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

# Consistency of a linear system

Mathematics III · Chapter 1: Determinants and Matrices

- Asked in: 2070 Asar, 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-eight
- 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 if consistent.

## Short answer

The system is consistent with infinitely many solutions. Let z = k. Then y = 1 − 2k, x = 2 + k, for any real k.

## 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. The system is consistent when:
   A. |A| ≠ 0 only
   B. rank A = rank [A|B]
   C. rank A > rank [A|B]
   D. B = 0
   Answer: B. Rouché–Capelli condition. (Key point: Consistency test)

2. After elimination, the last row is:
   A. (0 0 0 | 0)
   B. (0 0 0 | 1)
   C. (0 0 1 | 0)
   D. (1 0 0 | 0)
   Answer: A. R₃−R₂ gives zeros. (Key point: Reduced form)

3. Both ranks are 2 with 3 unknowns, so:
   A. Unique solution
   B. No solution
   C. Exactly two solutions
   D. Infinitely many solutions
   Answer: D. Rank < unknowns gives infinitely many. (Key point: Number of solutions)

4. Solution with z = k:
   A. x = 2−k, y = 1+2k, z = k
   B. x = 3, y = 0, z = 0
   C. x = 2+k, y = 1−2k, z = k
   D. x = 1+k, y = 2−k, z = k
   Answer: C. y = 1−2k, x = 3−y−k = 2+k. (Key point: Solution)

## Close variants set in other exams

- [Consistency of a linear system](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Aconsistency-of-a-linear-system) — Mathematics III, chapter 1, asked 10×
- [Consistency of a linear system](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Aconsistency-of-a-linear-system-three) — Mathematics III, chapter 1, asked 9×
- [Consistency of a linear system](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Aconsistency-of-a-linear-system-six) — Mathematics III, chapter 1, asked 10×
- [Consistency of a linear system](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Aconsistency-of-a-linear-system-seven) — Mathematics III, chapter 1, asked 10×

## Related in Mathematics III

- [Consistency of a linear system](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Aconsistency-of-a-linear-system-five) — Mathematics III, chapter 1, asked 2×
- [Consistency of a linear system](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Aconsistency-of-a-linear-system-two) — Mathematics III, chapter 1, asked 8×
- [Consistency of a linear system](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Aconsistency-of-a-linear-system-four) — Mathematics III, chapter 1, asked 7×
- [Consistency of a linear system](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Aconsistency-of-a-linear-system-nine) — Mathematics III, chapter 1
- [Solve ODE by Laplace transform y''+3y'+2y=e^(−t)](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Asolve-ode-by-laplace-transform-y-two) — Mathematics III, chapter 3, asked 18×
- [Solve ODE by Laplace transform y''+2y'−3y=sint](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Asolve-ode-by-laplace-transform-y) — Mathematics III, chapter 3, asked 14×

## Related in other subjects

- [Solve y''+3y'+2y=e^(2x) sin x](https://www.bctengineer.com/question.md?key=semester_1%2Fmathematics_i%3Ade-d2-3d-2-e2x-sinx) — Mathematics I, chapter 4, asked 2×
- [Gauss elimination: x+2y+3z=6](https://www.bctengineer.com/question.md?key=semester_4%2Fnumerical_method%3Agauss-elimination-x-2y-3z) — Numerical Method, chapter 3, asked 3×
- [Solve dy/dx = (x+2y−3)/(2x+y−3)](https://www.bctengineer.com/question.md?key=semester_1%2Fmathematics_i%3Ade-homogeneous-linear-x-2y-3) — Mathematics I, chapter 4
- [Controllable canonical form of (z⁻¹+2z⁻²)/(1+4z⁻¹+3z⁻²)](https://www.bctengineer.com/question.md?key=semester_6%2Fdigital_control_system%3Accf-z1-2z2-4z1-3z2) — Digital Control System, chapter 5, asked 15×