$4\mathrm{.}6-9.$ Consider the following problem.

Maximize $,Z=x_1+4x_2+2x_3,$

subject to

$4x_1+x_2+2x_{3}5$

$-x_1+x_2+2x_{3}10$

and

$x_{2}0,x_{3}0$

$($no nonnegativity constraint for $x_1).$

$($a$)$ Reformulate this problem so all variables have nonnegativity constraints.

D$,$I $($b$)$ Work through the simplex method step by step to solve the problem.

C $($c$)$ Use a software package based on the simplex method to solve the problem.