Definition Two systems of linear equations are said to be equivalent to one another if they have the same sets of solutions.

Example The systems below are all equivalent to one another:

Indeed, the solution for all of these systems may be described in the form: (x, y, z ) = (3,2,1).

Why are we interested in the concept of equivalent systems? The reason for this is that systems of equations are often presented in such a way that the solutions are not immediately apparent. Thus in order to find the solutions of the system we need to transform it to another system for which the solutions are easier to find - while taking care not to add or subtract solutions along the way! In the example above, only the second system is presented in a way that makes it convenient to state the solution set; the first and third are much less conveniently written.

How do we make the transformation from one system to an equivalent system? This is done via the following basic operations and combinations of them:

1. Multiplying an equation by a non-zero constant.
2. Adding one equation (or a non-zero multiple of it) to another.
3. Interchanging between equations.

• Subtract 2*(Row1) from Row2 and place the result in the second row; subtract Row1 from Row2 and place in the third row. Leave Row1 as is.

x +   y        = 2          y +   z = 0          y + 2z = 4

•  Subtract Row2 from Row3, and place the result in row3. Leave Row1 and Row2 as they are.

x +   y        = 2          y +   z = 0                  z = 4