In describing the solution of a system of linear equations with many solutions, why do we use a free variable as a parameter to describe the other variables in the solution? Why do we not we use a leading variable? Since by the commutative property of addition we can swap between the free and leading variables, e.g. x + y + z = x + z + y; the solution set will essentially be the same (albeit having different orders).
Definitions:
- A free variable is a parameter that is not a leading variable.
- A leading variable is the first variable that has a non-zero coefficient in reduced form.
- These definitions are most easily understood with respect to the Echelon form of a system of linear equations expressed as a Matrix. See this link for details.
For example:
Let $S$ be the solution set of the system $\begin{align*} x+y+z &= 3\\ y-z &= 4 \end{align*}$
Using the free variable $z$ as the parameter $S = \{(-2z-1, z+4, z)\mid z\in\mathbb{R}\}.$
Using the leading variable $y$ as the parameter $S = \{(-2y+7, y, y-4)\mid y\in\mathbb{R}\}.$