Thursday, April 13, 2023

AIRIL AHMAD

LINEAR EQUATIONS

AIRIL AHMAD

Thursday, April 13, 2023

AIRIL AHMAD

Determine the value of a variable given the other variables

Example 1:Given 6m + n = 27, find the value of n, when m=3

Solution: 6 m + n = 27

6 + n = 27

(3)

18 + n = 27n = 27

- 18

n = 9 #

Thursday, April 13, 2023

AIRIL AHMAD

Example 2:

6

3m

Given 6m + n = 7, find the value of m, when n=4

Solution: 6 m + n = 7

6 m + = 7

(4)

6 m + 4 = 76m = 7

- 4

6m = 3 #

m= 36

1

2

2

1m

Thursday, April 13, 2023

AIRIL AHMAD

Exercise Find the value of y Find the value of x

1. 2x + 3y = 16 , x = 2

2. 4x + 2y =7 , x = 5

3. -3x + 2y = 9 , x = -3

4. -2x +3y = -3 , x = -1

5. 8x + 2y -10 =0 , x = 4

6. 3x + 2y = 15 , y = 3

7. 3x +4y = 8 , y = 2

8. 5x + 2y = 4 , y = -3

9. 2x - 3y = -3 , y = -1

10. 2x + 3y -10 =0 , y = 4

Thursday, April 13, 2023

AIRIL AHMAD

Find the value of x

6. 3x + 2y = 15 , y = 3

7. 3x +4y = 8 , y = 2

8. 5x + 2y = 4 , y = -3

9. 2x - 3y = -3 , y = -1

10. 2x + 3y -10 =0 , y = 4

Thursday, April 13, 2023

AIRIL AHMAD

Exercise Find the value of x

1. 3x + 2y = 15 , y = 31. 3x +4y = 8 , y = 2

Thursday, April 13, 2023

AIRIL AHMAD

Exercise Find the value of y

;34

3yx11. x = 4

12. x = 5

13. x = -3

;63

1

5

2 yx

;64

3yx

Thursday, April 13, 2023

AIRIL AHMAD

Exercise Find the value of y

;2432 yxyx 14. x = 4

15. x = -10;4 5

4yx

Thursday, April 13, 2023

AIRIL AHMAD

Exercise Find the value of x

;13

2xy16. y = 3

17. y = 6;3

210

5

3yx

Thursday, April 13, 2023

AIRIL AHMAD

Exercise Find the value of x

;3824 yx 18. y = -2

19. y = -3

20. y = -8

;01028 yx

;124

3

3

2 yx

Thursday, April 13, 2023

AIRIL AHMAD

SIMULTANEOUS LINEAR EQUATIONS

AIRIL AHMAD

Thursday, April 13, 2023

AIRIL AHMAD

Example 1:

a + b = 10 and a – b = 2

a + b = 10

a – b = 2

…(1)

…(2)(+)

2a

= 12

a = 12

2

a = 6

When a=6 , then:

a + b = 10 ….(1)

+ b = 10

6

b = 10 - 6

b = 4 #

Thursday, April 13, 2023

AIRIL AHMAD

Example 2:

3m - n = 7

m + n = 5

…(1)

…(2)(+)

4m

= 12

m = 12

4

m = 3

When m=3 , then:

m + n = 5 ….(2)

+ n = 5

3

n = 5 - 3

n = 2 #

3m - n = 7 and m +n = 5

Thursday, April 13, 2023

AIRIL AHMAD

Example 3:

2e +3f = 1

5e - 3f = 13

…(1)

…(2)(+)

7e

= 14

e = 14

7

e = 2

When e=2 , then:

2e + 3f = 1 ….(1)

2 + 3f = 1

(2)

3f = 1 - 4

f = -3

#

2e + 3f=1 and 5e–3f = 13

3

f = -1

Thursday, April 13, 2023

AIRIL AHMAD

Example 4:

2x + y = 4

2x – y = 16

…(1)

…(2)(+)

4x

= 20

x = 20

4

x = 5

When x=5 , then:

2x + y = 4 ….(1)

2 + y = 4

(5)

y = 4 - 10

y = -6#

2x + y = 4 and 2x - y = 16

Thursday, April 13, 2023

AIRIL AHMAD

Exercise

1. x + y =4 and x – y = 2

2. x + y = 8 and x – y = 4

3. 3x + y = 20 and x – y = 4

4. 5x – 4y =19 and 7x + 4y=17

Thursday, April 13, 2023

AIRIL AHMAD

SIMULTANEOUS LINEAR EQUATIONS

(SUBTRACTION)

AIRIL AHMAD

Thursday, April 13, 2023

AIRIL AHMAD

Example 1:

2a + b =10 and a + b =2

2a + b = 10

a + b = 2

…(1)

…(2)(-)

a

= 8

When a=8 , then:

a + b = 2 ….(2)

+ b = 2

8

b = 2 - 8

b = -6#

Thursday, April 13, 2023

AIRIL AHMAD

Example 2:

3m + n = 7

m + n = 5

…(1)

…(2)(-)

2m

= 2

m = 2

2

m = 1

When m=1 , then:

m + n = 5 ….(2)

+ n = 5

1

n = 5 - 1

n = 4 #

3m + n = 7 and m +n = 5

Thursday, April 13, 2023

AIRIL AHMAD

Exercise1. 5e + 3f = 13 and 2e + 3f = 1

2. 4x + y = 16 and 2x + y = 4

3. 2x + y = 10 and x + y = 7

4. 2x + y = 12 and x + y = 7

Thursday, April 13, 2023

AIRIL AHMAD

1. 5e + 3f = 13 and 2e + 3f = 1

Thursday, April 13, 2023

AIRIL AHMAD

SIMULTANEOUS LINEAR EQUATIONS

(MULTIPLICATION PART 1)

AIRIL AHMAD

Thursday, April 13, 2023

AIRIL AHMAD

Example 1:

a + b = 5 and 2a + 3b = 15

a + b = 5

2a + 3b = 15

…(1)

…(2)

(-)

a

= 0

When a=0 , then:

a + b = 5 ….(1)

+ b = 5

0

b = 5 - 0

b = 5#

x 3

3a + 3b = 15 ....(3)

2a + 3b = 15 …(2)

(-)(+)

x 3 x 3 x 3

Thursday, April 13, 2023

AIRIL AHMAD

Example 2:

a + 4b = 3 and 5a - 8b = 1

a + 4b = 3

5a - 8b = 1

…(1)

…(2)

(+)

7a

= 7

When a=1 , then:

a + 4b = 3 ….(1)

+ 4b = 3

1

4b = 3 - 1

4b = 2

#

x 2

2a + 8b = 6 ....(3)

5a - 8b = 1 …(2)

(-)(+)

x 2 x 2 x 2

a = 1

b =2

1

4

2

Thursday, April 13, 2023

AIRIL AHMAD

(3)Example 3:

2a + 5b = 16 and a - b = 1

2a + 5b = 16

a - b = 1

…(1)

…(2)

(+)

7a

= 21

When a=3 , then:

2a + 5b = 16 ….(1)

2 + 5b = 16

6 + 5b = 16

5b = 10

#

x 5

2a + 5b = 16 ....(1)

5a - 5b = 5 …(3)

(-)(+) x 5 x 5 x 5

a = 3

b = 25

10

5b = 16 - 6

Thursday, April 13, 2023

AIRIL AHMAD

Exercise1. 2e + f = -3 and 3e + 4f = -7

2. 4x + 3y = 3 and x + y = 0

3. x + y = 3 and 3x - 5y = 17

4. 3m - n = 3 and m + 3n = 11

Thursday, April 13, 2023

AIRIL AHMAD

1. x + y = 3 and 3x - 5y = 17

Thursday, April 13, 2023

AIRIL AHMAD

SIMULTANEOUS LINEAR EQUATIONS

(MULTIPLICATION PART2)

