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$Solve the below LP using SIMPLEX method. {:[" Maximize ",-x_(1)-x_(2)+4x_(3),],[" subject to ",x_(1)+x_(2)+2x_(3) <= 9],[,x_(1)+x_(2)-x_(3) <= 2],[,-x_(1)+x_(2)+x_(3) <= 4],[,x_(1)",",x_(2)",",x_(3) >= 0.]:}$

$2$ . Solve the below LP using SIMPLEX method. $\newline$ $\begin{array}{cccc} \text { Maximize } & -x_{1}-x_{2}+4 x_{3} & \\ \text { subject to } & x_{1}+x_{2}+2 x_{3} \leq 9 \\ & x_{1}+x_{2}-x_{3} \leq 2 \\ & -x_{1}+x_{2}+x_{3} \leq 4 \\ & x_{1}, & x_{2}, & x_{3} \geq 0 . \end{array}$

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Math Problems

Grade 7

One-step inequalities: word problems

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2

. Solve the below LP using SIMPLEX method.

\newline

\begin{array}{cccc} \text { Maximize } & -x_{1}-x_{2}+4 x_{3} & \\ \text { subject to } & x_{1}+x_{2}+2 x_{3} \leq 9 \\ & x_{1}+x_{2}-x_{3} \leq 2 \\ & -x_{1}+x_{2}+x_{3} \leq 4 \\ & x_{1}, & x_{2}, & x_{3} \geq 0 . \end{array}

Set up simplex tableau: Set up the initial simplex tableau with slack variables to convert inequalities to equations. $\newline$ Add slack variables $s_1$ , $s_2$ , and $s_3$ to the constraints. $\newline$ Objective function: $-x_1 - x_2 + 4x_3$ becomes $-x_1 - x_2 + 4x_3 + 0s_1 + 0s_2 + 0s_3$ . $\newline$ Constraints: $\newline$ $x_1 + x_2 + 2x_3 + s_1 = 9$ , $\newline$ $x_1 + x_2 - x_3 + s_2 = 2$ , $\newline$ $-x_1 + x_2 + x_3 + s_3 = 4$ .
Write initial tableau: Write the initial simplex tableau. $\newline$ \begin{array}{|c|c|c|c|c|c|c|c|} \hline & x_1 & x_2 & x_3 & s_1 & s_2 & s_3 & \text{RHS} \ \hline z & -1 & -1 & 4 & 0 & 0 & 0 & 0 \ 1 & 1 & 1 & 2 & 1 & 0 & 0 & 9 \ 2 & 1 & 1 & -1 & 0 & 1 & 0 & 2 \ 3 & -1 & 1 & 1 & 0 & 0 & 1 & 4 \ \hline \end{array}
Identify entering and leaving variables: Identify the entering variable (most negative coefficient in the objective function row) and the leaving variable (smallest positive ratio of RHS to the coefficient of entering variable in the constraint rows). $\newline$ Entering variable: $x_3$ (coefficient $4$ in $z$ -row). $\newline$ Leaving variable: $s_2$ (smallest positive ratio of $2$ to $-1$ in row $2$ ).
Perform pivot operation: Perform pivot operation to make $x_3$ the basic variable in place of $s_2$ . Pivot on the element at the intersection of the entering column ( $x_3$ ) and the leaving row ( $s_2$ ).
Update tableau: Update the tableau after pivoting. $\newline$ \begin{array}{|c|c|c|c|c|c|c|c|} \hline & x_1 & x_2 & x_3 & s_1 & s_2 & s_3 & \text{RHS} \ \hline z & 3 & 1 & 0 & 0 & 4 & 0 & 8 \ 1 & 3 & 3 & 0 & 1 & -2 & 0 & 5 \ 2 & 1 & 1 & 1 & 0 & 1 & 0 & 2 \ 3 & 0 & 2 & 0 & 0 & 1 & 1 & 6 \ \hline \end{array}