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\begin{document}
\noindent Name: \rule{7cm}{.2pt} \vspace{.3cm} \hfill Date:  \rule{3.5cm}{.2pt}
\vspace{.3cm}

\noindent\hrulefill
\begin{center}
{\bf \Large{MA 162}} \\\vspace{.1cm} Week 4 Recitation Worksheet (Tuesday)
\end{center}
\vspace{-.3cm}
\hrulefill

\begin{center}
\fbox{\textbf{You must show all work to receive full credit.}}
\end{center}

\begin{enumerate}
%1
\item Solve the given system of equations using either the substitution method or the elimination method.  If the system has no solution, state \emph{no solution}.  If there are infinitely many solutions, \emph{parameterize the set of solutions}.
\begin{equation*}
\left\{\begin{alignedat}{2}
-4x &{}-{}& 8y &= \phantom{-}11\\
3x &{}+{}& 6y &= -8
\end{alignedat}\right.
\end{equation*}\vfill


%2
\item Solve the given system of equations using either the substitution method or the elimination method.  If the system has no solution, state \emph{no solution}.  If there are infinitely many solutions, \emph{parameterize the set of solutions}.
\begin{equation*}
\left\{\begin{alignedat}{2}
-4x &{}-{}& 8y &= \phantom{-}11\\
3x &{}+{}& 6y &= -\frac{33}{4}
\end{alignedat}\right.
\end{equation*}\vfill


\newpage
%3
\item Determine all values of $h$ and $k$ for which the given system has infinitely many solutions.
\begin{equation*}
\left\{\begin{alignedat}{2}
-5x &{}-{}& 3y &= h\\
-2x &{}+{}& ky &= 9
\end{alignedat}\right.
\end{equation*}\vfill\vfill


%4
\item We have now seen two methods for solving a system of linear equations: (1) the substitution method and (2) the elimination method.  The systems that we have considered so far generally involve two equations and two variables.  Imagine you are asked to solve a system of \emph{four equations} with two variables.  Not sure how to solve such a system, you decide to graph each of the lines corresponding to the four equations.  Below is the resulting graph (lines are labeled $l_1$, $l_2$, $l_3$, and $l_4$).
\begin{center}
\begin{tikzpicture}[scale=0.5]
\draw[help lines] (-5.5,-5.5) grid (5.5,5.5);
\draw[ultra thick,<->] (-6,0) -- (6,0) node[right] {$x$};
\draw[ultra thick,<->] (0,-6) -- (0,6) node[above] {$y$};
\foreach \x in {1,2,...,5} { \draw[thick] ( \x,0.2) -- ( \x,-0.2) ( \x,0) +(0,-0.10) node[below] {\footnotesize $\x$}; }
\foreach \y in {1,2,...,5} { \draw[thick] (0.2, \y) -- (-0.2, \y) (0, \y) +(-0.10,0) node[left]  {\footnotesize $\y$}; }
\foreach \x in {1,2,...,5} { \draw[thick] (-\x,0.2) -- (-\x,-0.2) (-\x,0) +(0,-0.10) node[below] {\footnotesize $-\x$}; }
\foreach \y in {1,2,...,5} { \draw[thick] (0.2,-\y) -- (-0.2,-\y) (0,-\y) +(-0.10,0) node[left]  {\footnotesize $-\y$}; }

%\draw[draw = blue, thick,<->] (-5.5,-5/3) -- (5.5,17/3);
%\draw[draw = red, thick,<->] (-5.5,5/3) -- (5.5,-17/3);
%\draw[draw = violet, thick,<->] (-5.5,-17/3) -- (5.5,5/3);
%\draw[draw = orange, thick,<->] (-5.5,17/3) -- (5.5,-5/3);

\draw[draw = black, thick,<->] (-5.5,-5/3) -- (5.5,17/3);
\draw[draw = black, thick,<->] (-5.5,5/3) -- (5.5,-17/3);
\draw[draw = black, thick,<->] (-5.5,-17/3) -- (5.5,5/3);
\draw[draw = black, thick,<->] (-5.5,17/3) -- (5.5,-5/3);

\node[label=center:{$l_1$}] at (-5.5-0.3,17/3+0.3) {};
\node[label=center:{$l_2$}] at (-5.5-0.3,5/3+0.3) {};
\node[label=center:{$l_3$}] at (-5.5-0.3,-5/3-0.3) {};
\node[label=center:{$l_4$}] at (-5.5-0.3,-17/3-0.3) {};
\end{tikzpicture}
\end{center}
How many solutions does the system of equations have?  \textbf{Explain your reasoning.}\vfill


\newpage
%5
\item An online shop sells knitted hats and sweaters for cats.  Each hat sells for $\$10$ and each sweater sells for $\$22$.   The shop sold a total of $55$ items last month, with sales totaling $\$1{,}066$.  How many sweaters did the shop sell last month?\vfill


%6
\item Jake's Surf Shop rents surfboards for $\$11.75$ plus $\$3.25$ per hour.  Rita's rents them for $\$8.00$ plus $\$4.50$ per hour.  Which is cheaper for which time intervals?\vfill


\newpage
%7
\item A contractor purchased a backhoe for $\$40{,}000$.  Fuel and standard maintenance cost $\$6.57$ per hour, and the operator is paid $\$14.98$ per hour. 
\begin{parts}
\part Write a cost function $C$ for the cost of operating the backhoe for $x$ hours.  Make sure to include any fixed costs.\vfill
\part If customers pay $\$33.87$ per hour for the contractor's backhoe service, write the revenue function $R$ for the amount of revenue gained from $x$ hours of use.\vfill
\part Write the profit function $P$ for the amount of profit gained from $x$ hours of use.\vfill
\part Use part (c) to determine the number of hours the backhoe must be used to break even.\vspace{1cm}\vfill
\end{parts}


%8
\item The supply equation for a product is $x = 250p - 7500$ and the demand equation is $x = -110p + 11220$, where $p$ is the price in dollars and $x$ is the number of items.  At what price will the supply equal the demand, and how many items will be produced at that price?\vfill\vfill\vfill
\end{enumerate}
\end{document}