AIRIL AHMAD

Thursday, April 13, 2023

AIRIL AHMAD

Example 1:

3a + 2b = 9 and 2a + 3b = 11

3a + 2b = 9

2a + 3b = 11

…(1)

…(2)

(-)

5a

= 5

When a=1 , then:

3a + 2b = 9 ….(1)

3 + 2b = 9

(1)

2b = 9 - 3

b = 3#

x 3

9a + 6b = 27 ....(3)

4a + 6b = 22 …(4)

(-)(+)

x 3 x 3 x 3

x 2 x 2 x 2 x 2

a = 1

3 + 2b = 9

2b = 6

Thursday, April 13, 2023

AIRIL AHMAD

Thursday, April 13, 2023

AIRIL AHMAD

Example 2:

3a + 5b = 21 and 5a - 3b = 1

3a + 5b = 21

5a - 3b = 1

…(1)

…(2)

(+)

34a

= 68

When a=2 , then:

3a + 5b = 21 ….(1)

3 + 5b = 21

(2)

5b = 21 - 6

b = 3#

x 3

9a + 15b = 63 ....(3)

25a - 15b = 5 …(4)

(-)(+)

x 3 x 3 x 3

x 5 x 5 x 5 x 5

a = 2

6 + 5b = 21

5b = 15

Thursday, April 13, 2023

AIRIL AHMAD

Example 3:

2a - 3b = 4 and 7a - 4b = 1

2a - 3b = 4

7a - 4b = 1

…(1)

…(2)

(-)

-13a

= 13

When a = -1 , then:

2a - 3b = 4 ….(1)

2 - 3b = 4

(-1)

-3b = 4 + 2

b =

#

x 4

8a - 12b = 16 ...(3)

21a - 12b = 3 …(4)

(-)(+)

x 4 x 4 x 4

x 3 x 3 x 3 x 3

a = -1

-2 - 3b = 4

-3b = 6

3

6

b = -2

+

Thursday, April 13, 2023

AIRIL AHMAD

Exercise1. 2e + 3f = 7 and 5e - 4f = 6

2. 2x + 3y = 8 and 3x - 2y = -1

3. 3x - 2y = 1 and 5x - 3y = 3

4. 2m - 5n = 8 and 3m - 7n = 11

Thursday, April 13, 2023

AIRIL AHMAD

SIMULTANEOUS LINEAR EQUATIONS (PROBLEM SOLVING)

AIRIL AHMAD

Thursday, April 13, 2023

AIRIL AHMAD

Example 1: The sum of two numbers is 50.

If the difference is 12,

find the two numbers.

A + B = 50

A - B = 12

…(1)

…(2)(+)

2A = 62

A = 62

2A = 31

When A=31 , then:

A + B = 50 ….(1)

31

B = 50 - 31

B = 19 #

+ B = 50

Thursday, April 13, 2023

AIRIL AHMAD

Example 2: 2 apples and 3 lemons cost RM2.70. 2 apples and 1 lemon cost RM1.70.Find the cost of an apple and a lemon.

Thursday, April 13, 2023

AIRIL AHMAD

Example 3: 2 durians and 3 apples cost RM11.50. 1 durian and 2 apples cost RM6.00.Find the cost of a durian and apple.

Thursday, April 13, 2023

AIRIL AHMAD

Example 4: 3 pens and 2 pencils cost RM13.00. 2 pens and 2 pencils cost RM9.00.Find the cost of a pen and pencil.

Example 5: The cost of 6 benches and 4 chairs is

RM132 . While the cost of 5 benches

and 2 chairs is RM 78.Find the cost of

each chair and bench.

Example 6: In a ranch, there are goats and chickens. If there are 22 heads and 58 legs, find the numbers of goat and chicken.

Thursday, April 13, 2023

AIRIL AHMAD

Example 6:

Example 7:

In a ranch, there are goats and chickens. If there are 22 heads and 58 legs, find the numbers of goat and chicken.

