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Questions: Algebra BusinessCalculus
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Use the simplex method to maximize \(\displaystyle p = 3 x + 13 y\) subject to \(\displaystyle \begin{cases}61 x + 41 y \leq 2501 \\ 2 x + 79 y \leq 158 \\x \geq 0, y \geq 0 \end{cases}\)
Adding the slack variables \(\displaystyle s\) and \(\displaystyle t\) to the inequalities gives:
\begin{equation*}\begin{cases}61 x + 41 y +s = 2501 \\ 2 x + 79 y+t = 158 \\ - 3 x - 13 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
\(\displaystyle s\) & \(\displaystyle 61\) & \(\displaystyle 41\) & \(\displaystyle 1\) & \(\displaystyle 0\) & \(\displaystyle 0\) & \(\displaystyle 2501\) \\ \hline
\(\displaystyle t\) & \(\displaystyle 2\) & \(\displaystyle 79\) & \(\displaystyle 0\) & \(\displaystyle 1\) & \(\displaystyle 0\) & \(\displaystyle 158\) \\ \hline
\(\displaystyle p\) & \(\displaystyle -3\) & \(\displaystyle -13\) & \(\displaystyle 0\) & \(\displaystyle 0\) & \(\displaystyle 1\) & \(\displaystyle 0\) \\ \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:
\(\displaystyle 79R_1-41R_2\)
\(\displaystyle 79R_3+13R_2\)
\begin{tabular}{|c|c|c|c|c|c|c|}\hline
\(\displaystyle \) & \(\displaystyle x\) & \(\displaystyle y\) & \(\displaystyle s\) & \(\displaystyle t\) & \(\displaystyle p\) & \(\displaystyle \) \\ \hline
\(\displaystyle s\) & \(\displaystyle 4737\) & \(\displaystyle 0\) & \(\displaystyle 79\) & \(\displaystyle -41\) & \(\displaystyle 0\) & \(\displaystyle 191101\) \\ \hline
\(\displaystyle y\) & \(\displaystyle 2\) & \(\displaystyle 79\) & \(\displaystyle 0\) & \(\displaystyle 1\) & \(\displaystyle 0\) & \(\displaystyle 158\) \\ \hline
\(\displaystyle p\) & \(\displaystyle -211\) & \(\displaystyle 0\) & \(\displaystyle 0\) & \(\displaystyle 13\) & \(\displaystyle 79\) & \(\displaystyle 2054\) \\ \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:
\(\displaystyle 4737R_2-2R_1\)
\(\displaystyle 4737R_3+211R_1\)
\begin{tabular}{|c|c|c|c|c|c|c|}\hline
\(\displaystyle \) & \(\displaystyle x\) & \(\displaystyle y\) & \(\displaystyle s\) & \(\displaystyle t\) & \(\displaystyle p\) & \(\displaystyle \) \\ \hline
\(\displaystyle x\) & \(\displaystyle 4737\) & \(\displaystyle 0\) & \(\displaystyle 79\) & \(\displaystyle -41\) & \(\displaystyle 0\) & \(\displaystyle 191101\) \\ \hline
\(\displaystyle y\) & \(\displaystyle 0\) & \(\displaystyle 374223\) & \(\displaystyle -158\) & \(\displaystyle 4819\) & \(\displaystyle 0\) & \(\displaystyle 366244\) \\ \hline
\(\displaystyle p\) & \(\displaystyle 0\) & \(\displaystyle 0\) & \(\displaystyle 16669\) & \(\displaystyle 52930\) & \(\displaystyle 374223\) & \(\displaystyle 50052109\) \\ \hline
\end{tabular}
There are no negative values in row \(\displaystyle p\) and this is the final tableau.The value of \(\displaystyle x\) is \(\displaystyle \frac{191101}{4737}\). The value of \(\displaystyle y\) is \(\displaystyle \frac{4636}{4737}\). The max value is \(\displaystyle p = \frac{633571}{4737}\)
\begin{question}Use the simplex method to maximize $p = 3 x + 13 y$ subject to $\begin{cases}61 x + 41 y \leq 2501 \\ 2 x + 79 y \leq 158 \\x \geq 0, y \geq 0 \end{cases}$
