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Questions: Algebra BusinessCalculus
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Use the simplex method to maximize \(\displaystyle p = 16 x + 17 y\) subject to \(\displaystyle \begin{cases}66 x + 43 y \leq 2838 \\ 11 x + 46 y \leq 506 \\x \geq 0, y \geq 0 \end{cases}\)
Adding the slack variables \(\displaystyle s\) and \(\displaystyle t\) to the inequalities gives:
\begin{equation*}\begin{cases}66 x + 43 y +s = 2838 \\ 11 x + 46 y+t = 506 \\ - 16 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
\(\displaystyle s\) & \(\displaystyle 66\) & \(\displaystyle 43\) & \(\displaystyle 1\) & \(\displaystyle 0\) & \(\displaystyle 0\) & \(\displaystyle 2838\) \\ \hline
\(\displaystyle t\) & \(\displaystyle 11\) & \(\displaystyle 46\) & \(\displaystyle 0\) & \(\displaystyle 1\) & \(\displaystyle 0\) & \(\displaystyle 506\) \\ \hline
\(\displaystyle p\) & \(\displaystyle -16\) & \(\displaystyle -17\) & \(\displaystyle 0\) & \(\displaystyle 0\) & \(\displaystyle 1\) & \(\displaystyle 0\) \\ \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:
\(\displaystyle 46R_1-43R_2\)
\(\displaystyle 46R_3+17R_2\)
\begin{tabular}{|c|c|c|c|c|c|c|}\hline
\(\displaystyle \) & \(\displaystyle x\) & \(\displaystyle y\) & \(\displaystyle s\) & \(\displaystyle t\) & \(\displaystyle p\) & \(\displaystyle \) \\ \hline
\(\displaystyle s\) & \(\displaystyle 2563\) & \(\displaystyle 0\) & \(\displaystyle 46\) & \(\displaystyle -43\) & \(\displaystyle 0\) & \(\displaystyle 108790\) \\ \hline
\(\displaystyle y\) & \(\displaystyle 11\) & \(\displaystyle 46\) & \(\displaystyle 0\) & \(\displaystyle 1\) & \(\displaystyle 0\) & \(\displaystyle 506\) \\ \hline
\(\displaystyle p\) & \(\displaystyle -549\) & \(\displaystyle 0\) & \(\displaystyle 0\) & \(\displaystyle 17\) & \(\displaystyle 46\) & \(\displaystyle 8602\) \\ \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:
\(\displaystyle 233R_2-1R_1\)
\(\displaystyle 2563R_3+549R_1\)
\begin{tabular}{|c|c|c|c|c|c|c|}\hline
\(\displaystyle \) & \(\displaystyle x\) & \(\displaystyle y\) & \(\displaystyle s\) & \(\displaystyle t\) & \(\displaystyle p\) & \(\displaystyle \) \\ \hline
\(\displaystyle x\) & \(\displaystyle 2563\) & \(\displaystyle 0\) & \(\displaystyle 46\) & \(\displaystyle -43\) & \(\displaystyle 0\) & \(\displaystyle 108790\) \\ \hline
\(\displaystyle y\) & \(\displaystyle 0\) & \(\displaystyle 10718\) & \(\displaystyle -46\) & \(\displaystyle 276\) & \(\displaystyle 0\) & \(\displaystyle 9108\) \\ \hline
\(\displaystyle p\) & \(\displaystyle 0\) & \(\displaystyle 0\) & \(\displaystyle 25254\) & \(\displaystyle 19964\) & \(\displaystyle 117898\) & \(\displaystyle 81772636\) \\ \hline
\end{tabular}
There are no negative values in row \(\displaystyle p\) and this is the final tableau.The value of \(\displaystyle x\) is \(\displaystyle \frac{9890}{233}\). The value of \(\displaystyle y\) is \(\displaystyle \frac{198}{233}\). The max value is \(\displaystyle p = \frac{161606}{233}\)
\begin{question}Use the simplex method to maximize $p = 16 x + 17 y$ subject to $\begin{cases}66 x + 43 y \leq 2838 \\ 11 x + 46 y \leq 506 \\x \geq 0, y \geq 0 \end{cases}$