In an aquarium, there are ducks and octopus. If there are 30 heads and 90 legs, find the numbers of duck and octopus.

Thursday, April 13, 2023

AIRIL AHMAD

Example 5: The cost of 6 benches and 14 chairs is

RM132 . While the cost of 5 benches

and 2 chairs is RM 78.Find the cost of

each chair and bench.

Thursday, April 13, 2023

AIRIL AHMAD

Example 6: A sport store supplies 24 netballs and 16 volley balls to one school for RM275.60 and delivers 12 netballs and 32 volley balls to another school for RM211. If delivery is free, how much did the supplier charge for each type of ball?

Thursday, April 13, 2023 AIRIL AHMAD

SIMULTANEOUS LINEAR EQUATIONS

(SUBSTITUTION)

AIRIL AHMAD

Thursday, April 13, 2023 AIRIL AHMAD

- 6

(2)Example 1:

a + 3b = 3 and b = 2

a + 3b = 3

b = 2

…(1)

…(2)

Substitute b=2 in …(1)

a + 3b = 3

a + 3 = 3

a + 6 = 3

a = 3

a = -3

Thursday, April 13, 2023 AIRIL AHMAD

+ 6

(-3)Example 2:

2a - 3b = 6 and a = -3

2a - 3b = 6

a = -3

…(1)

…(2)

Substitute a = -3 in …(1)

2a - 3b = 6

2 – 3b = 6

- 6 – 3b = 6

-3b = 6

-b = 12

b = -12

Thursday, April 13, 2023 AIRIL AHMAD

(2b)Example 3:

2a + b = 15 and a = 2b

2a + b = 15

a = 2b

…(1)

…(2)

Substitute a = 2b in …(1)

2a + b = 15

2 + b = 15

4b + b = 15

5b = 15

b = 15

b = 3

5

a = 2

a = 6

(3)

Thursday, April 13, 2023 AIRIL AHMAD

(4)

(3b)Example 4:

2a - 3b = 12 and a = 3b

2a - 3b = 12

a = 3b

…(1)

…(2)

Substitute a = 3b in …(1)

2a – 3b = 12

2 -3b = 12

6b - 3b = 12

3b = 12

b = 12

b = 4

3

a = 3

a = 12

Thursday, April 13, 2023 AIRIL AHMAD

+ b

(2 + b)Example 5:

…(1)

…(2)

Substitute a = 2 + b in …(1)

a + b = 10

+ b = 102 + b + b = 10

2 + 2b = 10

2b = 10 - 2

b = 82

a + b = 10 and a – b = 2

a + b = 10

a – b = 2

a – b = 2

By using 2nd equation …

a = 2

2b = 8

b = 4a = 2 + 4a = 6

Thursday, April 13, 2023 AIRIL AHMAD

- m

(5 - m)Example 6:

…(1)

…(2)

Substitute n = 5 - m in …(1)

3m - n = 7

3m - = 73m - 5 + m = 7

4m – 5 = 7

4m = 7 + 5

m = 12 4

m + n = 5

By using 2nd equation …

n = 5

4m = 12

m = 3

3m - n = 7 and m +n = 5

3m - n = 7

m + n = 5

n = 5 – 3 n = 2

Thursday, April 13, 2023 AIRIL AHMAD

- 2g

(4 – 2g)Example 7:

…(1)

…(2)

Substitute h = 4 – 2g in (2)

2g - h = 16

2g - = 162g - 4 + 2g = 16

4g – 4 = 16

4g = 16 + 4

g = 20 4

2g + h = 4

By using 1st equation …

h = 4 4g = 20

g = 5

2g + h = 4 & 2g – h =16

2g + h = 4

2g – h = 16

h = 4 – 10 h = - 6

Thursday, April 13, 2023 AIRIL AHMAD

Exercise1. Use substitution method to solve the

following simultaneous equations.