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Questions: Algebra BusinessCalculus
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Use the simplex method to maximize \(\displaystyle p = 5 x + 17 y\) subject to \(\displaystyle \begin{cases}53 x + 33 y \leq 1749 \\ 10 x + 83 y \leq 830 \\x \geq 0, y \geq 0 \end{cases}\)
Adding the slack variables \(\displaystyle s\) and \(\displaystyle t\) to the inequalities gives:
\begin{equation*}\begin{cases}53 x + 33 y +s = 1749 \\ 10 x + 83 y+t = 830 \\ - 5 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
\(\displaystyle s\) & \(\displaystyle 53\) & \(\displaystyle 33\) & \(\displaystyle 1\) & \(\displaystyle 0\) & \(\displaystyle 0\) & \(\displaystyle 1749\) \\ \hline
\(\displaystyle t\) & \(\displaystyle 10\) & \(\displaystyle 83\) & \(\displaystyle 0\) & \(\displaystyle 1\) & \(\displaystyle 0\) & \(\displaystyle 830\) \\ \hline
\(\displaystyle p\) & \(\displaystyle -5\) & \(\displaystyle -17\) & \(\displaystyle 0\) & \(\displaystyle 0\) & \(\displaystyle 1\) & \(\displaystyle 0\) \\ \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:
\(\displaystyle 83R_1-33R_2\)
\(\displaystyle 83R_3+17R_2\)
\begin{tabular}{|c|c|c|c|c|c|c|}\hline
\(\displaystyle \) & \(\displaystyle x\) & \(\displaystyle y\) & \(\displaystyle s\) & \(\displaystyle t\) & \(\displaystyle p\) & \(\displaystyle \) \\ \hline
\(\displaystyle s\) & \(\displaystyle 4069\) & \(\displaystyle 0\) & \(\displaystyle 83\) & \(\displaystyle -33\) & \(\displaystyle 0\) & \(\displaystyle 117777\) \\ \hline
\(\displaystyle y\) & \(\displaystyle 10\) & \(\displaystyle 83\) & \(\displaystyle 0\) & \(\displaystyle 1\) & \(\displaystyle 0\) & \(\displaystyle 830\) \\ \hline
\(\displaystyle p\) & \(\displaystyle -245\) & \(\displaystyle 0\) & \(\displaystyle 0\) & \(\displaystyle 17\) & \(\displaystyle 83\) & \(\displaystyle 14110\) \\ \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:
\(\displaystyle 4069R_2-10R_1\)
\(\displaystyle 4069R_3+245R_1\)
\begin{tabular}{|c|c|c|c|c|c|c|}\hline
\(\displaystyle \) & \(\displaystyle x\) & \(\displaystyle y\) & \(\displaystyle s\) & \(\displaystyle t\) & \(\displaystyle p\) & \(\displaystyle \) \\ \hline
\(\displaystyle x\) & \(\displaystyle 4069\) & \(\displaystyle 0\) & \(\displaystyle 83\) & \(\displaystyle -33\) & \(\displaystyle 0\) & \(\displaystyle 117777\) \\ \hline
\(\displaystyle y\) & \(\displaystyle 0\) & \(\displaystyle 337727\) & \(\displaystyle -830\) & \(\displaystyle 4399\) & \(\displaystyle 0\) & \(\displaystyle 2199500\) \\ \hline
\(\displaystyle p\) & \(\displaystyle 0\) & \(\displaystyle 0\) & \(\displaystyle 20335\) & \(\displaystyle 61088\) & \(\displaystyle 337727\) & \(\displaystyle 86268955\) \\ \hline
\end{tabular}
There are no negative values in row \(\displaystyle p\) and this is the final tableau.The value of \(\displaystyle x\) is \(\displaystyle \frac{117777}{4069}\). The value of \(\displaystyle y\) is \(\displaystyle \frac{26500}{4069}\). The max value is \(\displaystyle p = \frac{1039385}{4069}\)
\begin{question}Use the simplex method to maximize $p = 5 x + 17 y$ subject to $\begin{cases}53 x + 33 y \leq 1749 \\ 10 x + 83 y \leq 830 \\x \geq 0, y \geq 0 \end{cases}$
\soln{9cm}{Adding the slack variables $s$ and $t$ to the inequalities gives:\newline\begin{equation*}\begin{cases}53 x + 33 y +s = 1749 \\ 10 x + 83 y+t = 830 \\ - 5 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
