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# Definition of solution set of a system of linear equations

Put content here**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.
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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
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The basic variables are expressed as formulas involving the free variables.
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**Example:** Suppose solving a system yields:
$$\begin{cases}\nx_1 = 3 + 2x_3 \\\nx_2 = 1 - x_3 \\\nx_3 \text{ is free}\n\end{cases}$$
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The parametric form is:
$$\begin{cases}\nx_1 = 3 + 2s \\\nx_2 = 1 - s \\\nx_3 = s
\end{cases}$$
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where $s \in \mathbb{R}$ is the free parameter.

# Parents

* Basic terminology