1. Dr J Frost (jfrost@tiffin.kingston.sch.uk) www.drfrostmaths.com
KS3: Straight Lines
Dr J Frost
Last modified: 14th October 2015

2. Lines and their Equations
Part 1
Lines and their Equations
To print: Yr8StraightLines-Ex1LinesAndTheirEquations

3. 𝑥=2 What is the equation of this line?y
What is the equation of this line?
And more importantly, why is it that? 4 3 2 1 -1 -2 -3 -4 x ?
𝑥=2
For any point we pick on the line, the 𝑥 value is always 2.

4. L J L J J L L J J Lines and Equations of Lines 2,0 −1,3 On the line? ?
A line consists of all points which satisfy some equation in terms of 𝑥 and/or 𝑦.
On the line?
𝑦=3
𝑥+𝑦=2
𝑦=3𝑥+1 ? ? ?
2,0 L J L ? ? ?
−1,3 J J L ? ? ?
1 4 , 7 4 L J J

5. y
What and why? 4 3 2 1 -1 -2 -3 -4 x ?
𝑦=−1

6. y
What and why? 4 3 2 1 -1 -2 -3 -4 x
For any point we pick on the line, the 𝑥 value is always equal to the 𝑦 value.
𝑦=𝑥 ?

7. y
What and why? 4 3 2 1 -1 -2 -3 -4 x ?
𝑦=−𝑥

8. y
Exercise 1 - Example 8 6 4 2 -2 -4 -6 -8
Use the axis to sketch the line with equation 𝑦=2𝑥−1 x
Pick two suitable values of 𝑥 suitable far apart (say -3 and 4)
Use the equation to work out what 𝑦 would be for each. Plot these points.
If you know the line is a straight line, we can just join them up.

9. Exercise 1 – Example 2 𝒙 −2 −1 1 2 𝒚 𝟑 𝟐 𝟏 𝟐 𝟎 − 𝟏 𝟐 ? ? ? ?
Complete the table of values for 𝑥+2𝑦=1. 𝒙 −2 −1 1 2 𝒚
𝟑 𝟐  
  𝟏 𝟐 𝟎 
− 𝟏 𝟐   ? ? ? ?
If 𝑥=−2 just sub it into the equation:
−2+2𝑦=1 2𝑦=3 𝑦= 3 2

10. Exercise 1 – Question 1 𝑥+𝑦=2 y 8 6 4 2 -2 -4 -6 -8

11. Exercise 1 – Question 2 𝑦=− 1 2 𝑥+1 y 8 6 4 2 -2 -4 -6 -8

12. Exercise 1 – Question 3
𝑦=4𝑥−2

13. Exercise 1 – Question 4 ? ? ?
Click to Reveal

14. Exercise 1 – Question 5 𝟑,−𝟐 𝟏,𝟐 𝟐, 𝟏 𝟐 −𝟏,𝟐 ? ? ? ? ? ? ? ?   
Put a tick or cross to determine whether each of the following points are on the line with the given equation.
𝒚=𝟏−𝒙
𝒙+𝟐𝒚=𝟑
𝟑,−𝟐    
𝟏,𝟐
𝟐, 𝟏 𝟐 
−𝟏,𝟐 ? ? ? ? ? ? ? ?

15. Exercise 1 – Question 6 3,11  7,−2  −3,10  3 4 , 4 5 ? ? ? ?
For the given equation of a line and point, indicate whether the point is above the line, on the line or below the line. (Hint: Find out what 𝑦 is on the line for the given 𝑥)
Below the line
On the line
Above the line
𝑦=3𝑥+4
3,11
  
𝑥+𝑦=5
7,−2  
𝑦=3−2𝑥
−3,10  
2𝑥+3𝑦=4
3 4 , 4 5 ? ? ? ?

