---
title: "Consistency of a linear system"
subject: "Mathematics III"
chapter: 1
key: "semester_3/mathematics_iii:consistency-of-a-linear-system-two"
source: https://www.bctengineer.com/q/semester_3/mathematics_iii/consistency-of-a-linear-system-two
site: BCT Engineering
---

# Consistency of a linear system

Mathematics III · Chapter 1: Determinants and Matrices

- Asked in: 2080 Baishakh, 5 marks; 2078 Bhadra, 5 marks (asked 2 times)
- Video: [Consistency of system of linear equations using rank method](https://www.youtube.com/watch?v=povS4KL3nR8) — maths voice
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/mathematics_iii/consistency-of-a-linear-system-two
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-1

## Question

Test the consistency of the system of equations x + 2y − z = 0, 2x + 3y + z = 10, 3x − y − 7z = 1 and solve completely if found consistent.

## Short answer

The system is consistent because rank A = rank [A|B] = 3. Unique solution: x = 31/5, y = −43/25, z = 69/25.

## Answer

A system AX = B is consistent if rank A = rank [A|B].

```text
[A|B] = [1  2 −1 |  0]
        [2  3  1 | 10]
        [3 −1 −7 |  1]
```

**Step 1.** R₂ → R₂ − 2R₁, R₃ → R₃ − 3R₁:

```text
[1  2 −1 |  0]
[0 −1  3 | 10]
[0 −7 −4 |  1]
```

**Step 2.** R₃ → R₃ − 7R₂:

```text
[1  2 −1 |   0]
[0 −1  3 |  10]
[0  0 −25 | −69]
```

#### Consistency
rank A = 3 = rank [A|B] = number of unknowns. (Also |A| = 25 ≠ 0.) So the system is **consistent** and has a **unique solution**.

#### Solution
- −25z = −69 ⇒ z = 69/25
- −y + 3z = 10 ⇒ y = 3(69/25) − 10 = (207 − 250)/25 = −43/25
- x + 2y − z = 0 ⇒ x = −2y + z = 86/25 + 69/25 = 155/25 = 31/5

**Check (eq. 3):** 3(31/5) − (−43/25) − 7(69/25) = (465 + 43 − 483)/25 = 25/25 = 1 ✓.

**Answer: x = 31/5, y = −43/25, z = 69/25.**

## Check yourself

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

1. The system is consistent when
   A. |A| = 0
   B. rank A > rank [A|B]
   C. rank A = rank [A|B]
   D. B = 0
   Answer: C. Equal ranks mean consistency. (Key point: Criterion)

2. Why is the solution unique?
   A. Rank is 2
   B. Rank is 3, equal to the number of unknowns
   C. The determinant is 0
   D. There are fewer equations than unknowns
   Answer: B. Rank = 3 = n gives one solution. (Key point: Unique solution)

3. What is z?
   A. 69/25
   B. 25/69
   C. -69/25
   D. 31/5
   Answer: A. From -25z = -69. (Key point: Solution)

4. x equals
   A. 155/5 with y = 43/25
   B. -31/5
   C. 69/25
   D. 31/5
   Answer: D. x = -2y + z = 86/25 + 69/25 = 31/5. (Key point: Back-substitution)

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