Problem $1(25$ points$).$ Solve the following LPs using the two$-$phase simplex method.

ma$\backslash$xi mize,$6$x$\_(1)+$x$\_(2),$

subject to $-$x$\_(1)+3$x

$,$x$\_(1)-3$x$\_(2),=6$

$,$x$\_(1)+$x$\_(2),>=1$

$,$x$\_(1),$x$\_(2),>=0$

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$.$

Problem $2(25$ points$).$ Solve the following LPs using the big$-$M method.

ma$\backslash$xi mize$6$x$\_(1)+$x$\_(2)$

subject to $-$x$\_(1)+3$x

x$\_(1)-3$x$\_(2)=6$

x$\_(1)+$x$\_(2)>=1$

x$\_(1),$x$\_(2)>=0$