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# Definition of free/independent variable in a linear system

Put content here**Definition:** In the echelon form of a linear system, a **free variable** (also called an **independent variable**) is a variable that does not correspond to any pivot column.
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Free variables are not determined by the system — they can be assigned any value. Once free variables are assigned values, the basic variables are uniquely determined through back-substitution.
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**Properties:**
- The number of free variables = total variables $-$ number of pivots = $n - \text{rank}(A)$
- Each free variable adds one dimension to the solution space
- Free variables are used as parameters in the parametric form of the solution
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**Example:** In the echelon form:
$$\begin{bmatrix} \boxed{2} & 3 & 1 & | & 5 \\ 0 & \boxed{1} & -2 & | & 3 \\ 0 & 0 & 0 & | & 0 \end{bmatrix}$$
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$x_3$ is a free variable (no pivot in column 3). It can take any value.

# Parents

* Linear systems of equations