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Questions: Algebra BusinessCalculus
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Use the simplex method to maximize \(\displaystyle p = 17 x + 9 y\) subject to \(\displaystyle \begin{cases}96 x + 23 y \leq 2208 \\ 78 x + 70 y \leq 5460 \\x \geq 0, y \geq 0 \end{cases}\)
Adding the slack variables \(\displaystyle s\) and \(\displaystyle t\) to the inequalities gives:
\begin{equation*}\begin{cases}96 x + 23 y +s = 2208 \\ 78 x + 70 y+t = 5460 \\ - 17 x - 9 y+p =0 \end{cases}\end{equation*}This gives the first tableau:\begin{tabular}{|c|c|c|c|c|c|c|}\hline
\(\displaystyle \) & \(\displaystyle x\) & \(\displaystyle y\) & \(\displaystyle s\) & \(\displaystyle t\) & \(\displaystyle p\) & \(\displaystyle \) \\ \hline
\(\displaystyle s\) & \(\displaystyle 96\) & \(\displaystyle 23\) & \(\displaystyle 1\) & \(\displaystyle 0\) & \(\displaystyle 0\) & \(\displaystyle 2208\) \\ \hline
\(\displaystyle t\) & \(\displaystyle 78\) & \(\displaystyle 70\) & \(\displaystyle 0\) & \(\displaystyle 1\) & \(\displaystyle 0\) & \(\displaystyle 5460\) \\ \hline
\(\displaystyle p\) & \(\displaystyle -17\) & \(\displaystyle -9\) & \(\displaystyle 0\) & \(\displaystyle 0\) & \(\displaystyle 1\) & \(\displaystyle 0\) \\ \hline
\end{tabular}
The pivot row is \(\displaystyle s\) and the pivot column is \(\displaystyle x\) . The departing variable is \(\displaystyle s\) and the incoming variable is \(\displaystyle x\). Pivoting using the row operations:
\(\displaystyle 16R_2-13R_1\)
\(\displaystyle 96R_3+17R_1\)
\begin{tabular}{|c|c|c|c|c|c|c|}\hline
\(\displaystyle \) & \(\displaystyle x\) & \(\displaystyle y\) & \(\displaystyle s\) & \(\displaystyle t\) & \(\displaystyle p\) & \(\displaystyle \) \\ \hline
\(\displaystyle x\) & \(\displaystyle 96\) & \(\displaystyle 23\) & \(\displaystyle 1\) & \(\displaystyle 0\) & \(\displaystyle 0\) & \(\displaystyle 2208\) \\ \hline
\(\displaystyle t\) & \(\displaystyle 0\) & \(\displaystyle 821\) & \(\displaystyle -13\) & \(\displaystyle 16\) & \(\displaystyle 0\) & \(\displaystyle 58656\) \\ \hline
\(\displaystyle p\) & \(\displaystyle 0\) & \(\displaystyle -473\) & \(\displaystyle 17\) & \(\displaystyle 0\) & \(\displaystyle 96\) & \(\displaystyle 37536\) \\ \hline
\end{tabular}
The pivot row is \(\displaystyle t\) and the pivot column is \(\displaystyle y\) . The departing variable is \(\displaystyle t\) and the incoming variable is \(\displaystyle y\). Pivoting using the row operations:
\(\displaystyle 821R_1-23R_2\)
\(\displaystyle 821R_3+473R_2\)
\begin{tabular}{|c|c|c|c|c|c|c|}\hline
\(\displaystyle \) & \(\displaystyle x\) & \(\displaystyle y\) & \(\displaystyle s\) & \(\displaystyle t\) & \(\displaystyle p\) & \(\displaystyle \) \\ \hline
