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

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

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Grade 7

One-step inequalities: word problems

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2

. Solve the below LP using SIMPLEX method.

\newline

\begin{array}{rrrl} \text { Maximize } & -\mathrm{x}_{1}-\mathrm{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}, \quad x_{2}, \quad x_{3} \geq 0 . \end{array}

Set up simplex tableau: Set up the initial simplex tableau. Introduce slack variables $s_1$ , $s_2$ , and $s_3$ for the constraints to turn inequalities into equations.
Identify entering and leaving variables: The initial simplex tableau is: $\newline$ \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 \ s_1 & 1 & 1 & 2 & 1 & 0 & 0 & 9 \ s_2 & 1 & 1 & -1 & 0 & 1 & 0 & 2 \ s_3 & -1 & 1 & 1 & 0 & 0 & 1 & 4 \ \hline \end{array}
Update tableau after row operations: Identify the entering variable. It's the one with the most negative coefficient in the objective function row. Here, it's $x_3$ .
Identify new entering variable: Identify the leaving variable using the minimum ratio test. Divide the RHS by the corresponding coefficients of $x_3$ in the constraint rows.
Update tableau after row operations: The ratios are $\frac{9}{2}$ for $s_1$ , undefined for $s_2$ (since $-1$ is negative), and $\frac{4}{1}$ for $s_3$ . The smallest positive ratio is $4$ , so $s_3$ will leave.
Update tableau after row operations: The ratios are $\frac{9}{2}$ for $s_1$ , undefined for $s_2$ (since $-1$ is negative), and $\frac{4}{1}$ for $s_3$ . The smallest positive ratio is $4$ , so $s_3$ will leave.Perform row operations to make $x_3$ a basic variable and update the tableau.
Update tableau after row operations: The ratios are $\frac{9}{2}$ for $s_1$ , undefined for $s_2$ (since $-1$ is negative), and $\frac{4}{1}$ for $s_3$ . The smallest positive ratio is $4$ , so $s_3$ will leave.Perform row operations to make $x_3$ a basic variable and update the tableau.The updated simplex tableau after making $x_3$ a basic variable is: $\newline$ | | $s_1$ $0$ | $s_1$ $1$ | $x_3$ | $s_1$ | $s_2$ | $s_3$ | RHS | $\newline$ |---|-------|-------|-------|-------|-------|-------|-----| $\newline$ | $s_1$ $6$ | $s_1$ $7$ | $s_1$ $8$ | $s_1$ $9$ | $s_1$ $9$ | $s_1$ $9$ | $4$ | $s_2$ $3$ | $\newline$ | $s_1$ | $s_2$ $5$ | $-1$ | $s_1$ $9$ | $s_2$ $8$ | $s_1$ $9$ | $-1$ $0$ | $s_2$ $8$ | $\newline$ | $s_2$ | $-1$ $3$ | $s_1$ $9$ | $s_1$ $9$ | $s_1$ $9$ | $s_2$ $8$ | $s_2$ $8$ | $-1$ $9$ | $\newline$ | $x_3$ | $-1$ | $s_2$ $8$ | $s_2$ $8$ | $s_1$ $9$ | $s_1$ $9$ | $s_2$ $8$ | $4$ |
