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.
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