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Questions: Algebra BusinessCalculus
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Use the simplex method to maximize \(\displaystyle p = 22 x + 17 y\) subject to \(\displaystyle \begin{cases}74 x + 39 y \leq 2886 \\ 21 x + 67 y \leq 1407 \\x \geq 0, y \geq 0 \end{cases}\)
Adding the slack variables \(\displaystyle s\) and \(\displaystyle t\) to the inequalities gives:
\begin{equation*}\begin{cases}74 x + 39 y +s = 2886 \\ 21 x + 67 y+t = 1407 \\ - 22 x - 17 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
\(\displaystyle s\) & \(\displaystyle 74\) & \(\displaystyle 39\) & \(\displaystyle 1\) & \(\displaystyle 0\) & \(\displaystyle 0\) & \(\displaystyle 2886\) \\ \hline
\(\displaystyle t\) & \(\displaystyle 21\) & \(\displaystyle 67\) & \(\displaystyle 0\) & \(\displaystyle 1\) & \(\displaystyle 0\) & \(\displaystyle 1407\) \\ \hline
\(\displaystyle p\) & \(\displaystyle -22\) & \(\displaystyle -17\) & \(\displaystyle 0\) & \(\displaystyle 0\) & \(\displaystyle 1\) & \(\displaystyle 0\) \\ \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:
\(\displaystyle 74R_2-21R_1\)
\(\displaystyle 37R_3+11R_1\)
\begin{tabular}{|c|c|c|c|c|c|c|}\hline
\(\displaystyle \) & \(\displaystyle x\) & \(\displaystyle y\) & \(\displaystyle s\) & \(\displaystyle t\) & \(\displaystyle p\) & \(\displaystyle \) \\ \hline
\(\displaystyle x\) & \(\displaystyle 74\) & \(\displaystyle 39\) & \(\displaystyle 1\) & \(\displaystyle 0\) & \(\displaystyle 0\) & \(\displaystyle 2886\) \\ \hline
\(\displaystyle t\) & \(\displaystyle 0\) & \(\displaystyle 4139\) & \(\displaystyle -21\) & \(\displaystyle 74\) & \(\displaystyle 0\) & \(\displaystyle 43512\) \\ \hline
\(\displaystyle p\) & \(\displaystyle 0\) & \(\displaystyle -200\) & \(\displaystyle 11\) & \(\displaystyle 0\) & \(\displaystyle 37\) & \(\displaystyle 31746\) \\ \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:
\(\displaystyle 4139R_1-39R_2\)
\(\displaystyle 4139R_3+200R_2\)
\begin{tabular}{|c|c|c|c|c|c|c|}\hline
\(\displaystyle \) & \(\displaystyle x\) & \(\displaystyle y\) & \(\displaystyle s\) & \(\displaystyle t\) & \(\displaystyle p\) & \(\displaystyle \) \\ \hline
\(\displaystyle x\) & \(\displaystyle 306286\) & \(\displaystyle 0\) & \(\displaystyle 4958\) & \(\displaystyle -2886\) & \(\displaystyle 0\) & \(\displaystyle 10248186\) \\ \hline
\(\displaystyle y\) & \(\displaystyle 0\) & \(\displaystyle 4139\) & \(\displaystyle -21\) & \(\displaystyle 74\) & \(\displaystyle 0\) & \(\displaystyle 43512\) \\ \hline
\(\displaystyle p\) & \(\displaystyle 0\) & \(\displaystyle 0\) & \(\displaystyle 41329\) & \(\displaystyle 14800\) & \(\displaystyle 153143\) & \(\displaystyle 140099094\) \\ \hline
\end{tabular}
There are no negative values in row \(\displaystyle p\) and this is the final tableau.The value of \(\displaystyle x\) is \(\displaystyle \frac{138489}{4139}\). The value of \(\displaystyle y\) is \(\displaystyle \frac{43512}{4139}\). The max value is \(\displaystyle p = \frac{3786462}{4139}\)
\begin{question}Use the simplex method to maximize $p = 22 x + 17 y$ subject to $\begin{cases}74 x + 39 y \leq 2886 \\ 21 x + 67 y \leq 1407 \\x \geq 0, y \geq 0 \end{cases}$