\(\displaystyle x\) & \(\displaystyle 78816\) & \(\displaystyle 0\) & \(\displaystyle 1120\) & \(\displaystyle -368\) & \(\displaystyle 0\) & \(\displaystyle 463680\) \\ \hline
\(\displaystyle y\) & \(\displaystyle 0\) & \(\displaystyle 821\) & \(\displaystyle -13\) & \(\displaystyle 16\) & \(\displaystyle 0\) & \(\displaystyle 58656\) \\ \hline
\(\displaystyle p\) & \(\displaystyle 0\) & \(\displaystyle 0\) & \(\displaystyle 7808\) & \(\displaystyle 7568\) & \(\displaystyle 78816\) & \(\displaystyle 58561344\) \\ \hline
\end{tabular}
There are no negative values in row \(\displaystyle p\) and this is the final tableau.The value of \(\displaystyle x\) is \(\displaystyle \frac{4830}{821}\). The value of \(\displaystyle y\) is \(\displaystyle \frac{58656}{821}\). The max value is \(\displaystyle p = \frac{610014}{821}\)
\begin{question}Use the simplex method to maximize $p = 17 x + 9 y$ subject to $\begin{cases}96 x + 23 y \leq 2208 \\ 78 x + 70 y \leq 5460 \\x \geq 0, y \geq 0 \end{cases}$
\soln{9cm}{Adding the slack variables $s$ and $t$ to the inequalities gives:\newline\begin{equation*}\begin{cases}96 x + 23 y +s = 2208 \\ 78 x + 70 y+t = 5460 \\ - 17 x - 9 y+p =0 \end{cases}\end{equation*}This gives the first tableau:\begin{tabular}{|c|c|c|c|c|c|c|}\hline
$$ & $x$ & $y$ & $s$ & $t$ & $p$ & $$ \\ \hline
$s$ & $96$ & $23$ & $1$ & $0$ & $0$ & $2208$ \\ \hline
$t$ & $78$ & $70$ & $0$ & $1$ & $0$ & $5460$ \\ \hline
$p$ & $-17$ & $-9$ & $0$ & $0$ & $1$ & $0$ \\ \hline
\end{tabular}\newline
The pivot row is $s$ and the pivot column is $x$ . The departing variable is $s$ and the incoming variable is $x$. Pivoting using the row operations:\newline
$16R_2-13R_1$\newline
$96R_3+17R_1$\newline
\begin{tabular}{|c|c|c|c|c|c|c|}\hline
$$ & $x$ & $y$ & $s$ & $t$ & $p$ & $$ \\ \hline
$x$ & $96$ & $23$ & $1$ & $0$ & $0$ & $2208$ \\ \hline
$t$ & $0$ & $821$ & $-13$ & $16$ & $0$ & $58656$ \\ \hline
$p$ & $0$ & $-473$ & $17$ & $0$ & $96$ & $37536$ \\ \hline
\end{tabular}\newline
The pivot row is $t$ and the pivot column is $y$ . The departing variable is $t$ and the incoming variable is $y$. Pivoting using the row operations:\newline
$821R_1-23R_2$\newline
$821R_3+473R_2$\newline
\begin{tabular}{|c|c|c|c|c|c|c|}\hline
$$ & $x$ & $y$ & $s$ & $t$ & $p$ & $$ \\ \hline
$x$ & $78816$ & $0$ & $1120$ & $-368$ & $0$ & $463680$ \\ \hline
$y$ & $0$ & $821$ & $-13$ & $16$ & $0$ & $58656$ \\ \hline
$p$ & $0$ & $0$ & $7808$ & $7568$ & $78816$ & $58561344$ \\ \hline
\end{tabular}\newline
There are no negative values in row $p$ and this is the final tableau.The value of $x$ is $\frac{4830}{821}$. The value of $y$ is $\frac{58656}{821}$. The max value is $p = \frac{610014}{821}$ }
\end{question}
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\begin{document}\begin{question}(10pts) The question goes here!