\soln{9cm}{Adding the slack variables $s$ and $t$ to the inequalities gives:\newline\begin{equation*}\begin{cases}66 x + 43 y +s = 2838 \\ 11 x + 46 y+t = 506 \\ - 16 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
$s$ & $66$ & $43$ & $1$ & $0$ & $0$ & $2838$ \\ \hline
$t$ & $11$ & $46$ & $0$ & $1$ & $0$ & $506$ \\ \hline
$p$ & $-16$ & $-17$ & $0$ & $0$ & $1$ & $0$ \\ \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
$46R_1-43R_2$\newline
$46R_3+17R_2$\newline
\begin{tabular}{|c|c|c|c|c|c|c|}\hline
$$ & $x$ & $y$ & $s$ & $t$ & $p$ & $$ \\ \hline
$s$ & $2563$ & $0$ & $46$ & $-43$ & $0$ & $108790$ \\ \hline
$y$ & $11$ & $46$ & $0$ & $1$ & $0$ & $506$ \\ \hline
$p$ & $-549$ & $0$ & $0$ & $17$ & $46$ & $8602$ \\ \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
$233R_2-1R_1$\newline
$2563R_3+549R_1$\newline
\begin{tabular}{|c|c|c|c|c|c|c|}\hline
$$ & $x$ & $y$ & $s$ & $t$ & $p$ & $$ \\ \hline
$x$ & $2563$ & $0$ & $46$ & $-43$ & $0$ & $108790$ \\ \hline
$y$ & $0$ & $10718$ & $-46$ & $276$ & $0$ & $9108$ \\ \hline
$p$ & $0$ & $0$ & $25254$ & $19964$ & $117898$ & $81772636$ \\ \hline
\end{tabular}\newline
There are no negative values in row $p$ and this is the final tableau.The value of $x$ is $\frac{9890}{233}$. The value of $y$ is $\frac{198}{233}$. The max value is $p = \frac{161606}{233}$ }
\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 = 16 x + 17 y " src="/equation_images/%20%5Cdisplaystyle%20p%20%3D%2016%20x%20%2B%2017%20y%20" alt="LaTeX: \displaystyle p = 16 x + 17 y " data-equation-content=" \displaystyle p = 16 x + 17 y " /> subject to <img class="equation_image" title=" \displaystyle \begin{cases}66 x + 43 y \leq 2838 \\ 11 x + 46 y \leq 506 \\x \geq 0, y \geq 0 \end{cases} " src="/equation_images/%20%5Cdisplaystyle%20%5Cbegin%7Bcases%7D66%20x%20%2B%2043%20y%20%5Cleq%202838%20%5C%5C%2011%20x%20%2B%2046%20y%20%5Cleq%20506%20%5C%5Cx%20%5Cgeq%200%2C%20y%20%5Cgeq%200%20%5Cend%7Bcases%7D%20" alt="LaTeX: \displaystyle \begin{cases}66 x + 43 y \leq 2838 \\ 11 x + 46 y \leq 506 \\x \geq 0, y \geq 0 \end{cases} " data-equation-content=" \displaystyle \begin{cases}66 x + 43 y \leq 2838 \\ 11 x + 46 y \leq 506 \\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}66 x + 43 y +s = 2838 \\ 11 x + 46 y+t = 506 \\ - 16 x - 17 y+p =0 \end{cases} " src="/equation_images/%20%5Cbegin%7Bcases%7D66%20x%20%2B%2043%20y%20%2Bs%20%3D%202838%20%5C%5C%2011%20x%20%2B%2046%20y%2Bt%20%3D%20506%20%5C%5C%20-%2016%20x%20-%2017%20y%2Bp%20%3D0%20%5Cend%7Bcases%7D%20" alt="LaTeX: \begin{cases}66 x + 43 y +s = 2838 \\ 11 x + 46 y+t = 506 \\ - 16 x - 17 y+p =0 \end{cases} " data-equation-content=" \begin{cases}66 x + 43 y +s = 2838 \\ 11 x + 46 y+t = 506 \\ - 16 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 66 " src="/equation_images/%20%5Cdisplaystyle%2066%20" alt="LaTeX: \displaystyle 66 " data-equation-content=" \displaystyle 66 " /> & <img class="equation_image" title=" \displaystyle 43 " src="/equation_images/%20%5Cdisplaystyle%2043%20" alt="LaTeX: \displaystyle 43 " data-equation-content=" \displaystyle 43 " /> & <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 2838 " src="/equation_images/%20%5Cdisplaystyle%202838%20" alt="LaTeX: \displaystyle 2838 " data-equation-content=" \displaystyle 2838 " /> \\ \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 