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The usual Cauchy matrix approach starts from a known plain wave factor vector inline image and known dressed Cauchy matrix inline image. In this paper, we start from a determining matrix equation set with undetermined inline image and inline image. From the determining equation set we can build shift relations for some defined scalar functions and then derive lattice equations. The determining equation set admits more choices for inline image and inline image and in the paper we give explicit formulae for all possible inline image and inline image. As applications, we get more solutions than usual multisoliton solutions for many lattice equations including the lattice potential KdV equation, the lattice potential modified KdV equation, the lattice Schwarzian KdV equation, NQC equation, and some lattice equations in ABS list.