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We give an algorithm for deciding E-unification problems for linear standard equational theories (linear equations with all shared variables at a depth less than two) and varity 1 goals (linear equations with no shared variables). We show that the algorithm halts in quadratic time for the non-uniform E-unification problem, and linear time if the equational theory is varity 1. The algorithm is still polynomial for the uniform problem. The size of the complete set of unifiers is exponential, but membership in that set can be determined in polynomial time. For any goal (not just varity 1) we give a NEXPTIME algorithm.