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Questions: Algebra BusinessCalculus
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Use the simplex method to maximize \(\displaystyle p = 25 x + 21 y\) subject to \(\displaystyle \begin{cases}79 x + 46 y \leq 3634 \\ 42 x + 94 y \leq 3948 \\x \geq 0, y \geq 0 \end{cases}\)
Adding the slack variables \(\displaystyle s\) and \(\displaystyle t\) to the inequalities gives:
\begin{equation*}\begin{cases}79 x + 46 y +s = 3634 \\ 42 x + 94 y+t = 3948 \\ - 25 x - 21 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
\(\displaystyle s\) & \(\displaystyle 79\) & \(\displaystyle 46\) & \(\displaystyle 1\) & \(\displaystyle 0\) & \(\displaystyle 0\) & \(\displaystyle 3634\) \\ \hline
\(\displaystyle t\) & \(\displaystyle 42\) & \(\displaystyle 94\) & \(\displaystyle 0\) & \(\displaystyle 1\) & \(\displaystyle 0\) & \(\displaystyle 3948\) \\ \hline
\(\displaystyle p\) & \(\displaystyle -25\) & \(\displaystyle -21\) & \(\displaystyle 0\) & \(\displaystyle 0\) & \(\displaystyle 1\) & \(\displaystyle 0\) \\ \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:
\(\displaystyle 79R_2-42R_1\)
\(\displaystyle 79R_3+25R_1\)
\begin{tabular}{|c|c|c|c|c|c|c|}\hline
\(\displaystyle \) & \(\displaystyle x\) & \(\displaystyle y\) & \(\displaystyle s\) & \(\displaystyle t\) & \(\displaystyle p\) & \(\displaystyle \) \\ \hline
\(\displaystyle x\) & \(\displaystyle 79\) & \(\displaystyle 46\) & \(\displaystyle 1\) & \(\displaystyle 0\) & \(\displaystyle 0\) & \(\displaystyle 3634\) \\ \hline
\(\displaystyle t\) & \(\displaystyle 0\) & \(\displaystyle 5494\) & \(\displaystyle -42\) & \(\displaystyle 79\) & \(\displaystyle 0\) & \(\displaystyle 159264\) \\ \hline
\(\displaystyle p\) & \(\displaystyle 0\) & \(\displaystyle -509\) & \(\displaystyle 25\) & \(\displaystyle 0\) & \(\displaystyle 79\) & \(\displaystyle 90850\) \\ \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:
\(\displaystyle 2747R_1-23R_2\)
\(\displaystyle 5494R_3+509R_2\)
\begin{tabular}{|c|c|c|c|c|c|c|}\hline
\(\displaystyle \) & \(\displaystyle x\) & \(\displaystyle y\) & \(\displaystyle s\) & \(\displaystyle t\) & \(\displaystyle p\) & \(\displaystyle \) \\ \hline
\(\displaystyle x\) & \(\displaystyle 217013\) & \(\displaystyle 0\) & \(\displaystyle 3713\) & \(\displaystyle -1817\) & \(\displaystyle 0\) & \(\displaystyle 6319526\) \\ \hline
\(\displaystyle y\) & \(\displaystyle 0\) & \(\displaystyle 5494\) & \(\displaystyle -42\) & \(\displaystyle 79\) & \(\displaystyle 0\) & \(\displaystyle 159264\) \\ \hline
\(\displaystyle p\) & \(\displaystyle 0\) & \(\displaystyle 0\) & \(\displaystyle 115972\) & \(\displaystyle 40211\) & \(\displaystyle 434026\) & \(\displaystyle 580195276\) \\ \hline
\end{tabular}
There are no negative values in row \(\displaystyle p\) and this is the final tableau.The value of \(\displaystyle x\) is \(\displaystyle \frac{79994}{2747}\). The value of \(\displaystyle y\) is \(\displaystyle \frac{79632}{2747}\). The max value is \(\displaystyle p = \frac{3672122}{2747}\)
\begin{question}Use the simplex method to maximize $p = 25 x + 21 y$ subject to $\begin{cases}79 x + 46 y \leq 3634 \\ 42 x + 94 y \leq 3948 \\x \geq 0, y \geq 0 \end{cases}$
\soln{9cm}{Adding the slack variables $s$ and $t$ to the inequalities gives:\newline\begin{equation*}\begin{cases}79 x + 46 y +s = 3634 \\ 42 x + 94 y+t = 3948 \\ - 25 x - 21 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
