The paper aims at unifying the various mathematical tools used in the analysis of linear economic models. Using the theorem of the Separating Hyperplane as stated in Gale's 'The Theory of Linear Economic Models', McGraw-Hill, 1960, a wide class of results are obtained: the Frobenius Theorem, the various alternative forms of the productivity conditions for a Leontief Matrix, the non-substitution theorem, and properties of the singular jacobian of excess demand functions under gross substitution. The concept of a dominant diagonal matrix plays an important role in the entire analysis.