Problem $2($Duality$)($ pts$.)$

Consider the following linear program:

max z $=2$x$1+5$x$2$

subject to x$1+3$x$2<=6$

$2$x$1+4$x$2<=9$

x$1,$ x$2>=0.$

$1.$ Write the dual of the above linear program.

$2.$ Match the complementary basic solution pairs.

$($P$)$ basic solutions $($D$)$ basic solutions

$(0,0)(1,0\mathrm{.}5)$

$(0,2)(0,1\mathrm{.}25)$

$(1,2)(0,0)$

$(1\mathrm{.}5,1\mathrm{.}5)(1,2/3)$

$3.$ From the complementary pairs, identify one pair where weak duality holds.

$4.$ Identify the optimal primal and dual solutions.