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\newtheorem{theorem}{Theorem}
\newtheorem{acknowledgement}[theorem]{Acknowledgement}
\newtheorem{algorithm}[theorem]{Algorithm}
\newtheorem{axiom}[theorem]{Axiom}
\newtheorem{case}[theorem]{Case}
\newtheorem{claim}[theorem]{Claim}
\newtheorem{conclusion}[theorem]{Conclusion}
\newtheorem{condition}[theorem]{Condition}
\newtheorem{conjecture}[theorem]{Conjecture}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{criterion}[theorem]{Criterion}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}
\newtheorem{exercise}{Exercise}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{notation}[theorem]{Notation}
\newtheorem{problem}[theorem]{Problem}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{solution}[theorem]{Solution}
\newtheorem{summary}[theorem]{Summary}
\newenvironment{proof}[1][Proof]{\noindent\textbf{#1.} }{\ \rule{0.5em}{0.5em}}
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\begin{definition}
Let an equation in $x$ and $y$ be given. \ The graph of the equation in $x$
and $y$ is the set of all points $P\left( x,y\right) $ \ whose coordinates
are solutions of the equation.
\end{definition}

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\begin{definition}
Let an equation in $x$ and $y$ be given. \ The graph of the equation in $x$
and $y$ is the set of all points $P\left( x,y\right) $ \ whose coordinates
are solutions of the equation.
\end{definition}

\bigskip 

Equation: \ $7-x=2y+1$.

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\begin{definition}
Let an equation in $x$ and $y$ be given. \ The graph of the equation in $x$
and $y$ is the set of all points $P\left( x,y\right) $ \ whose coordinates
are solutions of the equation.
\end{definition}

\bigskip 

Equation: \ $7-x=2y+1$.\bigskip 

The graph of this equation is the set of all points $P\left( x,y\right) $,
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\begin{definition}
Let an equation in $x$ and $y$ be given. \ The graph of the equation in $x$
and $y$ is the set of all points $P\left( x,y\right) $ \ whose coordinates
are solutions of the equation.
\end{definition}

\bigskip 

Equation: \ $7-x=2y+1$.\bigskip 

The graph of this equation is the set of all points $P\left( x,y\right) $,
for which the statement $7-x=2y+1$ \ is true.\bigskip 

Consider $A\left( -2,4\right) $. \ Is this point on the graph of the
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\begin{definition}
Let an equation in $x$ and $y$ be given. \ The graph of the equation in $x$
and $y$ is the set of all points $P\left( x,y\right) $ \ whose coordinates
are solutions of the equation.
\end{definition}

\bigskip 

Equation: \ $7-x=2y+1$.\bigskip 

The graph of this equation is the set of all points $P\left( x,y\right) $,
for which the statement $7-x=2y+1$ \ is true.\bigskip 

Consider $A\left( -2,4\right) $. \ Is this point on the graph of the
equation $7-x=2y+1$?\bigskip 

Is \ $x=-2$ \ and \ $y=4$ a solution of the equation \ $7-x=2y+1$?\pagebreak 

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\begin{definition}
Let an equation in $x$ and $y$ be given. \ The graph of the equation in $x$
and $y$ is the set of all points $P\left( x,y\right) $ \ whose coordinates
are solutions of the equation.
\end{definition}

\bigskip 

Equation: \ $7-x=2y+1$.\bigskip 

The graph of this equation is the set of all points $P\left( x,y\right) $,
for which the statement $7-x=2y+1$ \ is true.\bigskip 

Consider $A\left( -2,4\right) $. \ Is this point on the graph of the
equation $7-x=2y+1$?\bigskip 

Is \ $x=-2$ \ and \ $y=4$ a solution of the equation \ $7-x=2y+1$?%
\begin{equation*}
\text{LHS}=7-\left( -2\right) =7+2=9
\end{equation*}

\pagebreak 

\begin{definition}
Let an equation in $x$ and $y$ be given. \ The graph of the equation in $x$
and $y$ is the set of all points $P\left( x,y\right) $ \ whose coordinates
are solutions of the equation.
\end{definition}

\bigskip 

Equation: \ $7-x=2y+1$.\bigskip 

The graph of this equation is the set of all points $P\left( x,y\right) $,
for which the statement $7-x=2y+1$ \ is true.\bigskip 

Consider $A\left( -2,4\right) $. \ Is this point on the graph of the
equation $7-x=2y+1$?

