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A local generic position method is proposed to isolate the real roots of a bivariate polynomial system ∑={f(x,y),g(x,y)}. In this method, the roots of the system are represented as linear combinations of the roots of two univariate polynomial equations t(x)=0 and T(X)=0: {x = α, y = β -- α/s | α ε V(t(x)), β ε V(T(X)), ||β -- α| S}, where s, S are constants satisfying certain conditions. The multiplicities of the roots of Σ=0 are the same as that of the corresponding roots of T(X)=0. This representation leads to an efficient and stable algorithm to isolate the real roots of Σ.