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Problems PreCal 1508 PLTL Workshop, October 14, 2011\\
PLs: Alex Knaust, Edith Mejia. Lecturer: Yi-Yu Liao\\

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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 Solve the system of linear equations, and check the solutions algebraically\\
\begin{center}
\begin{inparaenum}[a)]
\item
$\displaystyle \left\{\begin{array}{lr}
	2x + 4y + z = 1\\
	x - 2y - 3z = 2\\
	x + y - z = -1 \\
	\end{array}
	\right.\quad\quad\quad
$
\item
$\displaystyle \left\{\begin{array}{lr}
	5x  -3y  + 2z = 3\\
	2x + 4y - z = 7\\
	x - 11y + 4z = 3 \\
	-x -2y + \frac{1}{2}z = -\frac{7}{2}\\
	\end{array}
	\right.\quad\quad\quad
$
\item
$ \left\{\begin{array}{lr}
	2x + y -3z = 4\\
	4x +\quad 2z = 10\\
	-2x +3y - 13z = -8 \\
	\end{array}
	\right.
$
\newline
\end{inparaenum}
\end{center}

\item Find the equation of the circle ($0 = x^2 + y^2 + Dx + Ey + F$) that passes through the points $(0,0),(5,5),(10,0)$.

\item Find values of $x$ that satisfy the following equations
\begin{enumerate}
\item $\log_6{(x + 2)} - \log_6{x} = \log_6{(x + 5)}$
\item $\log 4x - \log(12 + \sqrt{x}) = 2$
\item $\frac{500}{2 + e^{2x}} = 20$
\item $e^{2x} + 9e^x + 36 = 0$
\end{enumerate}

\item Two cheeseburgers and one small order of French fries contain a total of 830 Calories. Three cheeseburgers and two small orders of French fries contain a total of 1360 calories. Find the Caloric content of each item.

\item Find any solutions of the following system of equations
\[\left\{\begin{array}{lr}
	x^2 - y = 0\\
	2x + y = 0\\
	\end{array}\right.
\]




\item Are the following true or false? Explain why/give a counterexample
\begin{enumerate}
\item If two lines do not have exactly one point of intersection, then they must be parallel.
\item $2\cdot \log{(x + 3)} = \log{(2x - 4)} \Longleftrightarrow (x+3)^2 = 2x - 4$
\item The inverse function of $f(x) = x^4 + 2$ is $f^{-1}(x) = \sqrt[4]{x - 2}$
\item $a < b \Rightarrow \log_b{a} < 1$
\item If a system of equations consists of a parabola and a circle, it can have at most 3 solutions
\item $\frac{\log_b{x}}{\log_b{y}} = \log_b x - \log_b y$
\end{enumerate}



\end{enumerate}



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