This paper presents a new algorithm for solving the static traffic assignment problem that operates in the space of path flows. The algorithm uses a sequential equilibration technique by which origin-destination (O-D) pairs are equilibrated one at a time iteratively. This slope-based multipath algorithm (SMPA) inherits some insights from the gradient projection (GP) algorithm of Jayakrishnan et al., Dial's Algorithm B, and the recent GP method of Florian et al. However, the flow update mechanism of the SMPA is new, with different search directions for paths with higher and lower costs than the average cost for an O-D pair. It uses the slopes of cost functions in such a way as to bring the costs of different paths closer and enable faster convergence. Further, it does not require a line search for finding the move size. Computational experiments using the Sioux Falls, South Dakota, and Borman Corridor, Indiana, networks provide insights into the SMPA and compare its performance relative to the GP algorithm of Florian et al. The results illustrate that the SMPA has a superior rate of convergence.