Word Count : 589

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We are asked to set up the Gauss-Seidel iteration for the system: [ \begin{bmatrix} 1 & 1 & 1 \ 4 & 3 & -1 \ 3 & 5 & 3 \end{bmatrix} \begin{bmatrix} x \ y \ z \end{bmatrix} ======================================= \begin{bmatrix} 1 \ 6 \ 4 \end{bmatrix}. ] --- The system of equations is: 1. (x + y + z = 1) 2. (4x + 3y - z = 6) 3. (3x + 5y + 3z = 4) --- ### Step 1: Rewrite each variable in terms of the others (for Gauss-Seidel) From Equation 1: [ x = 1 - y - z ] From Equation 2: [ 4x + 3y - z = 6 \implies z = 4x + 3y - 6 ] From Equation 3: [ 3x + 5y + 3z = 4 \implies z = \frac{4 - 3x - 5y}{3} ] For Gauss-Seidel, we __________ ___ _____ ____ _________ ________ ___ __________ ___ ________.
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