16. Exercise 1 – Question N1
The equation of a line is 𝑎𝑥+𝑏𝑦=𝑐. If the 𝑥 value of some point on the line is 𝑑, what is the full coordinate of the point, in terms of 𝑎, 𝑏, 𝑐, 𝑑?
If 𝒙=𝒅, then 𝒂𝒅+𝒃𝒚=𝒄. Rearranging, 𝒚= 𝒄−𝒂𝒅 𝒃 .
So coordinate is 𝒅, 𝒄−𝒂𝒅 𝒃 ?

17. Exercise 1 – Question N2
What is the area of the region enclosed between the line with equation 2𝑥+7𝑦=3, the 𝑥 axis, and the 𝑦 axis?
We can set 𝒙=𝟎 to find where the lines cuts the 𝒚 axis:
𝟎+𝟕𝒚=𝟑 𝒚= 𝟑 𝟕
Similarly when 𝒚=𝟎:
𝟐𝒙+𝟎=𝟑 𝒙= 𝟑 𝟐
We have a triangle between the points 𝟎,𝟎 , 𝟎, 𝟑 𝟕 , 𝟑 𝟐 ,𝟎 .
Area is 𝟏 𝟐 × 𝟑 𝟕 × 𝟑 𝟐 = 𝟗 𝟐𝟖 . ?

18. Intercepts with the axis
Part 1b
Intercepts with the axis

19. Intercepts
We want to find the coordinates of the points where the line ‘intercepts’ the axes.
What do we know about any point on the 𝑦-axis?
How then can we work out the coordinate of the 𝑦-intercept?
𝒙=𝟎
So 𝒚=𝟐 𝟎 +𝟔=𝟔
Point is 𝟎,𝟔 𝑦
𝑦=2𝑥+6 ?
What do we know about any point on the 𝑥-axis?
How then can we work out the coordinate of the 𝑥-intercept?
𝒚=𝟎
So 𝟎=𝟐𝒙+𝟔 Point is −𝟑,𝟎 𝑥 ?

20. One more example ? ? ? Determine where the line 𝑥+2𝑦=3 crosses the:
𝑦-axis: Let 𝒙=𝟎 𝟐𝒚=𝟑 → 𝒚= 𝟑 𝟐 𝟎, 𝟑 𝟐
𝑥-axis: Let 𝒚=𝟎 𝒙+𝟎=𝟑 𝟑,𝟎 ? ?
What mistakes do you think it’s easy to make?
Mixing up x/y: Putting answer as (𝟎,𝟑) rather than (𝟑,𝟎).
Setting 𝒚=𝟎 to find the 𝒚-intercept, or 𝒙=𝟎 to find the 𝒙-intercept. ?

21. Test Your Understanding
Copy and complete this table.
The point where the line crosses the:
Equation
𝒚-axis
𝒙-axis
𝑦=3𝑥+1
0,1
− 1 3 ,0
𝑦=4𝑥−2
0,−2
1 2 ,0
𝑦= 1 2 𝑥−1
0,−1
2,0
2𝑥+3𝑦=4
0, 4 3
2, 0 ? ? ? ? ? ? ? ?

22. Part 2
Gradient

23. y
Sketch 𝑦=2𝑥−1 4 3 2 1 -1 -2 -3 -4 x
Do you notice any connection between how 𝑦 increases each time and the equation? 𝒙 -1 1 2 𝑦 -3 3 ? ? ? ?

24. y
Sketch 𝑦=−1𝑥+2 4 3 2 1 -1 -2 -3 -4 x
Do you notice any connection between how 𝑦 increases each time and the equation? 𝒙 -1 1 2 𝑦 3 ? ? ? ?

25. y
Sketch 𝑦= 1 2 𝑥+1 4 3 2 1 -1 -2 -3 -4 x
Do you notice any connection between how 𝑦 increases each time and the equation? 𝒙 -1 1 2 𝑦
0.5
1.5 ? ? ? ?