\soln{9cm}{Adding the slack variables $s$ and $t$ to the inequalities gives:\newline\begin{equation*}\begin{cases}61 x + 41 y +s = 2501 \\ 2 x + 79 y+t = 158 \\ - 3 x - 13 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
$s$ & $61$ & $41$ & $1$ & $0$ & $0$ & $2501$ \\ \hline
$t$ & $2$ & $79$ & $0$ & $1$ & $0$ & $158$ \\ \hline
$p$ & $-3$ & $-13$ & $0$ & $0$ & $1$ & $0$ \\ \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
$79R_1-41R_2$\newline
$79R_3+13R_2$\newline
\begin{tabular}{|c|c|c|c|c|c|c|}\hline
$$ & $x$ & $y$ & $s$ & $t$ & $p$ & $$ \\ \hline
$s$ & $4737$ & $0$ & $79$ & $-41$ & $0$ & $191101$ \\ \hline
$y$ & $2$ & $79$ & $0$ & $1$ & $0$ & $158$ \\ \hline
$p$ & $-211$ & $0$ & $0$ & $13$ & $79$ & $2054$ \\ \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
$4737R_2-2R_1$\newline
$4737R_3+211R_1$\newline
\begin{tabular}{|c|c|c|c|c|c|c|}\hline
$$ & $x$ & $y$ & $s$ & $t$ & $p$ & $$ \\ \hline
$x$ & $4737$ & $0$ & $79$ & $-41$ & $0$ & $191101$ \\ \hline
$y$ & $0$ & $374223$ & $-158$ & $4819$ & $0$ & $366244$ \\ \hline
$p$ & $0$ & $0$ & $16669$ & $52930$ & $374223$ & $50052109$ \\ \hline
\end{tabular}\newline
There are no negative values in row $p$ and this is the final tableau.The value of $x$ is $\frac{191101}{4737}$. The value of $y$ is $\frac{4636}{4737}$. The max value is $p = \frac{633571}{4737}$ }
\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 = 3 x + 13 y " src="/equation_images/%20%5Cdisplaystyle%20p%20%3D%203%20x%20%2B%2013%20y%20" alt="LaTeX: \displaystyle p = 3 x + 13 y " data-equation-content=" \displaystyle p = 3 x + 13 y " /> subject to <img class="equation_image" title=" \displaystyle \begin{cases}61 x + 41 y \leq 2501 \\ 2 x + 79 y \leq 158 \\x \geq 0, y \geq 0 \end{cases} " src="/equation_images/%20%5Cdisplaystyle%20%5Cbegin%7Bcases%7D61%20x%20%2B%2041%20y%20%5Cleq%202501%20%5C%5C%202%20x%20%2B%2079%20y%20%5Cleq%20158%20%5C%5Cx%20%5Cgeq%200%2C%20y%20%5Cgeq%200%20%5Cend%7Bcases%7D%20" alt="LaTeX: \displaystyle \begin{cases}61 x + 41 y \leq 2501 \\ 2 x + 79 y \leq 158 \\x \geq 0, y \geq 0 \end{cases} " data-equation-content=" \displaystyle \begin{cases}61 x + 41 y \leq 2501 \\ 2 x + 79 y \leq 158 \\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}61 x + 41 y +s = 2501 \\ 2 x + 79 y+t = 158 \\ - 3 x - 13 y+p =0 \end{cases} " src="/equation_images/%20%5Cbegin%7Bcases%7D61%20x%20%2B%2041%20y%20%2Bs%20%3D%202501%20%5C%5C%202%20x%20%2B%2079%20y%2Bt%20%3D%20158%20%5C%5C%20-%203%20x%20-%2013%20y%2Bp%20%3D0%20%5Cend%7Bcases%7D%20" alt="LaTeX: \begin{cases}61 x + 41 y +s = 2501 \\ 2 x + 79 y+t = 158 \\ - 3 x - 13 y+p =0 \end{cases} " data-equation-content=" \begin{cases}61 x + 41 y +s = 2501 \\ 2 x + 79 y+t = 158 \\ - 3 x - 13 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 61 " src="/equation_images/%20%5Cdisplaystyle%2061%20" alt="LaTeX: \displaystyle 61 " data-equation-content=" \displaystyle 61 " /> & <img class="equation_image" title=" \displaystyle 41 " src="/equation_images/%20%5Cdisplaystyle%2041%20" alt="LaTeX: \displaystyle 41 " data-equation-content=" \displaystyle 41 " /> & <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 2501 " src="/equation_images/%20%5Cdisplaystyle%202501%20" alt="LaTeX: \displaystyle 2501 " data-equation-content=" \displaystyle 