The document discusses solving simultaneous linear equations through addition, subtraction and multiplication methods. It provides examples of finding the value of one variable given the value of the other variable from two simultaneous linear equations. It also includes exercises for the reader to practice finding the value of variables using the different methods.

1. Friday, January 25, 2013
LINEAR EQUATIONS
AIRIL AHMAD
AIRIL AHMAD

2. Friday, January 25, 2013
Determine the value of a variable
given the other variables
Example 1:
Given 6m + n = 27, find the value of n, when m=3
(3)
Solution: 6 m + n = 27
6 + n = 27
- 18 + n = 27
n = 27
n=9#
AIRIL AHMAD

3. Friday, January 25, 2013
Example 2:
(4)
Given 6m + n = 7, find the value of m, when n=4
Solution: 6 m+n=7 1
3
m=
6m+ =7 62
6m+4 =7
- 1
6m = 7 m=
2
6
6m = 3 #
m= 3
AIRIL AHMAD

4. Friday, January 25, 2013
Exercise
Find the value of y Find the value of x
1. 2x + 3y = 16 , x = 2 6. 3x + 2y = 15 , y = 3
2. 4x + 2y =7 ,x=5 7. 3x +4y = 8 ,y=2
3. -3x + 2y = 9 , x = -3 8. 5x + 2y = 4 , y = -3
4. -2x +3y = -3 , x = -1 9. 2x - 3y = -3 , y = -1
5. 8x + 2y -10 =0 , x = 4 10. 2x + 3y -10 =0 , y = 4
AIRIL AHMAD

5. Friday, January 25, 2013
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
AIRIL AHMAD

6. Friday, January 25, 2013
Find the value of x
Exercise 1. 3x +4y = 8 ,y=2
1. 3x + 2y = 15 , y = 3
AIRIL AHMAD

7. Friday, January 25, 2013
Exercise
Find the value of y
3
11. x = y + 3; x=4
4
2 1
12. x + y = 6; x = 5
5 3
3
13. x = 6 y; x = -3
4
AIRIL AHMAD

8. Friday, January 25, 2013
Exercise
Find the value of y
14. 2 x − 3 y = 4 x + 2 y; x=4
4
15.
x = 4 y; x = -10
5
AIRIL AHMAD

9. Friday, January 25, 2013
Exercise
Find the value of x
2
16. y = x − 1; y=3
3
3 2
17. x = 10 − y; y=6
5 3
AIRIL AHMAD

10. Friday, January 25, 2013
Exercise
Find the value of x
18. 4 + 2 x = 8 − 3 y; y = -2
19. 8 x + 2 y − 10 = 0; y = -3
2 3
20. x − y = 12; y = -8
3 4
AIRIL AHMAD

11. Friday, January 25, 2013
SIMULTANEOUS
LINEAR EQUATIONS
AIRIL AHMAD
AIRIL AHMAD

12. Friday, January 25, 2013
Example 1:
a + b = 10 and a – b = 2 When a=6 , then:
6
a + b = 10 …(1) a + b = 10 ….(1)
(+) …(2) + b = 10
2a– b = 12
a =2 b = 10 - 6
b = 4 #
a = 12
2
a =6
AIRIL AHMAD

13. Friday, January 25, 2013
Example 2:
3m - n = 7 and m +n = When m=3 , then:
3
5
3m - n = 7 …(1) m+ n = 5 ….(2)
(+) …(2) + n = 5
4m + n = 12
m =5 n = 5-3
n = 2#
m = 12
4
m =3
AIRIL AHMAD

14. Friday, January 25, 2013
Example 3:
2e + 3f=1 and 5e–3f = 13 When e=2 , then:
(2)
2e +3f = 1 …(1) 2e + 3f = 1 ….(1)
2 + 3f = 1
(+) 5e - 3f = 13 …(2)
7e = 14 3f = 1 - 4
f = -3
e = 14
3
7
f = -1#
e =2
AIRIL AHMAD

15. Friday, January 25, 2013
Example 4:
2x + y = 4 and 2x - y = 16 When x=5 , then:
(5)
2x + y = 4 …(1) 2x +y = 4 ….(1)
2 +y = 4
(+) 2x – y = 16 …(2)
4x = 20 y = 4 - 10
y = -6
x = 20 #
4
x =5
AIRIL AHMAD

16. Friday, January 25, 2013
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
AIRIL AHMAD

17. Friday, January 25, 2013
SIMULTANEOUS
LINEAR EQUATIONS
(SUBTRACTION)
AIRIL AHMAD
AIRIL AHMAD

18. Friday, January 25, 2013
Example 1:
2a + b =10 and a + b =2 When a=8 , then:
8
2a + b = 10 …(1) a+ b = 2 ….(2)
+ b = 2
(-) a +b=2 …(2)
a =8 b = 2-8
b = -6#
AIRIL AHMAD