\soln{9cm}{The solution goes here.}
\end{question}\end{document}<p> <p>Use the simplex method to maximize <img class="equation_image" title=" \displaystyle p = 17 x + 9 y " src="/equation_images/%20%5Cdisplaystyle%20p%20%3D%2017%20x%20%2B%209%20y%20" alt="LaTeX: \displaystyle p = 17 x + 9 y " data-equation-content=" \displaystyle p = 17 x + 9 y " /> subject to <img class="equation_image" title=" \displaystyle \begin{cases}96 x + 23 y \leq 2208 \\ 78 x + 70 y \leq 5460 \\x \geq 0, y \geq 0 \end{cases} " src="/equation_images/%20%5Cdisplaystyle%20%5Cbegin%7Bcases%7D96%20x%20%2B%2023%20y%20%5Cleq%202208%20%5C%5C%2078%20x%20%2B%2070%20y%20%5Cleq%205460%20%5C%5Cx%20%5Cgeq%200%2C%20y%20%5Cgeq%200%20%5Cend%7Bcases%7D%20" alt="LaTeX: \displaystyle \begin{cases}96 x + 23 y \leq 2208 \\ 78 x + 70 y \leq 5460 \\x \geq 0, y \geq 0 \end{cases} " data-equation-content=" \displaystyle \begin{cases}96 x + 23 y \leq 2208 \\ 78 x + 70 y \leq 5460 \\x \geq 0, y \geq 0 \end{cases} " /> </p> </p><p> <p>Adding the slack variables <img class="equation_image" title=" \displaystyle s " src="/equation_images/%20%5Cdisplaystyle%20s%20" alt="LaTeX: \displaystyle s " data-equation-content=" \displaystyle s " /> and <img class="equation_image" title=" \displaystyle t " src="/equation_images/%20%5Cdisplaystyle%20t%20" alt="LaTeX: \displaystyle t " data-equation-content=" \displaystyle t " /> to the inequalities gives:<br> <img class="equation_image" title=" \begin{cases}96 x + 23 y +s = 2208 \\ 78 x + 70 y+t = 5460 \\ - 17 x - 9 y+p =0 \end{cases} " src="/equation_images/%20%5Cbegin%7Bcases%7D96%20x%20%2B%2023%20y%20%2Bs%20%3D%202208%20%5C%5C%2078%20x%20%2B%2070%20y%2Bt%20%3D%205460%20%5C%5C%20-%2017%20x%20-%209%20y%2Bp%20%3D0%20%5Cend%7Bcases%7D%20" alt="LaTeX: \begin{cases}96 x + 23 y +s = 2208 \\ 78 x + 70 y+t = 5460 \\ - 17 x - 9 y+p =0 \end{cases} " data-equation-content=" \begin{cases}96 x + 23 y +s = 2208 \\ 78 x + 70 y+t = 5460 \\ - 17 x - 9 y+p =0 \end{cases} " /> This gives the first tableau:\begin{tabular}{|c|c|c|c|c|c|c|}\hline
<img class="equation_image" title=" \displaystyle " src="/equation_images/%20%5Cdisplaystyle%20%20" alt="LaTeX: \displaystyle " data-equation-content=" \displaystyle " /> & <img class="equation_image" title=" \displaystyle x " src="/equation_images/%20%5Cdisplaystyle%20x%20" alt="LaTeX: \displaystyle x " data-equation-content=" \displaystyle x " /> & <img class="equation_image" title=" \displaystyle y " src="/equation_images/%20%5Cdisplaystyle%20y%20" alt="LaTeX: \displaystyle y " data-equation-content=" \displaystyle y " /> & <img class="equation_image" title=" \displaystyle s " src="/equation_images/%20%5Cdisplaystyle%20s%20" alt="LaTeX: \displaystyle s " data-equation-content=" \displaystyle s " /> & <img class="equation_image" title=" \displaystyle t " src="/equation_images/%20%5Cdisplaystyle%20t%20" alt="LaTeX: \displaystyle t " data-equation-content=" \displaystyle t " /> & <img class="equation_image" title=" \displaystyle p " src="/equation_images/%20%5Cdisplaystyle%20p%20" alt="LaTeX: \displaystyle p " data-equation-content=" \displaystyle p " /> & <img class="equation_image" title=" \displaystyle " src="/equation_images/%20%5Cdisplaystyle%20%20" alt="LaTeX: \displaystyle " data-equation-content=" \displaystyle " /> \\ \hline
