hours until it is full, at 100%? Keep your estimate; the worked example settles it. In the first figure, set a gradient and a point, and compare what you get from the point-slope form with the form you know from Lesson 3.
Through P(1, 3) with gradient 2: y − 3 = 2(x − 1). Expanding, y = 2x + 1, so c = 3 − 2 × 1 = 1.
Show me the steps
- m = 2, P(1, 3): y − 3 = 2(x − 1). Expand: y = 2x − 2 + 3 = 2x + 1. So c = 1.
- m = −1, P(2, 4): y − 4 = −(x − 2). Expand: y = −x + 2 + 4 = −x + 6. So c = 6.
- m = 1, P(−2, −3): y + 3 = x + 2. Subtracting 3 gives y = x + 2 − 3 = x − 1. So c = −1.
- m = 0, P(3, −2): y + 2 = 0, so y = −2. A horizontal line.
- m = 2, P(0, −1): y + 1 = 2x. The point is on the y-axis, so c = −1 straight away: y = 2x − 1.
A line with gradient m through the point (x1, y1) has the equation y − y1 = m(x − x1). It says: from the point, the change in y is m times the change in x. This is point-slope form (also called point-gradient form). Expand and tidy it to get y = mx + c, where c = y1 − m x1.
From two points: find the gradient first, then put one point in point-slope form or in y = mx + c. Either point gives the same line, and the other point is a check.
A common mistake: using the y of the point as c. For P(2, 7) on a line of gradient 4, c is not 7: it is the height at x = 0, which is 7 − 4 × 2 = −1. The point is 2 steps along from the y-axis, and the line has climbed 8 on the way.
Two points, then the equation
Pick a pair of points and follow the same three steps each time: the gradient, then c, then the equation. Look at what happens when the line is horizontal.
A(0, −1) and B(2, 3): rise 4, run 2, gradient 2. c = −1. The equation is y = 2x − 1.
Show me the steps
- (0, −1) and (2, 3): gradient 4/2 = 2. A is on the y-axis, so c = −1. y = 2x − 1.
- (1, 5) and (3, 1): gradient −4/2 = −2. Put (1, 5) in y = −2x + c: 5 = −2 + c, so c = 7. y = −2x + 7.
- (−2, −4) and (4, 5): gradient 9/6 = 3/2. Put (4, 5) in y = 3x/2 + c: 5 = 6 + c, so c = −1. y = 3x/2 − 1.
- (−3, 4) and (3, 0): gradient −4/6 = −2/3. Put (3, 0) in y = −2x/3 + c: 0 = −2 + c, so c = 2. y = −2x/3 + 2.
- (2, 6) and (5, 6): gradient 0/3 = 0, a horizontal line. y = 6.
Worked example: the solar lamp
A battery reads 30% at 1 hour and 50% at 3 hours. Write the rule for the charge y at x hours, and find when it is full.
- Gradient: (50 − 30) ÷ (3 − 1) = 20 ÷ 2 = 10 per cent per hour.
- Put (1, 30) in y = 10x + c: 30 = 10 + c, so c = 20. The rule is y = 10x + 20.
- Check with the other point: 10 × 3 + 20 = 50. Full when 10x + 20 = 100, so x = 8 hours.
Your go. The first step is given. A tank holds 150 litres after 2 hours and 90 litres after 5 hours. Write the rule for the volume y after x hours.
- Gradient: (90 − 150) ÷ (5 − 2) = −60 ÷ 3 = −20 litres per hour.
- Now find c by putting one point in y = −20x + c, then check with the other point.
Check the answer
150 = −20 × 2 + c gives c = 190, so y = −20x + 190. Check: −20 × 5 + 190 = 90.
Check before you go on
1. A line has gradient 3 and passes through the point (4, 5). What is its equation?
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. A line has gradient 2 and passes through (1, 7). In y = 2x + c, what is c?
3. A line has the equation y − 3 = 2(x − 4). What is y when x = 6?
Level 2 · Building
4. A line passes through (−2, 1) and (2, 9). In y = mx + c, what is c?
5. The line y = 3x + c passes through (−1, −2). What is c?
6. A glacier is 940 m long at year 2 and 880 m long at year 6, and the change is steady. How long was it at year 0, in metres?
Level 3 · Stretch
7. A line passes through (1, 4) and (4, −5). Where does it cross the y-axis? Give the value of y.
8. A line passes through (2, 5) and has the same y-intercept as y = 3x − 4. What is its gradient? Give it as a decimal.
Something to think about
A line passes through (2, 5) and (2, 9). What is the gradient, and what is its equation? Why does the method of this lesson break down, and what does that tell you about the line?
One way to think about it
The run is 2 − 2 = 0, so the gradient is undefined. The line is vertical: every point has x = 2, so the equation is x = 2. Point-slope form needs a gradient, and a vertical line has none, which is why it is the one case written differently.
Exit ticket
9. Which equation is the line through (1, 2) and (3, 8)?