n-1, not n.

a hyperplane is defined by a single equation of the form:

, where not all are 0.

the solution set consists of all elements of the solution space of the associated homogeneous system:

, plus any (particular) vector lying on the hyperplane

(in other words we have a coset, or translate, of the nullspace of the homogeneous system).

since the homogeneous system has rank 1, its solution space has dimension n-1 (by the rank-nullity theorem), therefore....