\soln{9cm}{Adding the slack variables $s$ and $t$ to the inequalities gives:\newline\begin{equation*}\begin{cases}74 x + 39 y +s = 2886 \\ 21 x + 67 y+t = 1407 \\ - 22 x - 17 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
$s$ & $74$ & $39$ & $1$ & $0$ & $0$ & $2886$ \\ \hline
$t$ & $21$ & $67$ & $0$ & $1$ & $0$ & $1407$ \\ \hline
$p$ & $-22$ & $-17$ & $0$ & $0$ & $1$ & $0$ \\ \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
$74R_2-21R_1$\newline
$37R_3+11R_1$\newline
\begin{tabular}{|c|c|c|c|c|c|c|}\hline
$$ & $x$ & $y$ & $s$ & $t$ & $p$ & $$ \\ \hline
$x$ & $74$ & $39$ & $1$ & $0$ & $0$ & $2886$ \\ \hline
$t$ & $0$ & $4139$ & $-21$ & $74$ & $0$ & $43512$ \\ \hline
$p$ & $0$ & $-200$ & $11$ & $0$ & $37$ & $31746$ \\ \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
$4139R_1-39R_2$\newline
$4139R_3+200R_2$\newline
\begin{tabular}{|c|c|c|c|c|c|c|}\hline
$$ & $x$ & $y$ & $s$ & $t$ & $p$ & $$ \\ \hline
$x$ & $306286$ & $0$ & $4958$ & $-2886$ & $0$ & $10248186$ \\ \hline
$y$ & $0$ & $4139$ & $-21$ & $74$ & $0$ & $43512$ \\ \hline
$p$ & $0$ & $0$ & $41329$ & $14800$ & $153143$ & $140099094$ \\ \hline
\end{tabular}\newline
There are no negative values in row $p$ and this is the final tableau.The value of $x$ is $\frac{138489}{4139}$. The value of $y$ is $\frac{43512}{4139}$. The max value is $p = \frac{3786462}{4139}$ }
\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 = 22 x + 17 y " src="/equation_images/%20%5Cdisplaystyle%20p%20%3D%2022%20x%20%2B%2017%20y%20" alt="LaTeX: \displaystyle p = 22 x + 17 y " data-equation-content=" \displaystyle p = 22 x + 17 y " /> subject to <img class="equation_image" title=" \displaystyle \begin{cases}74 x + 39 y \leq 2886 \\ 21 x + 67 y \leq 1407 \\x \geq 0, y \geq 0 \end{cases} " src="/equation_images/%20%5Cdisplaystyle%20%5Cbegin%7Bcases%7D74%20x%20%2B%2039%20y%20%5Cleq%202886%20%5C%5C%2021%20x%20%2B%2067%20y%20%5Cleq%201407%20%5C%5Cx%20%5Cgeq%200%2C%20y%20%5Cgeq%200%20%5Cend%7Bcases%7D%20" alt="LaTeX: \displaystyle \begin{cases}74 x + 39 y \leq 2886 \\ 21 x + 67 y \leq 1407 \\x \geq 0, y \geq 0 \end{cases} " data-equation-content=" \displaystyle \begin{cases}74 x + 39 y \leq 2886 \\ 21 x + 67 y \leq 1407 \\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}74 x + 39 y +s = 2886 \\ 21 x + 67 y+t = 1407 \\ - 22 x - 17 y+p =0 \end{cases} " src="/equation_images/%20%5Cbegin%7Bcases%7D74%20x%20%2B%2039%20y%20%2Bs%20%3D%202886%20%5C%5C%2021%20x%20%2B%2067%20y%2Bt%20%3D%201407%20%5C%5C%20-%2022%20x%20-%2017%20y%2Bp%20%3D0%20%5Cend%7Bcases%7D%20" alt="LaTeX: \begin{cases}74 x + 39 y +s = 2886 \\ 21 x + 67 y+t = 1407 \\ - 22 x - 17 y+p =0 \end{cases} " data-equation-content=" \begin{cases}74 x + 39 y +s = 2886 \\ 21 x + 67 y+t = 1407 \\ - 22 x - 17 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 74 " src="/equation_images/%20%5Cdisplaystyle%2074%20" alt="LaTeX: \displaystyle 74 " data-equation-content=" \displaystyle 74 " /> & <img class="equation_image" title=" \displaystyle 39 " src="/equation_images/%20%5Cdisplaystyle%2039%20" alt="LaTeX: \displaystyle 39 " data-equation-content=" \displaystyle 39 " /> & <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 2886 " src="/equation_images/%20%5Cdisplaystyle%202886%20" alt="LaTeX: \displaystyle 2886 " data-equation-content=" \displaystyle 2886 " /> \\ \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 21 " src="/equation_images/%20%5Cdisplaystyle%2021%20" alt="LaTeX: \displaystyle 21 " data-equation-content=" \displaystyle 21 " /> & <img class="equation_image" title=" \displaystyle 67 " src="/equation_images/%20%5Cdisplaystyle%2067%20" alt="LaTeX: \displaystyle 67 " data-equation-content=" \displaystyle 67 " /> & <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 1407 " src="/equation_images/%20%5Cdisplaystyle%201407%20" alt="LaTeX: \displaystyle 1407 " data-equation-content=" \displaystyle 1407 " /> \\ \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 -22 " src="/equation_images/%20%5Cdisplaystyle%20-22%20" alt="LaTeX: \displaystyle -22 " data-equation-content=" \displaystyle -22 " /> & <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 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 74R_2-21R_1 " src="/equation_images/%20%5Cdisplaystyle%2074R_2-21R_1%20" alt="LaTeX: \displaystyle 74R_2-21R_1 " data-equation-content=" \displaystyle 74R_2-21R_1 " /> <br>
<img class="equation_image" title=" \displaystyle 37R_3+11R_1 " src="/equation_images/%20%5Cdisplaystyle%2037R_3%2B11R_1%20" alt="LaTeX: \displaystyle 37R_3+11R_1 " data-equation-content=" \displaystyle 37R_3+11R_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 74 " src="/equation_images/%20%5Cdisplaystyle%2074%20" alt="LaTeX: \displaystyle 74 " data-equation-content=" \displaystyle 74 " /> & <img class="equation_image" title=" \displaystyle 39 " src="/equation_images/%20%5Cdisplaystyle%2039%20" alt="LaTeX: \displaystyle 39 " data-equation-content=" \displaystyle 39 " /> & <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 2886 " src="/equation_images/%20%5Cdisplaystyle%202886%20" alt="LaTeX: \displaystyle 2886 " data-equation-content=" \displaystyle 2886 " /> \\ \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 4139 " src="/equation_images/%20%5Cdisplaystyle%204139%20" alt="LaTeX: \displaystyle 4139 " data-equation-content=" \displaystyle 4139 " /> & <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 74 " src="/equation_images/%20%5Cdisplaystyle%2074%20" alt="LaTeX: \displaystyle 74 " data-equation-content=" \displaystyle 74 " /> & <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 43512 " src="/equation_images/%20%5Cdisplaystyle%2043512%20" alt="LaTeX: \displaystyle 43512 " data-equation-content=" \displaystyle 43512 " /> \\ \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 -200 " src="/equation_images/%20%5Cdisplaystyle%20-200%20" alt="LaTeX: \displaystyle -200 " data-equation-content=" \displaystyle -200 " /> & <img class="equation_image" title=" \displaystyle 11 " src="/equation_images/%20%5Cdisplaystyle%2011%20" alt="LaTeX: \displaystyle 11 " data-equation-content=" \displaystyle 11 " /> & <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 37 " src="/equation_images/%20%5Cdisplaystyle%2037%20" alt="LaTeX: \displaystyle 37 " data-equation-content=" \displaystyle 37 " /> & <img class="equation_image" title=" \displaystyle 31746 " src="/equation_images/%20%5Cdisplaystyle%2031746%20" alt="LaTeX: \displaystyle 31746 " data-equation-content=" \displaystyle 31746 " /> \\ \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 4139R_1-39R_2 " src="/equation_images/%20%5Cdisplaystyle%204139R_1-39R_2%20" alt="LaTeX: \displaystyle 4139R_1-39R_2 " data-equation-content=" \displaystyle 4139R_1-39R_2 " /> <br>