26. The equation of a straight line is of the form:!
The steepness of a line is known as the gradient.
It tells us what 𝑦 changes by as 𝑥 increases by 1. ?
The equation of a straight line is of the form:
𝒚=𝒎𝒙+𝒄
The gradient is 𝑚.
𝑐 is the ‘y-intercept’.
Gradient 1

27. On your printed sheet, identify the gradient of each line.4 D A 3 F C 2 B 1 E G x -1 H -2 -3
On your printed sheet, identify the gradient of each line. -4

28. y 4 D A 3 F C 2 B 1 E G x -1 H -2 -3 -4

29. y
Suppose we just had two points on the line and wanted to determine the gradient, but didn’t want to draw a grid.
𝟑, 𝟒 4 3
𝑦 has increased by 6. 2 1 x -1
−𝟏, −𝟐 -2
𝑥 has increased by 4. -3
So what does 𝑦 change by for each unit increase in 𝑥?
𝒎= 𝟔 𝟒 =𝟏.𝟓 ? -4

30. Gradient using two points!
Given two points on a line, the gradient is:
𝑚= 𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑦 𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑥
1, (3, 10)
𝑚=3 ?
𝑚=−2 ?
5, (8, 1)
𝑚=− 8 3 ?
2, (−1, 10)

31. Gradient using the Equation
We can get the gradient of a line using just its equation.
Rearrange into the form 𝒚=𝒎𝒙+𝒄, and then the gradient is 𝒎.
Examples
Test Your Understanding
𝑦+2𝑥=1 𝒚=−𝟐𝒙+𝟏 ∴𝒎=−𝟐
2𝑦=𝑥+1 𝒚= 𝟏 𝟐 𝒙+ 𝟏 𝟐 ∴𝒎= 𝟏 𝟐
𝑦=1+3𝑥 𝒎=𝟑
𝑥−𝑦=1 𝒚=𝒙−𝟏 ∴𝒎=𝟏
2𝑦+3𝑥=4 𝒚=− 𝟑 𝟐 𝒙+𝟐 ∴𝒎=− 𝟑 𝟐 ? ? ? ? ?

32. Exercise 2
Determine the gradient of the line which goes through the following points.
By rearranging the equations into the form 𝑦=𝑚𝑥+𝑐, determine the gradient of each line. 1 2
Point 1
Point 2
Gradient
(0,0)
2,2  𝟏
1,3
3,7  𝟐
0,5
4, 25  𝟓
−1,5
 −𝟏
4,3
10,6
  𝟏 𝟐
7,8
−4,−3
7,1
 − 𝟏 𝟐
6,5
8,1 −𝟐
5,10
𝟕 𝟒
−1,4
9,−5
− 𝟗 𝟏𝟎
1,0
−2,−4
𝟒 𝟑
1.5, −6.5
−1.75,4.3
−𝟑.𝟑𝟐𝟑 𝒕𝒐 𝟑𝒅𝒑
Equation
Gradient
𝑦=𝑥+1  𝟏
𝑦=2𝑥+3  𝟐
𝑦=−𝑥+2
 −𝟏
𝑦=1− 1 2 𝑥
 − 𝟏 𝟐
𝑦=2  𝟎
2𝑦=6𝑥−4  𝟑
4𝑦=5𝑥+1
 𝟏.𝟐𝟓
𝑥+𝑦=1
2𝑥+3𝑦=−4
 − 𝟐 𝟑
𝑥−3𝑦=4
𝟏 𝟑
𝑥+4𝑦=5
− 𝟏 𝟒 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?
Determine the gradient of the line with equation 𝑎𝑥+𝑏𝑦=1, in terms of the constants 𝑎 and 𝑏.
Rearranging: 𝒚=− 𝒂 𝒃 𝒙+ 𝟏 𝒃
So the gradient is − 𝒂 𝒃
Write an equation that ensures that three points 𝑥 1 , 𝑦 1 , 𝑥 2 , 𝑦 2 and 𝑥 3 , 𝑦 3 where 𝑥 1 < 𝑥 2 < 𝑥 3 , form a straight line (i.e. are “collinear”.
We just require that the gradient between points 1 and 2, and points 2 and 3 are the same, i.e.
𝒚 𝟐 − 𝒚 𝟏 𝒙 𝟐 − 𝒙 𝟏 = 𝒚 𝟑 − 𝒚 𝟐 𝒙 𝟑 − 𝒙 𝟐 N1 N2 ? ?

