# Theorem describing the vector form of sulutions to a linear system.

Put content here.**Theorem:** The general solution to a consistent linear system $Ax = b$ can be expressed in vector form as:
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$$x = p + c_1 v_1 + c_2 v_2 + \cdots + c_k v_k$$
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where:
- $p$ is a particular solution to $Ax = b$
- $v_1, v_2, \ldots, v_k$ are vectors that span the solution space of the homogeneous system $Ax = 0$
- $c_1, c_2, \ldots, c_k$ are free parameters (one per free variable)
- $k = n - \text{rank}(A)$ is the number of free variables
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**Geometric interpretation:** The solution set is an affine subspace — a translation of the null space by the particular solution $p$.
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**How to compute:**
1. Find any particular solution $p$ to $Ax = b$
2. Solve $Ax = 0$ to find the basis vectors $v_i$
3. Combine as shown above

# Parents

* Linear systems of equations