Solve the following linear program using the graphical method. Compute the value of the objective function and decision variables at optimalityQuestion $3$:

$($a$)$ Solve the following linear program using the graphical method. Compute the value of the

objective function and decision variables at optimality.

Minimize $Z=x_1+x_2$

such that $3x_1+2x_{2}4$

$,0x_{1}1$

$,0x_{2}2$

$($b$)$ If the objective function $"$ MinZ$=x_1+x_2"$ in part $($a$)$ changes to $"$ MinZ$=k{x}_1+x_2",$

$($i$)$ find the range of values of $k$ such that the minimum value of the new objective

function is $2.$

$($ii$)$ find the value of $k$ such that the minimum value of the new objective function

is $1.$