$s$ & $79$ & $46$ & $1$ & $0$ & $0$ & $3634$ \\ \hline
$t$ & $42$ & $94$ & $0$ & $1$ & $0$ & $3948$ \\ \hline
$p$ & $-25$ & $-21$ & $0$ & $0$ & $1$ & $0$ \\ \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
$79R_2-42R_1$\newline
$79R_3+25R_1$\newline
\begin{tabular}{|c|c|c|c|c|c|c|}\hline
$$ & $x$ & $y$ & $s$ & $t$ & $p$ & $$ \\ \hline
$x$ & $79$ & $46$ & $1$ & $0$ & $0$ & $3634$ \\ \hline
$t$ & $0$ & $5494$ & $-42$ & $79$ & $0$ & $159264$ \\ \hline
$p$ & $0$ & $-509$ & $25$ & $0$ & $79$ & $90850$ \\ \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
$2747R_1-23R_2$\newline
$5494R_3+509R_2$\newline
\begin{tabular}{|c|c|c|c|c|c|c|}\hline
$$ & $x$ & $y$ & $s$ & $t$ & $p$ & $$ \\ \hline
$x$ & $217013$ & $0$ & $3713$ & $-1817$ & $0$ & $6319526$ \\ \hline
$y$ & $0$ & $5494$ & $-42$ & $79$ & $0$ & $159264$ \\ \hline
$p$ & $0$ & $0$ & $115972$ & $40211$ & $434026$ & $580195276$ \\ \hline
\end{tabular}\newline
There are no negative values in row $p$ and this is the final tableau.The value of $x$ is $\frac{79994}{2747}$. The value of $y$ is $\frac{79632}{2747}$. The max value is $p = \frac{3672122}{2747}$ }
\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 = 25 x + 21 y " src="/equation_images/%20%5Cdisplaystyle%20p%20%3D%2025%20x%20%2B%2021%20y%20" alt="LaTeX: \displaystyle p = 25 x + 21 y " data-equation-content=" \displaystyle p = 25 x + 21 y " /> subject to <img class="equation_image" title=" \displaystyle \begin{cases}79 x + 46 y \leq 3634 \\ 42 x + 94 y \leq 3948 \\x \geq 0, y \geq 0 \end{cases} " src="/equation_images/%20%5Cdisplaystyle%20%5Cbegin%7Bcases%7D79%20x%20%2B%2046%20y%20%5Cleq%203634%20%5C%5C%2042%20x%20%2B%2094%20y%20%5Cleq%203948%20%5C%5Cx%20%5Cgeq%200%2C%20y%20%5Cgeq%200%20%5Cend%7Bcases%7D%20" alt="LaTeX: \displaystyle \begin{cases}79 x + 46 y \leq 3634 \\ 42 x + 94 y \leq 3948 \\x \geq 0, y \geq 0 \end{cases} " data-equation-content=" \displaystyle \begin{cases}79 x + 46 y \leq 3634 \\ 42 x + 94 y \leq 3948 \\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}79 x + 46 y +s = 3634 \\ 42 x + 94 y+t = 3948 \\ - 25 x - 21 y+p =0 \end{cases} " src="/equation_images/%20%5Cbegin%7Bcases%7D79%20x%20%2B%2046%20y%20%2Bs%20%3D%203634%20%5C%5C%2042%20x%20%2B%2094%20y%2Bt%20%3D%203948%20%5C%5C%20-%2025%20x%20-%2021%20y%2Bp%20%3D0%20%5Cend%7Bcases%7D%20" alt="LaTeX: \begin{cases}79 x + 46 y +s = 3634 \\ 42 x + 94 y+t = 3948 \\ - 25 x - 21 y+p =0 \end{cases} " data-equation-content=" \begin{cases}79 x + 46 y +s = 3634 \\ 42 x + 94 y+t = 3948 \\ - 25 x - 21 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 79 " src="/equation_images/%20%5Cdisplaystyle%2079%20" alt="LaTeX: \displaystyle 79 " data-equation-content=" \displaystyle 79 " /> & <img class="equation_image" title=" \displaystyle 46 " src="/equation_images/%20%5Cdisplaystyle%2046%20" alt="LaTeX: \displaystyle 46 " data-equation-content=" \displaystyle 46 " /> & <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 3634 " src="/equation_images/%20%5Cdisplaystyle%203634%20" alt="LaTeX: \displaystyle 3634 " data-equation-content=" \displaystyle 3634 " /> \\ \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 42 " src="/equation_images/%20%5Cdisplaystyle%2042%20" alt="LaTeX: \displaystyle 42 " data-equation-content=" \displaystyle 42 " /> & <img class="equation_image" title=" \displaystyle 94 " src="/equation_images/%20%5Cdisplaystyle%2094%20" alt="LaTeX: \displaystyle 