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.