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\title{MA162: Finite mathematics}
\subtitle{Systems of Linear Equations}
\author{Paul Koester}
\institute{University of Kentucky}
\date{March 24, 2014}

\begin{document}

\maketitle

\begin{frame}[allowframebreaks,t]{Cost, Revenue, Profit, Break-even}
You have \(100\) acre plot of land.  You plan to grow and sell corn.

Suppose the cost of growing \(x\) acres of corn is 
\[
  C(x) = \$17,000 + \$340 x
\]
\begin{enumerate}[(a.)]
  \item In plain English, what is the meaning of the vertical intercept?  \vfill
  \item In plain English, what is the meaning of the slope?   \vfill
\end{enumerate}
\end{frame}

\begin{frame}[allowframebreaks,t]{Cost, Revenue, Profit, Break-even}
Suppose the revenue generated by selling \(x\) acres worth of corn is 
\[
  R(x) = \$650.70 x
\]
\begin{enumerate}[(a.)]
  \item In plain English, what does the slope mean? \vfill
  \item In plain English, what does the vertical intercept mean? \vfill
\end{enumerate}
\end{frame}

\begin{frame}[allowframebreaks,t]{Cost, Revenue, Profit}
\begin{enumerate}[(a.)]
  \item If you grow and sell \(20\) acres of corn, which is greater, the cost or the revenue?  \vfill
  \item If you grow and sell \(70\) acres of corn, which is greater, the cost or the revenue? \vfill
  \item Determine the value of x at which the revenue and cost functions are equal.
\end{enumerate}
\end{frame}

% Estimates of seed/etc:   http://fbfm.ace.uiuc.edu/pdf%20files/Farm%20Business%20Results/Cost%20of%20Production%20for%20Grain/c&s_south.pdf
% Bushels per acre:        http://www.clover.okstate.edu/fourh/aitc/lessons/upper/cornmath.pdf
% Price:                   http://www.quotecorn.com/


\begin{frame}{Supply, Demand, and Equilibrium}
\begin{itemize}
  \item Your tech company is about to unveil its new \emph{Me-Pod}. \vfill  
  \item Your task is to determine how much to charge for this device. \vfill 
  \item Charge too little, then your company will sell a lot of these devices, but will generate very little 
        revenue on each individual sale. \vfill 
  \item Charge too much, then your company will generate a lot of revenue on each individual sale, but 
        very few of these devices will be sold. \vfill  
  \item What to do? \vfill
\end{itemize}
\end{frame}

\begin{frame}[t,allowframebreaks]{The Demand Function}
First, you ask your marketing team to determine how many of these devices your company might expect to sell at 
various prices. Their results: 
\begin{itemize}
  \item Expect to sell \(1000\) units per month if you charge \(\$200\) per Me-Pod. \vfill
  \item Expect to sell \(800\) units per month if you charge \(\$250\) per Me-Pod. \vfill
\end{itemize}
Let \(N\) denote the number of Me-Pods sold per month and let \(p\) denote the price per Me-Pod.
\vfill
You find \[N = -4p + 1800\]  
(Double check!  Make sure you can determine the line if given two points)\\

This is the \textit{demand function.}
\end{frame}

\begin{frame}[t,allowframebreaks]{The Supply Function}
Next, you ask your manufacturing division to determine how many of these devices they are willing to produce at 
various prices. Their answer: 
\begin{itemize}
  \item Willing to produce \(600\) units per month if they sell for \(\$200\) per Me-Pod.
  \item Willing to produce \(1000\) units per month if they sell for \(\$250\) per Me-Pod.
\end{itemize}
You find \[ N = 8p - 1000 \]

This is the \textit{supply function.}
\end{frame}

\begin{frame}[t]{Supply and Demand}
Plot the supply and demand functions on the same set of axes \vfill \vfill \vfill
Now find the point of intersection. \vfill
\end{frame}

\begin{frame}[t]{Two Linear Equations in Two Unknowns}
\begin{itemize}
 \item Many applied problems can be solved by determining the point where two lines cross. \vfill
 \item This is equivalent to solving a system of two linear equations in two unknown.  \vfill
 \item 
  \[
    \begin{array}{ccccc}
      ax &+& by &=& c \\
      dx &+& ey &=& f \\
    \end{array}
  \]
 \vfill
 \item Systems of equations can be solved using either
   \begin{itemize}
    \item substitution
    \item elimination
   \end{itemize}
 \vfill
 \item We will focus on elimination: 
           elimination provides an efficient method for solving very large systems (lots of equations and lots of unknowns)
 \end{itemize}
\end{frame}

\begin{frame}[t]{Special Cases:  Inconsistent Systems}
Find the solution to the system of equations:
\[
  \begin{array}{ccccc}
    x   &  -  &   y  &  =  &  4 \\
   -3x  &  +  &  3y  &  =  &  5 \\ 
  \end{array}
\]
\end{frame}

\begin{frame}[t]{Special Cases: Underdetermined Systems}
Find the solution to the system of equations:
\[
  \begin{array}{ccccc}
    3u   &  +  &  6v  &  =  &  9 \\
    2u   &  +  &  4v  &  =  &  6 \\ 
  \end{array}
\]
\end{frame}

\begin{frame}[t]{Bigger systems}
The management of Hartman Rent-A-Car has allocated \$2.25 million to buy a 
fleet of new automobiles consisting of compact, intermediate-size, and full-size cars. 
Compacts cost \$18,000 each, intermediate-size cars cost \$27,000 each, and full-size cars cost
\$36,000 each. 
If Hartman purchases twice as many compacts as intermediate-size cars and the total number of cars to be
purchased is 100, determine how many cars of each type will be purchased. (Assume that the entire budget will be used.)
\end{frame}




\end{document}
