Solving Systems of Linear Equations

Solving Linear Systems

When we have multiple linear equations, we look for a solution that satisfies all of them simultaneously.

The Three Possibilities

For any system of linear equations, there are only three possible outcomes:

Unique Solution (Intersects at one point)
Infinitely Many Solutions (Lines overlap)
No Solution (Parallel lines)

Solving Techniques

1. Cramer’s Rule

Uses determinants to solve systems where the coefficient matrix is invertible. It’s elegant but computationally expensive for large systems.

2. Inverse Matrix Method

If $AX = B$ and $A$ is invertible, then $X = A^{-1}B$ .

3. Gauss Elimination

The most robust method. It involves transforming the system’s augmented matrix into:

Row Echelon Form (REF)
Reduced Row Echelon Form (RREF)

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Homogeneous vs Non-Homogeneous

Homogeneous: $AX = 0$ . Always has at least the trivial solution ( $X=0$ ).
Non-Homogeneous: $AX = B$ (where $B \neq 0$ ).

Key Concept: Calculating the solution involves identifying

Free Variables

Independent vs Dependent variables.

All Chapters in this Book

Lesson 1

Vector and Matrices

Introduction to vectors, matrices, and their fundamental operations in linear algebra.

Lesson 2