TITLE: what do free variable and leading variables mean?
QUESTION [12 upvotes]: What do the leading variables and free variables in a matrix mean? I have the system below and am trying to understand which are which. I searched a lot for this, please help me ! $$w + x + y + z = 6 \qquad w + y + z = 4 \qquad w + y = 2$$

REPLY [3 votes]: If a set of linear equations can be expressed as let's say
a = 3x + 4y + 5z - 12
b = 2x + 8y + z - 11
c = 9x + 7y -z - 15
where

The left hand variables don't appear on the right side and vice versa.
On the left side, there is only one variable.

Then,

x, y and z can take values of any combination and are called free variables.
a, b and c are dependent on the above free variables (x, y and z) and cannot be any combination. a, b and c are called pivot or leading variables.

One more thing, depending upon how we form the above equations the pivot and free variable might be changed. For ex.,
a = x + y + z ==> a is pivot =>  x, y, z are free variables.
The same equation can be expressed as
x = a - y - z ==> x is pivot =>  a, y, z are free variables.
I thought of explaining the free/leading variables in a non-matrix way, since the original query was in a non-matrix way. Thanks to Gilbert Strang for getting me again started with Linear Algebra.