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

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.towards each other A B They meet in two hours The distance between them : 144 km + 96 km = 240 km 72 x 2 = 144 km 48 x 2 = 96 km

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.catch up a ) Find the distance between them. The distance between them :35 km - 15 km= 20 km 7 x 5 = 35 km 3 x 5 = 15 km BobbyMay b) How about if Bobby overtakes May in 3 hours ? The distance between them : 7 x 3 – 3 x 3 = 21 – 9 = 12 km

Learn how to set up equations to solve the problems

AB42 km They meet after 3 hours: x km : y km Let x be A’s speed and y be B’s speed After 1 hour, how far will A walk ? After 3 hours, how far will A walk ? How to equate the distances ? 3x + 3y = 42 x km y kmAfter 1 hour, how far will B walk ? 3x km After 3 hours, how far will B walk ?3y km 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. km/h

A B 22 km A will catch up with B after 9 hours : x km : y km Let x km/h be A’s speed and y km/h be B’s speed 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 9x km 9y km How to equate the distances ? or 9x = y9x – 9y = 22 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. Do worksheet : No.1-4

1) David and Mary are 16 km apart. If they walk towards each other, they will meet in 2 hours. Let x km/h be the speed of David and y km/h be the speed of Mary. Set up an equation with x and y. DavidMary 16 km They meet after 2 hours 2x + 2y = 16 : x km : y km

2) A man and a boy are 18 km apart. If they run in the same direction, the man will catch up with the boy in 4 hours. Let x km/h be the speed of the man and y km/h be the speed of the boy. Set up an equation with x and y. 4x km : y km : x km BoyMan 4y km 18 km 4x – 4y = 18 or 4x = y

3) 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. Train MTrain N 250 km After 50 minutes 50x km50y km 50x + 50y = 250 Let x km/h be the speed of train M and y km/h be the speed of train N.

4) 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. minibusAmy’s car 60 km After 7 hours 7y km 7x km 7y – 7x = 60 or 7y = x Do worksheet : No. 5,6

5) 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. Tommy’s bicycleMartin’s bicycle x km After 8 hours 8y km 8y – 120 = x or 8y = x = 120 km

6) 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. car bicycle 72 km xy km 12x km xy + 12x = 72 They meet after x hours

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. PQ480 km 3x km 3y km 3x + 3y = 480 PQ480 km 8y km 8x km 8y – 8x = 480 or 8y = 8x Let x km/h be the speed of car P and y km/h be the speed of car Q.  

Solve the simultaneous linear equations: …(1) …(2) The speed of car P is 50 km/h and the speed of car Q is 110 km/h. Substitute into (1), Do worksheet : No. 7

7) 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. Teddy’s bicycleAnn’s bicycle y km 36 km2x km x = y Teddy’s bicycleAnn’s bicycle y km 180 km 10x km 180 – 10x = y  

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

Solving simultaneous linear equations on the problems of circular relative motion

e.g.6) Cat A and cat B are running around a 640m circular park. Cat A runs faster. If they start together ( at the same time and position ) and they go in opposite directions, they willopposite directions meet in 35 seconds later. ABAB 20 seconds later 35x m 35y m 35x + 35y = 640 Let x m/s be cat A’s speed and y m/s be cat B’s speed. Do worksheet : No. 8

8) 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. Sammy Judy After 40 seconds 40y km40x km 40x + 40y = 600 Sketchpad

e.g.7) Dog A and dog B are running around a 640m circular park. Dog A runs faster. If they start together ( at the same time and position ) and they go in the same direction, dog A will overtake dog B in 1 minute and 15 seconds later.overtake ABAB 1 minute and 15 seconds later How far does dog A run in terms of x ? 75x m How far does dog B run in terms of y ? 75y m How to equate the distances ? 75x = 75y or 75x – 75y = 640 Do worksheet : No.9 Let x m/s be dog A’s speed and y m/s be dog B’s speed. ( Set up an equation with x and y.)

9) 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 5 minutes later 5x m 5y m 5x = 5y or 5x –5y = 400

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 6 minutes later 1.2 minutes later 6y m 6x m 1.2x m 1.2y m 6y – 6x = 900 or 6y = 6x x + 1.2y = 900

… (1) … (2) Substitute into (2), Susan’s speed is 300 m/min and Peter’s speed is 450 m/min. Do worksheet : No.10

10) 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. length Let x m/s be Ken’s speed and y m be the length of the circular park. Ken James Ken James 50 seconds later 50x m = 150m 50x = y 2.5 minutes later 150x m = 450m 150x = y or 150x – 450 = y

…(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.