1. Module 3: Constructing and interpreting linear graphs
Chapter 16

2. The gradient of a straight line
We already know from the core unit that the equation of a straight line can be stated by finding the gradient and the vertical intercept.
The gradient measures the slope of the line.
The value of the gradient is the ratio between the rise and the run.
The rise is the change in the vertical values between any two points.
The run is the change in the horizontal values between the same two points.

3. Example rise = 8 run = 4 rise 8 Gradient =  =  = 2 run 4 y x 8 4 48 6 4 2 2 4 6 8
run = 4 4 4
rise 8
Gradient =  =  = 2
run 4

4. Formula to calculate the gradient from two pointsy
(x2, y2) 
y2  y1
rise = y2  y1 x
(x1, y1) 
run = x2  x1
x2  x1
rise
y2  y1
Gradient =  =

x2  x1
run

5. Example (4, 4)  e.g., (x1, y1) = (0, 4) and (x2, y2) = (4, 4)
To calculate the gradient of the line, choose any two points on the line. y
(4, 4)  4
4  (4)
e.g., (x1, y1) = (0, 4) x
and (x2, y2) = (4, 4) 8 6 4 2 2 4 6 8
y2  y1
Gradient =

x2  x1
(0, 4)  4
4  (4)
4  0 =

4  0 8 =  = 2 4

6. Example e.g., (x1, y1) = (0, 4) and (x2, y2) = (4, 4) y2  y1
To calculate the gradient of the line, choose any two points on the line. y
 (0, 4) 4 8
e.g., (x1, y1) = (0, 4) x
and (x2, y2) = (4, 4) 8 6 4 2 2 4 6 8
y2  y1
Gradient =

x2  x1
 (4, 4) 4
4  4 4 =

4  0  8 =  = 2 4

7. Questions
Exercise 16A
Pages 402 – 403
All

8. Positive and negative gradientsy y
(x2, y2)
(x2, y2) x x
(x1, y1)
(x1, y1)
Positive gradient:
Negative gradient:
as x values increase
as x values increase
y values increase
y values decrease

9. The general equation of a straight line
The general equation of a straight line is y = mx + c, where m is the gradient of the line and c is equal to the y-axis intercept.
This form, expressing the relation in terms of y, is called the gradient form.

10. Example Find the gradient and y-axis intercept of the graph of
y = 3x − 4.
Gradient = 3 Y intercept = -4
y = -2x + 4.
y = 6x − 5.
2y = 8x + 6

11. Sketching the equation of a straight line
If we are given the rule of a straight line, we can sketch the graph using the gradient and the y-axis intercept.
Mark in the y intercept first
Draw the gradient from this point
Sketch the graph of y = 3x +1
Sketch the graph of 3y + 6x = 9
Sketch the graph of 3x – y = 6

12. Parallel Lines
Lines that have the same gradient will be parallel.

13. Questions
Exercise 16B
Pages 405
All

14. Finding the equation of a straight line given the gradient and the y intercept
y = mx + c
A line cuts the y axis at 3 and has a gradient of -2. What is its equation.
y = -2x + 3

15. Finding the equation of a straight line given the gradient and a point.
Find the equation of the line that passes through the point ( 3, 2 ) and has a gradient of -4.
y = mx + c
2 = -4 * 3 + c
2 = c
c = 14
y = -4x + 14

16. Finding the equation of a straight line given two points
Find the equation of the straight line passing through the points (1, -2) and (3, 2)
First, find the gradient
rise
y2  y1
Gradient =  =

run
x2  x1
2 - -2
Gradient =

3 - 1 4
Gradient =
 2
Gradient = 2

17. Finding the equation of a straight line given two points
Then use the gradient and one of the points to find the equation
m = 2 and ( 3 , 2 )
y = mx + c
2 = 2 * 3 + c
2 = 6 + c
c = -4
y = 2x – 4

18. Finding the equation of a straight line given the graph

26. Graphs of vertical and horizontal lines
All horizontal lines will have an equation of the form y = c where c is the vertical intercept.
All vertical lines will have an equation of the form x = k where k is the horizontal intercept

31. Questions
Exercise 16C
Page
All

32. The intercept form of a linear equation
Not all equations will be given to you the form y = mx + c
An alternative notation is, ax + by = c where a, b & c are constants.
The best way to sketch an equation when it is given in this form is to find the x and y intercepts.

