Problem $5$

The following tableau represents a specific simplex iteration. All variables are non$-$negative.

$\backslash$table$[[,x1,x2,x3,$X$4,$X$5,$X$6,$X$7,$X$8,$Solution$],[Z,0,-5,0,4,-1,-10,0,0,620],[x8,0,3,0,-2,-3,-1,5,1,12],[$X$3,0,1,1,3,1,0,3,0,6],[$X$1,1,-1,0,0,6,4,0,0,0]]$

a$)$ Categorize the variables as basic and non$-$basic, and provide the current values of all the variables.

b$)$ Assuming that the problem is maximization, identify all the variables that will improve the value of $Z.$ If each such variable enters the basis, determine the associated leaving variable $($if any$)$ and the associated changes in $z$ without using any Gauss$-$Jordan operations.

c$)$ Which non$-$basic variable$($s$)$ will not cause a change in the value of $z,$ when selected to enter?