Definition of solution set of a system of linear equations

Definition: The parametric form of the solution set expresses the solutions in terms of one or more free parameters that can take any real value.

When solving a linear system, variables are classified as:

• Basic (leading) variables — determined by the pivot positions in echelon form
• Free variables — assigned arbitrary parameter values

The basic variables are expressed as formulas involving the free variables.

Example: Suppose solving a system yields:
$$\begin{cases}\nx_1 = 3 + 2x_3 \\\nx_2 = 1 - x_3 \\\nx_3 \text{ is free}\n\end{cases}$$

The parametric form is:
$$\begin{cases}\nx_1 = 3 + 2s \\\nx_2 = 1 - s \\\nx_3 = s \end{cases}$$

where $s \in \mathbb{R}$ is the free parameter.