33. Example Sketch the equation of 3x – 2y = 6 3x – 2 * 0 = 6 3x = 6 x = 2
To find the x intercept, let y = 0
3x – 2 * 0 = 6
3x = 6
x = 2
To find the y intercept, let x = 0
3 * 0 – 2y = 6
- 2y = 6
y = -3

35. Exercises
Exercise 16D
Page 410
Question 1

36. Linear Models

37. Example Construct a rule for the Cost of hiring the taxi.
C = k
How much will a journey of 10km cost?
$19.20
How far can you travel with $30?
16.75km
Sketch the graph of the relationship between cost and the number of kilometres travelled for journeys from 0 to 20 kilometres.
What does the vertical intercept represent?
What does the gradient represent?

38. Exercise
Exercise 16E
Pages 411 – 412
All

39. Solving Simultaneous Equations by the graphical method
Simultaneous equations involves finding the point of intersection of two linear equations at the same time
Solve the following two equations simultaneously
y = 3x – 4
y = -2x + 1

40. Solving Simultaneous Equations by the graphical method

41. Solving Simultaneous Equations by the graphical method

42. Solving Simultaneous Equations by the graphical method

43. Solving Simultaneous Equations by the graphical method

44. Solving Simultaneous Equations by the graphical method
Solve graphically the simultaneous equations x – y = 5 and x = 2

45. Solving Simultaneous Equations using the calculator
Solve(y = 3x - 4 and y = -2x + 1 ,{x,y})

46. Using a Calculator to solve Simultaneous Equations
x + y = 3
2x – y = -9

47. Calculator
x + y = 3
2x – y = -9

48. Calculator
x + y = 3
2x – y = -9

49. Calculator
2x - 5y = 11
5x + 3y = 12

50. Questions
Exercise 16F Page 416, as many as you need.

51. Practical applications of simultaneous equations
The perimeter of a rectangle is 48 cm. If the length of the rectangle is three times the width, determine its dimensions.
Let w represent the width.
Let l represent the length
2l + 2w = 48
l = 3w
Use the substitution method to solve.

52. Practical applications of simultaneous equations
Two families went to the theatre.
The first family bought tickets for 3 adults and 5 children and paid $73.50
The second family bought tickets for 2 adults and 3 children and paid $46.50
Let a represent the cost of an adult ticket
Let c represent the cost of a child ticket.
Express these statements as equations.
3 a + 5 c = 73.50
2 a + 3 c = 46.50
Calculator

53. Questions
Exercise 16G Page all

54. Break Even Analysis You decide to go into business making calculators.
Each week you find that there are always fixed costs that are independent of the cost of making the actual calculator i.e. wages, rent, insurance, utilities, etc.
There are also ongoing costs to buy the parts required for calculator.

55. Break even analysis
You find that the fixed costs of producing the calculator is $1500 and the cost for the parts for each calculator is $30.
Represent this relationship as an equation where ‘C’ represents the total cost per week and ‘n’ represents the number of calculators made in that week.
C = n

56. Break even analysis
The calculators are sold which provides revenue for your business.
You decide to sell the calculators for $150.
Represent this relationship as an equation where ‘R’ represents the revenue and ‘n’ represents the number of calculators sold in that week.
R = 150n

57. Break even analysis
The break even point is when the revenue earned is equal to the costs incurred.
R = C
Before the break even point your business will be making a loss.
After the break even point your business will be making a profit

58. Break even analysis
The break even point can be found algebraically or graphically.

59. Algebraically Revenue = Cost 150n = 30n + 1500 120n = 1500 n = 12.5
The Revenue and Costs at the breakeven point will both be
$1875
The business will begin to make a profit when you sell
13 calculators.

60. Graphically

61. Graphically C = n

62. Graphically R = 150n

63. The Profit Function
The profit made is equal to the Revenue earned less the costs incurred.
Profit = Revenue – Costs
Profit = 150n – ( 30n )
Profit = 150n – 30n – 1500
Profit = 120n – 1500
Calculate the profit made when you sell 35 calculators.
Profit = 120 * 35 – 1500
Profit = $2700

64. Questions
Exercise 16H
Page 419
All