Definition of basic/dependent/leading variable in a linear system

Definition: In the echelon form of a linear system, a basic variable (also called a dependent or leading variable) is a variable that corresponds to a pivot column in the coefficient matrix.

A pivot is the first nonzero entry in each nonzero row of the echelon form. The column containing a pivot identifies the basic variable for that row.

Properties:

• Basic variables are determined by the pivot positions
• Their values are computed from the free variables through back-substitution
• The number of basic variables equals the rank of the matrix

Example: In the echelon form:
$$\begin{bmatrix} \boxed{2} & 3 & 1 & | & 5 \\ 0 & \boxed{1} & -2 & | & 3 \\ 0 & 0 & 0 & | & 0 \end{bmatrix}$$

$x_1$ and $x_2$ are basic variables (pivots in columns 1 and 2).