$($A$)$ Using slack$$ variables, determine the initial system for the linear programming problem.

$($B$)$ Write the simplex tableau and identify the first pivot and the entering and exiting variables.

$($C$)$ Use the simplex method to solve the problem.

Maximize

Upper P equals $24$ x $1$ plus x $2$P$=24$x$1+$x$2$

subject to

$4$ x $1$ plus x $2$ less than or equals $84$x$1+$x$2<=8$

x $1$ plus $8$ x $2$ less than or equals $8$x$1+8$x$2<=8$

x $1$ comma x $2$ greater than or equals $0$x$1,$ x$2>=0$

Question content area bottom

Part $1$

$($A$)$ Using slack$$ variables, determine the initial system for the linear programming problem. Use s $1$s$1$ for the first constraint and s $2$s$2$ for the second constraint.

enter your response here

First constraint

enter your response here

Second constraint

enter your response here

Objective function

x $1$ comma x $2$ comma s $1$ comma s $2$x$1,$ x$2,$ s$1,$ s$2$

greater than or equals$>=$

$0$

Part $2$

$($B$)$ Write the simplex tableau by filling in the blanks below.

x$11$

x$22$

s$11$

s$22$

P

s$11$

enter your response here

enter your response here

enter your response here

$0$

enter your response here

$88$

s$22$

enter your response here

enter your response here

$0$

enter your response here

enter your response here

$88$

P

enter your response here

enter your response here

enter your response here

enter your response here

$1$

enter your response here

Part $3$

The pivot element is located in the

$$

s $2$s$2$

s $1$s$1$

Upper PP

x $1$x$1$

x $2$x$2$

column and

$$

s $1$s$1$

Upper PP

s $2$s$2$

row.

Part $4$

The entering variable is

$$

x $2.$x$2.$

x $1.$x$1.$

P$.$P$.$

s $1.$s$1.$

s $2.$s$2.$

The exiting variable is

$$

x $1.$x$1.$

P$.$P$.$

s $2.$s$2.$

x $2.$x$2.$

s $1.$s$1.$

Part $5$

$($C$)$ Use the simplex method to solve the problem. Select the correct choice below$$ and, if$$ necessary, fill in the answer boxes to complete your choice.

A$.$

The maximum value is Pequals$=$enter your response here when x $1$x$1$equals$=$enter your response here and x $2$x$2$equals$=$enter your response here.

$($Type integers or simplified$$ fractions.$)$

B$.$

There is no optimal solution.

input field $1$input field $2$input field $3$Use s $1$ for the first constraint and s $2$ for the second constraint.Use s $1$ for the first constraint and s $2$ for the second constraint.Use s $1$ for the first constraint and s $2$ for the second constraint.Use s $1$ for the first constraint and s $2$ for the second constraint.Use s $1$ for the first constraint and s $2$ for the second constraint.Use s $1$ for the first constraint and s $2$ for the second constraint.Use s $1$ for the first constraint and s $2$ for the second constraint.Use s $1$ for the first constraint and s $2$ for the second constraint.Use s $1$ for the first constraint and s $2$ for the second constraint.Use s $1$ for the first constraint and s $2$ for the second constraint.Use s $1$ for the first constraint and s $2$ for the second constraint.Use s $1$ for the first constraint and s $2$ for the second constraint.Use s $1$ for the first constraint and s $2$ for the second constraint.The pivot element is located in thecolumn andcolumn androw.The entering variable isThe exiting variable isThe exiting variable isinput field $17$input field $18$input field $19$Select the correct choice below and, if necessary, fill in the answer boxes to complete your choice.