<img class="equation_image" title=" \displaystyle s " src="/equation_images/%20%5Cdisplaystyle%20s%20" alt="LaTeX: \displaystyle s " data-equation-content=" \displaystyle s " /> & <img class="equation_image" title=" \displaystyle 96 " src="/equation_images/%20%5Cdisplaystyle%2096%20" alt="LaTeX: \displaystyle 96 " data-equation-content=" \displaystyle 96 " /> & <img class="equation_image" title=" \displaystyle 23 " src="/equation_images/%20%5Cdisplaystyle%2023%20" alt="LaTeX: \displaystyle 23 " data-equation-content=" \displaystyle 23 " /> & <img class="equation_image" title=" \displaystyle 1 " src="/equation_images/%20%5Cdisplaystyle%201%20" alt="LaTeX: \displaystyle 1 " data-equation-content=" \displaystyle 1 " /> & <img class="equation_image" title=" \displaystyle 0 " src="/equation_images/%20%5Cdisplaystyle%200%20" alt="LaTeX: \displaystyle 0 " data-equation-content=" \displaystyle 0 " /> & <img class="equation_image" title=" \displaystyle 0 " src="/equation_images/%20%5Cdisplaystyle%200%20" alt="LaTeX: \displaystyle 0 " data-equation-content=" \displaystyle 0 " /> & <img class="equation_image" title=" \displaystyle 2208 " src="/equation_images/%20%5Cdisplaystyle%202208%20" alt="LaTeX: \displaystyle 2208 " data-equation-content=" \displaystyle 2208 " /> \\ \hline
<img class="equation_image" title=" \displaystyle t " src="/equation_images/%20%5Cdisplaystyle%20t%20" alt="LaTeX: \displaystyle t " data-equation-content=" \displaystyle t " /> & <img class="equation_image" title=" \displaystyle 78 " src="/equation_images/%20%5Cdisplaystyle%2078%20" alt="LaTeX: \displaystyle 78 " data-equation-content=" \displaystyle 78 " /> & <img class="equation_image" title=" \displaystyle 70 " src="/equation_images/%20%5Cdisplaystyle%2070%20" alt="LaTeX: \displaystyle 70 " data-equation-content=" \displaystyle 70 " /> & <img class="equation_image" title=" \displaystyle 0 " src="/equation_images/%20%5Cdisplaystyle%200%20" alt="LaTeX: \displaystyle 0 " data-equation-content=" \displaystyle 0 " /> & <img class="equation_image" title=" \displaystyle 1 " src="/equation_images/%20%5Cdisplaystyle%201%20" alt="LaTeX: \displaystyle 1 " data-equation-content=" \displaystyle 1 " /> & <img class="equation_image" title=" \displaystyle 0 " src="/equation_images/%20%5Cdisplaystyle%200%20" alt="LaTeX: \displaystyle 0 " data-equation-content=" \displaystyle 0 " /> & <img class="equation_image" title=" \displaystyle 5460 " src="/equation_images/%20%5Cdisplaystyle%205460%20" alt="LaTeX: \displaystyle 5460 " data-equation-content=" \displaystyle 5460 " /> \\ \hline
<img class="equation_image" title=" \displaystyle p " src="/equation_images/%20%5Cdisplaystyle%20p%20" alt="LaTeX: \displaystyle p " data-equation-content=" \displaystyle p " /> & <img class="equation_image" title=" \displaystyle -17 " src="/equation_images/%20%5Cdisplaystyle%20-17%20" alt="LaTeX: \displaystyle -17 " data-equation-content=" \displaystyle -17 " /> & <img class="equation_image" title=" \displaystyle -9 " src="/equation_images/%20%5Cdisplaystyle%20-9%20" alt="LaTeX: \displaystyle -9 " data-equation-content=" \displaystyle -9 " /> & <img class="equation_image" title=" \displaystyle 0 " src="/equation_images/%20%5Cdisplaystyle%200%20" alt="LaTeX: \displaystyle 0 " data-equation-content=" \displaystyle 0 " /> & <img class="equation_image" title=" \displaystyle 0 " src="/equation_images/%20%5Cdisplaystyle%200%20" alt="LaTeX: \displaystyle 0 " data-equation-content=" \displaystyle 0 " /> & <img class="equation_image" title=" \displaystyle 1 " src="/equation_images/%20%5Cdisplaystyle%201%20" alt="LaTeX: \displaystyle 1 " data-equation-content=" \displaystyle 1 " /> & <img class="equation_image" title=" \displaystyle 0 " src="/equation_images/%20%5Cdisplaystyle%200%20" alt="LaTeX: \displaystyle 0 " data-equation-content=" \displaystyle 0 " /> \\ \hline
\end{tabular}<br>
