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Lesson 9: Systems of Linear Equations – Introduction

1. What is a System of Linear Equations?

A system is a set of linear equations with the same variables.

Example (2 equations, 2 variables):

\[ \begin{cases} 2x + 3y = 8 \\ 4x - y = 2 \end{cases} \]

We want to find values of x and y that satisfy all equations simultaneously.

Exercise 1

Which is NOT x² a linear equation?

2. Matrix Form of a System

Any system Ax = b can be written using matrices.

Example:

\[ \begin{pmatrix} 2 & 3 \\ 4 & -1 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 8 \\ 2 \end{pmatrix} \]

A = coefficient matrix, x = variable vector, b = constant vector.

Exercise 2

Write this system in matrix form:
\[ \begin{pmatrix} 1 & 2 \\ 3 & -1 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 5 \\ 4 \end{pmatrix} \] x + y =
x + y =

3. Types of Solutions

A system can have:

We use Gaussian elimination to solve and classify systems.

Exercise 3

Which systems have exactly one solution? (Select all that apply)

Summary – Lesson 9

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