Solve the following LPs using the two$-$phase simplex method.

maximize $6$x$1+$ x$2$

subject to $$x$1+3$x$26$

x$13$x$2=6$

x$1+$ x$21$

x$1,$ x$20$

Part I $(10$ points$)$: formulate the auxiliary $($phase$-$I$)$ LP$.($introduce the slack and auxiliary variables for

each of the constraints$)$

Part II $(5$ points$)$: determine the initial basic and non$-$basic variables for the phase$-$I LP$.$

Part III $(5$ points$)$: solve the phase$-$I LP using simplex method.

Part IV $(5$ points$)$: starting from the optimal basic and non$-$basic variables for phase$-$I LP$,$ remove the

auxiliary variables, and use the rest basic and non$-$basic variables to start simplex for the original LP