The pivot row is <img class="equation_image" title=" \displaystyle s " src="/equation_images/%20%5Cdisplaystyle%20s%20" alt="LaTeX: \displaystyle s " data-equation-content=" \displaystyle s " /> and the pivot column is <img class="equation_image" title=" \displaystyle x " src="/equation_images/%20%5Cdisplaystyle%20x%20" alt="LaTeX: \displaystyle x " data-equation-content=" \displaystyle x " /> . The departing variable is <img class="equation_image" title=" \displaystyle s " src="/equation_images/%20%5Cdisplaystyle%20s%20" alt="LaTeX: \displaystyle s " data-equation-content=" \displaystyle s " /> and the incoming variable is <img class="equation_image" title=" \displaystyle x " src="/equation_images/%20%5Cdisplaystyle%20x%20" alt="LaTeX: \displaystyle x " data-equation-content=" \displaystyle x " /> . Pivoting using the row operations:<br>
<img class="equation_image" title=" \displaystyle 16R_2-13R_1 " src="/equation_images/%20%5Cdisplaystyle%2016R_2-13R_1%20" alt="LaTeX: \displaystyle 16R_2-13R_1 " data-equation-content=" \displaystyle 16R_2-13R_1 " /> <br>
<img class="equation_image" title=" \displaystyle 96R_3+17R_1 " src="/equation_images/%20%5Cdisplaystyle%2096R_3%2B17R_1%20" alt="LaTeX: \displaystyle 96R_3+17R_1 " data-equation-content=" \displaystyle 96R_3+17R_1 " /> <br>
\begin{tabular}{|c|c|c|c|c|c|c|}\hline
<img class="equation_image" title=" \displaystyle " src="/equation_images/%20%5Cdisplaystyle%20%20" alt="LaTeX: \displaystyle " data-equation-content=" \displaystyle " /> & <img class="equation_image" title=" \displaystyle x " src="/equation_images/%20%5Cdisplaystyle%20x%20" alt="LaTeX: \displaystyle x " data-equation-content=" \displaystyle x " /> & <img class="equation_image" title=" \displaystyle y " src="/equation_images/%20%5Cdisplaystyle%20y%20" alt="LaTeX: \displaystyle y " data-equation-content=" \displaystyle y " /> & <img class="equation_image" title=" \displaystyle s " src="/equation_images/%20%5Cdisplaystyle%20s%20" alt="LaTeX: \displaystyle s " data-equation-content=" \displaystyle s " /> & <img class="equation_image" title=" \displaystyle t " src="/equation_images/%20%5Cdisplaystyle%20t%20" alt="LaTeX: \displaystyle t " data-equation-content=" \displaystyle t " /> & <img class="equation_image" title=" \displaystyle p " src="/equation_images/%20%5Cdisplaystyle%20p%20" alt="LaTeX: \displaystyle p " data-equation-content=" \displaystyle p " /> & <img class="equation_image" title=" \displaystyle " src="/equation_images/%20%5Cdisplaystyle%20%20" alt="LaTeX: \displaystyle " data-equation-content=" \displaystyle " /> \\ \hline
<img class="equation_image" title=" \displaystyle x " src="/equation_images/%20%5Cdisplaystyle%20x%20" alt="LaTeX: \displaystyle x " data-equation-content=" \displaystyle x " /> & <img class="equation_image" title=" \displaystyle 96 " src="/equation_images/%20%5Cdisplaystyle%2096%20" alt="LaTeX: \displaystyle 96 " data-equation-content=" \displaystyle 96 " /> & <img class="equation_image" title=" \displaystyle 23 " src="/equation_images/%20%5Cdisplaystyle%2023%20" alt="LaTeX: \displaystyle 23 " data-equation-content=" \displaystyle 23 " /> & <img class="equation_image" title=" \displaystyle 1 " src="/equation_images/%20%5Cdisplaystyle%201%20" alt="LaTeX: \displaystyle 1 " data-equation-content=" \displaystyle 1 " /> & <img class="equation_image" title=" \displaystyle 0 " src="/equation_images/%20%5Cdisplaystyle%200%20" alt="LaTeX: \displaystyle 0 " data-equation-content=" \displaystyle 0 " /> & <img class="equation_image" title=" \displaystyle 0 " src="/equation_images/%20%5Cdisplaystyle%200%20" alt="LaTeX: \displaystyle 0 " data-equation-content=" \displaystyle 0 " /> & <img class="equation_image" title=" \displaystyle 2208 " src="/equation_images/%20%5Cdisplaystyle%202208%20" alt="LaTeX: \displaystyle 2208 " data-equation-content=" \displaystyle 2208 " /> \\ \hline
