$(30$ points$)$ Consider the linear program

minimize $2x_1+x_2$

subject $to$: $3x_1+x_{2}10,x_1+2x_{2}4,$

$x_{1}0,x_{2}0.$

$($a$)(5$ points$)$ Write its dual program.

$($b$)(20$ points$)$ Solve the dual program by hand with slack variables

and the one$-$phase simplex method.

$($c$)(5$ points$)$ Find the optimal solution of the primal program from

that of the dual program.