1. Solving simultaneous linear equations on the problems of
linear relative motion
Speed Formula:
Distance = Speed × Time

2. 72 x 2 = 144 km They meet in two hours 48 x 2 = 96 km A B
e.g.1 ) Two cars A and B are at a certain distance apart. The speed of car A is 72 km/h while the speed of car B is 48 km/h. If they start at the same time and they travel towards each other, they will meet in two hours. Find the distance between them.
They meet in two hours
72 x 2 = 144 km
48 x 2 = 96 km A B
The distance between them : 144 km + 96 km = 240 km

3. e. g. 2) May and Bobby are at a certain distance apart
e.g.2) May and Bobby are at a certain distance apart. The walking speed of May is 3km/h and that of Bobby is 7 km/h. If they walk in the same direction, Bobby will catch up with May in 5 hours.
Find the distance between them.
7 x 5 = 35 km
3 x 5 = 15 km
Bobby
May
The distance between them :
35 km - 15 km
= 20 km

4. Learn how to set up equations
to solve the problems

5. e.g.3 ) A and B are 42 km apart. If they walk towards each other,
they will meet after 3 hours. Set up an equation with two
unknown speeds.
Let x be A’s speed and y be B’s speed
km/h
They meet after 3 hours
: x km
: y km A B
42 km
After 1 hour, how far will A walk ?
x km
After 3 hours, how far will A walk ?
3x km
After 1 hour, how far will B walk ?
y km
After 3 hours, how far will B walk ?
3y km
3x + 3y = 42
How to equate the distances ?

6. e.g. 4) A and B are 22 km apart. If they walk in the same direction, A will
catch up with B after 9 hours. Set up an equation with two unknown speeds.
Let x km/h be A’s speed and y km/h be B’s speed
: x km
: y km
A will catch up with B after 9 hours
9x km
9y km A B
22 km
How far will A walk after 1 hour ?
x km
How far will A walk after 9 hours ?
9x km
How far will B walk after 1 hour ?
y km
How far will B walk after 9 hours ?
9y km
How to equate the distances ?
9x – 9y = 22
or 9x = y
Do worksheet : No.1-2

7. 1) Two trains M and N are 250 km apart
1) Two trains M and N are 250 km apart. If they start at the same time and they
travel towards each other, they will meet after 50 minutes. Set up an equation
with two unknown speeds.
Let x km/min be the speed of train M and y km/min be the
speed of train N.
After 50 minutes
50x km
50y km
Train M
Train N
250 km
Let x km/h be the speed of train M and y km/h be the
speed of train N.
50x + 50y = 250

8. 2) Jacky and Amy are 60 km apart. Jacky takes a minibus
2) Jacky and Amy are 60 km apart. Jacky takes a minibus. Amy travels by her car
in the same direction as the minibus and overtakes it after 7 hours. Set up an
equation with two unknown speeds.
Let x km/h be the speed of the minibus and y km/h be the
speed of Amy’s car.
After 7 hours
7y km
60 km
7x km
Amy’s car
minibus
7y – 7x = or 7y = x
Do worksheet : No. 3,4

9. 3) Tommy and Martin ride bicycles on the same road at constant speeds and they
are a certain distance apart. The speed of Martin’s bicycle is 15 km/h. If they
travel in the same direction, Tommy’s bicycle will catch up with Martin’s bicycle
in 8 hours.
a) Draw a diagram to show the situation.
b) Set up an equation with the unknown distance apart and the unknown speed of
Tommy’s bicycle.
Let x km be the distance apart and y km/h be the speed of Tommy’s
bicycle.
After 8 hours
8y km
= 120 km
x km
Tommy’s bicycle
Martin’s bicycle
8y – 120 = x or 8y = x + 120

10. 4) A car and a bicycle are 72 km apart
4) A car and a bicycle are 72 km apart. The speed of the bicycle is 12 km/h.
If they travel towards each other, they will meet after some time.
a) Draw a diagram to show the situation.
b) Set up an equation with the unknown time and the unknown speed of the car.
Let x hours be the time and y km/h be the speed of the car.
They meet after x hours
12x km
xy km
car
bicycle
72 km
xy + 12x = 72

11. e. g. 5) Two cars P and Q are 480 km apart
e.g.5) Two cars P and Q are 480 km apart. If they start at the same time and
travel towards each other, they will meet in three hours. If they travel
in the same direction, car Q will overtake car P in eight hours.
Find the speeds of cars P and Q.  
Let x km/h be the speed of car P and y km/h be the speed of car Q.
3x + 3y = 480
3x km
3y km P Q
480 km
8y km
8x km P Q
480 km
8y – 8x = or y = 8x + 480

12. Solve the simultaneous linear equations:
…(1)
Do worksheet : No. 5
…(2)
Substitute into (1),
The speed of car P is 50 km/h and the speed of car Q is 110 km/h.