<img class="equation_image" title=" \displaystyle t " src="/equation_images/%20%5Cdisplaystyle%20t%20" alt="LaTeX: \displaystyle t " data-equation-content=" \displaystyle t " /> & <img class="equation_image" title=" \displaystyle 0 " src="/equation_images/%20%5Cdisplaystyle%200%20" alt="LaTeX: \displaystyle 0 " data-equation-content=" \displaystyle 0 " /> & <img class="equation_image" title=" \displaystyle 821 " src="/equation_images/%20%5Cdisplaystyle%20821%20" alt="LaTeX: \displaystyle 821 " data-equation-content=" \displaystyle 821 " /> & <img class="equation_image" title=" \displaystyle -13 " src="/equation_images/%20%5Cdisplaystyle%20-13%20" alt="LaTeX: \displaystyle -13 " data-equation-content=" \displaystyle -13 " /> & <img class="equation_image" title=" \displaystyle 16 " src="/equation_images/%20%5Cdisplaystyle%2016%20" alt="LaTeX: \displaystyle 16 " data-equation-content=" \displaystyle 16 " /> & <img class="equation_image" title=" \displaystyle 0 " src="/equation_images/%20%5Cdisplaystyle%200%20" alt="LaTeX: \displaystyle 0 " data-equation-content=" \displaystyle 0 " /> & <img class="equation_image" title=" \displaystyle 58656 " src="/equation_images/%20%5Cdisplaystyle%2058656%20" alt="LaTeX: \displaystyle 58656 " data-equation-content=" \displaystyle 58656 " /> \\ \hline
<img class="equation_image" title=" \displaystyle p " src="/equation_images/%20%5Cdisplaystyle%20p%20" alt="LaTeX: \displaystyle p " data-equation-content=" \displaystyle p " /> & <img class="equation_image" title=" \displaystyle 0 " src="/equation_images/%20%5Cdisplaystyle%200%20" alt="LaTeX: \displaystyle 0 " data-equation-content=" \displaystyle 0 " /> & <img class="equation_image" title=" \displaystyle -473 " src="/equation_images/%20%5Cdisplaystyle%20-473%20" alt="LaTeX: \displaystyle -473 " data-equation-content=" \displaystyle -473 " /> & <img class="equation_image" title=" \displaystyle 17 " src="/equation_images/%20%5Cdisplaystyle%2017%20" alt="LaTeX: \displaystyle 17 " data-equation-content=" \displaystyle 17 " /> & <img class="equation_image" title=" \displaystyle 0 " src="/equation_images/%20%5Cdisplaystyle%200%20" alt="LaTeX: \displaystyle 0 " data-equation-content=" \displaystyle 0 " /> & <img class="equation_image" title=" \displaystyle 96 " src="/equation_images/%20%5Cdisplaystyle%2096%20" alt="LaTeX: \displaystyle 96 " data-equation-content=" \displaystyle 96 " /> & <img class="equation_image" title=" \displaystyle 37536 " src="/equation_images/%20%5Cdisplaystyle%2037536%20" alt="LaTeX: \displaystyle 37536 " data-equation-content=" \displaystyle 37536 " /> \\ \hline
\end{tabular}<br>
The pivot row is <img class="equation_image" title=" \displaystyle t " src="/equation_images/%20%5Cdisplaystyle%20t%20" alt="LaTeX: \displaystyle t " data-equation-content=" \displaystyle t " /> and the pivot column is <img class="equation_image" title=" \displaystyle y " src="/equation_images/%20%5Cdisplaystyle%20y%20" alt="LaTeX: \displaystyle y " data-equation-content=" \displaystyle y " /> . The departing variable is <img class="equation_image" title=" \displaystyle t " src="/equation_images/%20%5Cdisplaystyle%20t%20" alt="LaTeX: \displaystyle t " data-equation-content=" \displaystyle t " /> and the incoming variable is <img class="equation_image" title=" \displaystyle y " src="/equation_images/%20%5Cdisplaystyle%20y%20" alt="LaTeX: \displaystyle y " data-equation-content=" \displaystyle y " /> . Pivoting using the row operations:<br>
