a$)$ Write the optimal solution of P$,$ i$.$e$.$ values of all variables $($original and slack$)$ and z$-$

value and the optimal solution of its dual $($dual variables, surplus and dual objective

value$).$

From the column of BV and the RHS column : x$1=0,$ x$2=40,$ x$3=30,$ x$4=0,$ x$5=0,$ Z$=100$;

From the $(0)$ row: y $1=3,$ y $2=4,$ y $3=8,$ y $4=0,$ y $5=0,$ dual obj value$=100$

$($b$)$ Write the defining equations of the optimal corner point.

x$1=0,$ x$1-$ x$2+2$x$3=20,2$x$1+$ x$2-$ x$3=10$

$($c$)$ Are all resources used at the optimal solution? If not, determine the unused amounts

of the resources.

All resources are used because both slacks have zero values.