Exercise 3.2 [Gaussian Elimination and Gauss- Jordan]

EXERCISE 3.2 [Gaussian Elimination and Gauss- Jordan]

Solve the system by Gauss Elimination method and Gauss-Jordan method:

1. 2x + 3y z = 5, 4x+4y-3z = 3 and 2x-3y+2z = 2 [A.U CBT A/M 2011]

[Ans. x = 1, y = 2, z = 3]

2. 3x+4y+5z = 18, 2xy + 8z = 13, 5x-2y + 7z = 20

[Ans. x = 3, y=1, z= 1] [A.U CBT N/D 2011, A.U M/J 2012]

3. 4.12x - 9.68y + 2.01z = 4.93; 1.88x - 4.62y + 5.50z = 3.11 1.10x - 0.96y + 2.72z = 4.02

[Ans. 4.2075, 1.3327, 0.2468]

4. 2x + 4y + 8z = 41, 4x + 6y + 10z = 56, 6x + 8y + 10z = 64

[Ans. 1.5, 2.5, 3.5]

5. x + 0.5y + 0.33z = 1, 0.33x + 0.25y + 0.2z = 0, 0.5x + 0.33y + 0.25z = 0

 [Ans. 55.56, -277.78, 255.56]

6. 2x - y + 3z + w = 9, 3x + y - 4z + 3w = 3, 5x - 4y + 3z - 6w = 2, x - 2y - z + 2w = -2

[Ans. 1, 2, 2, 2]

7. 10x + y + z = 18.141, x + 10y+z = 28.140, x + y + 10z = 38.139

[Ans. 1.234, 2.348, 3.455]

8. 6x - y + z = 13, x + y + z = 9, 10x + y - z = 19.

[Ans , 2, 3, 4]

9.  x - 3y - z = -30, 2x – y - 3z = 5, 5x-y-2z = 142

[Ans. 39.2, 16.7, 19]

10. 2x + 2y – z + w = 4, 4x + 3y z + 2w = 6, 8x + 5y - 3z + 4w = 12, 3x + 3y-2z+2w = 6.

[Ans. 1, 1, -1, -1]

11. 2x + y = 3, 7x-3y = 4

[Ans. 1, 1]

12. x + y + z = 9, 2x - 3y + 4z = 13, 3x + 4y + 5z = 40[A.U A/M 2011]

[Ans. x = 1, y = 3, z = 5]

13. 2.38x₁ +1.95x₂ - 3.27x₃ + 1.58x₄ = 2.16

3.21x₁ - 0.86x₂ + 2.42x₃ - 3.20x₄ = 3.28

1.44x₁ + 2.95x₂ - 2.14x₃ + 1.86x₄ = 1.42

4.17x₁ + 3.6x₂ - 1.68x₃ - 2.26x₄ = 5.21

[Ans. x₁ = 0.8072, x₂ = 0.2372, x₃ = - 0.1046, x₄ = - 0.3581]

14. x₁ + x₂ + x₃ + x₄ = 2;

x₁ + x₂ + 3x₃ - 2x₄ = -6

2x₁ + 3x₂ – x₃ + 2x₄ = 7;

x₁ + 2x₂ + x₃ – x₄ = -2

[Ans. x₁ = 1, x₂ = 0, x₃ = -1 , x₄ = - 2]

15. 2x₁ – x₂ + x₃ = 0.3; - 4x₁ + 3x₂ – 2x₃ = - 1.4

3x₁ - 8x₂ + 3x₃ = 0.1

[Ans. x₁ = 1.6, x₂ =  -0.8, x₃ = -3.7]

16. w + x + y = 3; -3w - 17x + y + 2z = 1; 4w + 8y - 5z = 1- 5x - 2y + z = 1

 [Ans. x = 0, y = 1, z = 3, w = 2]

17. 5x₁ – x₂ = 9, -x₁ + 5x₂ – x₃ =4, - x₂ + 5x₃ = - 6

[Ans. x = 2, y = 1, z = -1]

18. 2x - y + 3z = 8, -x + 2y + z = 4, 3x + y - 4z = 0

[Ans. x = 2, y = 2, z = 2]

 [A.U N/D 2007, N/D 2014] R2AE [A.U N/D 2016 R13]

19. x₁  - x₂ +  x₃ = 1, -3x₁ + 2x₂ - 3x₃ = -6, 2x₁ - 5x₂ + 4x₃ = 5

[Ans. x = -2, y = 3, z = 6]  [A.U A/M 2014 R-13]

20. 2x + y + 4z = 12, 8x - 3y + 2z = 20, 4x + 11y - z = 33

[Ans. x = 3, y = 2, z = 1]

[A.U N/D 2019 R-13]

21. x + 2y + z = 3, 2x + 3y + 3z = 10, 3x - y + 2z = 13

[Ans. x = 2, y = -1, z = 3]