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\begin{eqnarray*}
\text{LHS} &=&7-\left( -2\right) =7+2=9 \\
\text{RHS} &=&2\left( 4\right) +1=9
\end{eqnarray*}%
\pagebreak 

\begin{definition}
Let an equation in $x$ and $y$ be given. \ The graph of the equation in $x$
and $y$ is the set of all points $P\left( x,y\right) $ \ whose coordinates
are solutions of the equation.
\end{definition}

\bigskip 

Equation: \ $7-x=2y+1$.\bigskip 

The graph of this equation is the set of all points $P\left( x,y\right) $,
for which the statement $7-x=2y+1$ \ is true.\bigskip 

Consider $A\left( -2,4\right) $. \ Is this point on the graph of the
equation $7-x=2y+1$?

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\begin{eqnarray*}
\text{LHS} &=&7-\left( -2\right) =7+2=9 \\
\text{RHS} &=&2\left( 4\right) +1=9
\end{eqnarray*}

$x=-2$ \ $y=4$ \ is \ a solution.Since the coordinates are a solution of the
equation, $\left( -2,4\right) $ \ is on the graph of the equation $7-x=2y+1$%
.\pagebreak 

\begin{definition}
Let an equation in $x$ and $y$ be given. \ The graph of the equation in $x$
and $y$ is the set of all points $P\left( x,y\right) $ \ whose coordinates
are solutions of the equation.
\end{definition}

\bigskip 

Equation: \ $7-x=2y+1$.\bigskip 

The graph of this equation is the set of all points $P\left( x,y\right) $,
for which the statement $7-x=2y+1$ \ is true.\bigskip 

Consider $A\left( -2,4\right) $. \ Is this point on the graph of the
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\begin{eqnarray*}
\text{LHS} &=&7-\left( -2\right) =7+2=9 \\
\text{RHS} &=&2\left( 4\right) +1=9
\end{eqnarray*}%
Since the coordinates are a solution of the equation, $\left( -2,4\right) $
\ is on the graph of the equation $7-x=2y+1$.\bigskip 

\bigskip 

\bigskip 

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\pagebreak 

\begin{definition}
Let an equation in $x$ and $y$ be given. \ The graph of the equation in $x$
and $y$ is the set of all points $P\left( x,y\right) $ \ whose coordinates
are solutions of the equation.
\end{definition}

\bigskip 

Equation: \ $7-x=2y+1$.\bigskip 

The graph of this equation is the set of all points $P\left( x,y\right) $,
for which the statement $7-x=2y+1$ \ is true.\bigskip 

Consider $A\left( -2,4\right) $.

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\bigskip 

Let an equation in $x$ and $y$ be given. \ The graph of the equation in $x$
and $y$ is the set of all points $P\left( x,y\right) $ \ whose coordinates
are solutions of the equation.

\bigskip 

Equation: \ $7-x=2y+1$.

The graph of this equation is the set of all points $P\left( x,y\right) $,
for which the statement $7-x=2y+1$ \ is true.\bigskip 

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\begin{definition}
%TCIMACRO{\TeXButton{Large}{\Large}}%
%BeginExpansion
\Large%
%EndExpansion
Let an equation in $x$ and $y$ be given. \ The graph of the equation in $x$
and $y$ is the set of all points $P\left( x,y\right) $ \ whose coordinates
are solutions of the equation.
\end{definition}

\bigskip 

Equation: \ $7-x=2y+1$.

The graph of this equation is the set of all points $P\left( x,y\right) $,
for which the statement $7-x=2y+1$ \ is true.

