# Rates of substitution

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.