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To solve the system of equations using the Gauss elimination method with pivoting, we'll follow these steps: 1. Write the augmented matrix of the system. 2. Perform row operations to transform the matrix into upper triangular form. 3. Back-substitute to find the values of the variables. ### Step 1: Write the Augmented Matrix The system of equations is: \[ \begin{aligned} 2x_1 + 3x_2 + 4x_3 + x_4 &= 3 \\ x_1 + 2x_2 + x_4 &= 2 \\ 2x_1 + 3x_2 + x_3 - x_4 &= 1 \\ x_1 - 2x_2 - x_3 + 4x_4 &= 5 \end{aligned} \] The augmented matrix is: \[ \begin{bmatrix} 2 & 3 & 4 & 1 & | & 3 \\ 1 & 2 & 0 & 1 & | & 2 \\ 2 & 3 & 1 & -1 & | & 1 \\ 1 & -2 & -1 & 4 & | & 5 \end{bmatrix} \] ### Step 2: Perform Row Operations #### Step 2.1: Pivot the First Column The largest absolute value in the first column is 2 in the __________ ________ ______ _________ _______ __________ ____ _____.
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