In this paper, the delay-dependent stability of linear, high-order delay differential systems is investigated. First, two bounds of the unstable eigenvalues of the systems are derived. The two bounds are based on the spectral radius and the norms of the parameter matrices of the systems, respectively. We emphasise that the bounds of the unstable eigenvalues involve only the spectral radius and norms of the matrices of lower size. They can be obtained with much less computational effort and work well in practice for large problems. Then, using the argument principle, a computable stability criterion is presented which is a necessary and sufficient condition for the delay-dependent stability of the systems. Furthermore, a numerical algorithm is provided for checking the delay-dependent stability of the systems. Numerical examples are given to illustrate the main results.