Let W be a real symmetric matrix associated with a weighted 2-connected planar graph. It is important to study a fast algorithm to solve the linear system Wx = c, since the system has many various applicaions, for example to solve partial defferencial equations numerically. In this paper, a new algorithm for the solution of a linear system of equations by Δ-Y transformations is proposed, and a sufficient condition for using this algorithm is proved. We show that this algorithm solves in O (n3/2) time a linear system associated with a planar graph which is embedded a cylinder graph with n vertices.