<img class="equation_image" title=" \displaystyle 821R_1-23R_2 " src="/equation_images/%20%5Cdisplaystyle%20821R_1-23R_2%20" alt="LaTeX: \displaystyle 821R_1-23R_2 " data-equation-content=" \displaystyle 821R_1-23R_2 " /> <br>
<img class="equation_image" title=" \displaystyle 821R_3+473R_2 " src="/equation_images/%20%5Cdisplaystyle%20821R_3%2B473R_2%20" alt="LaTeX: \displaystyle 821R_3+473R_2 " data-equation-content=" \displaystyle 821R_3+473R_2 " /> <br>
\begin{tabular}{|c|c|c|c|c|c|c|}\hline
<img class="equation_image" title=" \displaystyle " src="/equation_images/%20%5Cdisplaystyle%20%20" alt="LaTeX: \displaystyle " data-equation-content=" \displaystyle " /> & <img class="equation_image" title=" \displaystyle x " src="/equation_images/%20%5Cdisplaystyle%20x%20" alt="LaTeX: \displaystyle x " data-equation-content=" \displaystyle x " /> & <img class="equation_image" title=" \displaystyle y " src="/equation_images/%20%5Cdisplaystyle%20y%20" alt="LaTeX: \displaystyle y " data-equation-content=" \displaystyle y " /> & <img class="equation_image" title=" \displaystyle s " src="/equation_images/%20%5Cdisplaystyle%20s%20" alt="LaTeX: \displaystyle s " data-equation-content=" \displaystyle s " /> & <img class="equation_image" title=" \displaystyle t " src="/equation_images/%20%5Cdisplaystyle%20t%20" alt="LaTeX: \displaystyle t " data-equation-content=" \displaystyle t " /> & <img class="equation_image" title=" \displaystyle p " src="/equation_images/%20%5Cdisplaystyle%20p%20" alt="LaTeX: \displaystyle p " data-equation-content=" \displaystyle p " /> & <img class="equation_image" title=" \displaystyle " src="/equation_images/%20%5Cdisplaystyle%20%20" alt="LaTeX: \displaystyle " data-equation-content=" \displaystyle " /> \\ \hline
<img class="equation_image" title=" \displaystyle x " src="/equation_images/%20%5Cdisplaystyle%20x%20" alt="LaTeX: \displaystyle x " data-equation-content=" \displaystyle x " /> & <img class="equation_image" title=" \displaystyle 78816 " src="/equation_images/%20%5Cdisplaystyle%2078816%20" alt="LaTeX: \displaystyle 78816 " data-equation-content=" \displaystyle 78816 " /> & <img class="equation_image" title=" \displaystyle 0 " src="/equation_images/%20%5Cdisplaystyle%200%20" alt="LaTeX: \displaystyle 0 " data-equation-content=" \displaystyle 0 " /> & <img class="equation_image" title=" \displaystyle 1120 " src="/equation_images/%20%5Cdisplaystyle%201120%20" alt="LaTeX: \displaystyle 1120 " data-equation-content=" \displaystyle 1120 " /> & <img class="equation_image" title=" \displaystyle -368 " src="/equation_images/%20%5Cdisplaystyle%20-368%20" alt="LaTeX: \displaystyle -368 " data-equation-content=" \displaystyle -368 " /> & <img class="equation_image" title=" \displaystyle 0 " src="/equation_images/%20%5Cdisplaystyle%200%20" alt="LaTeX: \displaystyle 0 " data-equation-content=" \displaystyle 0 " /> & <img class="equation_image" title=" \displaystyle 463680 " src="/equation_images/%20%5Cdisplaystyle%20463680%20" alt="LaTeX: \displaystyle 463680 " data-equation-content=" \displaystyle 463680 " /> \\ \hline
