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A new hybrid digital/analog scheme is proposed for lossy transmission of a Gaussian source over a bandwidth-matched Gaussian broadcast channel with source side information available at each receiver. The proposed scheme combines two schemes that were previously shown to achieve optimal point-to-point distortion/power tradeoff simultaneously at all receivers under two distinct conditions stated in terms of channel and side information quality parameters. For the two-receiver case, the combined scheme is shown to achieve the same kind of optimality for the entire region in the parameter space sandwiched between those two conditions. Crucial to this result is a new degree of freedom discovered in designing point-to-point hybrid digital/analog schemes with side information. When superimposed with analog transmission, the proposed scheme outperforms all previously known schemes even outside the optimality region in the parameter space.