In this paper, we study the solvability analysis of arbitrarily high-order linear differential-algebraic equations (DAEs) with time-varying coefficients, using the algebraic-behavior approach. We propose a concept of strangeness-index and construct condensed forms for high-order linear DAEs. We also discuss other structural properties like the existence and uniqueness of a solution, consistency and smoothness requirements for an initial vector and for an inhomogeneity. Our work extends the algebraic approach for DAEs and combines this approach with the behavior approach to establish a reformulation algorithm that reveals an underlying ordinary differential equation (ODE) and all hidden constraints in the DAE. This direct treatment of the system addresses the limitations of the classical approach to transforming the system into a first-order DAE. We illustrate our theoretical results with applications in mechanical systems and electrical circuits. This comprehensive study into general high-order systems is a natural progression from the extensive study of first and second-order DAEs.