If we perform one of the following operations on a system of linear equations:

  - list the equations in a different order; or
  - multiply one of the equations by a non-zero real number; or
  - replace equation $j$ by "equation $j$ ${}+{}$ $c\times {}$ (equation $i$)", where $c$ is a non-zero real number,

then the new system will have exactly the same solutions as the original system. These are called **elementary operations** on the linear system.

==== Why do elementary operations leave the solutions of systems unchanged? ====

  * we are doing the same thing to the left hand side and the right hand side of each equation, so any solution to the original system will also be a solution to the new system; and
  * these operations are reversible, using operations of the same type, so any solution to the new system will also be a solution to the original system.