# Parametric form of the solution set of a system of linear equations

Put content here**Definition:** The **parametric vector form** expresses the solution set of a linear system as a vector equation with free parameters:
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$$x = p + s_1 v_1 + s_2 v_2 + \cdots + s_k v_k$$
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where:
- $p$ is a **particular solution** (a fixed vector)
- $v_1, v_2, \ldots, v_k$ are **direction vectors** (one per free variable)
- $s_1, s_2, \ldots, s_k$ are free parameters in $\mathbb{R}$
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**Example:** For the system with solution $x_1 = 3 + 2x_3$, $x_2 = 1 - x_3$, $x_3$ free:
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$$\begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} = \begin{bmatrix} 3 \\ 1 \\ 0 \end{bmatrix} + x_3 \begin{bmatrix} 2 \\ -1 \\ 1 \end{bmatrix}$$
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Geometrically, this describes a line through $(3, 1, 0)$ in the direction $(2, -1, 1)$.

# Parents

* Linear systems of equations