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\course \hfill
Review  1
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The following topics and skills will be examined in the first midterm test.
\begin{itemize}
\item Solving $2\times 2$ systems.
\item Demand and supply curves.
\item Converting between augmented matrices and systems of equations.
\item Rewriting systems of equations as matrix equations.
\item Writing packaging problems, RST games and similar problems as systems of equations.
\item Solving $3\times 3$ and $2\times 2$ systems by the methods of
 Gauss-Jordan elimination and matrix inversion.
\item Systems of equations with no solutions and with many solutions.
\item Matrix addition, multiplication and multiplication of a matrix
by a real number.
\item Finding the inverses of $2\times 2$ and $3\times 3 $ matrices.
\comment{
\item Graphing the feasible set of a system of linear inequalities in
two variables.  Finding corner points
\item Formulating written descriptions of feasible sets as linear
inequalities. 
 }
\end{itemize}
Below are some sample questions to help you prepare for the exam.
Be sure that you understand all of the topics from the above list, not
just the topics covered by the questions below!
 In your answers, please be sure to 1) Check answers when
possible. 2) Clearly indicate your answer and the reasoning used to
arrive at that answer. Unsupported answers will receive no credit. 3)
Label all variables and equations. 

\begin{enumerate}
\item  The following information is available regarding the supply,
demand and price of new and improved $\beta$-widgets.

\begin{tabular}{rr|r}
Supply &Demand & \\ \hline
100 & 60 & \$10\\
125 & 35 & \$12\\
150 & 10 & \$15 
\end{tabular}
\begin{enumerate}
\item Plot this information on a graph.
\item Estimate  the equilibrium price.
\item If the price is \$15, is there shortage or surplus?

\end{enumerate}


\item Suppose that a company produce packages of nuts which contain
cashews, peanuts and filberts.
The D-package contains 4 ounces of cashews, 3 oz.~of peanuts and 1 oz.
of filberts. The L-package contains 1 oz.~of cashews, 3 oz.~of peanuts
and 1 oz.~of filberts.  The E-package contains 2 oz.~of cashews, 3
oz.~of peanut and no filberts.  Suppose that next month they expect
to have 100 pounds of cashews, 200 pounds of peanuts and 150 pounds of
filberts.  Set up a system of equations whose solution tells how many
of each package they can make while using up all of the available
nuts.  Do not solve this system. 


\item
\begin{enumerate} \item Find the inverse of $\left(\begin{array}{rrr}
3&1&1\\
1&1&1\\
2&2&0
\end{array}\right)$
\item Use your answer to part a to solve the following system.
 \begin{eqnarray*}
3x+y+z&=&2\\
x+y+z&=&1\\
2x+2y &= &4
\end{eqnarray*}
\end{enumerate}

\item 
\begin{enumerate}
\item
Solve 
\begin{eqnarray*}
x+y-z &=& 2\\
y+z&=& 2\\
x+3y+z &=& 6
\end{eqnarray*}
\item Solve
\begin{eqnarray*}
x+2y&=& 1
\\
2x+y&=&1
\end{eqnarray*}

\end{enumerate}

\item For each of the following matrix operations, either give the
result or explain why it is impossible.
$$
\left( \begin{array}{rr} 1 & 2 \\ 4 & -1 \end{array}\right)
\left( \begin{array}{r} 2 \\3 \end{array}\right), \quad
\left( \begin{array}{rr} 1 & 2 \\ 4 & -1 \end{array}\right) +
\left( \begin{array}{r} 2 \\ 3 \end{array}\right), \quad
\left( \begin{array}{r} 2 \\ -1 \end{array}\right)
\left( \begin{array}{rr} 2& 3\end{array}\right).
$$





\end{enumerate}
{\small \today}
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