what do free variable and leading variables mean?
The terms "leading variable" and "free variable" are usually defined for the matrix representing a system, and only when the matrix is in row-echelon form.
The augmented matrix for your system is $$ \left( \begin{array}{c c c c|c} 1&1&1&1&6\\ 1&0&1&1&4\\ 1&0&1&0&2\\ \end{array}\right) $$ Notice that each column corresponds to a variable.
Do you know how to use elementary row operations to bring this matrix to row-echelon form?
In row-echelon form, the first non-zero entry in each row (if there is one) is a $1$, and the column it is in corresponds to a leading variable; the columns that don't have that kind of $1$ correspond to the free variables. Essentially, columns that don't have a leading variable, have a free variable.
So for example if a row-echelon form is given by $$ \left( \begin{array}{c c c c c|c} 1&2&3&4&5&6\\ 0&0&1&7&8&9\\ 0&0&0&1&\pi&\sqrt2\\ \end{array}\right) $$ then the 1st, 3rd, and 4th variables are leading variables while the 2nd and 5th variables are free variables.
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.