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