---
title: "Gauss-Seidel: 10x+y-z=11.19"
subject: "Numerical Method"
chapter: 3
key: "semester_4/numerical_method:gauss-seidel-10x-y-z"
source: https://www.bctengineer.com/q/semester_4/numerical_method/gauss-seidel-10x-y-z
site: BCT Engineering
---

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

Numerical Method · Chapter 3: Solution of system of linear algebraic equations

- Asked in: 2073 Bhadra, 8 marks
- Video: [Gauss Seidel Method | Solve 10x + y + z = 12 System | VTU Numerical Methods](https://www.youtube.com/watch?v=wtR6akToudg) — Mathematics Tutor
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/numerical_method/gauss-seidel-10x-y-z
- Whole chapter: https://www.bctengineer.com/md/semester_4/numerical_method/chapter-3

## Question

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

## 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.

## Answer

**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
```text
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
| Iter | x | y | z |
|---|---|---|---|
| 0 | 0.0000 | 0.0000 | 0.0000 |
| 1 | 1.1190 | 2.6961 | 3.4033 |
| 2 | 1.1897 | 2.3487 | 3.4451 |
| 3 | 1.2286 | 2.3406 | 3.4498 |
| 4 | 1.2299 | 2.3400 | 3.4500 |

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

**Answer: x ≈ 1.23, y ≈ 2.34, z ≈ 3.45**

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. What makes Gauss-Seidel different from simply recomputing every unknown from the previous iteration's values?
   A. Newly computed values are used immediately within the same iteration
   B. All unknowns are updated together using only old values
   C. The equations are solved by row operations instead of iteration
   D. Only the unknown with the largest coefficient is updated each time
   Answer: A. Gauss-Seidel solves each equation for its diagonal unknown and uses the newest values straight away, e.g. x(k+1) in the formula for y(k+1). (Key point: Gauss-Seidel idea)

2. Why is convergence guaranteed for 10x + y - z = 11.19, x + 10y + z = 28.08, -x + y + 10z = 35.61?
   A. The right-hand sides are all positive
   B. The coefficient matrix is symmetric
   C. The system is diagonally dominant: 10 > 1 + 1 in every equation
   D. The starting guess is zero
   Answer: C. Diagonal dominance (diagonal coefficient larger than the sum of the others) makes Gauss-Seidel converge. (Key point: Convergence condition)

3. With x(0) = y(0) = z(0) = 0, what is y(1)?
   A. 2.8080
   B. 2.6961
   C. 2.9199
   D. 1.1190
   Answer: B. x(1) = 11.19/10 = 1.119. Then y(1) = (28.08 - 1.119 - 0)/10 = 2.6961, using the new x immediately. (Key point: First iteration)

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