33. Summary
The gradient of a line is the steepness: how much 𝑦 changes as 𝑥 increases by 1.
We’ve seen 3 ways in which we can calculate the gradient:
a. Counting Squares
b. Using the equation
c. Using two points
𝑦=4− 3 2 𝑥
1, 4 , 4, 13
𝒎=− 𝟑 𝟐 ?
𝒎=−𝟑 ? ?
𝒎=𝟑

34. Part 3
𝑦=𝑚𝑥+𝑐

35. Recap D What was the gradient of these lines? A F C B E G H y 4 3 2 1x -1 H -2 -3 -4

36. y-intercept y 4 D A 3 F C 2 B 1 E G x -1 H -2
The y-intercept is the point at which the line crosses the 𝑦-axis.
It is the 𝑐 in 𝑦=𝑚𝑥+𝑐 (why?) -3 -4

37. Now determine the full equation of each line.y 4 D A 3 F C 2 B 1 E G x -1 H -2 -3
Now determine the full equation of each line. -4

38. Test Your Understanding
A line has the equation 𝑥+2𝑦=5. What is the 𝑦-intercept of the line?
𝟐𝒚=−𝒙+𝟓 𝒚=− 𝟏 𝟐 𝒙+ 𝟓 𝟐
So 𝒚-intercept is 𝟓 𝟐 . ?

39. Card Sort!

40. Exercise 3
Copy and complete the following table. 2
Gradient
𝒚-intercept
Equation
1 2 2
2𝑦=𝑥+4 −1 1
𝑥+𝑦=1
2 3
− 4 3
2𝑥−3𝑦=4
1 4
− 3 4
𝑦= 𝑥−3 4 4
−12
𝑦=4(𝑥−3) 1 ? ?
Gradient
𝒚-intercept
Equation 2 1
𝑦=2𝑥+1 4 −3
𝑦=4𝑥−3 −1
𝑦=𝑥−1
1 2
𝑦= 1 2 𝑥+1
𝑦=𝑥
𝑦=1 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?
A line has equation 2𝑥+3𝑦=𝑎. The area enclosed between this line, the 𝑥-axis and the 𝑦-axis is 1.
Determine 𝑎.
Intercepts are 𝒂 𝟐 and 𝒂 𝟑 .
𝟏 𝟐 × 𝒂 𝟐 × 𝒂 𝟑 =𝟏 → 𝒂 𝟐 𝟏𝟐 =𝟏 𝒂= 𝟏𝟐 3
The equation of a line is 3𝑦=𝑥+𝑎. If the 𝑦-intercept is 6, what is 𝑎?
𝟏 𝟑 𝒂=𝟔 → 𝒂=𝟏𝟖
The equation of a line is 𝑥−2𝑦=𝑎. If the 𝑦-intercept is 8, what is 𝑎?
𝒚= 𝟏 𝟐 𝒙− 𝟏 𝟐 𝒂 → − 𝟏 𝟐 𝒂=𝟖 𝒂=−𝟏𝟔 N ? ? 4 ?

41. Part 4
Parallel lines

42. Puzzle
Preliminary Question: What will be the same about the equations of two lines if they are parallel?
They have the same gradient. ?
(This was in a Year 8 End of Year exam) 𝑳
The diagram shows three points 𝐴 −1,5 , 𝐵 (−2,1) and 𝐶 (0,5).
A line 𝐿 is parallel to 𝐴𝐵 and passes through 𝐶.
Find the equation of the line 𝐿.
𝐶 (0,5)
𝐴 (−1,5)
𝑦=−2𝑥+5 ?
𝐵 (2,−1)

