# Equivalent systems of linear equations

## Prerequisites

### Equivalent systems of linear equations

#### Introduction

• Given a system of linear equations:
• We can solve this to get:
• Given a second system:
• We solve this, and find that the solution is the same:
• Two systems that have the same solution set are described as "equivalent"

#### Operations that produce equivalent systems

• Let's start with a generic system:
• Exchanging the order of the equations: Has no effect on the solution set, and produces an equivalent system.
• Multiplying the equations by a (non-zero) real number: Has no effect on the solution set, and produces an equivalent system.
• Adding one equation to another: Has no effect on the solution set, and produces an equivalent system.

#### Uses

• Producing an equivalent system is useful because it can be easier to solve than the original.
• For example:
• We can subtract the second equation from the first to get:
• And we can subject the third equation from the second to get:
• This has solved the system.