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

# Consistency of a linear system

Mathematics III · Chapter 1: Determinants and Matrices

- Asked in: 2074 Asoj, 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-five
- 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, 2x + 3y + 4z = 7 by using the rank of matrix method and solve if consistent.

## Short answer

rank A = rank[A|B] = 2 < 3, so consistent with infinitely many solutions. Solution: x = 2+k, y = 1−2k, z = k (k ∈ ℝ).

## Answer

A system AX = B is consistent if rank A = rank [A|B]. A unique solution needs this rank to equal the number of unknowns; otherwise there are infinitely many.

```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. After reduction, rank A and rank [A|B] are:
   A. 3 and 3
   B. 2 and 3
   C. 1 and 2
   D. 2 and 2
   Answer: D. Both have two non-zero rows, so the system is consistent. (Key point: Ranks)

2. Rank 2 with 3 unknowns means:
   A. Unique solution
   B. No solution
   C. Infinitely many solutions with 1 free variable
   D. Infinitely many solutions with 2 free variables
   Answer: C. Free variables = 3 - 2 = 1. (Key point: Solution type)

3. Which operation makes the third row zero?
   A. R3 -> R3 + R2
   B. R3 -> R3 - R2
   C. R3 -> R3 - R1
   D. R3 -> 2R3
   Answer: B. After step 1 rows 2 and 3 are identical, so subtract. (Key point: Elimination step)

4. With z = k, the solution is:
   A. x = 2 + k, y = 1 - 2k, z = k
   B. x = 2 - k, y = 1 + 2k, z = k
   C. x = 3, y = 1, z = k
   D. x = 2 + k, y = 1 + 2k, z = k
   Answer: A. y = 1 - 2k from y + 2z = 1, and x = 3 - y - z = 2 + k. (Key point: General solution)

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