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Use the standard maximum form of a linear program convention $\max c^{T}u$ subject to $Mu\le b$ and $u\ge 0$. Split the unrestricted variables as $x=x^{+}-x^{-}$ and $y=y^{+}-y^{-}$, with all four new variables nonnegative. The resulting linear program isIf equality-constraint standard form is preferred, add a nonnegative slack variable to each inequality.

Give the two inequalities nonnegative dual variables $\alpha ,\beta$. The dual linear program isThe paired inequalities encode equalities for the two original unrestricted variables. Equivalently, the dual linear program has $2\alpha =2$, $-\alpha +\beta =3$, $\beta \ge 4$, $5\alpha +\beta \ge 7$ and $\alpha ,\beta \ge 0$. Thus its only feasible point is $(\alpha ,\beta )=(1,4)$, of objective function value $13$.

This convention for standard maximum form of a linear program requires a maximizing linear function, upper-bound linear inequalities, and nonnegative variables. An unrestricted variable becomes the difference of two nonnegative variables; a lower-bound inequality changes sign. The corresponding dual linear program is $\min \{b^{T}y:A^{T}y\ge c,y\ge 0\}$. Adding nonnegative slack variables instead gives the equivalent equality-constraint standard form. Since standard-form terminology varies, state the chosen convention.