Apply the Gauss Seidel method, to solve the following system of linear equations up to the two iterations. \(15x+3y-2z=85,\ x-2y+8z=5,\ 2x+10y+z=51\).

Rearrange for diagonal dominance: \(15x+3y-2z=85\); \(2x+10y+z=51\); \(x-2y+8z=5\).

\(x=\frac{1}{15}(85-3y+2z),\quad y=\frac{1}{10}(51-2x-z),\quad z=\frac18(5-x+2y)\)

Start with \(y=z=0\), always using the latest values.

Iteration 1:
\(x_1=\frac{85}{15}=5.6667\)
\(y_1=\frac{51-2(5.6667)-0}{10}=3.9667\)
\(z_1=\frac{5-5.6667+2(3.9667)}{8}=0.9083\)

Iteration 2:
\(x_2=\frac{85-3(3.9667)+2(0.9083)}{15}=4.9944\)
\(y_2=\frac{51-2(4.9944)-0.9083}{10}=4.0103\)
\(z_2=\frac{5-4.9944+2(4.0103)}{8}=1.0033\)

After two iterations: \(x\approx4.9944,\ y\approx4.0103,\ z\approx1.0033\) (approaching the exact solution \(x=5,\ y=4,\ z=1\), which satisfies all three equations).