Lecture 5 — System of Linear Equations

1. Introduction

A system of linear equations is a collection of two or more linear equations involving the same set of variables. The goal is to find the values of these variables that satisfy all equations simultaneously.

2. General Form

The general form of a system of linear equations is:

a₁₁x₁ + a₁₂x₂ + ... + a_1nx_n = b₁
a₂₁x₁ + a₂₂x₂ + ... + a_2nx_n = b₂
...
a_m1x₁ + a_m2x₂ + ... + a_mnx_n = b_m

3. Flow Diagram of Solving System

Define Equations

→

Represent in Matrix Form

→

Apply Method (Substitution / Elimination / Matrix)

→

Find Solution

4. Types of Solutions

Unique Solution: Only one set of values satisfies the equations.
No Solution: Equations are inconsistent.
Infinitely Many Solutions: Equations represent the same line/plane.

5. Example

Consider the system:

2x + y = 5
x - y = 1

Solution using elimination:

From 2nd equation: x = 1 + y
Substitute into 1st: 2(1+y) + y = 5 → 2 + 2y + y = 5 → 3y = 3 → y = 1
Then, x = 1 + 1 = 2
Solution: (x, y) = (2, 1)

6. Methods of Solving

Graphical Method
Substitution Method
Elimination Method
Matrix Method (AX = B)

7. Homogeneous and Non-Homogeneous Systems

(a) Homogeneous System

A system is called homogeneous if all constant terms are zero. General form:

AX = 0

- Always has at least one solution: the trivial solution (X = 0).
- If rank(A) = n (variables), only trivial solution exists.
- If rank(A) < n, infinitely many non-trivial solutions exist.

Example:

x + y = 0
2x + 2y = 0

Here, both equations are multiples → rank(A) = 1, variables = 2 → infinite non-trivial solutions.

(b) Non-Homogeneous System

A system is called non-homogeneous if the constants are not all zero. General form:

AX = B

The existence of solutions depends on rank conditions:

If rank(A) = rank([A|B]) = n → unique solution.
If rank(A) = rank([A|B]) < n → infinite solutions.
If rank(A) ≠ rank([A|B]) → no solution (inconsistent system).

Example:

x + y = 5
2x + 2y = 10

Here, rank(A) = rank([A|B]) = 1 < 2 → infinitely many solutions.

8. Flow Diagram: Rank and Solution Types

The existence of solutions depends on the comparison of rank(A) and rank([A|B]):

Start: System AX = B

↓

Check rank(A) and rank([A|B])

↓

Case 1: rank(A) = rank([A|B]) (Consistent)

↓

If rank = n (variables)

↓

Unique Solution

If rank < n

↓

Infinite Solutions

Case 2: rank(A) ≠ rank([A|B])

↓

Inconsistent System (No Solution)

Lecture 5: System of Linear Equations