Question

Gauss Seidel method to solve 25x + 2y + 2z = 69, 2x + 10y +z = 63, X +Y+z=43​

Answer 1

Answer:

25x + 2y + 2z = 69, 2x + 10y +z = 63, X +Y+z=43

Answer 2

Given,

                  [tex]25x + 2y+ 2z = 69\\ 2x+10y+z = 63\\ x+y+z = 43[/tex]

To find: solve by gauss seidel method.

explanation:

First we make conditions by given equation,

                            |25|>|2|+|2| for x

                            |10|>|2|+|1| for y

                            |1| = |1|+|1|  for z

Now the equation,

                            [tex]x = \frac{1}{25}(69-2y-2z)\\ y = \frac{1}{10}(63-2x-z)\\ z = (43-x-y) [/tex]

1st iteration, x=0, y=0, z=0

                           [tex]x_{1} = \frac{1}{25} (69) = 2.7\\ y_{2} = \frac{1}{10}(63) = 6.3\\ z_{3} = (43) = 43[/tex]

2nd iteration, x = 2.7, y = 6.3, z = 43

                           [tex]x= \frac{1}{25}(69- 12.6-86) = -1.18 \\ y= \frac{1}{10}(63- 5.4-43) = 1.46 \\ z= (43-2.7-6.3) = 34[/tex]

3rd iteration, x= -1.18, y = 1.46, z = 34

                           $x = \frac{1}{25}(69-2.92-68)= -0.07\\ y = \frac{1}{10}(63+2.36-34)= 3.13\\z = (43+1.18-1.46) = 42.72$

4th iteration, x = -0.07, y = 3.13, z = 42.72

                           $x = \frac{1}{25}(69-6.26-85.14) = -0.896\\ y = \frac{1}{10}(63+0.14-42.72) = 2.04\\z = (43+0.07-42.72) = 0.35$

Hence the process continues till we get the consequent same values.