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This two part paper proposes a methodology of designing and analyzing an inverse controller for stable fuzzy interval linear systems according to fuzzy interval arithmetic. The first part aims at revisiting the solving of first order fuzzy linear equations. The general solution is expressed in a compact form which is directly used for deriving the condition of the solution existence. The suggested method is inspired from the a-cut principle where the fuzzy interval arithmetic is defined in terms of well-established interval arithmetic operations on closed intervals of real numbers. The so-built solution is useful for the inverse controller synthesis dealt with in part II of this paper. The validity of the proposed method is illustrated by a simulation example.