Clearly, Linear System II is easier to solve because the last equation directly gives us the value of \(z\text{.}\) We can then substitute this value back into the second equation to find \(y\text{,}\) and finally use both \(y\) and \(z\) in the first equation to solve for \(x\text{.}\) This step-by-step substitution process is straightforward and efficient.
In contrast, Linear System I does not provide any immediate values for the variables. We would need to use methods such as substitution or elimination, which can be more complex and time-consuming. Therefore, having a system in a form where variables can be easily isolated, as seen in Linear System II, significantly simplifies the solving process.
In this chapter, we will explore how to systematically transform any linear system into an equivalent form that is as straightforward to solve as Linear System II. This transformation is achieved through equation operations, leading us to the concept of reduced row echelon form (RREF), which allows us to read off solutions directly.
There are three elementary equation operations that can be performed on the linear system without changing the solution of the corresponding linear system:
Interchange two equations. For example, swapping Row 1 and Row 2:
