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Questions: Algebra BusinessCalculus
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Use the simplex method to maximize \(\displaystyle p = 6 x + 13 y\) subject to \(\displaystyle \begin{cases}40 x + 25 y \leq 1000 \\ 18 x + 81 y \leq 1458 \\x \geq 0, y \geq 0 \end{cases}\)
Adding the slack variables \(\displaystyle s\) and \(\displaystyle t\) to the inequalities gives:
\begin{equation*}\begin{cases}40 x + 25 y +s = 1000 \\ 18 x + 81 y+t = 1458 \\ - 6 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
\(\displaystyle s\) & \(\displaystyle 40\) & \(\displaystyle 25\) & \(\displaystyle 1\) & \(\displaystyle 0\) & \(\displaystyle 0\) & \(\displaystyle 1000\) \\ \hline
\(\displaystyle t\) & \(\displaystyle 18\) & \(\displaystyle 81\) & \(\displaystyle 0\) & \(\displaystyle 1\) & \(\displaystyle 0\) & \(\displaystyle 1458\) \\ \hline
\(\displaystyle p\) & \(\displaystyle -6\) & \(\displaystyle -13\) & \(\displaystyle 0\) & \(\displaystyle 0\) & \(\displaystyle 1\) & \(\displaystyle 0\) \\ \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:
\(\displaystyle 81R_1-25R_2\)
\(\displaystyle 81R_3+13R_2\)
\begin{tabular}{|c|c|c|c|c|c|c|}\hline
\(\displaystyle \) & \(\displaystyle x\) & \(\displaystyle y\) & \(\displaystyle s\) & \(\displaystyle t\) & \(\displaystyle p\) & \(\displaystyle \) \\ \hline
\(\displaystyle s\) & \(\displaystyle 2790\) & \(\displaystyle 0\) & \(\displaystyle 81\) & \(\displaystyle -25\) & \(\displaystyle 0\) & \(\displaystyle 44550\) \\ \hline
\(\displaystyle y\) & \(\displaystyle 18\) & \(\displaystyle 81\) & \(\displaystyle 0\) & \(\displaystyle 1\) & \(\displaystyle 0\) & \(\displaystyle 1458\) \\ \hline
\(\displaystyle p\) & \(\displaystyle -252\) & \(\displaystyle 0\) & \(\displaystyle 0\) & \(\displaystyle 13\) & \(\displaystyle 81\) & \(\displaystyle 18954\) \\ \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:
\(\displaystyle 155R_2-1R_1\)
\(\displaystyle 155R_3+14R_1\)
\begin{tabular}{|c|c|c|c|c|c|c|}\hline
\(\displaystyle \) & \(\displaystyle x\) & \(\displaystyle y\) & \(\displaystyle s\) & \(\displaystyle t\) & \(\displaystyle p\) & \(\displaystyle \) \\ \hline
\(\displaystyle x\) & \(\displaystyle 2790\) & \(\displaystyle 0\) & \(\displaystyle 81\) & \(\displaystyle -25\) & \(\displaystyle 0\) & \(\displaystyle 44550\) \\ \hline
\(\displaystyle y\) & \(\displaystyle 0\) & \(\displaystyle 12555\) & \(\displaystyle -81\) & \(\displaystyle 180\) & \(\displaystyle 0\) & \(\displaystyle 181440\) \\ \hline
\(\displaystyle p\) & \(\displaystyle 0\) & \(\displaystyle 0\) & \(\displaystyle 1134\) & \(\displaystyle 1665\) & \(\displaystyle 12555\) & \(\displaystyle 3561570\) \\ \hline
\end{tabular}
There are no negative values in row \(\displaystyle p\) and this is the final tableau.The value of \(\displaystyle x\) is \(\displaystyle \frac{495}{31}\). The value of \(\displaystyle y\) is \(\displaystyle \frac{448}{31}\). The max value is \(\displaystyle p = \frac{8794}{31}\)
\begin{question}Use the simplex method to maximize $p = 6 x + 13 y$ subject to $\begin{cases}40 x + 25 y \leq 1000 \\ 18 x + 81 y \leq 1458 \\x \geq 0, y \geq 0 \end{cases}$
\soln{9cm}{Adding the slack variables $s$ and $t$ to the inequalities gives:\newline\begin{equation*}\begin{cases}40 x + 25 y +s = 1000 \\ 18 x + 81 y+t = 1458 \\ - 6 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