$s$ & $53$ & $33$ & $1$ & $0$ & $0$ & $1749$ \\ \hline
$t$ & $10$ & $83$ & $0$ & $1$ & $0$ & $830$ \\ \hline
$p$ & $-5$ & $-17$ & $0$ & $0$ & $1$ & $0$ \\ \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
$83R_1-33R_2$\newline
$83R_3+17R_2$\newline
\begin{tabular}{|c|c|c|c|c|c|c|}\hline
$$ & $x$ & $y$ & $s$ & $t$ & $p$ & $$ \\ \hline
$s$ & $4069$ & $0$ & $83$ & $-33$ & $0$ & $117777$ \\ \hline
$y$ & $10$ & $83$ & $0$ & $1$ & $0$ & $830$ \\ \hline
$p$ & $-245$ & $0$ & $0$ & $17$ & $83$ & $14110$ \\ \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
$4069R_2-10R_1$\newline
$4069R_3+245R_1$\newline
\begin{tabular}{|c|c|c|c|c|c|c|}\hline
$$ & $x$ & $y$ & $s$ & $t$ & $p$ & $$ \\ \hline
$x$ & $4069$ & $0$ & $83$ & $-33$ & $0$ & $117777$ \\ \hline
$y$ & $0$ & $337727$ & $-830$ & $4399$ & $0$ & $2199500$ \\ \hline
$p$ & $0$ & $0$ & $20335$ & $61088$ & $337727$ & $86268955$ \\ \hline
\end{tabular}\newline
There are no negative values in row $p$ and this is the final tableau.The value of $x$ is $\frac{117777}{4069}$. The value of $y$ is $\frac{26500}{4069}$. The max value is $p = \frac{1039385}{4069}$ }
\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 = 5 x + 17 y " src="/equation_images/%20%5Cdisplaystyle%20p%20%3D%205%20x%20%2B%2017%20y%20" alt="LaTeX: \displaystyle p = 5 x + 17 y " data-equation-content=" \displaystyle p = 5 x + 17 y " /> subject to <img class="equation_image" title=" \displaystyle \begin{cases}53 x + 33 y \leq 1749 \\ 10 x + 83 y \leq 830 \\x \geq 0, y \geq 0 \end{cases} " src="/equation_images/%20%5Cdisplaystyle%20%5Cbegin%7Bcases%7D53%20x%20%2B%2033%20y%20%5Cleq%201749%20%5C%5C%2010%20x%20%2B%2083%20y%20%5Cleq%20830%20%5C%5Cx%20%5Cgeq%200%2C%20y%20%5Cgeq%200%20%5Cend%7Bcases%7D%20" alt="LaTeX: \displaystyle \begin{cases}53 x + 33 y \leq 1749 \\ 10 x + 83 y \leq 830 \\x \geq 0, y \geq 0 \end{cases} " data-equation-content=" \displaystyle \begin{cases}53 x + 33 y \leq 1749 \\ 10 x + 83 y \leq 830 \\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}53 x + 33 y +s = 1749 \\ 10 x + 83 y+t = 830 \\ - 5 x - 17 y+p =0 \end{cases} " src="/equation_images/%20%5Cbegin%7Bcases%7D53%20x%20%2B%2033%20y%20%2Bs%20%3D%201749%20%5C%5C%2010%20x%20%2B%2083%20y%2Bt%20%3D%20830%20%5C%5C%20-%205%20x%20-%2017%20y%2Bp%20%3D0%20%5Cend%7Bcases%7D%20" alt="LaTeX: \begin{cases}53 x + 33 y +s = 1749 \\ 10 x + 83 y+t = 830 \\ - 5 x - 17 y+p =0 \end{cases} " data-equation-content=" \begin{cases}53 x + 33 y +s = 1749 \\ 10 x + 83 y+t = 830 \\ - 5 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 53 " src="/equation_images/%20%5Cdisplaystyle%2053%20" alt="LaTeX: \displaystyle 53 " data-equation-content=" \displaystyle 53 " /> & <img class="equation_image" title=" \displaystyle 33 " src="/equation_images/%20%5Cdisplaystyle%2033%20" alt="LaTeX: \displaystyle 33 " data-equation-content=" \displaystyle 33 " /> & <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 1749 " src="/equation_images/%20%5Cdisplaystyle%201749%20" alt="LaTeX: \displaystyle 1749 " data-equation-content=" \displaystyle 1749 " /> \\ \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 10 " src="/equation_images/%20%5Cdisplaystyle%2010%20" alt="LaTeX: \displaystyle 10 " data-equation-content=" \displaystyle 10 " /> & <img class="equation_image" title=" \displaystyle 83 " src="/equation_images/%20%5Cdisplaystyle%2083%20" alt="LaTeX: \displaystyle 83 " data-equation-content=" \displaystyle 83 " /> & <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 830 " src="/equation_images/%20%5Cdisplaystyle%20830%20" alt="LaTeX: \displaystyle 830 " data-equation-content=" \displaystyle 830 " /> \\ \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 -5 " src="/equation_images/%20%5Cdisplaystyle%20-5%20" alt="LaTeX: \displaystyle -5 " data-equation-content=" \displaystyle -5 " /> & <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 83R_1-33R_2 " src="/equation_images/%20%5Cdisplaystyle%2083R_1-33R_2%20" alt="LaTeX: \displaystyle 83R_1-33R_2 " data-equation-content=" \displaystyle 83R_1-33R_2 " /> <br>
<img class="equation_image" title=" \displaystyle 83R_3+17R_2 " src="/equation_images/%20%5Cdisplaystyle%2083R_3%2B17R_2%20" alt="LaTeX: \displaystyle 83R_3+17R_2 " data-equation-content=" \displaystyle 83R_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 4069 " src="/equation_images/%20%5Cdisplaystyle%204069%20" alt="LaTeX: \displaystyle 4069 " data-equation-content=" \displaystyle 4069 " /> & <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 83 " src="/equation_images/%20%5Cdisplaystyle%2083%20" alt="LaTeX: \displaystyle 83 " data-equation-content=" \displaystyle 83 " /> & <img class="equation_image" title=" \displaystyle -33 " src="/equation_images/%20%5Cdisplaystyle%20-33%20" alt="LaTeX: \displaystyle -33 " data-equation-content=" \displaystyle -33 " /> & <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 117777 " src="/equation_images/%20%5Cdisplaystyle%20117777%20" alt="LaTeX: \displaystyle 117777 " data-equation-content=" \displaystyle 117777 " /> \\ \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 10 " src="/equation_images/%20%5Cdisplaystyle%2010%20" alt="LaTeX: \displaystyle 10 " data-equation-content=" \displaystyle 10 " /> & <img class="equation_image" title=" \displaystyle 83 " src="/equation_images/%20%5Cdisplaystyle%2083%20" alt="LaTeX: \displaystyle 83 " data-equation-content=" \displaystyle 83 " /> & <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 830 " src="/equation_images/%20%5Cdisplaystyle%20830%20" alt="LaTeX: \displaystyle 830 " data-equation-content=" \displaystyle 830 " /> \\ \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 -245 " src="/equation_images/%20%5Cdisplaystyle%20-245%20" alt="LaTeX: \displaystyle -245 " data-equation-content=" \displaystyle -245 " /> & <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 83 " src="/equation_images/%20%5Cdisplaystyle%2083%20" alt="LaTeX: \displaystyle 83 " data-equation-content=" \displaystyle 83 " /> & <img class="equation_image" title=" \displaystyle 14110 " src="/equation_images/%20%5Cdisplaystyle%2014110%20" alt="LaTeX: \displaystyle 14110 " data-equation-content=" \displaystyle 14110 " /> \\ \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 4069R_2-10R_1 " src="/equation_images/%20%5Cdisplaystyle%204069R_2-10R_1%20" alt="LaTeX: \displaystyle 4069R_2-10R_1 " data-equation-content=" \displaystyle 4069R_2-10R_1 " /> <br>
