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Gauss elimination: x+2y+3z=6

Short answer

Forward elimination reduces the system to upper triangular form; back substitution then gives x = -2.3750, y = -0.1250, z = 2.8750.

Solve the following system of linear equations using Gauss elimination method. x + 2y + 3z = 6 2x - 3y + 5z = 10 2x - y + 3z = 4

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

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]:

 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:

 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:

 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

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

✓ Answer checked by Claude Opus 5.5 (high effort)

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