---
title: "Iterative method: 4 equations"
subject: "Numerical Method"
chapter: 3
key: "semester_4/numerical_method:iterative-method-four-equations"
source: https://www.bctengineer.com/q/semester_4/numerical_method/iterative-method-four-equations
site: BCT Engineering
---

# Iterative method: 4 equations

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

- Asked in: 2068 Bhadra, 8 marks
- Video: [Gauss Seidel Method | Iterative Method | Numerical Methods | Problems](https://www.youtube.com/watch?v=gxy6VI1hEfs) — Dr.Gajendra Purohit and 2 more
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/numerical_method/iterative-method-four-equations
- Whole chapter: https://www.bctengineer.com/md/semester_4/numerical_method/chapter-3

## Question

Solve the following set of linear equations using a suitable iterative method.
2x + y + z - 2w = -10
4x + 2z + w = 8
3x + 2y + 2z = 7
x + 3y + 2z - w = -5

## Short answer

The Gauss-Seidel method is used after reordering equations for better diagonal dominance. Starting from zero, after about 19 iterations the solution converges to x≈5.000, y≈6.000, z≈-10.000, w≈8.000.

## 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
As given, the system is not diagonally dominant (for example, the coefficient of y in the second equation is 0 and the x coefficient in the first equation is only 2). Reorder the equations so that the largest available coefficient is on the diagonal:

```text
4x      + 2z +  w =   8     (given eq. 2)
3x + 2y + 2z      =   7     (given eq. 3)
 x + 3y + 2z −  w =  −5     (given eq. 4)
2x +  y +  z − 2w = −10     (given eq. 1)
```

The diagonal entries are 4, 2, 2, −2. Only the first row is strictly dominant, but for this order the Gauss-Seidel iteration matrix has spectral radius 0.59 < 1 (checked), so the iteration converges.

#### Step 2: Iteration formulas
```text
x = (8 − 2z − w) / 4
y = (7 − 3x − 2z) / 2
z = (−5 − x − 3y + w) / 2
w = (10 + 2x + y + z) / 2
```

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

#### Step 3: Iterations
| Iter | x | y | z | w |
|---|---|---|---|---|
| 0 | 0.000 | 0.000 | 0.000 | 0.000 |
| 1 | 2.000 | 0.500 | -4.250 | 5.125 |
| 2 | 2.844 | 3.484 | -6.586 | 6.293 |
| 3 | 3.720 | 4.506 | -7.973 | 6.986 |
| 4 | 4.240 | 5.113 | -8.796 | 7.398 |
| 5 | 4.549 | 5.473 | -9.285 | 7.643 |
| 6 | 4.732 | 5.687 | -9.576 | 7.788 |
| 7 | 4.841 | 5.814 | -9.748 | 7.874 |
| 8 | 4.906 | 5.890 | -9.850 | 7.925 |
| 9 | 4.944 | 5.935 | -9.911 | 7.956 |
| 10 | 4.967 | 5.961 | -9.947 | 7.974 |
| 11 | 4.980 | 5.977 | -9.969 | 7.984 |
| 12 | 4.988 | 5.986 | -9.981 | 7.991 |
| 13 | 4.993 | 5.992 | -9.989 | 7.994 |
| 14 | 4.996 | 5.995 | -9.993 | 7.997 |
| 15 | 4.998 | 5.997 | -9.996 | 7.998 |
| 16 | 4.999 | 5.998 | -9.998 | 7.999 |
| 17 | 4.999 | 5.999 | -9.999 | 7.999 |
| 18 | 4.999 | 5.999 | -9.999 | 8.000 |
| 19 | 5.000 | 6.000 | -10.000 | 8.000 |

Between the last two iterations every unknown changes by less than 0.0005, so we stop. (Check: x = 5, y = 6, z = −10, w = 8 satisfy all four equations exactly.)

**Answer: x ≈ 5.000, y ≈ 6.000, z ≈ -10.000, w ≈ 8.000**

## Check yourself

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

1. Why are the equations re-ordered before iterating?
   A. To put the largest available coefficients on the diagonal
   B. To make the right-hand sides positive
   C. To eliminate w
   D. To make the system symmetric
   Answer: A. Iterative methods work best when the diagonal is as dominant as possible. (Key point: Reordering)

2. Only the first row is strictly dominant, yet the iteration is used. Why is that acceptable?
   A. Diagonal dominance is never needed
   B. The Gauss-Seidel iteration matrix has spectral radius 0.59 < 1, so it converges
   C. Because zero is a good starting guess
   D. Because there are four equations
   Answer: B. Diagonal dominance is sufficient but not necessary; spectral radius below 1 guarantees convergence. (Key point: Convergence despite weak dominance)

3. Which formula for w comes from the given equation 2x + y + z - 2w = -10?
   A. w = (8 - 2z - w)/4
   B. w = (-10 - 2x - y - z)/2
   C. w = (10 + 2x + y + z)/2
   D. w = (7 - 3x - 2z)/2
   Answer: C. Solving for w: 2w = 2x + y + z + 10. (Key point: Formula for w)

4. What is the converged solution?
   A. x = 5, y = 6, z = -10, w = 8
   B. x = 8, y = 5, z = 6, w = -10
   C. x = 5, y = 6, z = 10, w = 8
   D. x = 2, y = 0.5, z = -4.25, w = 5.125
   Answer: A. Check: 2(5)+6-10-16 = -10, and the other equations also hold. (Key point: Solution)

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