$1.$ The following is the final basic solution for phase $1$ in a linear programming problem, where $\backslash ($ a$\_\{1\}\backslash )$ and $\backslash ($ a$\_\{2\}\backslash )$ are the artificial variables for the two constraints:

Find conditions on the parameters a$,$ b$,$ c$,$ d$,$ e$,$ f$,$ g$,$ h$,$ i$,$ and j such that the following statements are true. You need not mention those parameters that can take on arbitrary positive or negative values. You should attempt to find the most general conditions possible.

$($a$)$ A basic feasible solution to the original problem has been found.

$($b$)$ The problem is infeasible.

$($c$)$ The problem is feasible, but some artificial variables are still in the basis. However, by performing update operations a basic feasible solution to the original problem can be found.

$($d$)$ The problem is feasible, but it has a redundant constraint.

$($e$)$ For the case $\backslash ($ a$=4,$ b$=1,$ c$=0,$ d$=0,$ e$=0,$ f$=-4,$ g$=0,$ h$=-1,$ i$=1\backslash ),$ and $\backslash ($ j$=0\backslash ),$ determine whether the system is feasible. If so$,$ find an initial basic solution for phase $2.$ Assume that the objective is to minimize $\backslash (\backslash$mathrm$\{$z$\}=\backslash )\backslash (\backslash$mathrm$\{$X$\}\_\{1\}+\backslash$mathrm$\{$x$\}\_\{4\}\backslash ).$