- Asked 4/13
- 2069 Poush · 8 marks
Gauss elimination, partial pivoting, 4 unknowns
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.
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
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
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]:
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:
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:
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:
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:
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:
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
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
✓ Answer checked by Claude Opus 5.5 (high effort)
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