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The point $A\left( -1,4\right) $ \ is on the graph.

\begin{enumerate}
\item 

\item 

\item 

\item 

\item Consider the graph shown below. \FRAME{dtbpF}{3.5129in}{3.5129in}{0pt}{%
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points \ $A\left( -4,1\right) $,\ \ $B\left( 2,4\right) $, \ $C\left(
4,5\right) $, \ and \ $D\left( -1,-5\right) $ \ are on the graph.\bigskip 

Determine which of the following equations belongs to this graph.\bigskip 

a) \ $8y=5x^{2}+14x-16$

b) \ $2y-3=x+3$

c) \ $6\left( 1-y\right) =\left( x-1\right) \left( x^{2}-x-20\right) $%
\pagebreak 
\end{enumerate}

a) \ Consider the equation \ $8y=5x^{2}+14x-16$ \ and points $A\left(
-4,1\right) $,\ \ $B\left( 2,4\right) $, \ and \ $C\left( 4,5\right) .$%
\bigskip \bigskip

Is the point $A\left( -4,1\right) $ \ on the graph of \ $8y=5x^{2}+14x-16$?

Is the point $B\left( 2,4\right) $ \ on the graph of \ $8y=5x^{2}+14x-16$?

Is the point $C\left( 4,5\right) $ \ on the graph of \ $8y=5x^{2}+14x-16$%
?\bigskip

We start with point $A\left( -4,1\right) $. \ Is this point on the graph of $%
8y=5x^{2}+14x-16$?\bigskip

\pagebreak

a) \ Consider the equation \ $8y=5x^{2}+14x-16$ \ and points $A\left(
-4,1\right) $,\ \ $B\left( 2,4\right) $, \ and \ $C\left( 4,5\right) .$%
\bigskip \bigskip

Is the point $A\left( -4,1\right) $ \ on the graph of \ $8y=5x^{2}+14x-16$?

Is the point $B\left( 2,4\right) $ \ on the graph of \ $8y=5x^{2}+14x-16$?

Is the point $C\left( 4,5\right) $ \ on the graph of \ $8y=5x^{2}+14x-16$%
?\bigskip

We start with point $A\left( -4,1\right) $. \ Is this point on the graph of $%
8y=5x^{2}+14x-16$?\bigskip \bigskip

Is $x=-4$ \ and $y=1$ \ a solution of \ $8y=5x^{2}+14x-16$?\pagebreak

a) \ Consider the equation \ $8y=5x^{2}+14x-16$ \ and points $A\left(
-4,1\right) $,\ \ $B\left( 2,4\right) $, \ and \ $C\left( 4,5\right) .$%
\bigskip \bigskip

Is the point $A\left( -4,1\right) $ \ on the graph of \ $8y=5x^{2}+14x-16$?

Is the point $B\left( 2,4\right) $ \ on the graph of \ $8y=5x^{2}+14x-16$?

Is the point $C\left( 4,5\right) $ \ on the graph of \ $8y=5x^{2}+14x-16$%
?\bigskip

We start with point $A\left( -4,1\right) $. \ Is this point on the graph of $%
8y=5x^{2}+14x-16$?\bigskip \bigskip

Is $x=-4$ \ and $y=1$ \ a solution of \ $8y=5x^{2}+14x-16$?\bigskip

\begin{equation*}
\text{LHS}=8\left( 1\right) =8
\end{equation*}

\pagebreak

a) \ Consider the equation \ $8y=5x^{2}+14x-16$ \ and points $A\left(
-4,1\right) $,\ \ $B\left( 2,4\right) $, \ and \ $C\left( 4,5\right) .$%
\bigskip \bigskip

Is the point $A\left( -4,1\right) $ \ on the graph of \ $8y=5x^{2}+14x-16$?

Is the point $B\left( 2,4\right) $ \ on the graph of \ $8y=5x^{2}+14x-16$?

Is the point $C\left( 4,5\right) $ \ on the graph of \ $8y=5x^{2}+14x-16$%
?\bigskip

We start with point $A\left( -4,1\right) $. \ Is this point on the graph of $%
8y=5x^{2}+14x-16$?\bigskip \bigskip

Is $x=-4$ \ and $y=1$ \ a solution of \ $8y=5x^{2}+14x-16$?\bigskip

\begin{eqnarray*}
\text{LHS} &=&8\left( 1\right) =8 \\
\text{RHS} &=&5\left( -4\right) ^{2}+14\left( -4\right) -16=5\cdot
16+14\left( -4\right) -16 \\
&=&80-56-16=24-16=8
\end{eqnarray*}%
\bigskip

\pagebreak

a) \ Consider the equation \ $8y=5x^{2}+14x-16$ \ and points $A\left(
-4,1\right) $,\ \ $B\left( 2,4\right) $, \ and \ $C\left( 4,5\right) .$%
\bigskip \bigskip 

Is the point $A\left( -4,1\right) $ \ on the graph of \ $8y=5x^{2}+14x-16$?

