This note states without proof some results on operator equations of the form \[ a_{i1}x_ 1+...+a_{in}x_ n=b_ i,\quad i=1,...n \] with unknown \(x_ i\), where the \(a_{ij}\), \(b_ i\) and \(x_ i\) are bounded linear operators in the complex Banach space X, and \(a_{ij}\) and \(b_ i\) commute with \(a_{k\ell}\) and \(b_ k\) for \(i\neq k\). Let \(\delta_ 1=(a_{ij})^ n_{ij=1}\), \(\Delta_ 0:=\det \delta_ 1\) and \(\Delta_ 1,...,\Delta_ n\) be the determinants of the operator matrices obtained from \(\Delta_ 1\) by replacing the jth column by the column \(b_ 1,...,b_ n\), \(j=1,...,n\). The main result is that under certain invertibility conditions the Taylor joint spectrum of \((\Delta_ 0^{-1}\Delta_ 1,...,\Delta_ 0^{-1}\Delta_ n)\) is given by \[ \sigma (\Delta_ 0^{-1}\Delta_ 1,...,\Delta_ 0^{-1}\Delta_ n)=\{\lambda \in {\mathbb{C}}^ n| 0\in \sigma (b_ 1- \sum^{n}_{j=1}\lambda_ ja_{1j},...,b_ n- \sum^{n}_{j=1}\lambda_ ja_{nj}\}\}. \] Some applications to the theory of multiparameter systems of operators \[ P_ i(\lambda)=B_ i- \lambda_ 1A_{i1}-...-\lambda_ nA_{in},\quad i=1,...,n \] are given. The spectrum of \(P(\lambda)=(P_ 1(\lambda),...,P_ n(\lambda))\) is defined to be the set \(\sigma\) (P) of \(\lambda \in {\mathbb{C}}^ n\) such that each of the operators \(P_ i(\lambda)\) is not invertible. Topics like nonemptiness of the spectrum, the essential spectrum, the Fredholm property and the index of the system P(\(\lambda)\) are discussed.