Employ the Gauss-Seidel method, solve the system. 10𝑥 + 𝑦 + 𝑧 = 12 2𝑥 + 2𝑦 + 10𝑧 = 14 2𝑥 + 10𝑦 + z=13

Rewrite

$10x+y+z=12$

$2x+10y+z=13$

$2x+2y+10z=14$

$x_{n+1}=\dfrac{1}{10}(12-y_n-z_n)$

$y_{n+1}=\dfrac{1}{10}(13-2x_{n+1}-z_n)$

$z_{n+1}=\dfrac{1}{10}(14-2x_{n+1}-2y_{n+1})$

Initial gauss $(x,y,z)=(0.5,0.5,0.5)$

1st Approximation

$x_{1}=\dfrac{1}{10}(12-0.5-0.5)=1.1$

$y_{1}=\dfrac{1}{10}(13-2(1.1)-0.5)=1.03$

$z_{1}=\dfrac{1}{10}(14-2(1.1)-2(1.03))=0.974$

2nd Approximation

$x_{2}=\dfrac{1}{10}(12-1.03-0.974)=0.9996$

$y_{2}=\dfrac{1}{10}(13-2(0.9996)-0.974)=1.00268$

$z_{2}=\dfrac{1}{10}(14-2(0.9996)-2(1.00268))=0.999544$

3rd Approximation

$x_{3}=\dfrac{1}{10}(12-1.00268-0.999544)=0.9997776$

$y_{3}=\dfrac{1}{10}(13-2(0.9997776)-0.999544)=1.00009008$

$z_{3}=\dfrac{1}{10}(14-2(0.9997776)-2(1.00009008))=1.000026464$

Solution

$x=1.000, y=1.000, z=1.000$