Is the point $B\left( 2,4\right) $ \ on the graph of \ $8y=5x^{2}+14x-16$?

Is the point $C\left( 4,5\right) $ \ on the graph of \ $8y=5x^{2}+14x-16$%
?\bigskip

We start with point $A\left( -4,1\right) $. \ Is this point on the graph of $%
8y=5x^{2}+14x-16$?\bigskip \bigskip

Is $x=-4$ \ and $y=1$ \ a solution of \ $8y=5x^{2}+14x-16$?\bigskip

\begin{eqnarray*}
\text{LHS} &=&8\left( 1\right) =8 \\
\text{RHS} &=&5\left( -4\right) ^{2}+14\left( -4\right) -16=5\cdot
16+14\left( -4\right) -16 \\
&=&80-56-16=24-16=8
\end{eqnarray*}%
\bigskip

Since LHS$~=~$RHS$,$ \ the point $A\left( -4,1\right) $ \ is on the graph of 
$\ 8y=5x^{2}+14x-16$.\pagebreak 

a) \ Consider the equation \ $8y=5x^{2}+14x-16$ \ and points $A\left(
-4,1\right) $,\ \ $B\left( 2,4\right) $, \ and \ $C\left( 4,5\right) .$%
\bigskip \bigskip

Is the point $A\left( -4,1\right) $ \ on the graph of \ $8y=5x^{2}+14x-16$%
?~~~%
%TCIMACRO{\TeXButton{red}{\color{red}}}%
%BeginExpansion
\color{red}%
%EndExpansion
yes%
%TCIMACRO{\TeXButton{black}{\color{black}}}%
%BeginExpansion
\color{black}%
%EndExpansion

Is the point $B\left( 2,4\right) $ \ on the graph of \ $8y=5x^{2}+14x-16$?

Is the point $C\left( 4,5\right) $ \ on the graph of \ $8y=5x^{2}+14x-16$%
?\bigskip

\pagebreak 

a) \ Consider the equation \ $8y=5x^{2}+14x-16$ \ and points $A\left(
-4,1\right) $,\ \ $B\left( 2,4\right) $, \ and \ $C\left( 4,5\right) .$%
\bigskip \bigskip

Is the point $A\left( -4,1\right) $ \ on the graph of \ $8y=5x^{2}+14x-16$%
?~~~%
%TCIMACRO{\TeXButton{red}{\color{red}}}%
%BeginExpansion
\color{red}%
%EndExpansion
yes%
%TCIMACRO{\TeXButton{black}{\color{black}}}%
%BeginExpansion
\color{black}%
%EndExpansion

Is the point $B\left( 2,4\right) $ \ on the graph of \ $8y=5x^{2}+14x-16$?

Is the point $C\left( 4,5\right) $ \ on the graph of \ $8y=5x^{2}+14x-16$%
?\bigskip

We now consider $B$.\bigskip

\pagebreak 

a) \ Consider the equation \ $8y=5x^{2}+14x-16$ \ and points $A\left(
-4,1\right) $,\ \ $B\left( 2,4\right) $, \ and \ $C\left( 4,5\right) .$%
\bigskip \bigskip 

Is the point $A\left( -4,1\right) $ \ on the graph of \ $8y=5x^{2}+14x-16$%
?~~~%
%TCIMACRO{\TeXButton{red}{\color{red}}}%
%BeginExpansion
\color{red}%
%EndExpansion
yes%
%TCIMACRO{\TeXButton{black}{\color{black}}}%
%BeginExpansion
\color{black}%
%EndExpansion

Is the point $B\left( 2,4\right) $ \ on the graph of \ $8y=5x^{2}+14x-16$?

