Equivalent Systems of Linear Equations

What do we mean when we say that two systems of linear equations are equivalent to one another?

Example The systems below are all equivalent to one another:

x + y - z = 4
2x + y     = 8
 x - z = 2

x + y + z = 6
    y + 2z = 5

x + 3z = 6

3x + 2y - z = 12

3x + y - z =  10

y + z = 3

 

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.
Example Consider the following system of 3 equations in 3 unknowns:
 
  x +   y         =
2x + 3y +   z = 4 
  x + 2y + 2z = 6
Our goal is to apply the operations given above to transform this system into an equivalent system from which it is easy to find the solutions. We now do this step by step.  From the last form of the system we can deduce the following unique solution to the system:
                                   z = 4,  y = -4, and x = 2-(-4) = 6
Equivalently, we say that the unique solution to this system is (x, y, z) = (6, -4, 4).