94 " data-equation-content=" \displaystyle 94 " /> & <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 3948 " src="/equation_images/%20%5Cdisplaystyle%203948%20" alt="LaTeX: \displaystyle 3948 " data-equation-content=" \displaystyle 3948 " /> \\ \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 -25 " src="/equation_images/%20%5Cdisplaystyle%20-25%20" alt="LaTeX: \displaystyle -25 " data-equation-content=" \displaystyle -25 " /> & <img class="equation_image" title=" \displaystyle -21 " src="/equation_images/%20%5Cdisplaystyle%20-21%20" alt="LaTeX: \displaystyle -21 " data-equation-content=" \displaystyle -21 " /> & <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 79R_2-42R_1 " src="/equation_images/%20%5Cdisplaystyle%2079R_2-42R_1%20" alt="LaTeX: \displaystyle 79R_2-42R_1 " data-equation-content=" \displaystyle 79R_2-42R_1 " /> <br>
<img class="equation_image" title=" \displaystyle 79R_3+25R_1 " src="/equation_images/%20%5Cdisplaystyle%2079R_3%2B25R_1%20" alt="LaTeX: \displaystyle 79R_3+25R_1 " data-equation-content=" \displaystyle 79R_3+25R_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 79 " src="/equation_images/%20%5Cdisplaystyle%2079%20" alt="LaTeX: \displaystyle 79 " data-equation-content=" \displaystyle 79 " /> & <img class="equation_image" title=" \displaystyle 46 " src="/equation_images/%20%5Cdisplaystyle%2046%20" alt="LaTeX: \displaystyle 46 " data-equation-content=" \displaystyle 46 " /> & <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 3634 " src="/equation_images/%20%5Cdisplaystyle%203634%20" alt="LaTeX: \displaystyle 3634 " data-equation-content=" \displaystyle 3634 " /> \\ \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 5494 " src="/equation_images/%20%5Cdisplaystyle%205494%20" alt="LaTeX: \displaystyle 5494 " data-equation-content=" \displaystyle 5494 " /> & <img class="equation_image" title=" \displaystyle -42 " src="/equation_images/%20%5Cdisplaystyle%20-42%20" alt="LaTeX: \displaystyle -42 " data-equation-content=" \displaystyle -42 " /> & <img class="equation_image" title=" \displaystyle 79 " src="/equation_images/%20%5Cdisplaystyle%2079%20" alt="LaTeX: \displaystyle 79 " data-equation-content=" \displaystyle 79 " /> & <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 159264 " src="/equation_images/%20%5Cdisplaystyle%20159264%20" alt="LaTeX: \displaystyle 159264 " data-equation-content=" \displaystyle 159264 " /> \\ \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 -509 " src="/equation_images/%20%5Cdisplaystyle%20-509%20" alt="LaTeX: \displaystyle -509 " data-equation-content=" \displaystyle -509 " /> & <img class="equation_image" title=" \displaystyle 25 " src="/equation_images/%20%5Cdisplaystyle%2025%20" alt="LaTeX: \displaystyle 25 " data-equation-content=" \displaystyle 25 " /> & <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 79 " src="/equation_images/%20%5Cdisplaystyle%2079%20" alt="LaTeX: \displaystyle 79 " data-equation-content=" \displaystyle 79 " /> & <img class="equation_image" title=" \displaystyle 90850 " src="/equation_images/%20%5Cdisplaystyle%2090850%20" alt="LaTeX: \displaystyle 90850 " data-equation-content=" \displaystyle 90850 " /> \\ \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 2747R_1-23R_2 " src="/equation_images/%20%5Cdisplaystyle%202747R_1-23R_2%20" alt="LaTeX: \displaystyle 2747R_1-23R_2 " data-equation-content=" \displaystyle 2747R_1-23R_2 " /> <br>
