  When there are no constraints upon the solutions of the equation
$\mathbf{A}\mathbf{\xi}= \mathbf{y},$ where $\mathbf{A}$ is a $K\times
N-$matrix, $\mathbf{\xi}\in\mathbb{R}^N$ and $\mathbf{y}\in\mathbb{R}^K$ a
given vector, the description of the set of solutions as $\mathbf{y}$ varies in
$\mathbb{R}^K$ is well known. But this is not so when the solutions are
required to satisfy $\mathbf{\xi} \in \mathcal{K}\prod_{i\leq j\leq N}
[a_j,b_j],$ for finite $a_j\leq b_j: 1\leq j\leq N.$ Here we provide a
description of the set of solutions as a surface in the constraint set,
parameterized by the Lagrange multipliers that come up in a related
optimization problem in which $\mathbf{A}\mathbf{\xi} = \mathbf{y}$ appears as
a constraint. It is the dependence of the Lagrange multipliers on the data
vector $\mathbf{y}$ that determines how the solution changes as the datum
changes. The geometry on the solutions is inherited from a Riemannian geometry
on the set of constraints induced by the Hessian of an entropy of the
Fermi-Dirac type which is the objective in the restatement of the optimization
problem mentioned above. We prove that the set of solutions is contained in
$\ker(\mathbf{A})^\perp$ in the metric defined as the Hessian of the entropy.