$s$ & $40$ & $25$ & $1$ & $0$ & $0$ & $1000$ \\ \hline
$t$ & $18$ & $81$ & $0$ & $1$ & $0$ & $1458$ \\ \hline
$p$ & $-6$ & $-13$ & $0$ & $0$ & $1$ & $0$ \\ \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
$81R_1-25R_2$\newline
$81R_3+13R_2$\newline
\begin{tabular}{|c|c|c|c|c|c|c|}\hline
$$ & $x$ & $y$ & $s$ & $t$ & $p$ & $$ \\ \hline
$s$ & $2790$ & $0$ & $81$ & $-25$ & $0$ & $44550$ \\ \hline
$y$ & $18$ & $81$ & $0$ & $1$ & $0$ & $1458$ \\ \hline
$p$ & $-252$ & $0$ & $0$ & $13$ & $81$ & $18954$ \\ \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
$155R_2-1R_1$\newline
$155R_3+14R_1$\newline
\begin{tabular}{|c|c|c|c|c|c|c|}\hline
$$ & $x$ & $y$ & $s$ & $t$ & $p$ & $$ \\ \hline
$x$ & $2790$ & $0$ & $81$ & $-25$ & $0$ & $44550$ \\ \hline
$y$ & $0$ & $12555$ & $-81$ & $180$ & $0$ & $181440$ \\ \hline
$p$ & $0$ & $0$ & $1134$ & $1665$ & $12555$ & $3561570$ \\ \hline
\end{tabular}\newline
There are no negative values in row $p$ and this is the final tableau.The value of $x$ is $\frac{495}{31}$. The value of $y$ is $\frac{448}{31}$. The max value is $p = \frac{8794}{31}$ }
\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 = 6 x + 13 y " src="/equation_images/%20%5Cdisplaystyle%20p%20%3D%206%20x%20%2B%2013%20y%20" alt="LaTeX: \displaystyle p = 6 x + 13 y " data-equation-content=" \displaystyle p = 6 x + 13 y " /> subject to <img class="equation_image" title=" \displaystyle \begin{cases}40 x + 25 y \leq 1000 \\ 18 x + 81 y \leq 1458 \\x \geq 0, y \geq 0 \end{cases} " src="/equation_images/%20%5Cdisplaystyle%20%5Cbegin%7Bcases%7D40%20x%20%2B%2025%20y%20%5Cleq%201000%20%5C%5C%2018%20x%20%2B%2081%20y%20%5Cleq%201458%20%5C%5Cx%20%5Cgeq%200%2C%20y%20%5Cgeq%200%20%5Cend%7Bcases%7D%20" alt="LaTeX: \displaystyle \begin{cases}40 x + 25 y \leq 1000 \\ 18 x + 81 y \leq 1458 \\x \geq 0, y \geq 0 \end{cases} " data-equation-content=" \displaystyle \begin{cases}40 x + 25 y \leq 1000 \\ 18 x + 81 y \leq 1458 \\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}40 x + 25 y +s = 1000 \\ 18 x + 81 y+t = 1458 \\ - 6 x - 13 y+p =0 \end{cases} " src="/equation_images/%20%5Cbegin%7Bcases%7D40%20x%20%2B%2025%20y%20%2Bs%20%3D%201000%20%5C%5C%2018%20x%20%2B%2081%20y%2Bt%20%3D%201458%20%5C%5C%20-%206%20x%20-%2013%20y%2Bp%20%3D0%20%5Cend%7Bcases%7D%20" alt="LaTeX: \begin{cases}40 x + 25 y +s = 1000 \\ 18 x + 81 y+t = 1458 \\ - 6 x - 13 y+p =0 \end{cases} " data-equation-content=" \begin{cases}40 x + 25 y +s = 1000 \\ 18 x + 81 y+t = 1458 \\ - 6 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 40 " src="/equation_images/%20%5Cdisplaystyle%2040%20" alt="LaTeX: \displaystyle 40 " data-equation-content=" \displaystyle 40 " /> & <img class="equation_image" title=" \displaystyle 25 " src="/equation_images/%20%5Cdisplaystyle%2025%20" alt="LaTeX: \displaystyle 25 " data-equation-content=" \displaystyle 25 " /> & <img class="equation_image" title=" \displaystyle 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 1000 " src="/equation_images/%20%5Cdisplaystyle%201000%20" alt="LaTeX: \displaystyle 1000 " data-equation-content=" \displaystyle 1000 " /> \\ \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 18 " src="/equation_images/%20%5Cdisplaystyle%2018%20" alt="LaTeX: \displaystyle 18 " data-equation-content=" \displaystyle 18 " /> & <img class="equation_image" title=" \displaystyle 81 " src="/equation_images/%20%5Cdisplaystyle%2081%20" alt="LaTeX: \displaystyle 81 " data-equation-content=" \displaystyle 81 " /> & <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 1458 " src="/equation_images/%20%5Cdisplaystyle%201458%20" alt="LaTeX: \displaystyle 1458 " data-equation-content=" \displaystyle 1458 " /> \\ \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 -6 " src="/equation_images/%20%5Cdisplaystyle%20-6%20" alt="LaTeX: \displaystyle -6 " data-equation-content=" \displaystyle -6 " /> & <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 81R_1-25R_2 " src="/equation_images/%20%5Cdisplaystyle%2081R_1-25R_2%20" alt="LaTeX: \displaystyle 81R_1-25R_2 " data-equation-content=" \displaystyle 81R_1-25R_2 " /> <br>
<img class="equation_image" title=" \displaystyle 81R_3+13R_2 " src="/equation_images/%20%5Cdisplaystyle%2081R_3%2B13R_2%20" alt="LaTeX: \displaystyle 81R_3+13R_2 " data-equation-content=" \displaystyle 81R_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 2790 " src="/equation_images/%20%5Cdisplaystyle%202790%20" alt="LaTeX: \displaystyle 2790 " data-equation-content=" \displaystyle 2790 " /> & <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 81 " src="/equation_images/%20%5Cdisplaystyle%2081%20" alt="LaTeX: \displaystyle 81 " data-equation-content=" \displaystyle 81 " /> & <img class="equation_image" title=" \displaystyle -25 " src="/equation_images/%20%5Cdisplaystyle%20-25%20" alt="LaTeX: \displaystyle -25 " data-equation-content=" \displaystyle -25 " /> & <img class="equation_image" title=" \displaystyle 0 " src="/equation_images/%20%5Cdisplaystyle%200%20" alt="LaTeX: \displaystyle 0 " data-equation-content=" \displaystyle 0 " /> & <img class="equation_image" title=" \displaystyle 44550 " src="/equation_images/%20%5Cdisplaystyle%2044550%20" alt="LaTeX: \displaystyle 44550 " data-equation-content=" \displaystyle 44550 " /> \\ \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 18 " src="/equation_images/%20%5Cdisplaystyle%2018%20" alt="LaTeX: \displaystyle 18 " data-equation-content=" \displaystyle 18 " /> & <img class="equation_image" title=" \displaystyle 81 " src="/equation_images/%20%5Cdisplaystyle%2081%20" alt="LaTeX: \displaystyle 81 " data-equation-content=" \displaystyle 81 " /> & <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 1458 " src="/equation_images/%20%5Cdisplaystyle%201458%20" alt="LaTeX: \displaystyle 1458 " data-equation-content=" \displaystyle 1458 " /> \\ \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 -252 " src="/equation_images/%20%5Cdisplaystyle%20-252%20" alt="LaTeX: \displaystyle -252 " data-equation-content=" \displaystyle -252 " /> & <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 81 " src="/equation_images/%20%5Cdisplaystyle%2081%20" alt="LaTeX: \displaystyle 81 " data-equation-content=" \displaystyle 81 " /> & <img class="equation_image" title=" \displaystyle 18954 " src="/equation_images/%20%5Cdisplaystyle%2018954%20" alt="LaTeX: \displaystyle 18954 " data-equation-content=" \displaystyle 18954 " /> \\ \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 155R_2-1R_1 " src="/equation_images/%20%5Cdisplaystyle%20155R_2-1R_1%20" alt="LaTeX: \displaystyle 155R_2-1R_1 " data-equation-content=" \displaystyle 155R_2-1R_1 " /> <br>
