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Questions: Algebra BusinessCalculus
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Use the simplex method to maximize \(\displaystyle p = 14 x + 5 y\) subject to \(\displaystyle \begin{cases}59 x + 25 y \leq 1475 \\ 20 x + 73 y \leq 1460 \\x \geq 0, y \geq 0 \end{cases}\)
Adding the slack variables \(\displaystyle s\) and \(\displaystyle t\) to the inequalities gives:
\begin{equation*}\begin{cases}59 x + 25 y +s = 1475 \\ 20 x + 73 y+t = 1460 \\ - 14 x - 5 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
\(\displaystyle s\) & \(\displaystyle 59\) & \(\displaystyle 25\) & \(\displaystyle 1\) & \(\displaystyle 0\) & \(\displaystyle 0\) & \(\displaystyle 1475\) \\ \hline
\(\displaystyle t\) & \(\displaystyle 20\) & \(\displaystyle 73\) & \(\displaystyle 0\) & \(\displaystyle 1\) & \(\displaystyle 0\) & \(\displaystyle 1460\) \\ \hline
\(\displaystyle p\) & \(\displaystyle -14\) & \(\displaystyle -5\) & \(\displaystyle 0\) & \(\displaystyle 0\) & \(\displaystyle 1\) & \(\displaystyle 0\) \\ \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:
\(\displaystyle 59R_2-20R_1\)
\(\displaystyle 59R_3+14R_1\)
\begin{tabular}{|c|c|c|c|c|c|c|}\hline
\(\displaystyle \) & \(\displaystyle x\) & \(\displaystyle y\) & \(\displaystyle s\) & \(\displaystyle t\) & \(\displaystyle p\) & \(\displaystyle \) \\ \hline
\(\displaystyle x\) & \(\displaystyle 59\) & \(\displaystyle 25\) & \(\displaystyle 1\) & \(\displaystyle 0\) & \(\displaystyle 0\) & \(\displaystyle 1475\) \\ \hline
\(\displaystyle t\) & \(\displaystyle 0\) & \(\displaystyle 3807\) & \(\displaystyle -20\) & \(\displaystyle 59\) & \(\displaystyle 0\) & \(\displaystyle 56640\) \\ \hline
\(\displaystyle p\) & \(\displaystyle 0\) & \(\displaystyle 55\) & \(\displaystyle 14\) & \(\displaystyle 0\) & \(\displaystyle 59\) & \(\displaystyle 20650\) \\ \hline
\end{tabular}
There are no negative values in row \(\displaystyle p\) and this is the final tableau.The value of \(\displaystyle x\) is \(\displaystyle 25\). The value of \(\displaystyle t\) is \(\displaystyle 960\). The max value is \(\displaystyle p = 350\)
\begin{question}Use the simplex method to maximize $p = 14 x + 5 y$ subject to $\begin{cases}59 x + 25 y \leq 1475 \\ 20 x + 73 y \leq 1460 \\x \geq 0, y \geq 0 \end{cases}$
\soln{9cm}{Adding the slack variables $s$ and $t$ to the inequalities gives:\newline\begin{equation*}\begin{cases}59 x + 25 y +s = 1475 \\ 20 x + 73 y+t = 1460 \\ - 14 x - 5 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
$s$ & $59$ & $25$ & $1$ & $0$ & $0$ & $1475$ \\ \hline
$t$ & $20$ & $73$ & $0$ & $1$ & $0$ & $1460$ \\ \hline
$p$ & $-14$ & $-5$ & $0$ & $0$ & $1$ & $0$ \\ \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
$59R_2-20R_1$\newline
$59R_3+14R_1$\newline
\begin{tabular}{|c|c|c|c|c|c|c|}\hline
$$ & $x$ & $y$ & $s$ & $t$ & $p$ & $$ \\ \hline
$x$ & $59$ & $25$ & $1$ & $0$ & $0$ & $1475$ \\ \hline
$t$ & $0$ & $3807$ & $-20$ & $59$ & $0$ & $56640$ \\ \hline
$p$ & $0$ & $55$ & $14$ & $0$ & $59$ & $20650$ \\ \hline
\end{tabular}\newline
There are no negative values in row $p$ and this is the final tableau.The value of $x$ is $25$. The value of $t$ is $960$. The max value is $p = 350$ }
\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 = 14 x + 5 y " src="/equation_images/%20%5Cdisplaystyle%20p%20%3D%2014%20x%20%2B%205%20y%20" alt="LaTeX: \displaystyle p = 14 x + 5 y " data-equation-content=" \displaystyle p = 14 x + 5 y " /> subject to <img class="equation_image" title=" \displaystyle \begin{cases}59 x + 25 y \leq 1475 \\ 20 x + 73 y \leq 1460 \\x \geq 0, y \geq 0 \end{cases} " src="/equation_images/%20%5Cdisplaystyle%20%5Cbegin%7Bcases%7D59%20x%20%2B%2025%20y%20%5Cleq%201475%20%5C%5C%2020%20x%20%2B%2073%20y%20%5Cleq%201460%20%5C%5Cx%20%5Cgeq%200%2C%20y%20%5Cgeq%200%20%5Cend%7Bcases%7D%20" alt="LaTeX: \displaystyle \begin{cases}59 x + 25 y \leq 1475 \\ 20 x + 73 y \leq 1460 \\x \geq 0, y \geq 0 \end{cases} " data-equation-content=" \displaystyle \begin{cases}59 x + 25 y \leq 1475 \\ 20 x + 73 y \leq 1460 \\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}59 x + 25 y +s = 1475 \\ 20 x + 73 y+t = 1460 \\ - 14 x - 5 y+p =0 \end{cases} " src="/equation_images/%20%5Cbegin%7Bcases%7D59%20x%20%2B%2025%20y%20%2Bs%20%3D%201475%20%5C%5C%2020%20x%20%2B%2073%20y%2Bt%20%3D%201460%20%5C%5C%20-%2014%20x%20-%205%20y%2Bp%20%3D0%20%5Cend%7Bcases%7D%20" alt="LaTeX: \begin{cases}59 x + 25 y +s = 1475 \\ 20 x + 73 y+t = 1460 \\ - 14 x - 5 y+p =0 \end{cases} " data-equation-content=" \begin{cases}59 x + 25 y +s = 1475 \\ 20 x + 73 y+t = 1460 \\ - 14 x - 5 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 59 " src="/equation_images/%20%5Cdisplaystyle%2059%20" alt="LaTeX: \displaystyle 59 " data-equation-content=" \displaystyle 59 " /> & <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 1475 " src="/equation_images/%20%5Cdisplaystyle%201475%20" alt="LaTeX: \displaystyle 1475 " data-equation-content=" \displaystyle 1475 " /> \\ \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 20 " src="/equation_images/%20%5Cdisplaystyle%2020%20" alt="LaTeX: \displaystyle 20 " data-equation-content=" \displaystyle 20 " /> & <img class="equation_image" title=" \displaystyle 73 " src="/equation_images/%20%5Cdisplaystyle%2073%20" alt="LaTeX: \displaystyle 73 " data-equation-content=" \displaystyle 73 " /> & <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 1460 " src="/equation_images/%20%5Cdisplaystyle%201460%20" alt="LaTeX: \displaystyle 1460 " data-equation-content=" \displaystyle 1460 " /> \\ \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 -14 " src="/equation_images/%20%5Cdisplaystyle%20-14%20" alt="LaTeX: \displaystyle -14 " data-equation-content=" \displaystyle -14 " /> & <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 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 59R_2-20R_1 " src="/equation_images/%20%5Cdisplaystyle%2059R_2-20R_1%20" alt="LaTeX: \displaystyle 59R_2-20R_1 " data-equation-content=" \displaystyle 59R_2-20R_1 " /> <br>
<img class="equation_image" title=" \displaystyle 59R_3+14R_1 " src="/equation_images/%20%5Cdisplaystyle%2059R_3%2B14R_1%20" alt="LaTeX: \displaystyle 59R_3+14R_1 " data-equation-content=" \displaystyle 59R_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 59 " src="/equation_images/%20%5Cdisplaystyle%2059%20" alt="LaTeX: \displaystyle 59 " data-equation-content=" \displaystyle 59 " /> & <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 1475 " src="/equation_images/%20%5Cdisplaystyle%201475%20" alt="LaTeX: \displaystyle 1475 " data-equation-content=" \displaystyle 1475 " /> \\ \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 3807 " src="/equation_images/%20%5Cdisplaystyle%203807%20" alt="LaTeX: \displaystyle 3807 " data-equation-content=" \displaystyle 3807 " /> & <img class="equation_image" title=" \displaystyle -20 " src="/equation_images/%20%5Cdisplaystyle%20-20%20" alt="LaTeX: \displaystyle -20 " data-equation-content=" \displaystyle -20 " /> & <img class="equation_image" title=" \displaystyle 59 " src="/equation_images/%20%5Cdisplaystyle%2059%20" alt="LaTeX: \displaystyle 59 " data-equation-content=" \displaystyle 59 " /> & <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 56640 " src="/equation_images/%20%5Cdisplaystyle%2056640%20" alt="LaTeX: \displaystyle 56640 " data-equation-content=" \displaystyle 56640 " /> \\ \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 55 " src="/equation_images/%20%5Cdisplaystyle%2055%20" alt="LaTeX: \displaystyle 55 " data-equation-content=" \displaystyle 55 " /> & <img class="equation_image" title=" \displaystyle 14 " src="/equation_images/%20%5Cdisplaystyle%2014%20" alt="LaTeX: \displaystyle 14 " data-equation-content=" \displaystyle 14 " /> & <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 59 " src="/equation_images/%20%5Cdisplaystyle%2059%20" alt="LaTeX: \displaystyle 59 " data-equation-content=" \displaystyle 59 " /> & <img class="equation_image" title=" \displaystyle 20650 " src="/equation_images/%20%5Cdisplaystyle%2020650%20" alt="LaTeX: \displaystyle 20650 " data-equation-content=" \displaystyle 20650 " /> \\ \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 25 " src="/equation_images/%20%5Cdisplaystyle%2025%20" alt="LaTeX: \displaystyle 25 " data-equation-content=" \displaystyle 25 " /> . The value of <img class="equation_image" title=" \displaystyle t " src="/equation_images/%20%5Cdisplaystyle%20t%20" alt="LaTeX: \displaystyle t " data-equation-content=" \displaystyle t " /> is <img class="equation_image" title=" \displaystyle 960 " src="/equation_images/%20%5Cdisplaystyle%20960%20" alt="LaTeX: \displaystyle 960 " data-equation-content=" \displaystyle 960 " /> . The max value is <img class="equation_image" title=" \displaystyle p = 350 " src="/equation_images/%20%5Cdisplaystyle%20p%20%3D%20350%20" alt="LaTeX: \displaystyle p = 350 " data-equation-content=" \displaystyle p = 350 " /> </p> </p>