Rates of substitution

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Arises when equations are split into dependent (or basic) variables and independent (or nonbasic) variables. In linear programming, we rewrite LaTeX: Ax=b as LaTeX: Bu + Nv = b, where LaTeX: u is the vector of basic variables and LaTeX: v is the vector of nonbasics. Then, the original equations are equivalent to LaTeX: \textstyle u = b' + Mv, where LaTeX: \textstyle b' = B^{-1} b and LaTeX: \textstyle M = -B^{-1}N. This implies that the rate of substitution between LaTeX: u_i and LaTeX: v_j is LaTeX: M_{(i, j)} because it describes the marginal rate at which LaTeX: u_i must change in response to a change in LaTeX: v_j to maintain the equations with all other LaTeX: v's held fixed.

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