Question - Solving a System of Linear Equations Using the Gauss-Seidel Method

To use the Gauss-Seidel method, we first express each equation in the form \(x = ... , y = ... , z = ...\).

From \(5x + 2y + z = 12\), we get \[x = \frac{12 - 2y - z}{5}\]

From \(x + 4y + 2z = 15\), we get \[y = \frac{15 - x - 2z}{4}\]

From \(x + 2y + 5z = 20\), we get \[z = \frac{20 - x - 2y}{5}\]

We then assume initial values for \(x, y, z\), usually zeros: \(x_0 = 0, y_0 = 0, z_0 = 0\).

Iterate the equations using the previous values:

\[x_{n+1} = \frac{12 - 2y_n - z_n}{5}\]

\[y_{n+1} = \frac{15 - x_{n+1} - 2z_n}{4}\]

\[z_{n+1} = \frac{20 - x_{n+1} - 2y_{n+1}}{5}\]

The iterations are repeated until the solutions converge to a set level of accuracy.