<img class="equation_image" title=" \displaystyle 4139R_3+200R_2 " src="/equation_images/%20%5Cdisplaystyle%204139R_3%2B200R_2%20" alt="LaTeX: \displaystyle 4139R_3+200R_2 " data-equation-content=" \displaystyle 4139R_3+200R_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 306286 " src="/equation_images/%20%5Cdisplaystyle%20306286%20" alt="LaTeX: \displaystyle 306286 " data-equation-content=" \displaystyle 306286 " /> & <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 4958 " src="/equation_images/%20%5Cdisplaystyle%204958%20" alt="LaTeX: \displaystyle 4958 " data-equation-content=" \displaystyle 4958 " /> & <img class="equation_image" title=" \displaystyle -2886 " src="/equation_images/%20%5Cdisplaystyle%20-2886%20" alt="LaTeX: \displaystyle -2886 " data-equation-content=" \displaystyle -2886 " /> & <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 10248186 " src="/equation_images/%20%5Cdisplaystyle%2010248186%20" alt="LaTeX: \displaystyle 10248186 " data-equation-content=" \displaystyle 10248186 " /> \\ \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 4139 " src="/equation_images/%20%5Cdisplaystyle%204139%20" alt="LaTeX: \displaystyle 4139 " data-equation-content=" \displaystyle 4139 " /> & <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 74 " src="/equation_images/%20%5Cdisplaystyle%2074%20" alt="LaTeX: \displaystyle 74 " data-equation-content=" \displaystyle 74 " /> & <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 43512 " src="/equation_images/%20%5Cdisplaystyle%2043512%20" alt="LaTeX: \displaystyle 43512 " data-equation-content=" \displaystyle 43512 " /> \\ \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 41329 " src="/equation_images/%20%5Cdisplaystyle%2041329%20" alt="LaTeX: \displaystyle 41329 " data-equation-content=" \displaystyle 41329 " /> & <img class="equation_image" title=" \displaystyle 14800 " src="/equation_images/%20%5Cdisplaystyle%2014800%20" alt="LaTeX: \displaystyle 14800 " data-equation-content=" \displaystyle 14800 " /> & <img class="equation_image" title=" \displaystyle 153143 " src="/equation_images/%20%5Cdisplaystyle%20153143%20" alt="LaTeX: \displaystyle 153143 " data-equation-content=" \displaystyle 153143 " /> & <img class="equation_image" title=" \displaystyle 140099094 " src="/equation_images/%20%5Cdisplaystyle%20140099094%20" alt="LaTeX: \displaystyle 140099094 " data-equation-content=" \displaystyle 140099094 " /> \\ \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{138489}{4139} " src="/equation_images/%20%5Cdisplaystyle%20%5Cfrac%7B138489%7D%7B4139%7D%20" alt="LaTeX: \displaystyle \frac{138489}{4139} " data-equation-content=" \displaystyle \frac{138489}{4139} " /> . 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{43512}{4139} " src="/equation_images/%20%5Cdisplaystyle%20%5Cfrac%7B43512%7D%7B4139%7D%20" alt="LaTeX: \displaystyle \frac{43512}{4139} " data-equation-content=" \displaystyle \frac{43512}{4139} " /> . The max value is <img class="equation_image" title=" \displaystyle p = \frac{3786462}{4139} " src="/equation_images/%20%5Cdisplaystyle%20p%20%3D%20%5Cfrac%7B3786462%7D%7B4139%7D%20" alt="LaTeX: \displaystyle p = \frac{3786462}{4139} " data-equation-content=" \displaystyle p = \frac{3786462}{4139} " /> </p> </p>