Mathematics - Mathematics Pre-Diploma Studies SL (MYP 5 Plus) - First Edition

COORDINATES AND LINES (Chapter 17) 353

5 Given the points A(1, 4), B(¡1, 0), C(6, 3) and D(t, ¡1), find t if:

a AB is parallel to CD b AC is parallel to DB

c AB is perpendicular to CD d AD is perpendicular to BC

L USING GRADIENTS

In real life gradients occur in many situations, and can be interpreted
in a variety of ways.

For example, the sign alongside would indicate to motor vehicle
drivers that there is an uphill climb ahead.

Consider the situation in the graph alongside where a distance (km)
motor vehicle travels at a constant speed for a distance
of 600 km in 8 hours. 600
Clearly, the gradient of the line = vertical step 400
200
horizontal step
24
= 600
8

= 75

distance 600 km km h¡1. 68
time 8 hours time (hours)

However, speed = = = 75

So, in a graph of distance against time, the gradient can be interpreted as the speed.

In the following exercise we will consider a number of problems where gradient can be
interpreted as a rate.

EXERCISE 17L

1 50 distance (m) time (s) The graph alongside indicates the distances
20 and corresponding times as Inge swims
25 freestyle over 50 metres.
10
a Find the gradient of the line.

b Interpret the gradient found in a.

c Is the speed of the swimmer constant or
variable? What evidence do you have for
your answer?

2 The graph alongside indicates the distances travelled distance (km) D
by a train. Determine: (6,¡630)
(5,¡560)
a the average speed for the whole trip C
b the average speed from
(1,¡65) B
i A to B ii B to C A (2,¡180)

c the time interval over which the speed was time (hours)
greatest.

354 COORDINATES AND LINES (Chapter 17)

3 wage ($) The graph alongside indicates the
wages paid to taxi drivers.

(12,¡256) a What does the intercept on the
vertical axis mean?
(7,¡166)
(4,¡112) b Find the gradient of the line.
What does this gradient mean?
40
5 10 hours c Determine the wage for working:
15
i 6 hours ii 15 hours.

d If no payment is made for not working, but the same payment shown in the graph
is made for 8 hours’ work, what is the new rate of pay?

4 The graphs alongside indicate the fuel consumption distance travelled (km)
and distance travelled at speeds of 60 km h¡1 (graph (29, 350)
A) and 90 km h¡1 (graph B). A (33, 320)
B
a Find the gradient of each line.
fuel consumption(litres)
b What do these slopes mean?

c If fuel costs $1:24 per litre, how much more
would it cost to travel 1000 km at 90 km h¡1
compared with 60 km h¡1?

5 The graph alongside indicates the courier charge for charge ($)
different distances travelled.
C(20,¡27)
a What does the value at A indicate?
b Find the gradients of the line segments AB and B(8,¡15)
3A
BC. What do these gradients indicate?
distance (km)
c If a straight line segment was drawn from A to
C, find its gradient. What would this gradient
mean?

REVIEW SET 17A

1 Plot the following points on the number plane:
A(1, 3) B(¡2, 0) C(¡2, ¡3) D(2, ¡1)

2 Find the distance between the following sets of points:

a P(4, 0) and Q(0, ¡3) b R(2, ¡5) and S(¡1, ¡3)

3 Find the coordinates of the midpoint of the line segment joining A(8, ¡3) and
B(2, 1).

4 Find the gradients of the lines in the following graphs: y x
ab 30

B 8

A

COORDINATES AND LINES (Chapter 17) 355

5 A company manufactures saws. The set-up costs for the plant and machinery are
$2000. The total cost $C to produce x saws is given by C = 2000 + 4x dollars.

a What are the independent and dependent variables?
b Make a table of values for the cost $C of producing x saws, where x = 0, 100,

200, 300, 400, and plot the graph of C against x.

c Is the relationship linear?
d What does the value of C when x = 0 represent?
e What will it cost to make 650 saws?

6 For the rule y = 3x + 2: x0 1 2 3 4

a copy and complete the table y

b plot the points on the table and draw a straight line through them.

7 Find the equation of the line with gradient ¡2 and y-intercept 3.

8 Find the gradient and y-intercept of the line x = 1 ¡ 3y.

9 Find the equation of the line with gradient 2 which passes through (¡3, 4).
3

10 Use axes intercepts to draw a sketch graph of 3x ¡ 2y = 6.

11 Find k if (¡3, ¡1) lies on the line 4x ¡ y = k.

12 Find the equation of the line with zero gradient that passes through (5, ¡4).

13 Find t given that the line joining A(3, 4) and B(1, t) is parallel to a line with

gradient 3 .
5

14 The graph alongside shows the distance 120 distance (km) C
travelled by a train over a 2 hour journey 80
between two cities. B

a Find the average speed from:

i O to A ii A to B 40 time (h)
iii B to C A 23

b Compare your answers to a with the 0
gradients of the line segments: 01

i OA ii AB iii BC

c Find the average speed for the whole journey.