Is the point $C\left( 4,5\right) $ \ on the graph of \ $8y=5x^{2}+14x-16$%
?\bigskip

We now consider $B$.\bigskip

Is the point $B\left( 2,4\right) $ \ on the graph of \ $8y=5x^{2}+14x-16$%
?\bigskip 

\pagebreak 

a) \ Consider the equation \ $8y=5x^{2}+14x-16$ \ and points $A\left(
-4,1\right) $,\ \ $B\left( 2,4\right) $, \ and \ $C\left( 4,5\right) .$%
\bigskip \bigskip 

Is the point $A\left( -4,1\right) $ \ on the graph of \ $8y=5x^{2}+14x-16$%
?~~~%
%TCIMACRO{\TeXButton{red}{\color{red}}}%
%BeginExpansion
\color{red}%
%EndExpansion
yes%
%TCIMACRO{\TeXButton{black}{\color{black}}}%
%BeginExpansion
\color{black}%
%EndExpansion

Is the point $B\left( 2,4\right) $ \ on the graph of \ $8y=5x^{2}+14x-16$?

Is the point $C\left( 4,5\right) $ \ on the graph of \ $8y=5x^{2}+14x-16$%
?\bigskip

We now consider $B$.\bigskip

Is the point $B\left( 2,4\right) $ \ on the graph of \ $8y=5x^{2}+14x-16$%
?\bigskip 

Is $x=2$ \ and $y=4$. a solution of $8y=5x^{2}+14x-16$?\bigskip 

\pagebreak 

a) \ Consider the equation \ $8y=5x^{2}+14x-16$ \ and points $A\left(
-4,1\right) $,\ \ $B\left( 2,4\right) $, \ and \ $C\left( 4,5\right) .$%
\bigskip \bigskip 

Is the point $A\left( -4,1\right) $ \ on the graph of \ $8y=5x^{2}+14x-16$%
?~~~%
%TCIMACRO{\TeXButton{red}{\color{red}}}%
%BeginExpansion
\color{red}%
%EndExpansion
yes%
%TCIMACRO{\TeXButton{black}{\color{black}}}%
%BeginExpansion
\color{black}%
%EndExpansion

Is the point $B\left( 2,4\right) $ \ on the graph of \ $8y=5x^{2}+14x-16$?

Is the point $C\left( 4,5\right) $ \ on the graph of \ $8y=5x^{2}+14x-16$%
?\bigskip

We now consider $B$.\bigskip

Is the point $B\left( 2,4\right) $ \ on the graph of \ $8y=5x^{2}+14x-16$%
?\bigskip 

Is $x=2$ \ and $y=4$. a solution of $8y=5x^{2}+14x-16$?\bigskip 
\begin{equation*}
\text{LHS}=8\left( 4\right) =32
\end{equation*}

\pagebreak 

a) \ Consider the equation \ $8y=5x^{2}+14x-16$ \ and points $A\left(
-4,1\right) $,\ \ $B\left( 2,4\right) $, \ and \ $C\left( 4,5\right) .$%
\bigskip \bigskip 

Is the point $A\left( -4,1\right) $ \ on the graph of \ $8y=5x^{2}+14x-16$%
?~~~%
%TCIMACRO{\TeXButton{red}{\color{red}}}%
%BeginExpansion
\color{red}%
%EndExpansion
yes%
%TCIMACRO{\TeXButton{black}{\color{black}}}%
%BeginExpansion
\color{black}%
%EndExpansion

Is the point $B\left( 2,4\right) $ \ on the graph of \ $8y=5x^{2}+14x-16$?

Is the point $C\left( 4,5\right) $ \ on the graph of \ $8y=5x^{2}+14x-16$%
?\bigskip

We now consider $B$.\bigskip

Is the point $B\left( 2,4\right) $ \ on the graph of \ $8y=5x^{2}+14x-16$%
?\bigskip 

Is $x=2$ \ and $y=4$. a solution of $8y=5x^{2}+14x-16$?\bigskip 
\begin{eqnarray*}
\text{LHS} &=&8\left( 4\right) =32 \\
\text{RHS} &=&5\left( 2\right) ^{2}+14\left( 2\right) -16=5\cdot 4+14\cdot
2-16=20+28-16 \\
&=&48-16=32
\end{eqnarray*}%
\pagebreak 

a) \ Consider the equation \ $8y=5x^{2}+14x-16$ \ and points $A\left(
-4,1\right) $,\ \ $B\left( 2,4\right) $, \ and \ $C\left( 4,5\right) .$%
\bigskip \bigskip 

Is the point $A\left( -4,1\right) $ \ on the graph of \ $8y=5x^{2}+14x-16$%
?~~~%
%TCIMACRO{\TeXButton{red}{\color{red}}}%
%BeginExpansion
\color{red}%
%EndExpansion
yes%
%TCIMACRO{\TeXButton{black}{\color{black}}}%
%BeginExpansion
\color{black}%
%EndExpansion

Is the point $B\left( 2,4\right) $ \ on the graph of \ $8y=5x^{2}+14x-16$?

