The Laplace transforms with respect to time of the governing equations of one-dimensional thermoelasticity are obtained, with the displacement and temperature fields coupled. Elimination of the Laplace transform of either field variable between the resulting linear, simultaneous, and coupled ordinary differential equations results in ordinary differential equations in the other field variable. The characteristic equations are identical for both variables. Depending on the sequence in which the elimination is carried out, the solutions are obtained in two different forms. The equivalence of these forms is established by using the properties of the characteristic roots. (This process provides an alternative to the state approach recently developed by the authors.) Applications to problems pertaining to a half-space and a layer of finite thickness are presented.