---
title: "Gauss-Seidel: 20x+y-2z=17"
subject: "Numerical Method"
chapter: 3
key: "semester_4/numerical_method:gauss-seidel-20x-y-2z"
source: https://www.bctengineer.com/q/semester_4/numerical_method/gauss-seidel-20x-y-2z
site: BCT Engineering
---

# Gauss-Seidel: 20x+y-2z=17

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

- Asked in: 2071 Magh, 8 marks
- Video: [Gauss Seidel Method | Solve 20x + y − 2z = 17 System | VTU Numerical Methods](https://www.youtube.com/watch?v=teBBG5Q78gk) — 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-20x-y-2z
- Whole chapter: https://www.bctengineer.com/md/semester_4/numerical_method/chapter-3

## Question

Use Gauss Seidel method to solve the following equations:
20x + y - 2z = 17
3x + 20y - z = -18
2x - 3y + 20z = 25

## Short answer

The system is diagonally dominant, so Gauss-Seidel converges quickly from zero initial guess. After 3 iterations the solution is x = 1.0000, y = -1.0000, z = 1.0000.

## 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
Diagonal coefficients: 20 > 1 + 2, 20 > 3 + 1, 20 > 2 + 3. The system is diagonally dominant, so the method converges.

#### Step 2: Iteration formulas
```text
x(k+1) = (17 − y(k) + 2z(k)) / 20
y(k+1) = (−18 − 3x(k+1) + z(k)) / 20
z(k+1) = (25 − 2x(k+1) + 3y(k+1)) / 20
```

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 | 0.8500 | -1.0275 | 1.0109 |
| 2 | 1.0025 | -0.9998 | 0.9998 |
| 3 | 1.0000 | -1.0000 | 1.0000 |
| 4 | 1.0000 | -1.0000 | 1.0000 |
| 5 | 1.0000 | -1.0000 | 1.0000 |

Values of two successive iterations agree to 4 decimal places, so we stop.

**Answer: x ≈ 1.0000, y ≈ -1.0000, z ≈ 1.0000**

## Check yourself

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

1. Why does Gauss-Seidel converge for 20x + y - 2z = 17, 3x + 20y - z = -18, 2x - 3y + 20z = 25?
   A. The answer is a whole number
   B. In each equation 20 exceeds the sum of the other coefficients' magnitudes
   C. The coefficient of x is the same in every equation
   D. The determinant is 20
   Answer: B. 20 > 1+2, 20 > 3+1, 20 > 2+3, so the system is diagonally dominant. (Key point: Diagonal dominance)

2. Which formula is correct for y(k+1)?
   A. y(k+1) = (-18 - 3x(k+1) + z(k))/20
   B. y(k+1) = (-18 - 3x(k) + z(k))/20
   C. y(k+1) = (17 - x(k+1) + 2z(k))/20
   D. y(k+1) = (-18 + 3x(k+1) - z(k))/20
   Answer: A. Gauss-Seidel uses the newly computed x(k+1) but the old z(k) in the second equation. (Key point: Use of new values)

3. From zeros, what is y(1)?
   A. -0.9000
   B. -1.0500
   C. -1.0275
   D. 0.8500
   Answer: C. x(1) = 17/20 = 0.85; y(1) = (-18 - 3(0.85))/20 = -1.0275. (Key point: First iteration)

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