2501 " /> \\ \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 2 " src="/equation_images/%20%5Cdisplaystyle%202%20" alt="LaTeX: \displaystyle 2 " data-equation-content=" \displaystyle 2 " /> & <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 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 158 " src="/equation_images/%20%5Cdisplaystyle%20158%20" alt="LaTeX: \displaystyle 158 " data-equation-content=" \displaystyle 158 " /> \\ \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 -3 " src="/equation_images/%20%5Cdisplaystyle%20-3%20" alt="LaTeX: \displaystyle -3 " data-equation-content=" \displaystyle -3 " /> & <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 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 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 79R_1-41R_2 " src="/equation_images/%20%5Cdisplaystyle%2079R_1-41R_2%20" alt="LaTeX: \displaystyle 79R_1-41R_2 " data-equation-content=" \displaystyle 79R_1-41R_2 " /> <br>
<img class="equation_image" title=" \displaystyle 79R_3+13R_2 " src="/equation_images/%20%5Cdisplaystyle%2079R_3%2B13R_2%20" alt="LaTeX: \displaystyle 79R_3+13R_2 " data-equation-content=" \displaystyle 79R_3+13R_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 s " src="/equation_images/%20%5Cdisplaystyle%20s%20" alt="LaTeX: \displaystyle s " data-equation-content=" \displaystyle s " /> & <img class="equation_image" title=" \displaystyle 4737 " src="/equation_images/%20%5Cdisplaystyle%204737%20" alt="LaTeX: \displaystyle 4737 " data-equation-content=" \displaystyle 4737 " /> & <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 -41 " src="/equation_images/%20%5Cdisplaystyle%20-41%20" alt="LaTeX: \displaystyle -41 " data-equation-content=" \displaystyle -41 " /> & <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 191101 " src="/equation_images/%20%5Cdisplaystyle%20191101%20" alt="LaTeX: \displaystyle 191101 " data-equation-content=" \displaystyle 191101 " /> \\ \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 2 " src="/equation_images/%20%5Cdisplaystyle%202%20" alt="LaTeX: \displaystyle 2 " data-equation-content=" \displaystyle 2 " /> & <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 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 158 " src="/equation_images/%20%5Cdisplaystyle%20158%20" alt="LaTeX: \displaystyle 158 " data-equation-content=" \displaystyle 158 " /> \\ \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 -211 " src="/equation_images/%20%5Cdisplaystyle%20-211%20" alt="LaTeX: \displaystyle -211 " data-equation-content=" \displaystyle -211 " /> & <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 13 " src="/equation_images/%20%5Cdisplaystyle%2013%20" alt="LaTeX: \displaystyle 13 " data-equation-content=" \displaystyle 13 " /> & <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 2054 " src="/equation_images/%20%5Cdisplaystyle%202054%20" alt="LaTeX: \displaystyle 2054 " data-equation-content=" \displaystyle 2054 " /> \\ \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 4737R_2-2R_1 " src="/equation_images/%20%5Cdisplaystyle%204737R_2-2R_1%20" alt="LaTeX: \displaystyle 4737R_2-2R_1 " data-equation-content=" \displaystyle 4737R_2-2R_1 " /> <br>
