(I)
2x1+3x2+7x3 = 12 -----(1)
x1-4x2+5x3 = 2 -----(2)
4x1+5x2-12x3= -3 ----(3)

Gauss Elimination Method

From (1) & (2)
2x1+3x2+7x3=12 ---(1) × 1
x1-4x2+5x3 = 2 ---(2) × 2

2x1 +3x2+7x3= 12
2x1 -8x2+10x3= 4
- + - -
11x2-ex3=8 ----(4)

From (I) & (3)
2x1+3x2+7x3=12 ---(1)×2
4x1+5x2-12x3= -3 ---- (3) × 1

4x1+6x2+14x3 = 24
4x1+5x2-12x3 = -3
- - + +
x2+26x3 = 27 --(5)

From (4) and (5)
11x2-3x3 = 8 ---(4) × 1
x2+26x3 = 27 ---(5) ×11

11x2-3x3 = 8
11x2+286x3 = 297
- - -
-289x3 = -289
x3 = 1

put the value of x3 in (4)
11x2-3×1=8
11x2 = 8 + 3
x2 = 1

Put the value of x2in(1)
2x1+3+7 = 12
2x1= 12-10
x1 = 1

x1=1
x2=1
x3=1 Ans:

Guss-Scidal Method

(i)
2x1+3x2+7x3 = 12 ---(1)
x1-4x3+x3 = 2 ---(2)
4x1+5x2-12x3 = -3 ----(3)

From (1)
2x1 = 12-3x2-7x3
x1=1/2 (12-3x2-7x3)

From (2)
-4x2 = 2-x1-5x3
x2 = ¼ (x1+5x3-2)

From (3)
-12x3= [-3-4x1-5x2]
x3 = 1/12 [3+4x1+5x2]

1st Iteration

x1 = 0, x2 = 0, x3 = 0
[ x2=0, x3=0]

From (1)
x1 = ½ [12] = 6
From (2)
x1=6 x3=0
x2 = ¼ [6-2] = ¼ × 4 = 1

From (3)
x1 = 6, x2 = 1

x3 = 1/12 [3 + 4×6 + 5×1]
= 1/12 [ 3+24+5]
= 32/12 = 2.666

x1=6, x2=1, x3=2.666

2nd Iteration

From (1)
x2=1, x3=2.666
x1 = ½ [12-3-7×2.666]
= ½ [12-21.662] = -4.831
From (2)
x1 = -4.833 x3 = 2.666
x2 = ¼ [-4.831 + 5 × 2.666-2]
= 1.4 [13,33 – 6,833] = 1.624

From (3)
x1 = -4.833 x2 = 1.624
x3 = 1/12 [3-4×4.833 + 5 ×1.624]
= 1/12 [ 11.12 – 19.332]
= -0.684

x1 = -4.8333, x2 = 1.624, x3 = -0.684

3rd Iteration

x2= 1.624, x3 = -0.684
from (1)
x1=1/2 [12-3×1.624 + 7×0.684]
= ½ [16.788 – 4.872]
= 5.958

From (2)
x1 = 5.958 x3 = 0.684
x2 = ¼ [ 5.958 + 5 × -0.684 – 2]
= ¼ [5.958-5.42]
= 0.1345

From (3)
x1 = 5.958, x2 = 0.1345
x3 = 1/12 [3+4×5.958 + 5 × 0.1345]
= 1/12 [26.832 – 0.67] = 2.291

x1 = 5.958, x2 = 0.1345, x3 = 2.291

4th Iteration

from (1) x2 = 0.1345 x3 = 2.291
x1 = ½ [12 – 3 × 0.1345 – 7 × 2.291]
= ½ [12 – 0.402 – 16.037]
= -2.2195

From (2)
x1 = -2.2195 x3 = 3.337
x2 = ¼ [-2.2195 + 5 × 2.291 – 2]
= ¼ [11.455 – 4.2195] = 1.8086

