1. **Problem Statement:** We are given a system of simultaneous linear equations representing reservoir simulation: $$a_i P_{i-1} + b_i P_i + C_i P_{i+1} = d_i, \quad i=1,2,\ldots,N$$ This can be written in matrix form as: $$\begin{bmatrix} a_{11} & b_{12} & C_{13} \\ a_{21} & b_{22} & C_{23} \\ a_{31} & b_{32} & C_{33} \end{bmatrix} \begin{bmatrix} P_{i-1} \\ P_i \\ P_{i+1} \end{bmatrix} = \begin{bmatrix} d_1 \\ d_2 \\ d_3 \end{bmatrix}$$ 2. **Goal:** Solve for the vector $\mathbf{P} = [P_{i-1}, P_i, P_{i+1}]^T$ using Gaussian elimination. 3. **Gaussian Elimination Setup:** Form the augmented matrix: $$\left[ \begin{array}{ccc|c} a_{11} & b_{12} & C_{13} & d_1 \\ a_{21} & b_{22} & C_{23} & d_2 \\ a_{31} & b_{32} & C_{33} & d_3 \end{array} \right]$$ 4. **Method Explanation:** Gaussian elimination transforms this augmented matrix into an upper triangular form by: - Using row operations to eliminate the lower left elements. - This involves multiplying a row by a scalar and subtracting it from another row. 5. **Step-by-step:** - For example, eliminate $a_{21}$ by replacing row 2 with $\text{row}_2 - \frac{a_{21}}{a_{11}} \times \text{row}_1$: $$\text{row}_2 \to \text{row}_2 - \frac{a_{21}}{a_{11}} \times \text{row}_1$$ - Show intermediate cancellation: $$\text{row}_2 = \left[ a_{21} \quad b_{22} \quad C_{23} \quad d_2 \right] - \frac{a_{21}}{a_{11}} \times \left[ a_{11} \quad b_{12} \quad C_{13} \quad d_1 \right]$$ $$= \left[ \cancel{a_{21}} \quad b_{22} - \frac{a_{21}}{a_{11}} b_{12} \quad C_{23} - \frac{a_{21}}{a_{11}} C_{13} \quad d_2 - \frac{a_{21}}{a_{11}} d_1 \right]$$ - Repeat for $a_{31}$ in row 3. 6. **After elimination:** The matrix is upper triangular, then use back substitution to solve for $P_{i+1}$, $P_i$, and $P_{i-1}$. 7. **Summary:** Gaussian elimination systematically reduces the system to a simpler form to find the unknown pressures $P_i$ in the reservoir simulation. This method is fundamental in solving linear systems arising in engineering and physics.