11 " src="/equation_images/%20%5Cdisplaystyle%2011%20" alt="LaTeX: \displaystyle 11 " data-equation-content=" \displaystyle 11 " /> & <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 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 506 " src="/equation_images/%20%5Cdisplaystyle%20506%20" alt="LaTeX: \displaystyle 506 " data-equation-content=" \displaystyle 506 " /> \\ \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 -16 " src="/equation_images/%20%5Cdisplaystyle%20-16%20" alt="LaTeX: \displaystyle -16 " data-equation-content=" \displaystyle -16 " /> & <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 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 46R_1-43R_2 " src="/equation_images/%20%5Cdisplaystyle%2046R_1-43R_2%20" alt="LaTeX: \displaystyle 46R_1-43R_2 " data-equation-content=" \displaystyle 46R_1-43R_2 " /> <br>
<img class="equation_image" title=" \displaystyle 46R_3+17R_2 " src="/equation_images/%20%5Cdisplaystyle%2046R_3%2B17R_2%20" alt="LaTeX: \displaystyle 46R_3+17R_2 " data-equation-content=" \displaystyle 46R_3+17R_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 2563 " src="/equation_images/%20%5Cdisplaystyle%202563%20" alt="LaTeX: \displaystyle 2563 " data-equation-content=" \displaystyle 2563 " /> & <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 46 " src="/equation_images/%20%5Cdisplaystyle%2046%20" alt="LaTeX: \displaystyle 46 " data-equation-content=" \displaystyle 46 " /> & <img class="equation_image" title=" \displaystyle -43 " src="/equation_images/%20%5Cdisplaystyle%20-43%20" alt="LaTeX: \displaystyle -43 " data-equation-content=" \displaystyle -43 " /> & <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 108790 " src="/equation_images/%20%5Cdisplaystyle%20108790%20" alt="LaTeX: \displaystyle 108790 " data-equation-content=" \displaystyle 108790 " /> \\ \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 11 " src="/equation_images/%20%5Cdisplaystyle%2011%20" alt="LaTeX: \displaystyle 11 " data-equation-content=" \displaystyle 11 " /> & <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 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 506 " src="/equation_images/%20%5Cdisplaystyle%20506%20" alt="LaTeX: \displaystyle 506 " data-equation-content=" \displaystyle 506 " /> \\ \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 -549 " src="/equation_images/%20%5Cdisplaystyle%20-549%20" alt="LaTeX: \displaystyle -549 " data-equation-content=" \displaystyle -549 " /> & <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 17 " src="/equation_images/%20%5Cdisplaystyle%2017%20" alt="LaTeX: \displaystyle 17 " data-equation-content=" \displaystyle 17 " /> & <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 8602 " src="/equation_images/%20%5Cdisplaystyle%208602%20" alt="LaTeX: \displaystyle 8602 " data-equation-content=" \displaystyle 8602 " /> \\ \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 233R_2-1R_1 " src="/equation_images/%20%5Cdisplaystyle%20233R_2-1R_1%20" alt="LaTeX: \displaystyle 233R_2-1R_1 " data-equation-content=" \displaystyle 233R_2-1R_1 " /> <br>
