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:
For example:
Let S be the solution set of the system
x+y+z=3y−z=4
Using the free variable z as the parameter
S={(−2z−1,z+4,z)∣z∈R}.
Using the leading variable y as the parameter
S={(−2y+7,y,y−4)∣y∈R}.