`Consider the linear system
`

`We can represent it in matrix form as
`

`If we multiply one equation by a constant (for example, multiply
first equation by ) we obtain an equivalent system,
i.e. a system with the same solution. In our example,
we can see that
`

`If we add two equations together, and replace the second
equation by the result, we obtain an equivalent system also.
For example, first equation plus the second give
`

`In conclusion,
multiplying one equation by a constant
or replacing
one equation by the sum of itself plus another equation
lead to equivalent systems.
`

`If we combine these two operations into a single
step we conclude that we can replace one equation
by the sum of itself plus a multiple of another equation
without modifying the solution of the system. For example, multiplying
the first equation of the system () by and adding
it to the second equation leads to the equivalent system
`