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

# Consistency of a linear system

Mathematics III · Chapter 1: Determinants and Matrices

- Asked in: 2072 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-six
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-1

## Question

Test the consistency of the system x + y + z = 3, x + 2y + 3z = 4 and 2x + 3y + 4z = 7 and solve completely if found consistent.

## Short answer

System is consistent because rank A = rank[A|B] = 2, with infinite solutions. Solution: x = 2+k, y = 1−2k, z = k (k ∈ ℝ).

## Answer

The system AX = B is consistent when rank A = rank [A|B]; we reduce the augmented matrix to echelon form to compare the two ranks.

```text
[A|B] = [1 1 1 | 3]
        [1 2 3 | 4]
        [2 3 4 | 7]
```

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

```text
[1 1 1 | 3]
[0 1 2 | 1]
[0 1 2 | 1]
```

**Step 2.** R₃ → R₃ − R₂:

```text
[1 1 1 | 3]
[0 1 2 | 1]
[0 0 0 | 0]
```

#### Consistency
rank A = 2 and rank [A|B] = 2. The ranks are equal, so the system is **consistent**. Since the rank (2) is less than the number of unknowns (3), there are **infinitely many solutions**, with 3 − 2 = 1 free variable.

#### Solution
Put z = k.
- y + 2k = 1 ⇒ y = 1 − 2k
- x + y + z = 3 ⇒ x = 3 − (1 − 2k) − k = 2 + k

**Answer: x = 2 + k, y = 1 − 2k, z = k, for any real k.**

Check with k = 1: (3, −1, 1): 3 − 1 + 1 = 3 ✓; 3 − 2 + 3 = 4 ✓; 6 − 3 + 4 = 7 ✓.

## Check yourself

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

1. Which condition is checked?
   A. |A| ≠ 0
   B. A is symmetric
   C. rank A = rank [A|B]
   D. rank A = 3
   Answer: C. Consistency means the ranks of A and [A|B] are equal. (Key point: Consistency test)

2. Here rank A = rank [A|B] = 2 with 3 unknowns. The system has:
   A. Infinitely many solutions
   B. A unique solution
   C. No solution
   D. Exactly two solutions
   Answer: A. Rank less than the number of unknowns gives infinitely many solutions. (Key point: Result of reduction)

3. How many free parameters?
   A. 2
   B. 0
   C. 3
   D. 1
   Answer: D. 3 - 2 = 1. (Key point: Free variable)

4. Taking z = k gives:
   A. x = 2 + k, y = 1 - 2k
   B. x = 3 - k, y = 1
   C. x = 2 - k, y = 1 + 2k
   D. x = 1, y = 1
   Answer: A. y + 2k = 1 and x + y + z = 3. (Key point: Solution)

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