FIR Filter Deconvolution
A few comments:


An overdetermined system (with more equations than unknowns) can have exact solutions.
An overdetermined system can have approximate solutions; for example, in the least-squares sense, where $\mathbf{x}=(H^TH)^{-1}H^T\mathbf{y}$ minimizes $||H\mathbf{x}-\mathbf{y}||$.
Deconvolution is usually performed in the frequency domain, where $X(f) = Y(f)/H(f)$.