43. Test Your Understanding
The diagram shows three points 𝐴 −1,5 , 𝐵 (−2,1) and 𝐶 (0,5).
A line 𝐿 is parallel to 𝐴𝐵 and passes through 𝐶.
Find the equation of the line 𝐿. 𝑳
𝐶 (0, 4)
𝐵 (4,3)
𝑦= 1 2 𝑥+4 ?
𝐴 (−6, −2)

44. Equation given a gradient and point
The gradient of a line is 3. It goes through the point (4, 10). What is the equation of the line? E1 ?
𝒚=𝟑𝒙−𝟐
Start with 𝒚=𝟑𝒙+𝒄 (where 𝒄 is to be determined)
Substituting: 𝟏𝟎= 𝟑×𝟒 +𝒄
Therefore 𝒄=−𝟐 E2
The gradient of a line is -2. It goes through the point (5, 10). What is the equation of the line? ?
𝒚=−𝟐𝒙+𝟐𝟎

45. Exercise 4
Give the equation of a line which is parallel to 𝑦=3𝑥+2.
𝒚=𝟑𝒙+𝒄 (where c can be any number)
Give the equation of a line which passes through 0,−2 and is parallel to another line which passes through the points 1,4 and 5,24
𝒚=𝟓𝒙−𝟐
Give the equation of a line which passes through the point (0, 6) and has the gradient -2.
𝒚=−𝟐𝒙+𝟔
Which line has the greater gradient, 4𝑥−5𝑦=1 or 5𝑥−4𝑦=1?
First line rearranges to 𝒚= 𝟒 𝟓 𝒙− 𝟏 𝟓 , second to 𝒚= 𝟓 𝟒 𝒙− 𝟏 𝟒
So second line has the greater gradient. 4
Gradient
Goes through
Equation a 3
(4,5)
𝑦=3𝑥−7 b 5
(2,3)
𝑦=5𝑥−7 c −1
2,5
𝑦=−𝑥+7 d
1 2
10,11
𝑦= 1 2 𝑥+6 e
3 2
6,6
𝑦= 3 2 𝑥−3 f
3 4
1 3 , 4 5
𝑦= 3 4 𝑥 1 ? ? ? 2 ? ? ? ? 3 ? ?
A and B are straight lines. Line A has equation 2𝑦=3𝑥+8. Line B goes through the points −1,2 and 2,8 . Do lines A and B intersect?
Line A: 𝒚= 𝟑 𝟐 𝒙+𝟒 so 𝒎= 𝟑 𝟐 .
Line B: 𝒎= 𝟔 𝟑 =𝟐.
The gradients are different so the lines are not parallel, and therefore intersect. 4 N ? ?

46. Equation given two points
A straight line goes through the points (3, 6) and (5, 12). Determine the full equation of the line.
Gradient: 3
Equation: 𝒚=𝟑𝒙−𝟑 ?
(5,12)
Choose one of the two points and then use the previous method we saw when we have a gradient and point. ?
(3,6)
A straight line goes through the points (5, -2) and (1, 0). Determine the full equation of the line.
(5, -2)
Gradient: -0.5
Equation: 𝒚=− 𝟏 𝟐 𝒙+ 𝟏 𝟐 ?
(1,0) ?

47. Test Your Understanding
A line passes through the points (4,7) and 8,15 . Find the equation of the line. ?
𝑚= 8 4 =2 𝑦=2𝑥+𝑐 Using the point 4,7 :
7= 2×4 +𝑐 7=8+𝑐 𝑐=−1 𝑦=2𝑥−1
If you finish: A line passes through the points 6,9 and 12,6 . Give the coordinate of the point this line crosses the 𝑥-axis. ?
𝑚=− 3 6 =− 1 2 𝑦=− 1 2 𝑥+12
If 𝑦=0:
0=− 1 2 𝑥 → 𝑥=24 → ,0