Update tableau after row operations: The ratios are $\frac{9}{2}$ for $s_1$ , undefined for $s_2$ (since $-1$ is negative), and $\frac{4}{1}$ for $s_3$ . The smallest positive ratio is $4$ , so $s_3$ will leave.Perform row operations to make $x_3$ a basic variable and update the tableau.The updated simplex tableau after making $x_3$ a basic variable is: $\newline$ | | $s_1$ $0$ | $s_1$ $1$ | $x_3$ | $s_1$ | $s_2$ | $s_3$ | RHS | $\newline$ |---|-------|-------|-------|-------|-------|-------|-----| $\newline$ | $s_1$ $6$ | $s_1$ $7$ | $s_1$ $8$ | $s_1$ $9$ | $s_1$ $9$ | $s_1$ $9$ | $4$ | $s_2$ $3$ | $\newline$ | $s_1$ | $s_2$ $5$ | $-1$ | $s_1$ $9$ | $s_2$ $8$ | $s_1$ $9$ | $-1$ $0$ | $s_2$ $8$ | $\newline$ | $s_2$ | $-1$ $3$ | $s_1$ $9$ | $s_1$ $9$ | $s_1$ $9$ | $s_2$ $8$ | $s_2$ $8$ | $-1$ $9$ | $\newline$ | $x_3$ | $-1$ | $s_2$ $8$ | $s_2$ $8$ | $s_1$ $9$ | $s_1$ $9$ | $s_2$ $8$ | $4$ | $\newline$ Identify the new entering variable. It's $s_1$ $0$ with the most positive coefficient in the objective function row.
Update tableau after row operations: The ratios are $\frac{9}{2}$ for $s_1$ , undefined for $s_2$ (since $-1$ is negative), and $\frac{4}{1}$ for $s_3$ . The smallest positive ratio is $4$ , so $s_3$ will leave.Perform row operations to make $x_3$ a basic variable and update the tableau.The updated simplex tableau after making $x_3$ a basic variable is: $\newline$ | | $s_1$ $0$ | $s_1$ $1$ | $x_3$ | $s_1$ | $s_2$ | $s_3$ | RHS | $\newline$ |---|-------|-------|-------|-------|-------|-------|-----| $\newline$ | $s_1$ $6$ | $s_1$ $7$ | $s_1$ $8$ | $s_1$ $9$ | $s_1$ $9$ | $s_1$ $9$ | $4$ | $s_2$ $3$ | $\newline$ | $s_1$ | $s_2$ $5$ | $-1$ | $s_1$ $9$ | $s_2$ $8$ | $s_1$ $9$ | $-1$ $0$ | $s_2$ $8$ | $\newline$ | $s_2$ | $-1$ $3$ | $s_1$ $9$ | $s_1$ $9$ | $s_1$ $9$ | $s_2$ $8$ | $s_2$ $8$ | $-1$ $9$ | $\newline$ | $x_3$ | $-1$ | $s_2$ $8$ | $s_2$ $8$ | $s_1$ $9$ | $s_1$ $9$ | $s_2$ $8$ | $4$ | $\newline$ Identify the new entering variable. It's $s_1$ $0$ with the most positive coefficient in the objective function row.Perform the minimum ratio test again for $s_1$ $0$ . The ratios are $s_3$ $0$ for $s_1$ and $s_3$ $2$ for $s_2$ . The smallest positive ratio is $s_3$ $0$ , so $s_1$ will leave.
Update tableau after row operations: The ratios are $\frac{9}{2}$ for $s_1$ , undefined for $s_2$ (since $-1$ is negative), and $\frac{4}{1}$ for $s_3$ . The smallest positive ratio is $4$ , so $s_3$ will leave.Perform row operations to make $x_3$ a basic variable and update the tableau.The updated simplex tableau after making $x_3$ a basic variable is: $\newline$ | | $s_1$ $0$ | $s_1$ $1$ | $x_3$ | $s_1$ | $s_2$ | $s_3$ | RHS | $\newline$ |---|-------|-------|-------|-------|-------|-------|-----| $\newline$ | $s_1$ $6$ | $s_1$ $7$ | $s_1$ $8$ | $s_1$ $9$ | $s_1$ $9$ | $s_1$ $9$ | $4$ | $s_2$ $3$ | $\newline$ | $s_1$ | $s_2$ $5$ | $-1$ | $s_1$ $9$ | $s_2$ $8$ | $s_1$ $9$ | $-1$ $0$ | $s_2$ $8$ | $\newline$ | $s_2$ | $-1$ $3$ | $s_1$ $9$ | $s_1$ $9$ | $s_1$ $9$ | $s_2$ $8$ | $s_2$ $8$ | $-1$ $9$ | $\newline$ | $x_3$ | $-1$ | $s_2$ $8$ | $s_2$ $8$ | $s_1$ $9$ | $s_1$ $9$ | $s_2$ $8$ | $4$ | $\newline$ Identify the new entering variable. It's $s_1$ $0$ with the most positive coefficient in the objective function row.Perform the minimum ratio test again for $s_1$ $0$ . The ratios are $s_3$ $0$ for $s_1$ and $s_3$ $2$ for $s_2$ . The smallest positive ratio is $s_3$ $0$ , so $s_1$ will leave.Perform row operations to make $s_1$ $0$ a basic variable and update the tableau.
Update tableau after row operations: The ratios are $\frac{9}{2}$ for $s_1$ , undefined for $s_2$ (since $-1$ is negative), and $\frac{4}{1}$ for $s_3$ . The smallest positive ratio is $4$ , so $s_3$ will leave.Perform row operations to make $x_3$ a basic variable and update the tableau.The updated simplex tableau after making $x_3$ a basic variable is: $\newline$ | | $s_1$ $0$ | $s_1$ $1$ | $x_3$ | $s_1$ | $s_2$ | $s_3$ | RHS | $\newline$ |---|-----|-----|-----|-----|-----|-----|-----| $\newline$ | $s_1$ $6$ | $s_1$ $7$ | $s_1$ $8$ | $s_1$ $9$ | $s_1$ $9$ | $s_1$ $9$ | $4$ | $s_2$ $3$ | $\newline$ | $s_1$ | $s_2$ $5$ | $-1$ | $s_1$ $9$ | $s_2$ $8$ | $s_1$ $9$ | $-1$ $0$ | $s_2$ $8$ | $\newline$ | $s_2$ | $-1$ $3$ | $s_1$ $9$ | $s_1$ $9$ | $s_1$ $9$ | $s_2$ $8$ | $s_2$ $8$ | $-1$ $9$ | $\newline$ | $x_3$ | $-1$ | $s_2$ $8$ | $s_2$ $8$ | $s_1$ $9$ | $s_1$ $9$ | $s_2$ $8$ | $4$ | $\newline$ Identify the new entering variable. It's $s_1$ $0$ with the most positive coefficient in the objective function row.Perform the minimum ratio test again for $s_1$ $0$ . The ratios are $s_3$ $0$ for $s_1$ and $s_3$ $2$ for $s_2$ . The smallest positive ratio is $s_3$ $0$ , so $s_1$ will leave.Perform row operations to make $s_1$ $0$ a basic variable and update the tableau.The updated simplex tableau after making $s_1$ $0$ a basic variable is: $\newline$ | | $s_1$ $0$ | $s_1$ $1$ | $x_3$ | $s_1$ | $s_2$ | $s_3$ | RHS | $\newline$ |---|-----|-----|-----|-----|-----|-----|-----| $\newline$ | $s_1$ $6$ | $s_1$ $9$ | $4$ $6$ | $s_1$ $9$ | $s_1$ $7$ | $s_1$ $9$ | $s_2$ $8$ | $s_3$ $1$ | $\newline$ | $s_1$ $0$ | $s_2$ $8$ | $s_3$ $4$ | $s_1$ $9$ | $s_3$ $6$ | $s_1$ $9$ | $s_3$ $8$ | $s_3$ $6$ | $\newline$ | $s_2$ | $s_1$ $9$ | $x_3$ $2$ | $s_1$ $9$ | $s_3$ $8$ | $s_2$ $8$ | $x_3$ $6$ | $x_3$ $7$ | $\newline$ | $x_3$ | $s_1$ $9$ | $x_3$ $2$ | $s_2$ $8$ | $s_3$ $6$ | $s_1$ $9$ | $s_3$ $6$ | $x_3$ $5$ |