Exercise $2$

Let

$(P)\{\begin{array}{ccc}\text{m}\text{a}\text{x}\text{Z}& =,2x_1,-,x_2,+,& 2x_3,\\ S.t,& 2x_1,+,x_2,& \\ ,& x_1,+,2x_2,-,2x_3,& \\ ,& x_2,+,2x_3,,& 5\\ ,& x_1,x_2,x_3,,& 0\end{array}$

Write the dual problem $(D)$ of $(P).$

Use the Complementary slackness theorem to find the optimal solution $($if exist$)$ for $(D),$ if $\stackrel{}{x}=(5,0,\frac{5}{2})$ is optimal for $(P).$