--- title: "Gauss elimination: x+2y+3z=6" subject: "Numerical Method" chapter: 3 key: "semester_4/numerical_method:gauss-elimination-x-2y-3z" source: https://www.bctengineer.com/q/semester_4/numerical_method/gauss-elimination-x-2y-3z site: BCT Engineering --- # Gauss elimination: x+2y+3z=6 Numerical Method · Chapter 3: Solution of system of linear algebraic equations - Asked in: 2069 Bhadra, 8 marks - Video: [gauss elimination method, numerical methods, Gaussian Elimination, Solution of linear equation, math](https://www.youtube.com/watch?v=G0TsGZMstmg) — Civil Engineering Exam - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_4/numerical_method/gauss-elimination-x-2y-3z - Whole chapter: https://www.bctengineer.com/md/semester_4/numerical_method/chapter-3 ## Question Solve the following system of linear equations using Gauss elimination method. x + 2y + 3z = 6 2x - 3y + 5z = 10 2x - y + 3z = 4 ## Short answer Forward elimination reduces the system to upper triangular form; back substitution then gives x = -2.3750, y = -0.1250, z = 2.8750. ## Answer **Gauss elimination:** make the coefficient matrix upper triangular using row operations (forward elimination), then find the unknowns from the last equation upward (back substitution). #### Step 1: System ```text x + 2y + 3z = 6 2x - 3y + 5z = 10 2x - y + 3z = 4 ``` #### Step 2: Forward elimination (reduce to upper triangular form) Augmented matrix [A | b]: ```text 1.0000 2.0000 3.0000 | 6.0000 2.0000 -3.0000 5.0000 | 10.0000 2.0000 -1.0000 3.0000 | 4.0000 ``` R2 ← R2 − (2.0000) R1; R3 ← R3 − (2.0000) R1: ```text 1.0000 2.0000 3.0000 | 6.0000 0.0000 -7.0000 -1.0000 | -2.0000 0.0000 -5.0000 -3.0000 | -8.0000 ``` R3 ← R3 − (0.7143) R2: ```text 1.0000 2.0000 3.0000 | 6.0000 0.0000 -7.0000 -1.0000 | -2.0000 0.0000 0.0000 -2.2857 | -6.5714 ``` #### Step 3: Back substitution ```text z = (-6.5714) / -2.2857 = 2.8750 y = (-2.0000 − (-1.0000)(2.8750)) / -7.0000 = -0.1250 x = (6.0000 − (2.0000)(-0.1250) − (3.0000)(2.8750)) / 1.0000 = -2.3750 ``` **Answer: x = -2.3750, y = -0.1250, z = 2.8750** ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. What are the two phases of Gauss elimination? A. Forward elimination to upper triangular form, then back substitution B. Back substitution, then forward elimination C. Iteration then averaging D. Finding the inverse then multiplying Answer: A. Row operations create zeros below the diagonal, and the unknowns are found from the last equation upward. (Key point: Gauss elimination steps) 2. After R2 ← R2 - 2R1 and R3 ← R3 - 2R1, what are rows 2 and 3? A. [0 -7 -1 | 2] and [0 -5 -3 | 8] B. [0 -7 -1 | -2] and [0 -5 -3 | -8] C. [0 1 -1 | -2] and [0 3 -3 | -8] D. [0 -7 -1 | -2] and [0 -3 -5 | -8] Answer: B. Row 2: (2-2, -3-4, 5-6 | 10-12); row 3: (2-2, -1-4, 3-6 | 4-12). (Key point: First elimination) 3. What multiplier is used in R3 ← R3 - m·R2, and what is z? A. m = (-5)/(-7) = 0.7143 and z = 2.875 B. m = 7/5 = 1.4 and z = 2.875 C. m = 0.7143 and z = -2.875 D. m = 2 and z = 6 Answer: A. m = 5/7 = 0.7143 gives -2.2857 z = -6.5714, so z = 2.875; then y = -0.125 and x = -2.375. 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