Respuesta :
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]