<img class="equation_image" title=" \displaystyle 4069R_3+245R_1 " src="/equation_images/%20%5Cdisplaystyle%204069R_3%2B245R_1%20" alt="LaTeX: \displaystyle 4069R_3+245R_1 " data-equation-content=" \displaystyle 4069R_3+245R_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 4069 " src="/equation_images/%20%5Cdisplaystyle%204069%20" alt="LaTeX: \displaystyle 4069 " data-equation-content=" \displaystyle 4069 " /> & <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 83 " src="/equation_images/%20%5Cdisplaystyle%2083%20" alt="LaTeX: \displaystyle 83 " data-equation-content=" \displaystyle 83 " /> & <img class="equation_image" title=" \displaystyle -33 " src="/equation_images/%20%5Cdisplaystyle%20-33%20" alt="LaTeX: \displaystyle -33 " data-equation-content=" \displaystyle -33 " /> & <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 117777 " src="/equation_images/%20%5Cdisplaystyle%20117777%20" alt="LaTeX: \displaystyle 117777 " data-equation-content=" \displaystyle 117777 " /> \\ \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 337727 " src="/equation_images/%20%5Cdisplaystyle%20337727%20" alt="LaTeX: \displaystyle 337727 " data-equation-content=" \displaystyle 337727 " /> & <img class="equation_image" title=" \displaystyle -830 " src="/equation_images/%20%5Cdisplaystyle%20-830%20" alt="LaTeX: \displaystyle -830 " data-equation-content=" \displaystyle -830 " /> & <img class="equation_image" title=" \displaystyle 4399 " src="/equation_images/%20%5Cdisplaystyle%204399%20" alt="LaTeX: \displaystyle 4399 " data-equation-content=" \displaystyle 4399 " /> & <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 2199500 " src="/equation_images/%20%5Cdisplaystyle%202199500%20" alt="LaTeX: \displaystyle 2199500 " data-equation-content=" \displaystyle 2199500 " /> \\ \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 20335 " src="/equation_images/%20%5Cdisplaystyle%2020335%20" alt="LaTeX: \displaystyle 20335 " data-equation-content=" \displaystyle 20335 " /> & <img class="equation_image" title=" \displaystyle 61088 " src="/equation_images/%20%5Cdisplaystyle%2061088%20" alt="LaTeX: \displaystyle 61088 " data-equation-content=" \displaystyle 61088 " /> & <img class="equation_image" title=" \displaystyle 337727 " src="/equation_images/%20%5Cdisplaystyle%20337727%20" alt="LaTeX: \displaystyle 337727 " data-equation-content=" \displaystyle 337727 " /> & <img class="equation_image" title=" \displaystyle 86268955 " src="/equation_images/%20%5Cdisplaystyle%2086268955%20" alt="LaTeX: \displaystyle 86268955 " data-equation-content=" \displaystyle 86268955 " /> \\ \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{117777}{4069} " src="/equation_images/%20%5Cdisplaystyle%20%5Cfrac%7B117777%7D%7B4069%7D%20" alt="LaTeX: \displaystyle \frac{117777}{4069} " data-equation-content=" \displaystyle \frac{117777}{4069} " /> . 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{26500}{4069} " src="/equation_images/%20%5Cdisplaystyle%20%5Cfrac%7B26500%7D%7B4069%7D%20" alt="LaTeX: \displaystyle \frac{26500}{4069} " data-equation-content=" \displaystyle \frac{26500}{4069} " /> . The max value is <img class="equation_image" title=" \displaystyle p = \frac{1039385}{4069} " src="/equation_images/%20%5Cdisplaystyle%20p%20%3D%20%5Cfrac%7B1039385%7D%7B4069%7D%20" alt="LaTeX: \displaystyle p = \frac{1039385}{4069} " data-equation-content=" \displaystyle p = \frac{1039385}{4069} " /> </p> </p>