- 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.
- 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.