\documentclass[12pt,letterpaper]{article}
\usepackage[latin1]{inputenc}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsthm}
\usepackage{hyperref}
\usepackage{paralist}
\usepackage[all=normal,margins]{savetrees}
\begin{document}
\setlength{\plitemsep}{12ex}
\newtheorem{notation}{Notation}
\begin{center}
Problems PreCal 1508 PLTL Workshop, October 12, 2011\\
PLs: Alex Knaust, Edith Mejia. Lecturer: Yi-Yu Liao\\

\end{center}
Please do the problems that you feel will help your group the most first (you don't have to do them in order). All handouts are available at \url{http://alex.knaust.info/pltlfall2011/}
\begin{enumerate}

\item Determine whether each ordered pair is a solution to the system of equations
\item \begin{displaymath}
   \left\{
     \begin{array}{lr}
       -\log x + 3 = y\\
       \frac{1}{9}x + y = \frac{28}{9}
     \end{array}
   \right.
\end{displaymath}
\begin{inparaenum}[a]
\item $(1, 3)\quad$
\item $(10, 2)\quad$
\item $(9, \frac{37}{9})$
\end{inparaenum}

\item Solve the following systems, algebraically, if they can't be solved algebraically solve them graphically\\
\begin{inparaenum}[a)]
\item
$ \left\{\begin{array}{lr}
	x - y = -1\\
	x^2 - y = -4 \\
	\end{array}
	\right.\quad\quad\quad
$
\item$ \left\{\begin{array}{lr}
	3x - 5y = 7\\
	2x + y = 9 \\
	\end{array}
	\right.\quad\quad\quad
$
\item $\left\{\begin{array}{lr}
	x + 2z = 5 \\
	3x - y -z = 1\\
	6x - y + 5z = 16\\
	\end{array}
	\right.\quad\quad\quad
$\newline\newline\newline
\item $\left\{\begin{array}{lr}
	xe^x = y \\
	x + 2y = 4\\
	\end{array}
	\right.\quad\quad\quad
$
\item $\left\{\begin{array}{lr}
	y = -x \\
	y = x^3 + 3x^2 + 2x\\
	\end{array}
	\right.\quad\quad
$
\item $\left\{\begin{array}{lr}
	3\alpha -3\beta + 6\gamma = 6 \\
	\alpha + 2\beta - \gamma = 5\\
	5\alpha - 8\beta + 13\gamma = 7
	\end{array}
	\right.
$
\end{inparaenum}

\item Describe, geometrically, the possible solutions for a system of two equations.

\item Find a system of linear equations that has the pair $(3, 1)$ as a solution. (There are many correct answers)

\item Let $f$ be a \emph{one-to-one} function (with a known inverse function $f^{-1}$). How would you go about solving for $x$ in
\[3\cdot f\left(\frac{x}{2}\right) - 5 = 7\]

\item Discuss with your group : When solving a system of linear equations do you decide what to do intuitively or do you follow a methodology?

\item Find all values of $x$ that satisfy the equation : $\log_2(2x - 3) = \log_4(x + 1)$

\item Describe, geometrically, the possible solutions for a system of three equations.

\item How much more work is solving a system of three equations than solving a system of two equations? Does it take one more operation? Twice as many? Three times as many?


\end{enumerate}



\end{document}