4. In an LP in which there are several unrestricted variables, a transformation of the type Xj = xj - xj, xj, xj ;::: 0 will double the corresponding number of nonnegative variables.

We can, instead, replace k unrestricted variables with exactly k + 1 nonnegative variables by using the substitution Xj = xi - w, xi, w ;::: O. Use TORA, Solver, or AMPL to show that the two methods produce the same solution for the following LP:

Maximize z = ~2xt + 3X2 - 2xJ subject to 4Xt - X2 - 5xJ = 10 2Xl + 3X2 + 2X3 = 12