13. 5) Teddy and Ann are a certain distance apart
5) Teddy and Ann are a certain distance apart. They ride bicycles at uniform speeds.
The speed of Teddy’s bicycle is 18 km/h. If they ride towards each other, they will
meet in 2 hours. If they ride in the same direction, Teddy will overtake Ann in 10
hours. Find the speed of Ann’s bicycle and the original distance apart.
( Set up two simultaneous linear equations.)  
Let x km/h be the speed of Ann’s bicycle and y km be the
original distance apart.
36 km
2x km
36 + 2x = y
Teddy’s bicycle
Ann’s bicycle
y km
180 – 10x = y
180 km
y km
10x km
Teddy’s bicycle
Ann’s bicycle

14. Solve the simultaneous linear equations:
…(1)
…(2)
Substitute (1) into (2), Substitute x = 12 into (1),
y + 10x = x = y
36 + 2x + 10x = = y
12x = 180 – y = 60
12x = 144
x = 12
The speed of Ann’s bicycle is 12 km/h and the original distance
is 60 km.

15. Solving simultaneous linear equations on the problems of
circular relative motion

16. e.g.6) Cat A and cat B are running around a 640m circular track.
Cat A runs faster. If they start together ( at the same time
and position ) and they go in opposite directions, they will
meet in 35 seconds later.
Let x m/s be cat A’s speed and y m/s be cat B’s speed .
Can you draw the paths run by cats A and B ? A B
35x m
How far does cat A run
in terms of x?
35y m
How to equate the distances ?
35 seconds later
35x + 35y = 640

17. e. g. 7) Cat A and cat B are running around a 640m circular track
e.g.7) Cat A and cat B are running around a 640m circular track. Cat A runs
faster. If they start together ( at the same time and position ) and they go in the
same direction, cat A will catch up with cat B in 1 minute and 15 seconds later.
Let x m/s be dog A’s speed and y m/s be dog B’s speed .
How far does A run in terms of x? A B
75x m
How to equate the distances ?
75x – 75y = 640
75y m
1 minute and
15 seconds later
or 75x = 75y + 640
Do worksheet : No.6,7

18. 6) Sammy and Judy are practicing on a 600m circular track
6) Sammy and Judy are practicing on a 600m circular track. Sammy runs faster than
Judy.If they start together ( at the same time and position ) and they go in opposite
directions, they will meet 40 seconds later.
Let x m/s be Sammy’s speed and y m/s be Judy’s speed .
Set up an equation with x and y.
40x + 40y = 600
Sammy
Judy
40y km
40x km
After 40 seconds

19. 7) In the sports day, Kenneth and Sally join the 1500m running race and run on a
400m circular track. If they start together, Kenneth will overtake Sally 5 minutes
later.
Let x m/min be Kenneth’s speed and y m/min be Sally’s speed .
Set up an equation with x and y.
Kenneth
Judy
5x –5y = 400
5x m
or 5x = 5y + 400
5y m
5 minutes later

20. e.g.8) Susan and Peter are running on a 900m circular track outside the
playground. Peter runs faster than Susan. If they start together
and run in the same direction, Peter will catch up with Susan
6 minutes later. If they go in opposite directions, they will meet
1.2 minutes later. Find their speeds.
Let x m/min be Susan’s speed and y m/min be Peter’s speed .
Peter
Susan
Peter
Susan
1.2x m
6x m
6y m
1.2 minutes later
6 minutes later
1.2y m
6y – 6x = or 6y = 6x + 900
1.2x + 1.2y = 900

21. … (1)
… (2)
Do worksheet : No.8
Substitute into (2),
Susan’s speed is 300 m/min and Peter’s speed is 450 m/min.

22. 8) James and Ken are jogging round a circular park. Ken jogs faster
8) James and Ken are jogging round a circular park. Ken jogs faster. If they
start together and jog in opposite directions, they will meet 50 seconds later.
If they go in the same direction, Ken will overtake James 2.5 minutes later.
If James’ jogging speed is 3m/s, find the jogging speed of Ken and the length
of the circular park.
Let x m/s be Ken’s speed and y m be the length of the circular park.
Ken
James
Ken
James
150x m
50x m
= 150m
2.5 minutes later
50 seconds later
= 450m
50x = y
150x = y or 150x – 450 = y

23. …(1)
…(2)
Substitute (1) into (2),
Substitute into (1),
Ken’s speed is 6 m/s and the length of the circular park is 450m.

24. There are two people running on a circular track.
Harder Problem:
There are two people running on a circular track.
Write an equation to relate the distances travelled by the two persons
for the nth catch-up on the circular track.
Let x m be the distance travelled by the faster one,
y m be the distance travelled by the slower one
and z m be the circular track length.

25. Four Types of Relative Motion
What is the critical feature in setting up equations
to solve these relative motion problems?