<img class="equation_image" title=" \displaystyle y " src="/equation_images/%20%5Cdisplaystyle%20y%20" alt="LaTeX: \displaystyle y " data-equation-content=" \displaystyle y " /> & <img class="equation_image" title=" \displaystyle 0 " src="/equation_images/%20%5Cdisplaystyle%200%20" alt="LaTeX: \displaystyle 0 " data-equation-content=" \displaystyle 0 " /> & <img class="equation_image" title=" \displaystyle 821 " src="/equation_images/%20%5Cdisplaystyle%20821%20" alt="LaTeX: \displaystyle 821 " data-equation-content=" \displaystyle 821 " /> & <img class="equation_image" title=" \displaystyle -13 " src="/equation_images/%20%5Cdisplaystyle%20-13%20" alt="LaTeX: \displaystyle -13 " data-equation-content=" \displaystyle -13 " /> & <img class="equation_image" title=" \displaystyle 16 " src="/equation_images/%20%5Cdisplaystyle%2016%20" alt="LaTeX: \displaystyle 16 " data-equation-content=" \displaystyle 16 " /> & <img class="equation_image" title=" \displaystyle 0 " src="/equation_images/%20%5Cdisplaystyle%200%20" alt="LaTeX: \displaystyle 0 " data-equation-content=" \displaystyle 0 " /> & <img class="equation_image" title=" \displaystyle 58656 " src="/equation_images/%20%5Cdisplaystyle%2058656%20" alt="LaTeX: \displaystyle 58656 " data-equation-content=" \displaystyle 58656 " /> \\ \hline
<img class="equation_image" title=" \displaystyle p " src="/equation_images/%20%5Cdisplaystyle%20p%20" alt="LaTeX: \displaystyle p " data-equation-content=" \displaystyle p " /> & <img class="equation_image" title=" \displaystyle 0 " src="/equation_images/%20%5Cdisplaystyle%200%20" alt="LaTeX: \displaystyle 0 " data-equation-content=" \displaystyle 0 " /> & <img class="equation_image" title=" \displaystyle 0 " src="/equation_images/%20%5Cdisplaystyle%200%20" alt="LaTeX: \displaystyle 0 " data-equation-content=" \displaystyle 0 " /> & <img class="equation_image" title=" \displaystyle 7808 " src="/equation_images/%20%5Cdisplaystyle%207808%20" alt="LaTeX: \displaystyle 7808 " data-equation-content=" \displaystyle 7808 " /> & <img class="equation_image" title=" \displaystyle 7568 " src="/equation_images/%20%5Cdisplaystyle%207568%20" alt="LaTeX: \displaystyle 7568 " data-equation-content=" \displaystyle 7568 " /> & <img class="equation_image" title=" \displaystyle 78816 " src="/equation_images/%20%5Cdisplaystyle%2078816%20" alt="LaTeX: \displaystyle 78816 " data-equation-content=" \displaystyle 78816 " /> & <img class="equation_image" title=" \displaystyle 58561344 " src="/equation_images/%20%5Cdisplaystyle%2058561344%20" alt="LaTeX: \displaystyle 58561344 " data-equation-content=" \displaystyle 58561344 " /> \\ \hline
\end{tabular}<br>
There are no negative values in row <img class="equation_image" title=" \displaystyle p " src="/equation_images/%20%5Cdisplaystyle%20p%20" alt="LaTeX: \displaystyle p " data-equation-content=" \displaystyle p " /> and this is the final tableau.The value of <img class="equation_image" title=" \displaystyle x " src="/equation_images/%20%5Cdisplaystyle%20x%20" alt="LaTeX: \displaystyle x " data-equation-content=" \displaystyle x " /> is <img class="equation_image" title=" \displaystyle \frac{4830}{821} " src="/equation_images/%20%5Cdisplaystyle%20%5Cfrac%7B4830%7D%7B821%7D%20" alt="LaTeX: \displaystyle \frac{4830}{821} " data-equation-content=" \displaystyle \frac{4830}{821} " /> . The value of <img class="equation_image" title=" \displaystyle y " src="/equation_images/%20%5Cdisplaystyle%20y%20" alt="LaTeX: \displaystyle y " data-equation-content=" \displaystyle y " /> is <img class="equation_image" title=" \displaystyle \frac{58656}{821} " src="/equation_images/%20%5Cdisplaystyle%20%5Cfrac%7B58656%7D%7B821%7D%20" alt="LaTeX: \displaystyle \frac{58656}{821} " data-equation-content=" \displaystyle \frac{58656}{821} " /> . The max value is <img class="equation_image" title=" \displaystyle p = \frac{610014}{821} " src="/equation_images/%20%5Cdisplaystyle%20p%20%3D%20%5Cfrac%7B610014%7D%7B821%7D%20" alt="LaTeX: \displaystyle p = \frac{610014}{821} " data-equation-content=" \displaystyle p = \frac{610014}{821} " /> </p> </p>