Why does a homogeneous first-order linear PDE have $n-1$ functionally independent solutions?

Question

Why does a homogeneous first-order linear PDE$$\sum_{i=0}^n\xi_i(\mathbf{x})\frac{\partial u}{\partial x_i}=0,$$where $\mathbf{x} = (x_1, x_2, …, x_n)$, have $n-1$ functionally independent solutions for $u(\mathbf{x})$?