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Gauss-Seidel: 10x+y-z=11.19

Short answer

Gauss-Seidel iteration: rearrange equations for diagonal dominance, iterate using newest values. Starting from zero, after 4 iterations values stabilize to two decimal places: x=1.23, y=2.34, z=3.45.

Apply Gauss Seidel Iterative Method to solve the linear equations correct to 2 decimal places. 10x + y - z = 11.19 x + 10y + z = 28.08 -x + y + 10z = 35.61

Gauss-Seidel method: solve each equation for its diagonal unknown and use the newest values immediately in the same iteration. It converges when the system is diagonally dominant.

Step 1: Check dominance and rearrange

In each equation the diagonal coefficient (10) is larger than the sum of the others (1 + 1 = 2), so the system is diagonally dominant and Gauss-Seidel converges.

Step 2: Iteration formulas

x(k+1) = (11.19 − y(k) + z(k)) / 10
y(k+1) = (28.08 − x(k+1) − z(k)) / 10
z(k+1) = (35.61 + x(k+1) − y(k+1)) / 10

Start with x(0) = 0, y(0) = 0, z(0) = 0.

Step 3: Iterations

Iterxyz
00.00000.00000.0000
11.11902.69613.4033
21.18972.34873.4451
31.22862.34063.4498
41.22992.34003.4500

Values of two successive iterations agree to 2 decimal places, so we stop.

Answer: x ≈ 1.23, y ≈ 2.34, z ≈ 3.45

✓ Answer checked by Claude Opus 5.5 (high effort)

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