48. Exercise 5
Work out the gradient given the points on the line.
Point 1
Point 2
Full Equation Q1
(0,0)
(2,2) ?
𝑦=𝑥 Q2
(-5,0)
(0,-5) ?
𝑦=−𝑥−5 Q3
(1,-3)
(3,1) ?
𝑦=2𝑥−5 Q4
(-4,1)
(4, 5) ?
𝑦=0.5𝑥+3 Q5
(-3,7)
(2,2) ?
𝑦=−𝑥+4 Q6
(1,6)
(3,-2)
𝑦=−4𝑥+10 ? Q7
(-7,3)
(5,-1) ?
𝑦=− 1 3 𝑥+ 2 3 Q8
(4,9)
(-3,10) ?
𝑦=− 1 7 𝑥+ 67 7 9
A line goes through the point 1,6 and (2,4).
Find the equation of the line. 𝒚=−𝟐𝒙+𝟖
Hence find the point at which this line intercepts the 𝑥−axis. (𝟒,𝟎) N
A line goes through the points (𝑎,𝑏) and (0,𝑐). Determine the coordinate of the point the line crosses the 𝑥-axis, in terms of 𝑎, 𝑏, 𝑐.
𝒎= 𝒄−𝒃 𝒂 → 𝒚= 𝒄−𝒃 𝒂 𝒙+𝒄
𝟎= 𝒄−𝒃 𝒂 𝒙+𝒄 → 𝒙,𝟎 𝒘𝒉𝒆𝒓𝒆 𝒙=− 𝒂𝒄 𝒄−𝒃 𝒐𝒓 𝒂𝒄 𝒃−𝒄 ? ? ?

49. REVISION
Vote with your diaries! 𝐴 𝐵 𝐶 𝐷

50. The equation of a line is 𝒚=− 𝟏 𝟐 𝒙+𝟑
The equation of a line is 𝒚=− 𝟏 𝟐 𝒙+𝟑. What is the missing value of this point on the line?
𝟒 , ? −1 1 5 3

51. y 4 3 2 1 x -1 -2
𝑦= 1 2 𝑥+1
𝑦=1𝑥−2
𝑦=−2𝑥+1
𝑦=− 1 2 𝑥+1 -3 -4

52. 𝑥=3 𝑦=3 𝑦=3𝑥 𝑦=𝑥+3 y 4 3 2 1 x -5 -4 -3 -2 -1 0 1 2 3 4 5 6 -1 -2 -3

53. 𝑦= 𝑒 𝜋𝑖 𝑦=−1 𝑦=−𝑥 𝑦=𝑥 y 4 3 2 1 x -5 -4 -3 -2 -1 0 1 2 3 4 5 6 -1 -2

54. What is the equation of a line parallel to 𝒚=𝟑𝒙+𝟐 and goes through the point (𝟎,−𝟑)?
𝑦=3𝑥−3
𝑦=−3𝑥+2
𝑦=2𝑥−3
𝑦=−3

55. What is the gradient of the line which goes through the points (𝟐,𝟓) and 𝟒,𝟏 ?2 −2
1 2
− 1 2

56. What is the full equation of a line which has gradient 3 and passes through the point (2,5)?
𝑦=3𝑥+5
𝑦=3𝑥−1
𝑦=2𝑥+5
𝑦=5𝑥−6

57. What is the full equation of the line which goes through the point 𝟐,𝟕 , 𝟔,𝟗 ?
𝑦= 1 2 𝑥+7
𝑦=2𝑥−3
𝑦=2𝑥+3
𝑦= 1 2 𝑥+6

58. What is the y-intercept of the line 𝟑𝒚+𝒙=𝟏?1
1 3 3
− 1 3

59. What is the gradient of the line 𝒚=𝟏−𝟑𝒙?−3 1
−3𝑥 3

60. Give the coordinate of the point where the line 𝒚=𝟐𝒙−𝟒 crosses the 𝒙−axis.
(2,0)
(−2,0)
0,2
0,−2

61. Give the coordinate of the point where the line 𝒚=𝟐𝒙−𝟒 crosses the 𝒚−axis.
(4,0)
(−4,0)
0,4
0,−4