From (3)
x1 = -2.2195 x2 = 1.8086
x3 = 1/12 [3+4× -2.2195 + 5×1.8086]
= 1/12 [3-8.878 + 9.043]
= 1/12[12.043-8.878]

Gauss Elimination

(1) (ii)
x1 +| 2x2 + 3x3 = 8 ---(1)
2x1+4x2+9x3 = 19 ---(2)
4x1-6x2+3x3 = -5 ---(3)

From (1) and (2)
x1+2x2 + 3x3 = 8 ---(1)×2
2x14x2+9x3 = 19 ---(2)×1

2x1+4x2+6x3 = 16
2x1+4x2+9x3 = 19
- - - -
-3x3= -3
x3 = 1

From (1) and (3)
x1+2x2+3x3 = 8
4x1-6x2+3x3 = -5
- + - +
-3x1+8x2 = 13
3x1-8x2 = -13 ---(4)
from (1)
x1+2x2+3×1 = 8
x1+2x2 = 8-3
x1+2x2 = 5 ---(5)

From (4) and (5)
3x1-8x2 = -13 ---(4) × 1
x1+2x2 = 5 ---(5) ×3

 

3x1-8x2 = -13
3x1 + 6x2 = 15
- - -
-14x2 = -28
x2 =2

Put the value of x2 in (5)
x1+2×2 = 5
x1 = 5-4
x1= 1

x1 = 1
x2 = 2
x3 = 1

Gauss Seidal Method

x1+2x2+3x3 = 8 ---(1)
2x1+4x2+9x3 = 19 ---(2)
4x1-6x2+3x3 = -15 ---(3)

From (1)
x1+2x2+3x3 = 8
x1 = 8-2x2-3x3 ---(4)

From (2)
2x1+4x2+9x3 = 19
4x2 = 19-2x1-9x3
x2 = ¼ [19-2x1-9x3]

From (3)
4x1-6x2+3x3 = -5
3x3 = 6x2-5-3x3
x3 = 1/3 [6x2-5-4x1]

x1=x2=x3=0

1st Iteration

x1=8-2x2-3x3 = 8-0-0
x1=8

From (2)
x1 = 8, x3 = 0
x2 = ¼ [19-2×8-0]
= ¼ [19-16] = ¼ ×3 = 0.75

From (3)
x1 = 8 x2 = 0.75
x3 = 1/3 [6×0.75-5-4Ú8]
= 1/3 [4.5-5-32]
= 1/3 [-32.5] = -10.833

[x1-8, x2=0.75, x3 = 10.833]

2nd Iteration

From (1) x2 = 0.75, x3 = -10.833
x1 = 8-2×0.75 + 3 × 10.833
= 8-1.5 + 32.499
= 38.999

From (2)
x1=38.999, x3= -10.833

x2 = ¼ [19-2×38.999 + 9 ×10.833]
= ¼ [19-77.998 + 97.497]
= ¼ [116.497 – 77.998]
x2= 9.624

From (3) x1 = 38.999, x2 = 9.624
x3 = 1/3 [6×9.624 – 5- 4× 38.992]
= 1/3 [57.744-160.996]
= -34.417

x1 = 38.999, x2 = 9.624, x3 = 34.417

3rd Iteration

From (1) x2 = 9.624, x3 = -34.477
x1 = 8 – 2 ×624 – 3×(34.477)
= 8 – 19.248 + 103.251
= 92.003

From (2) x1 = 92.003 x3 = -34.417
= ¼ [19-2×92.003+9×34.417]
= ¼ [328.753 – 184.006]
= 36.186

From (3)
x1 = 92.003, x2 = 36.186
= 1/3 [6 × 36.186 – 5 –4 × 92.003]
= 51.965

x1 = 92.003, x2 = 36.186, x3 = 51.965

4th Iteration

(x+1) (x) (x-1) (x-2)
(x2-1)(x2-2x)
= x4-2x3-x2+2x