NOTE MATH FORM 3 - 11 linear equations
- 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.