REVIEW SET 17B

1 On different sets of axes, show all points with:

a x-coordinates equal to ¡3 b y-coordinates equal to 5

c positive x-coordinates and negative y-coordinates

2 Find the distance between V(¡5, ¡3) and W(¡2, 6).

3 If M(1, ¡1) is the midpoint of AB, and A is (¡3, 2), find the coordinates of B.

356 COORDINATES AND LINES (Chapter 17)

4 Find the gradient of the line segment joining:

a (5, ¡1) and (¡2, 6) b (5, 0) and (5, ¡2)

5 Jacques sells vacuum cleaners. Each week he is paid a basic salary of E150 plus E25
for each vacuum cleaner that he sells.

a What are the independent and dependent variables?

b Construct a table and draw a graph of income I against vacuum cleaners sold v,
where v = 0, 1, 2, 3, ..... 8.

c Is the relationship linear?

d Is it sensible to join the points with a straight line?

e For each vacuum sold, what will be the increase in income?

f i What is the fixed income? ii What is the variable income?

g Find Jacques’ income in a week when he sells 5 vacuum cleaners.

6 From a table of values, plot the graph of the line with equation y = 1 x ¡ 1.
2

7 Find the equations of the following graphs:

ay b yc y

3 20 (25, 60)
-1 x

x x

8 Find the equation of the line with gradient 4 and y-intercept ¡2.

9 Find the gradient and y-intercept of the line with equation:

a y = 5x ¡ 7 b y = 6 ¡ 3 x c y = 10x
2

10 Find the gradient of the line with equation 4x + 3y = 5.

11 Find the equation of the line through (1, ¡5) with gradient 1 .
3

12 a Find the gradient of the line with equation y = 2x ¡ 3.

b Find the equation of the line perpendicular to y = 2x¡3 which passes through
(4, 1).

13 The graph alongside shows the amount 400 charge ($) C
charged by a plumber according to the
time he takes to do a job. 300

a What does the value at A indicate? B
200
b Find the gradients of the line
segments AB and BC. What do 100 hours worked (h)
these gradients indicate? A 24 6

c If a straight line segment was drawn 0
from A to C, what would be its 0
gradient? What would this gradient
mean?

18Chapter

Simultaneous linear
equations

Contents: A The point of intersection
of linear graphs

B Simultaneous equations

C Algebraic methods for
solving simultaneous
equations

D Problem solving

E Using a graphics
calculator to solve
simultaneous equations

358 SIMULTANEOUS LINEAR EQUATIONS (Chapter 18)

OPENING PROBLEM

A farmer has only hens and pigs in an
enclosure. He said to his daughter Susan,
“You know we have only hens and pigs. I
counted 48 heads altogether and 122 legs.
Can you tell me how many of each animal type we
have?”

It was too dark for Susan to go out and count them, but
she thought for a while and gave the correct answer.
What answer did she obtain, and how did she do it?

Doing the work in this chapter should make it easier for you to solve this and other similar
problems.

Let us consider the graphs of two straight lines which are not parallel. These lines will meet
somewhere. The point where they meet is called the point of intersection.

Notice that the point of intersection is the only point of
point common to both lines. intersection

If (a, b) is the point of intersection then
(a, b) satisfies the equations of both lines.

At the point of intersection we have the simultaneous solution of both equations, since this
point satisfies both equations at the same time.

Note that not all line pairs meet at one point. Two other situations can occur:

parallel lines coincident lines

no solution infinitely many solutions.

A THE POINT OF INTERSECTION

OF LINEAR GRAPHS

Example 1 Self Tutor

Find the point of intersection of the lines with equations y = 2x + 5 and
y = ¡x ¡ 1:

For y = 2x + 5:
when x = 0, y = 5

SIMULTANEOUS LINEAR EQUATIONS (Chapter 18) 359

when y = 0, 2x + 5 = 0 y
y = 2x + 5
) 2x = ¡5
5
) x = ¡ 5
2

) (0, 5) and (¡ 5 , 0) lie on the graph.
2

For y = ¡x ¡ 1:

when x = 0, y = ¡1 -2\_Qw 1
when y = 0, ¡x ¡ 1 = 0 -4 -2 x

) x = ¡1 4
-1

y = -x - 1

) (0, ¡1) and (¡1, 0) lie on the graph. -3

) the graphs meet at the point (¡2, 1).

EXERCISE 18A

1 Find the point of intersection of the following pairs of lines by drawing the graphs of
the two lines on the same set of axes:

a y=x b y=x¡4 c y = 2x + 3
y = 4¡x y = ¡3x y = x+3

d y = ¡3x + 1 e y = 5 ¡ 2x f y = ¡5x ¡ 3
y = ¡x ¡ 1 y = x¡1 y = ¡2x + 3

2 Find the simultaneous solution of the following pairs of equations using graphical methods:

a y=x¡2 b y=x+1 c y=5¡x
y = 3x + 6 y = 7¡x y = x+4

d y = ¡x ¡ 2 e y = 3x + 2 f y = 5x + 1
y = 6x ¡ 9 y = 2x + 3 y = 2x ¡ 5

INVESTIGATION 1 USING A COMPUTER TO FIND
WHERE GRAPHS MEET

Click on the icon to run the graphing package. GRAPHING
Type in y = 3x + 2 and click on PLOT . PACKAGE

Now type in y = 5 ¡ x and click on PLOT .

Now click on Intersect . Move the cursor near the point of intersection and click.

Your answer should be ¡ 3 , 4 1¢ : For a fresh start click on CLEAR ALL .
4
4

What to do:

1 Show that the point of intersection of y = 2x + 1 and y = 7 ¡ 5x is at about
(0:857, 2:714).

2 Plot the graphs of 2x+3y = 5 and 3x¡4y = 10. Find the point of intersection
of the two lines.