<img class="equation_image" title=" \displaystyle 2563R_3+549R_1 " src="/equation_images/%20%5Cdisplaystyle%202563R_3%2B549R_1%20" alt="LaTeX: \displaystyle 2563R_3+549R_1 " data-equation-content=" \displaystyle 2563R_3+549R_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 2563 " src="/equation_images/%20%5Cdisplaystyle%202563%20" alt="LaTeX: \displaystyle 2563 " data-equation-content=" \displaystyle 2563 " /> & <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 46 " src="/equation_images/%20%5Cdisplaystyle%2046%20" alt="LaTeX: \displaystyle 46 " data-equation-content=" \displaystyle 46 " /> & <img class="equation_image" title=" \displaystyle -43 " src="/equation_images/%20%5Cdisplaystyle%20-43%20" alt="LaTeX: \displaystyle -43 " data-equation-content=" \displaystyle -43 " /> & <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 108790 " src="/equation_images/%20%5Cdisplaystyle%20108790%20" alt="LaTeX: \displaystyle 108790 " data-equation-content=" \displaystyle 108790 " /> \\ \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 10718 " src="/equation_images/%20%5Cdisplaystyle%2010718%20" alt="LaTeX: \displaystyle 10718 " data-equation-content=" \displaystyle 10718 " /> & <img class="equation_image" title=" \displaystyle -46 " src="/equation_images/%20%5Cdisplaystyle%20-46%20" alt="LaTeX: \displaystyle -46 " data-equation-content=" \displaystyle -46 " /> & <img class="equation_image" title=" \displaystyle 276 " src="/equation_images/%20%5Cdisplaystyle%20276%20" alt="LaTeX: \displaystyle 276 " data-equation-content=" \displaystyle 276 " /> & <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 9108 " src="/equation_images/%20%5Cdisplaystyle%209108%20" alt="LaTeX: \displaystyle 9108 " data-equation-content=" \displaystyle 9108 " /> \\ \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 25254 " src="/equation_images/%20%5Cdisplaystyle%2025254%20" alt="LaTeX: \displaystyle 25254 " data-equation-content=" \displaystyle 25254 " /> & <img class="equation_image" title=" \displaystyle 19964 " src="/equation_images/%20%5Cdisplaystyle%2019964%20" alt="LaTeX: \displaystyle 19964 " data-equation-content=" \displaystyle 19964 " /> & <img class="equation_image" title=" \displaystyle 117898 " src="/equation_images/%20%5Cdisplaystyle%20117898%20" alt="LaTeX: \displaystyle 117898 " data-equation-content=" \displaystyle 117898 " /> & <img class="equation_image" title=" \displaystyle 81772636 " src="/equation_images/%20%5Cdisplaystyle%2081772636%20" alt="LaTeX: \displaystyle 81772636 " data-equation-content=" \displaystyle 81772636 " /> \\ \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{9890}{233} " src="/equation_images/%20%5Cdisplaystyle%20%5Cfrac%7B9890%7D%7B233%7D%20" alt="LaTeX: \displaystyle \frac{9890}{233} " data-equation-content=" \displaystyle \frac{9890}{233} " /> . 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{198}{233} " src="/equation_images/%20%5Cdisplaystyle%20%5Cfrac%7B198%7D%7B233%7D%20" alt="LaTeX: \displaystyle \frac{198}{233} " data-equation-content=" \displaystyle \frac{198}{233} " /> . The max value is <img class="equation_image" title=" \displaystyle p = \frac{161606}{233} " src="/equation_images/%20%5Cdisplaystyle%20p%20%3D%20%5Cfrac%7B161606%7D%7B233%7D%20" alt="LaTeX: \displaystyle p = \frac{161606}{233} " data-equation-content=" \displaystyle p = \frac{161606}{233} " /> </p> </p>