$(1$ point$)$ For this LP $\backslash [\backslash$begin$\{$array$\}\{$ll$\}\backslash$max & z$=4$ x$\_\{1\}+$x$\_\{2\}\backslash \backslash \backslash$text $\{$ s$.$t$.\}$ & $3$ x$\_\{1\}+2$ x$\_\{2\}\backslash$leq $6\backslash \backslash$ & $6$ x$\_\{1\}+3$ x$\_\{2\}\backslash$leq $10\backslash \backslash$ & x$\_\{1\},$ x$\_\{2\}\backslash$geq $0\backslash$end$\{$array$\}\backslash ]$ Betsy says that Row $0$ of the optimal Primal tableau is: $\backslash [\backslash$left$[\backslash$begin$\{$array$\}\{$cccccc$\}$ z & x$\_\{1\}$ & x$\_\{2\}$ & s$\_\{1\}$ & s$\_\{2\}$ & $\backslash$text $\{$ rhs $\}\backslash \backslash \backslash$hline $1$ & $0$ & $2$ & $0$ & $\backslash$frac$\{2\}\{3\}$ & $\backslash$frac$\{20\}\{3\}\backslash$end$\{$array$\}\backslash$right$]\backslash ](1)$ Read

y$=[$

y

$1$

$$

$$

y

$2$

$$

$$

$]=$c

BV

$$

B

$1$

$=[$

$]$ from the optimal tableau and the basic variables are

$\{$s

$1$

$$

$,$x

$1$

$$

$\}$

$(2)$ Will the current optimal dual solution remain feasible if Change $4$ is considered where

c

$2$

$$

$=1$

and

a

$2$

$$

$=[$

$9$

$3$

$$

$]?($y$/$n$)$

$(3)$ Will the current optimal dual solution remain feasible if a different Change $4$ is considered, where

c

$2$

$$

$=1$

and

a

$2$

$$

$=[$

$21$

$1$

$$

$]?($y$/$n$)$

$(4)$ What is the $($possibly new$)$ value of

z

~

if the Change $4$ of item $(3)$ is made?