<img class="equation_image" title=" \displaystyle 155R_3+14R_1 " src="/equation_images/%20%5Cdisplaystyle%20155R_3%2B14R_1%20" alt="LaTeX: \displaystyle 155R_3+14R_1 " data-equation-content=" \displaystyle 155R_3+14R_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 2790 " src="/equation_images/%20%5Cdisplaystyle%202790%20" alt="LaTeX: \displaystyle 2790 " data-equation-content=" \displaystyle 2790 " /> & <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 81 " src="/equation_images/%20%5Cdisplaystyle%2081%20" alt="LaTeX: \displaystyle 81 " data-equation-content=" \displaystyle 81 " /> & <img class="equation_image" title=" \displaystyle -25 " src="/equation_images/%20%5Cdisplaystyle%20-25%20" alt="LaTeX: \displaystyle -25 " data-equation-content=" \displaystyle -25 " /> & <img class="equation_image" title=" \displaystyle 0 " src="/equation_images/%20%5Cdisplaystyle%200%20" alt="LaTeX: \displaystyle 0 " data-equation-content=" \displaystyle 0 " /> & <img class="equation_image" title=" \displaystyle 44550 " src="/equation_images/%20%5Cdisplaystyle%2044550%20" alt="LaTeX: \displaystyle 44550 " data-equation-content=" \displaystyle 44550 " /> \\ \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 12555 " src="/equation_images/%20%5Cdisplaystyle%2012555%20" alt="LaTeX: \displaystyle 12555 " data-equation-content=" \displaystyle 12555 " /> & <img class="equation_image" title=" \displaystyle -81 " src="/equation_images/%20%5Cdisplaystyle%20-81%20" alt="LaTeX: \displaystyle -81 " data-equation-content=" \displaystyle -81 " /> & <img class="equation_image" title=" \displaystyle 180 " src="/equation_images/%20%5Cdisplaystyle%20180%20" alt="LaTeX: \displaystyle 180 " data-equation-content=" \displaystyle 180 " /> & <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 181440 " src="/equation_images/%20%5Cdisplaystyle%20181440%20" alt="LaTeX: \displaystyle 181440 " data-equation-content=" \displaystyle 181440 " /> \\ \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 1134 " src="/equation_images/%20%5Cdisplaystyle%201134%20" alt="LaTeX: \displaystyle 1134 " data-equation-content=" \displaystyle 1134 " /> & <img class="equation_image" title=" \displaystyle 1665 " src="/equation_images/%20%5Cdisplaystyle%201665%20" alt="LaTeX: \displaystyle 1665 " data-equation-content=" \displaystyle 1665 " /> & <img class="equation_image" title=" \displaystyle 12555 " src="/equation_images/%20%5Cdisplaystyle%2012555%20" alt="LaTeX: \displaystyle 12555 " data-equation-content=" \displaystyle 12555 " /> & <img class="equation_image" title=" \displaystyle 3561570 " src="/equation_images/%20%5Cdisplaystyle%203561570%20" alt="LaTeX: \displaystyle 3561570 " data-equation-content=" \displaystyle 3561570 " /> \\ \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{495}{31} " src="/equation_images/%20%5Cdisplaystyle%20%5Cfrac%7B495%7D%7B31%7D%20" alt="LaTeX: \displaystyle \frac{495}{31} " data-equation-content=" \displaystyle \frac{495}{31} " /> . 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{448}{31} " src="/equation_images/%20%5Cdisplaystyle%20%5Cfrac%7B448%7D%7B31%7D%20" alt="LaTeX: \displaystyle \frac{448}{31} " data-equation-content=" \displaystyle \frac{448}{31} " /> . The max value is <img class="equation_image" title=" \displaystyle p = \frac{8794}{31} " src="/equation_images/%20%5Cdisplaystyle%20p%20%3D%20%5Cfrac%7B8794%7D%7B31%7D%20" alt="LaTeX: \displaystyle p = \frac{8794}{31} " data-equation-content=" \displaystyle p = \frac{8794}{31} " /> </p> </p>