<img class="equation_image" title=" \displaystyle 4737R_3+211R_1 " src="/equation_images/%20%5Cdisplaystyle%204737R_3%2B211R_1%20" alt="LaTeX: \displaystyle 4737R_3+211R_1 " data-equation-content=" \displaystyle 4737R_3+211R_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 4737 " src="/equation_images/%20%5Cdisplaystyle%204737%20" alt="LaTeX: \displaystyle 4737 " data-equation-content=" \displaystyle 4737 " /> & <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 -41 " src="/equation_images/%20%5Cdisplaystyle%20-41%20" alt="LaTeX: \displaystyle -41 " data-equation-content=" \displaystyle -41 " /> & <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 191101 " src="/equation_images/%20%5Cdisplaystyle%20191101%20" alt="LaTeX: \displaystyle 191101 " data-equation-content=" \displaystyle 191101 " /> \\ \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 374223 " src="/equation_images/%20%5Cdisplaystyle%20374223%20" alt="LaTeX: \displaystyle 374223 " data-equation-content=" \displaystyle 374223 " /> & <img class="equation_image" title=" \displaystyle -158 " src="/equation_images/%20%5Cdisplaystyle%20-158%20" alt="LaTeX: \displaystyle -158 " data-equation-content=" \displaystyle -158 " /> & <img class="equation_image" title=" \displaystyle 4819 " src="/equation_images/%20%5Cdisplaystyle%204819%20" alt="LaTeX: \displaystyle 4819 " data-equation-content=" \displaystyle 4819 " /> & <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 366244 " src="/equation_images/%20%5Cdisplaystyle%20366244%20" alt="LaTeX: \displaystyle 366244 " data-equation-content=" \displaystyle 366244 " /> \\ \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 16669 " src="/equation_images/%20%5Cdisplaystyle%2016669%20" alt="LaTeX: \displaystyle 16669 " data-equation-content=" \displaystyle 16669 " /> & <img class="equation_image" title=" \displaystyle 52930 " src="/equation_images/%20%5Cdisplaystyle%2052930%20" alt="LaTeX: \displaystyle 52930 " data-equation-content=" \displaystyle 52930 " /> & <img class="equation_image" title=" \displaystyle 374223 " src="/equation_images/%20%5Cdisplaystyle%20374223%20" alt="LaTeX: \displaystyle 374223 " data-equation-content=" \displaystyle 374223 " /> & <img class="equation_image" title=" \displaystyle 50052109 " src="/equation_images/%20%5Cdisplaystyle%2050052109%20" alt="LaTeX: \displaystyle 50052109 " data-equation-content=" \displaystyle 50052109 " /> \\ \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{191101}{4737} " src="/equation_images/%20%5Cdisplaystyle%20%5Cfrac%7B191101%7D%7B4737%7D%20" alt="LaTeX: \displaystyle \frac{191101}{4737} " data-equation-content=" \displaystyle \frac{191101}{4737} " /> . 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{4636}{4737} " src="/equation_images/%20%5Cdisplaystyle%20%5Cfrac%7B4636%7D%7B4737%7D%20" alt="LaTeX: \displaystyle \frac{4636}{4737} " data-equation-content=" \displaystyle \frac{4636}{4737} " /> . The max value is <img class="equation_image" title=" \displaystyle p = \frac{633571}{4737} " src="/equation_images/%20%5Cdisplaystyle%20p%20%3D%20%5Cfrac%7B633571%7D%7B4737%7D%20" alt="LaTeX: \displaystyle p = \frac{633571}{4737} " data-equation-content=" \displaystyle p = \frac{633571}{4737} " /> </p> </p>