In this paper, we derive new state transformations of linear systems with a time-varying delay in the state vector. We first provide a new algebraic and systematic method for computing forward state transformations to transform time-delay systems into a novel form where time-varying delay appears in the input and output vectors, but not in the state vector. In the new coordinate system, a Luenberger-type state observer with a guaranteed β-exponential stability margin can be designed. Then, a backward state transformation problem which allows us to reconstruct the original state vector of the system is investigated. By using both the forward and the backward state transformations, state observers for time-varying delay systems can be systematically designed. Conditions for ensuring the existence of the forward and backward state transformations and an effective algorithm for computing them are given in this paper. We illustrate our results by three examples and simulation results.