19. Friday, January 25, 2013
Example 2:
3m + n = 7 and m +n = 5 When m=1 , then:
1
3m + n = 7 …(1) m+ n = 5 ….(2)
+ n = 5
(-) m +n= 5 …(2)
2m = 2 n = 5-1
n = 4#
m = 2
2
m =1
AIRIL AHMAD

20. Friday, January 25, 2013
Exercise
1. 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
AIRIL AHMAD

21. Friday, January 25, 2013
1. 5e + 3f = 13 and 2e + 3f = 1
AIRIL AHMAD

22. Friday, January 25, 2013
SIMULTANEOUS LINEAR
EQUATIONS
(MULTIPLICATION PART 1)
AIRIL AHMAD
AIRIL AHMAD

23. Friday, January 25, 2013
Example 1:
a + b = 5 and 2a + 3b = 15 When a=0 , then:
0
x3 x3 x3
a +b = 5 …(1) x3 a+ b = 5 ….(1)
+ b = 5
(-)
(+) 2a + 3b = 15 …(2)
b = 5-0
3a + 3b = 15 ....(3)
(-) 2a + 3b = 15 …(2)
b = 5#
a = 0
AIRIL AHMAD

24. Friday, January 25, 2013
Example 2:
a + 4b = 3 and 5a - 8b = 1 1
When a=1 , then:
x2 x2 x2
a + 4b = 3 …(1) x 2 a + 4b = 3 ….(1)
+ 4b = 3
(-)
(+) 5a - 8b = 1 …(2)
4b = 3 - 1
2a + 8b = 6 ....(3)
(+) 5a - 8b = 1 …(2)
4b = 2
2 1
7a = 7 b = =
4 2
#
a = 1
AIRIL AHMAD

25. Friday, January 25, 2013
Example 3:
When a=3 , then:
(3)
2a + 5b = 16 and a - b = 1
2a + 5b = 16 ….(1)
2a + 5b = 16 …(1) 2 + 5b = 16
(-)
(+) x5 x5 x 5…(2) x5
6 + 5b = 16
a - b = 1
2a + 5b = 16 ....(1) 5b = 16 - 6
(+) 5a - 5b = 5 …(3) 5b = 10
10
7a = 21 b = =2
5 #
a = 3
AIRIL AHMAD

26. Friday, January 25, 2013
Exercise
1. 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
AIRIL AHMAD

27. Friday, January 25, 2013
1. x + y = 3 and 3x - 5y = 17
AIRIL AHMAD

28. Friday, January 25, 2013
SIMULTANEOUS
LINEAR EQUATIONS
(MULTIPLICATION PART2)
AIRIL AHMAD
AIRIL AHMAD

29. Friday, January 25, 2013
Example 1:
3a + 2b = 9 and 2a + 3b = 11 When a=1 , then:
(1)
x3 x3 x3
3a + 2b = 9 …(1) x 3 3a + 2b = 9 ….(1)
(+) 2a
x2 x2 x2 3 + 2b = 9
(-) + 3b = 11 …(2) x 2
3 + 2b = 9
9a + 6b = 27 ....(3)
(-) 4a + 6b = 22 …(4)
2b = 9 - 3
2b = 6
5a = 5 b = 3
#
a = 1
AIRIL AHMAD

30. Friday, January 25, 2013
AIRIL AHMAD

31. Friday, January 25, 2013
Example 2:
3a + 5b = 21 and 5a - 3b = 1 When a=2 , then:
(2)
x3 x3 x3
3a + 5b = 21 …(1) x 3 3a + 5b = 21 ….(1)
x5 x5 x5 3 + 5b = 21
(-)
(+) 5a - 3b = 1 …(2) x 5
6 + 5b = 21
9a + 15b = 63 ....(3)
(+) 25a - 15b = 5 …(4)
5b = 21 - 6
5b = 15
34a = 68 b = 3
#
a = 2
AIRIL AHMAD

32. Friday, January 25, 2013
Example 3: When a =(-1), then:
-1
2a - 3b = 4 and 7a - 4b = 1 2a - 3b = 4 ….(1)
x4 x4 x4 2 - 3b = 4
2a - 3b = 4 …(1) x 4
x3 x3 x3
-2 - 3b = 4
(-)
(+) 7a - 4b = 1 …(2) x 3
-3b = 4 + 2
8a - 12b = 16 ...(3) -3b = 6
(-)21a + 12b = 3
- …(4) 6
b =
−3
-13a = 13
b = -2 #
a = -1
AIRIL AHMAD

