NOTE MATH FORM 3 - 11 linear equations

Thursday, April 13, 2023

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LINEAR EQUATIONS

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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 #


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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


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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


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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


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Exercise Find the value of x

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


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Exercise Find the value of y

;34

3yx11. x = 4

12. x = 5

13. x = -3

;63

1

5

2 yx

;64

3yx


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Exercise Find the value of y

;2432 yxyx 14. x = 4

15. x = -10;4 5

4yx


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;13

2xy16. y = 3

17. y = 6;3

210

5

3yx


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;3824 yx 18. y = -2

19. y = -3

20. y = -8

;01028 yx

;124

3

3

2 yx


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SIMULTANEOUS LINEAR EQUATIONS

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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 #


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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


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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


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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


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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


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(SUBTRACTION)

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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#


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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


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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


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


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(MULTIPLICATION PART 1)

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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


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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


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(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


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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


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


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(MULTIPLICATION PART2)

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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


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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


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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

+


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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


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SIMULTANEOUS LINEAR EQUATIONS (PROBLEM SOLVING)

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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


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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.


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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.


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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.


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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.


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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.


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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


(SUBSTITUTION)

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- 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


+ 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


(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)


(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


+ 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


- 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


- 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


Exercise1. Use substitution method to solve the

following simultaneous equations.

NOTE MATH FORM 3 - 11 linear equations

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