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

# Consistency of a linear system

Mathematics III · Chapter 1: Determinants and Matrices

- Asked in: 2075 Chaitra, 5 marks
- 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-four
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-1

## Question

Test the consistency of the system by the matrix rank method and solve completely if found consistent: x + 2y − z = 3, 2x + 3y + z = 10, 3x − y − 7z = 1.

## Short answer

The augmented matrix reduces to row-echelon form with three pivots, so rank A = rank [A|B] = 3. The system is consistent with unique solution x = 19/5, y = 8/25, z = 36/25.

## Answer

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

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

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

```text
[1  2 −1 |   3]
[0 −1  3 |   4]
[0 −7 −4 |  −8]
```

**Step 2.** R₃ → R₃ − 7R₂:

```text
[1  2 −1 |   3]
[0 −1  3 |   4]
[0  0 −25 | −36]
```

#### Consistency
rank A = 3 = rank [A|B] = number of unknowns. The system is **consistent** with a **unique solution** (|A| = 25 ≠ 0).

#### Solution (back substitution)
- −25z = −36 ⇒ z = 36/25
- −y + 3z = 4 ⇒ y = 3(36/25) − 4 = (108 − 100)/25 = 8/25
- x + 2y − z = 3 ⇒ x = 3 − 16/25 + 36/25 = (75 + 20)/25 = 95/25 = 19/5

**Check (eq. 3):** 3(19/5) − 8/25 − 7(36/25) = (285 − 8 − 252)/25 = 1 ✓.

**Answer: x = 19/5, y = 8/25, z = 36/25.**

## Check yourself

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

1. A system AX = B is consistent if and only if:
   A. |A| = 0
   B. rank A = rank [A|B]
   C. rank A < rank [A|B]
   D. A is symmetric
   Answer: B. Equal ranks of A and the augmented matrix mean a solution exists. (Key point: Consistency criterion)

2. Here rank A = rank [A|B] = 3 = number of unknowns. The solution is:
   A. None
   B. Infinitely many
   C. Unique
   D. Trivial only
   Answer: C. Equal ranks that equal the number of unknowns give a unique solution. (Key point: Type of solution)

3. Which operation gives the last row (0, 0, -25 | -36)?
   A. R3 -> R3 + 7R2
   B. R3 -> R3 - 7R2
   C. R3 -> R3 - R2
   D. R3 -> 7R3
   Answer: B. From (0,-7,-4|-8) subtract 7(0,-1,3|4). (Key point: Elimination)

4. From -25z = -36, what is z?
   A. 36/25
   B. -36/25
   C. 25/36
   D. 19/5
   Answer: A. z = 36/25, then y = 8/25 and x = 19/5. (Key point: Back substitution)

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