Budget hyperplane in n dimensions
Let matrix $A = \begin{bmatrix} p_1 &amp; p_2 &amp; \ldots &amp; p_n \end{bmatrix}$. Let $\mathbf{x}^*$ be a fixed solution to $A \mathbf{x} = c$. Then for any vector $\mathbf{u}$ that belongs to the null space of $A$, we have $A \mathbf{u} = 0$ hence $\mathbf{x} = \mathbf{x}^* + \mathbf{u}$ is also a solution (furthermore, all solutions $\mathbf{x}$ can be written this way). The dimension of the set of solutions to $A\mathbf{x} = c$ is thus the dimension of the null space of $A$.

By the rank-nullity theorem, the rank of matrix $A$ plus the nullity (dimensionality of the null space) of matrix $A$ equals the number of columns of $A$:

$$ \operatorname{rank}(A) + \operatorname{nullity}(A) = n$$

Matrix A is rank 1 (assuming not all $p_i$ are zero) and hence the nullity is $n-1$.