Given an arbitrary initial value $x_0^-$ for the differential-algebraic equation $A\dot{x}(t)+Bx(t)=f(t)$, an initial value $x_0^+$ can be selected from among all consistent initial values by means of the Laplace transform. We show that this choice is the only one that fulfills some simple, physically reasonable assumptions on linear systems’ behavior. Our derivation is elementary compared to previous justifications of the above Laplace transform based method. We also characterize $x_0^+$ by means of a system of linear equations involving $A$, $B$, derivatives of $f$, and $x_0^-$, which gives a new method to numerically calculate $x_0^+$.