solve the system of equation by guess sidle method8x1 + x2 + x3 = 82x1 + 4x2 + x3 = 4x1 + 3x2 + 5x3 = 5

Answer: The solution is,[tex]x_1\approx 0.876[/tex][tex]x_2\approx 0.419[/tex][tex]x_3\approx 0.574[/tex]Step-by-step explanation:Given equations are,[tex]8x_1 + x_2 + x_3 = 8[/tex][tex]2x_1 + 4x_2 + x_3 = 4[/tex][tex]x_1 + 3x_2 + 5x_3 = 5[/tex],From the above equations,[tex]x_1=\frac{1}{8}(8-x_2-x_3)[/tex][tex]x_2=\frac{1}{4}(4-2x_1-x_3)[/tex][tex]x_3=\frac{1}{5}(5-x_1-3x_2)[/tex]First approximation,[tex]x_1(1)=\frac{1}{8}(8-(0)-(0))=1[/tex][tex]x_2(1)=\frac{1}{4}(4-2(1)-(0))=0.5[/tex][tex]x_3(1)=\frac{1}{5}(5-1-3(0.5))=0.5[/tex]Second approximation,[tex]x_1(2)=\frac{1}{8}(8-(0.5)-(0.5))=0.875[/tex][tex]x_2(2)=\frac{1}{4}(4-2(0.875)-(0.5))=0.4375[/tex][tex]x_3(2)=\frac{1}{5}((0.875)-3(0.4375))=0.5625[/tex]Third approximation,[tex]x_1(3)=\frac{1}{8}(8-(0.4375)-(0.5625))=0.875[/tex][tex]x_2(3)=\frac{1}{4}(4-2(0.875)-(0.5625))=0.421875[/tex][tex]x_3(3)=\frac{1}{5}(5-(0.875)-3(0.421875))=0.571875[/tex]Fourth approximation,[tex]x_1(4)=\frac{1}{8}(8-(0.421875)-(0.571875))=0.875781[/tex][tex]x_2(4)=\frac{1}{4}(4-2(0.875781)-(0.571875))=0.419141[/tex][tex]x_3(4)=\frac{1}{5}(5-(0.875781)-3(0.419141))=0.573359[/tex]Fifth approximation,[tex]x_1(5)=\frac{1}{8}(8-(0.419141)-(0.573359))=0.875938[/tex][tex]x_2(5)=\frac{1}{4}(4-2(0.875938)-(0.573359))=0.418691[/tex][tex]x_3(5)=\frac{1}{5}(5-(0.875938)-3(0.418691))=0.573598[/tex]Hence, by the Gauss Seidel method the solution of the given system is,[tex]x_1\approx 0.876[/tex][tex]x_2\approx 0.419[/tex][tex]x_3\approx 0.574[/tex]