Example

Consider the following system of equations:

Error Margin is 0.1

1. Rearranging the Equations:

• $$x = \frac{12 - y - z}{4}$$
• $$y = \frac{27 - x - 2z}{5}$$
• $$z = \frac{18 - 2x - y}{6}$$

2. Initial Guess:

Iterations:

First Iteration:

• Calculate $x_1$:
  $$x_1 = \frac{12 - 0 - 0}{4} = 3$$
• Calculate $y_1$:
  $$y_1 = \frac{27 - 3 - 0}{5} = 4.8$$
• Calculate $z_1$:
  $$z_1 = \frac{18 - 2(3) - 4.8}{6} = 1.2$$

Second Iteration:

• Calculate $x_2$:
  $$x_2 = \frac{12 - 4.8 -1.2 }{4} \approx 1.5$$
• Calculate $y_2$:
  $$y_2 = \frac{27 - 1.5 - 2(1.2)}{5} \approx 4.62$$
• Calculate $z_2$:
  $$z_2 = \frac{18 − 2(1.5) − 4.62}{6} \approx 1.73$$

Third Iteration:

• Calculate $x_3$:
  $$x_3 = \frac{12 - 4.62 - 1.73}{4} \approx 1.41$$
• Calculate $y_3$:
  $$y_3 = \frac{27 - 1.41 - 2(1.73)}{5} \approx 4.43$$
• Calculate $z_3$:
  $$z_3 = \frac{18 - 2(1.41) - 4.43}{6} \approx 1.79$$

Fourth Iteration:

• Calculate $x_4$:
  $$x_4 = \frac{12 - 4.43 - 1.79}{4} \approx 1.445$$
• Calculate $y_4$:
  $$y_4 = \frac{27 - 1.445 - 2(1.79)}{5} \approx 4.395$$
• Calculate $z_4$:
  $$z_4 = \frac{18 - 2(1.445) - 4.395}{6} \approx 1.785$$