--- title: "Gauss elimination, partial pivoting, 4 unknowns" subject: "Numerical Method" chapter: 3 key: "semester_4/numerical_method:gauss-elimination-four-pivoting" source: https://www.bctengineer.com/q/semester_4/numerical_method/gauss-elimination-four-pivoting site: BCT Engineering --- # Gauss elimination, partial pivoting, 4 unknowns Numerical Method · Chapter 3: Solution of system of linear algebraic equations - Asked in: 2069 Poush, 8 marks - Video: [Gaussian Elimination with Partial Pivoting](https://www.youtube.com/watch?v=3aO2eG9lGk4) — AF Math & Engineering - 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-four-pivoting - Whole chapter: https://www.bctengineer.com/md/semester_4/numerical_method/chapter-3 ## Question Solve the following system of linear equations using the Gauss Elimination method with partial pivoting. x + 2y - 12z + 8v = 27 5x + 4y + 7z - 2v = 4 -3x + 7y + 9z + 5v = 11 6x - 12y - 8z + 3v = 49 ## Short answer The system is solved by partial pivoting: at each step the row with largest pivot is swapped to the top. Forward elimination yields an upper triangular matrix; back substitution gives x=3.0000, y=-2.0000, z=1.0000, v=5.0000. ## Answer **Gauss elimination with partial pivoting:** at each stage the row having the largest absolute value in the pivot column is brought to the pivot position (row swap) before elimination. This reduces round-off error. Then back substitution is used. #### Step 1: System ```text x + 2y - 12z + 8v = 27 5x + 4y + 7z - 2v = 4 -3x + 7y + 9z + 5v = 11 6x - 12y - 8z + 3v = 49 ``` #### Step 2: Forward elimination (reduce to upper triangular form) Augmented matrix [A | b]: ```text 1.0000 2.0000 -12.0000 8.0000 | 27.0000 5.0000 4.0000 7.0000 -2.0000 | 4.0000 -3.0000 7.0000 9.0000 5.0000 | 11.0000 6.0000 -12.0000 -8.0000 3.0000 | 49.0000 ``` Column 1: largest pivot is in row 4, so swap R1 ↔ R4: ```text 6.0000 -12.0000 -8.0000 3.0000 | 49.0000 5.0000 4.0000 7.0000 -2.0000 | 4.0000 -3.0000 7.0000 9.0000 5.0000 | 11.0000 1.0000 2.0000 -12.0000 8.0000 | 27.0000 ``` R2 ← R2 − (0.8333) R1; R3 ← R3 − (-0.5000) R1; R4 ← R4 − (0.1667) R1: ```text 6.0000 -12.0000 -8.0000 3.0000 | 49.0000 0.0000 14.0000 13.6667 -4.5000 | -36.8333 0.0000 1.0000 5.0000 6.5000 | 35.5000 0.0000 4.0000 -10.6667 7.5000 | 18.8333 ``` R3 ← R3 − (0.0714) R2; R4 ← R4 − (0.2857) R2: ```text 6.0000 -12.0000 -8.0000 3.0000 | 49.0000 0.0000 14.0000 13.6667 -4.5000 | -36.8333 0.0000 0.0000 4.0238 6.8214 | 38.1310 0.0000 0.0000 -14.5714 8.7857 | 29.3571 ``` Column 3: largest pivot is in row 4, so swap R3 ↔ R4: ```text 6.0000 -12.0000 -8.0000 3.0000 | 49.0000 0.0000 14.0000 13.6667 -4.5000 | -36.8333 0.0000 0.0000 -14.5714 8.7857 | 29.3571 0.0000 0.0000 4.0238 6.8214 | 38.1310 ``` R4 ← R4 − (-0.2761) R3: ```text 6.0000 -12.0000 -8.0000 3.0000 | 49.0000 0.0000 14.0000 13.6667 -4.5000 | -36.8333 0.0000 0.0000 -14.5714 8.7857 | 29.3571 0.0000 0.0000 0.0000 9.2475 | 46.2377 ``` #### Step 3: Back substitution ```text v = (46.2377) / 9.2475 = 5.0000 z = (29.3571 − (8.7857)(5.0000)) / -14.5714 = 1.0000 y = (-36.8333 − (13.6667)(1.0000) − (-4.5000)(5.0000)) / 14.0000 = -2.0000 x = (49.0000 − (-12.0000)(-2.0000) − (-8.0000)(1.0000) − (3.0000)(5.0000)) / 6.0000 = 3.0000 ``` **Answer: x = 3.0000, y = -2.0000, z = 1.0000, v = 5.0000** ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. Why is R1 swapped with R4 in column 1? A. 6 has the largest absolute value in column 1, and a large pivot reduces round-off error B. 6 is the largest right-hand side C. R4 has the most negative entries D. R1 must always be swapped Answer: A. Partial pivoting brings the largest absolute value in the column to the pivot position. (Key point: Why swap rows) 2. After the swap, what multiplier removes the 5 in row 2? A. 6/5 = 1.2 B. 5/6 = 0.8333 C. -3/6 = -0.5 D. 1/6 = 0.1667 Answer: B. Multiplier = entry/pivot = 5/6 = 0.8333. (Key point: Multiplier) 3. In column 3, why are R3 and R4 swapped? A. Because 4.0238 is negative B. Because the right-hand side is smaller C. |-14.5714| is larger than |4.0238|, so it becomes the pivot D. Because R3 has a zero in column 2 Answer: C. The pivot is the entry of largest magnitude among those on and below the diagonal. (Key point: Second pivot swap) 4. Which unknown is found first in back substitution? A. v = 5, from the last row B. x = 3, from the first row C. y = -2, from the second row D. z = 1, from the third row Answer: A. The last row has only v: 9.2475 v = 46.2377 gives v = 5; then z, y, x follow. 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