\(\text{www.the}\beta\text{etafunction.com}\)
Home
Login
Questions: Algebra BusinessCalculus
Please login to create an exam or a quiz.
Use the simplex method to maximize \(\displaystyle p = 14 x + 9 y\) subject to \(\displaystyle \begin{cases}61 x + 43 y \leq 2623 \\ 9 x + 63 y \leq 567 \\x \geq 0, y \geq 0 \end{cases}\)
Adding the slack variables \(\displaystyle s\) and \(\displaystyle t\) to the inequalities gives:
\begin{equation*}\begin{cases}61 x + 43 y +s = 2623 \\ 9 x + 63 y+t = 567 \\ - 14 x - 9 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
\(\displaystyle s\) & \(\displaystyle 61\) & \(\displaystyle 43\) & \(\displaystyle 1\) & \(\displaystyle 0\) & \(\displaystyle 0\) & \(\displaystyle 2623\) \\ \hline
\(\displaystyle t\) & \(\displaystyle 9\) & \(\displaystyle 63\) & \(\displaystyle 0\) & \(\displaystyle 1\) & \(\displaystyle 0\) & \(\displaystyle 567\) \\ \hline
\(\displaystyle p\) & \(\displaystyle -14\) & \(\displaystyle -9\) & \(\displaystyle 0\) & \(\displaystyle 0\) & \(\displaystyle 1\) & \(\displaystyle 0\) \\ \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:
\(\displaystyle 61R_2-9R_1\)
\(\displaystyle 61R_3+14R_1\)
\begin{tabular}{|c|c|c|c|c|c|c|}\hline
\(\displaystyle \) & \(\displaystyle x\) & \(\displaystyle y\) & \(\displaystyle s\) & \(\displaystyle t\) & \(\displaystyle p\) & \(\displaystyle \) \\ \hline
\(\displaystyle x\) & \(\displaystyle 61\) & \(\displaystyle 43\) & \(\displaystyle 1\) & \(\displaystyle 0\) & \(\displaystyle 0\) & \(\displaystyle 2623\) \\ \hline
\(\displaystyle t\) & \(\displaystyle 0\) & \(\displaystyle 3456\) & \(\displaystyle -9\) & \(\displaystyle 61\) & \(\displaystyle 0\) & \(\displaystyle 10980\) \\ \hline
\(\displaystyle p\) & \(\displaystyle 0\) & \(\displaystyle 53\) & \(\displaystyle 14\) & \(\displaystyle 0\) & \(\displaystyle 61\) & \(\displaystyle 36722\) \\ \hline
\end{tabular}
There are no negative values in row \(\displaystyle p\) and this is the final tableau.The value of \(\displaystyle x\) is \(\displaystyle 43\). The value of \(\displaystyle t\) is \(\displaystyle 180\). The max value is \(\displaystyle p = 602\)
\begin{question}Use the simplex method to maximize $p = 14 x + 9 y$ subject to $\begin{cases}61 x + 43 y \leq 2623 \\ 9 x + 63 y \leq 567 \\x \geq 0, y \geq 0 \end{cases}$
\soln{9cm}{Adding the slack variables $s$ and $t$ to the inequalities gives:\newline\begin{equation*}\begin{cases}61 x + 43 y +s = 2623 \\ 9 x + 63 y+t = 567 \\ - 14 x - 9 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
$s$ & $61$ & $43$ & $1$ & $0$ & $0$ & $2623$ \\ \hline
$t$ & $9$ & $63$ & $0$ & $1$ & $0$ & $567$ \\ \hline
$p$ & $-14$ & $-9$ & $0$ & $0$ & $1$ & $0$ \\ \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
$61R_2-9R_1$\newline
$61R_3+14R_1$\newline
\begin{tabular}{|c|c|c|c|c|c|c|}\hline
$$ & $x$ & $y$ & $s$ & $t$ & $p$ & $$ \\ \hline
$x$ & $61$ & $43$ & $1$ & $0$ & $0$ & $2623$ \\ \hline
$t$ & $0$ & $3456$ & $-9$ & $61$ & $0$ & $10980$ \\ \hline
$p$ & $0$ & $53$ & $14$ & $0$ & $61$ & $36722$ \\ \hline
\end{tabular}\newline
There are no negative values in row $p$ and this is the final tableau.The value of $x$ is $43$. The value of $t$ is $180$. The max value is $p = 602$ }
\end{question}
\documentclass{article}
\usepackage{tikz}
\usepackage{amsmath}
\usepackage[margin=2cm]{geometry}
\usepackage{tcolorbox}
\newcounter{ExamNumber}
\newcounter{questioncount}
\stepcounter{questioncount}
\newenvironment{question}{{\noindent\bfseries Question \arabic{questioncount}.}}{\stepcounter{questioncount}}
\renewcommand{\labelenumi}{{\bfseries (\alph{enumi})}}
\newif\ifShowSolution
\newcommand{\soln}[2]{%
\ifShowSolution%
\noindent\begin{tcolorbox}[colframe=blue,title=Solution]#2\end{tcolorbox}\else%
\vspace{#1}%
\fi%
}%
\newcommand{\hideifShowSolution}[1]{%
\ifShowSolution%
%
\else%
#1%
\fi%
}%
\everymath{\displaystyle}
\ShowSolutiontrue
\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 + 9 y " src="/equation_images/%20%5Cdisplaystyle%20p%20%3D%2014%20x%20%2B%209%20y%20" alt="LaTeX: \displaystyle p = 14 x + 9 y " data-equation-content=" \displaystyle p = 14 x + 9 y " /> subject to <img class="equation_image" title=" \displaystyle \begin{cases}61 x + 43 y \leq 2623 \\ 9 x + 63 y \leq 567 \\x \geq 0, y \geq 0 \end{cases} " src="/equation_images/%20%5Cdisplaystyle%20%5Cbegin%7Bcases%7D61%20x%20%2B%2043%20y%20%5Cleq%202623%20%5C%5C%209%20x%20%2B%2063%20y%20%5Cleq%20567%20%5C%5Cx%20%5Cgeq%200%2C%20y%20%5Cgeq%200%20%5Cend%7Bcases%7D%20" alt="LaTeX: \displaystyle \begin{cases}61 x + 43 y \leq 2623 \\ 9 x + 63 y \leq 567 \\x \geq 0, y \geq 0 \end{cases} " data-equation-content=" \displaystyle \begin{cases}61 x + 43 y \leq 2623 \\ 9 x + 63 y \leq 567 \\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}61 x + 43 y +s = 2623 \\ 9 x + 63 y+t = 567 \\ - 14 x - 9 y+p =0 \end{cases} " src="/equation_images/%20%5Cbegin%7Bcases%7D61%20x%20%2B%2043%20y%20%2Bs%20%3D%202623%20%5C%5C%209%20x%20%2B%2063%20y%2Bt%20%3D%20567%20%5C%5C%20-%2014%20x%20-%209%20y%2Bp%20%3D0%20%5Cend%7Bcases%7D%20" alt="LaTeX: \begin{cases}61 x + 43 y +s = 2623 \\ 9 x + 63 y+t = 567 \\ - 14 x - 9 y+p =0 \end{cases} " data-equation-content=" \begin{cases}61 x + 43 y +s = 2623 \\ 9 x + 63 y+t = 567 \\ - 14 x - 9 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 61 " src="/equation_images/%20%5Cdisplaystyle%2061%20" alt="LaTeX: \displaystyle 61 " data-equation-content=" \displaystyle 61 " /> & <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 2623 " src="/equation_images/%20%5Cdisplaystyle%202623%20" alt="LaTeX: \displaystyle 2623 " data-equation-content=" \displaystyle 2623 " /> \\ \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 9 " src="/equation_images/%20%5Cdisplaystyle%209%20" alt="LaTeX: \displaystyle 9 " data-equation-content=" \displaystyle 9 " /> & <img class="equation_image" title=" \displaystyle 63 " src="/equation_images/%20%5Cdisplaystyle%2063%20" alt="LaTeX: \displaystyle 63 " data-equation-content=" \displaystyle 63 " /> & <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 567 " src="/equation_images/%20%5Cdisplaystyle%20567%20" alt="LaTeX: \displaystyle 567 " data-equation-content=" \displaystyle 567 " /> \\ \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 -9 " src="/equation_images/%20%5Cdisplaystyle%20-9%20" alt="LaTeX: \displaystyle -9 " data-equation-content=" \displaystyle -9 " /> & <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 61R_2-9R_1 " src="/equation_images/%20%5Cdisplaystyle%2061R_2-9R_1%20" alt="LaTeX: \displaystyle 61R_2-9R_1 " data-equation-content=" \displaystyle 61R_2-9R_1 " /> <br>
<img class="equation_image" title=" \displaystyle 61R_3+14R_1 " src="/equation_images/%20%5Cdisplaystyle%2061R_3%2B14R_1%20" alt="LaTeX: \displaystyle 61R_3+14R_1 " data-equation-content=" \displaystyle 61R_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 61 " src="/equation_images/%20%5Cdisplaystyle%2061%20" alt="LaTeX: \displaystyle 61 " data-equation-content=" \displaystyle 61 " /> & <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 2623 " src="/equation_images/%20%5Cdisplaystyle%202623%20" alt="LaTeX: \displaystyle 2623 " data-equation-content=" \displaystyle 2623 " /> \\ \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 3456 " src="/equation_images/%20%5Cdisplaystyle%203456%20" alt="LaTeX: \displaystyle 3456 " data-equation-content=" \displaystyle 3456 " /> & <img class="equation_image" title=" \displaystyle -9 " src="/equation_images/%20%5Cdisplaystyle%20-9%20" alt="LaTeX: \displaystyle -9 " data-equation-content=" \displaystyle -9 " /> & <img class="equation_image" title=" \displaystyle 61 " src="/equation_images/%20%5Cdisplaystyle%2061%20" alt="LaTeX: \displaystyle 61 " data-equation-content=" \displaystyle 61 " /> & <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 10980 " src="/equation_images/%20%5Cdisplaystyle%2010980%20" alt="LaTeX: \displaystyle 10980 " data-equation-content=" \displaystyle 10980 " /> \\ \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 53 " src="/equation_images/%20%5Cdisplaystyle%2053%20" alt="LaTeX: \displaystyle 53 " data-equation-content=" \displaystyle 53 " /> & <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 61 " src="/equation_images/%20%5Cdisplaystyle%2061%20" alt="LaTeX: \displaystyle 61 " data-equation-content=" \displaystyle 61 " /> & <img class="equation_image" title=" \displaystyle 36722 " src="/equation_images/%20%5Cdisplaystyle%2036722%20" alt="LaTeX: \displaystyle 36722 " data-equation-content=" \displaystyle 36722 " /> \\ \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 43 " src="/equation_images/%20%5Cdisplaystyle%2043%20" alt="LaTeX: \displaystyle 43 " data-equation-content=" \displaystyle 43 " /> . 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 180 " src="/equation_images/%20%5Cdisplaystyle%20180%20" alt="LaTeX: \displaystyle 180 " data-equation-content=" \displaystyle 180 " /> . The max value is <img class="equation_image" title=" \displaystyle p = 602 " src="/equation_images/%20%5Cdisplaystyle%20p%20%3D%20602%20" alt="LaTeX: \displaystyle p = 602 " data-equation-content=" \displaystyle p = 602 " /> </p> </p>