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
$$S = \{(-2y+7, y, y-4)\mid y\in\mathbb{R}\}.$$
Best Answer
Suppose you and I (and a few hundred other people) independently solve a system of linear equations, and we want to compare our answers. If we have taken Gaussian elimination to reduced row-echelon form, and have expressed our answers with the free variables as parameters, we can immediately compare our answers; if there is any difference at all, then (at least) one of us is wrong. If I used the free variables, and you used the leading variables, then we have some calculating to do before we can see whether we got the same answer.
So, it's really a matter of selecting a standard form so everyone can agree on an answer without having to do a lot more calculation to check.
EDIT: Lots of things that come later depend on solving systems of equations (e.g., finding a basis for the kernel of a linear transformation, finding a basis for an eigenspace of a matrix), and it continues to be useful to have a standard form.