<img class="equation_image" title=" \displaystyle 5494R_3+509R_2 " src="/equation_images/%20%5Cdisplaystyle%205494R_3%2B509R_2%20" alt="LaTeX: \displaystyle 5494R_3+509R_2 " data-equation-content=" \displaystyle 5494R_3+509R_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 217013 " src="/equation_images/%20%5Cdisplaystyle%20217013%20" alt="LaTeX: \displaystyle 217013 " data-equation-content=" \displaystyle 217013 " /> & <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 3713 " src="/equation_images/%20%5Cdisplaystyle%203713%20" alt="LaTeX: \displaystyle 3713 " data-equation-content=" \displaystyle 3713 " /> & <img class="equation_image" title=" \displaystyle -1817 " src="/equation_images/%20%5Cdisplaystyle%20-1817%20" alt="LaTeX: \displaystyle -1817 " data-equation-content=" \displaystyle -1817 " /> & <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 6319526 " src="/equation_images/%20%5Cdisplaystyle%206319526%20" alt="LaTeX: \displaystyle 6319526 " data-equation-content=" \displaystyle 6319526 " /> \\ \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 5494 " src="/equation_images/%20%5Cdisplaystyle%205494%20" alt="LaTeX: \displaystyle 5494 " data-equation-content=" \displaystyle 5494 " /> & <img class="equation_image" title=" \displaystyle -42 " src="/equation_images/%20%5Cdisplaystyle%20-42%20" alt="LaTeX: \displaystyle -42 " data-equation-content=" \displaystyle -42 " /> & <img class="equation_image" title=" \displaystyle 79 " src="/equation_images/%20%5Cdisplaystyle%2079%20" alt="LaTeX: \displaystyle 79 " data-equation-content=" \displaystyle 79 " /> & <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 159264 " src="/equation_images/%20%5Cdisplaystyle%20159264%20" alt="LaTeX: \displaystyle 159264 " data-equation-content=" \displaystyle 159264 " /> \\ \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 115972 " src="/equation_images/%20%5Cdisplaystyle%20115972%20" alt="LaTeX: \displaystyle 115972 " data-equation-content=" \displaystyle 115972 " /> & <img class="equation_image" title=" \displaystyle 40211 " src="/equation_images/%20%5Cdisplaystyle%2040211%20" alt="LaTeX: \displaystyle 40211 " data-equation-content=" \displaystyle 40211 " /> & <img class="equation_image" title=" \displaystyle 434026 " src="/equation_images/%20%5Cdisplaystyle%20434026%20" alt="LaTeX: \displaystyle 434026 " data-equation-content=" \displaystyle 434026 " /> & <img class="equation_image" title=" \displaystyle 580195276 " src="/equation_images/%20%5Cdisplaystyle%20580195276%20" alt="LaTeX: \displaystyle 580195276 " data-equation-content=" \displaystyle 580195276 " /> \\ \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{79994}{2747} " src="/equation_images/%20%5Cdisplaystyle%20%5Cfrac%7B79994%7D%7B2747%7D%20" alt="LaTeX: \displaystyle \frac{79994}{2747} " data-equation-content=" \displaystyle \frac{79994}{2747} " /> . 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{79632}{2747} " src="/equation_images/%20%5Cdisplaystyle%20%5Cfrac%7B79632%7D%7B2747%7D%20" alt="LaTeX: \displaystyle \frac{79632}{2747} " data-equation-content=" \displaystyle \frac{79632}{2747} " /> . The max value is <img class="equation_image" title=" \displaystyle p = \frac{3672122}{2747} " src="/equation_images/%20%5Cdisplaystyle%20p%20%3D%20%5Cfrac%7B3672122%7D%7B2747%7D%20" alt="LaTeX: \displaystyle p = \frac{3672122}{2747} " data-equation-content=" \displaystyle p = \frac{3672122}{2747} " /> </p> </p>