The method of symmetrical components is not effective for fault location in the case of untransposed lines, due to potential couplings between the sequence circuits. This paper proposes a non-iterative algorithm in the phase-coordinates for wide-area fault location on untransposed transmission networks. In doing so, first, an improved two-terminal method is suggested to accurately locate faults on untransposed lines. Next, an algorithm is proposed to infer voltage and current phasors at the faulted line ends without direct measurements, by taking advantage of the data provided by phasor measurement units (PMUs). Accordingly, the adverse effect of close instrument transformers transients on the estimation accuracy is minimized. Being highly nonlinear in terms of fault distance and impedance, the fault equations are derived and made linear in this paper by defining six suitable auxiliary variables. The resulting system of equations is solved using the least-squares method to obtain three-phase voltages and currents at the faulted line ends. A main feature of the proposed algorithm is that it only requires a limited number of current and voltage synchrophasors. An additional advantage of the proposed algorithm is that the faulted line is not required to be known a-priori. The proposed algorithm is validated using extensive simulation studies on the New England 39-bus test system, accounting for different fault locations, types and resistances.