33. Friday, January 25, 2013
Exercise
1. 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
AIRIL AHMAD

34. Friday, January 25, 2013
SIMULTANEOUS
LINEAR EQUATIONS
(PROBLEM SOLVING)
AIRIL AHMAD
AIRIL AHMAD

35. Friday, January 25, 2013
Example 1: The sum of two numbers is 50.
If the difference is 12,
find the two numbers.
A + B = 50 …(1) When A=31 , then:
31
(+) A - B = 12 …(2)
A+ B = 50 ….(1)
2A = 62 + B = 50
A = 62 B = 50 - 31
2 B = 19 #
A = 31
AIRIL AHMAD

36. Friday, January 25, 2013
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.
AIRIL AHMAD

37. Friday, January 25, 2013
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.
AIRIL AHMAD

38. Friday, January 25, 2013
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.
AIRIL AHMAD

39. Friday, January 25, 2013
In a ranch, there are goats and
Example 6: chickens. If there are 22 heads and
58 legs, find the numbers of goat
and chicken.
Example 7:In an aquarium, there are ducks and
octopus. If there are 30 heads and
90 legs, find the numbers of duck
and octopus.
AIRIL AHMAD

40. Friday, January 25, 2013
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.
AIRIL AHMAD

41. Friday, January 25, 2013
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?
AIRIL AHMAD

42. Friday, January 25, 2013 AIRIL AHMAD
SIMULTANEOUS
LINEAR EQUATIONS
(SUBSTITUTION)
AIRIL AHMAD

43. Friday, January 25, 2013 AIRIL AHMAD
Example 1: Substitute b=2 in …(1)
(2)
a + 3b = 3 and b = 2
a + 3b = 3
a + 3b = 3 …(1) a+3 =3
…(2) a- 6 =3
+
b=2 a =3
a = -3

44. Friday, January 25, 2013 AIRIL AHMAD
Example 2: Substitute a =(-3)in …(1)
-3
2a - 3b = 6 and a = -3
2a - 3b = 6
2a - 3b = 6 …(1) 2 – 3b = 6
a = -3 …(2)
+ 6 – 3b = 6
-
-3b = 6
-b = 12
b = -12

45. Friday, January 25, 2013 AIRIL AHMAD
Example 3: Substitute a =(2b)in …(1)
2b
2a + b = 15 and a = 2b 2a + b = 15
2 + b = 15
2a + b = 15 …(1)
4b + b = 15
a = 2b …(2)
5b = 15
a=2
b = 15
a=6
5
b =(3)
3

46. Friday, January 25, 2013 AIRIL AHMAD
Example 4: Substitute a =(3b)in …(1)
3b
2a - 3b = 12 and a = 3b 2a – 3b = 12
2 -3b = 12
2a - 3b = 12 …(1)
…(2)
6b - 3b = 12
a = 3b 3b = 12
a=3
a = 12 b = 12
3
b =(4)
4

47. Friday, January 25, 2013 AIRIL AHMAD
Example 5: Substitute a =(2 + b)in …(1)
2 b
a + b = 10 and a – b = 2 a + b = 10
+ b = 10
a + b = 10 …(1)
2 + b + b = 10
…(2)
2 + 2b = 10
By a – b = nd equation …
using 2 2 2b = 10 - 2
a–b=2
+ 2b = 8
b =8
a =2
a =2 + 4 2
a =6 b = 4

48. Friday, January 25, 2013 AIRIL AHMAD
Example 6: Substitute n =(5 - m)in …(1)
5 m
3m - n = 7 and m +n = 3m - n =7
5 3m - =7
3m - n = 7 …(1)
3m - 5 + m =7
…(2)
4m – 5 =7
Bym + n 2nd equation …
using = 5 4m =7+5
-m+n=5 4m = 12
m = 12
n =5
4
n =5–3 m = 3
n =2

49. Friday, January 25, 2013 AIRIL AHMAD
Example 7: Substitute h =(4 – 2g)in (2)
4 2g
2g + h = 4 & 2g – h =16
2g - h = 16
2g + h = 4 …(1) 2g - = 16
2g – h = 16 …(2) 2g - 4 + 2g = 16
By using 1st equation … 4g – 4 = 16
4g = 16 + 4
- 2g + h = 4
4g = 20
h =4 g = 20
h = 4 – 10 4
h =-6 g = 5

50. Friday, January 25, 2013 AIRIL AHMAD
Exercise
1. Use substitution method to solve the
following simultaneous equations.