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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.