Fill content

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

Put content here**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).

# Parents

* Linear systems of equations