Is the point $C\left( 4,5\right) $ \ on the graph of \ $8y=5x^{2}+14x-16$%
?\bigskip

We now consider $B$.\bigskip

Is the point $B\left( 2,4\right) $ \ on the graph of \ $8y=5x^{2}+14x-16$%
?\bigskip 

Is $x=2$ \ and $y=4$. a solution of $8y=5x^{2}+14x-16$?\bigskip 
\begin{eqnarray*}
\text{LHS} &=&8\left( 4\right) =32 \\
\text{RHS} &=&5\left( 2\right) ^{2}+14\left( 2\right) -16=5\cdot 4+14\cdot
2-16=20+28-16 \\
&=&48-16=32
\end{eqnarray*}%
Since LHS$~=~$RHS$,$ \ the point $B\left( 2,4\right) $ \ is on the graph of $%
8y=5x^{2}+14x-16$.\pagebreak 

Is the point $C\left( 4,5\right) $ \ on the graph of \ $8y=5x^{2}+14x-16$?

Solution: \ We need to determine whether the coordinates of $A$ are a
solution of the equation or not. \ Let $x=4$ \ and $y=5$. \ Is this pair a
solution of $8y=5x^{2}+14x-16$?%
\begin{eqnarray*}
\text{LHS} &=&8\left( 5\right) =40 \\
\text{RHS} &=&5\left( 4\right) ^{2}+14\left( 4\right) -16=5\cdot 16+14\cdot
4-16=80+56-16 \\
&=&136-16=120
\end{eqnarray*}%
Since LHS$~\not=~$RHS, \ the point $C\left( 4,5\right) $ \ is NOT on the
graph of $8y=5x^{2}+14x-16$.

Is it possible that the graph is that of \ $8y=5x^{2}+14x-16$? \ 

\begin{enumerate}
\item 
\begin{enumerate}
\item 
\begin{enumerate}
\item 
No, since $C$ is not on the graph of the equation of $8y=5x^{2}+14x-16$.
\end{enumerate}
\end{enumerate}
\end{enumerate}

b) \ Consider the equation \ \ $2y-3=x+3$.

\begin{enumerate}
\item 
\begin{enumerate}
\item 
\begin{enumerate}
\item Is the point $A\left( -4,1\right) $ \ on the graph of \ $2y-3=x+3$?

Solution: \ We need to determine whether the coordinates of $A$ are a
solution of the equation or not. \ Let $x=-4$ \ and $y=1$. \ Is this pair a
solution of $\ 2y-3=x+3$?%
\begin{eqnarray*}
\text{LHS} &=&2\left( 1\right) -3=2-3=-1 \\
\text{RHS} &=&-4+3=-1
\end{eqnarray*}%
Since LHS$~=~$RHS$,$ \ the point $A\left( -4,1\right) $ \ is on the graph of 
$2y-3=x+3$.

\item Is the point $A\left( 2,4\right) $ \ on the graph of \ $2y-3=x+3$?

Solution: \ We need to determine whether the coordinates of $A$ are a
solution of the equation or not. \ Let $x=2$ \ and $y=4$. \ Is this pair a
solution of $2y-3=x+3$?%
\begin{eqnarray*}
\text{LHS} &=&2\left( 4\right) -3=8-3=5 \\
\text{RHS} &=&2+3=5
\end{eqnarray*}%
Since LHS$~=~$RHS$,$ \ the point $B\left( 2,4\right) $ \ is on the graph of $%
2y-3=x+3$.

\item Is the point $C\left( 4,5\right) $ \ on the graph of \ $2y-3=x+3$?

Solution: \ We need to determine whether the coordinates of $A$ are a
solution of the equation or not. \ Let $x=4$ \ and $y=5$. \ Is this pair a
solution of $2y-3=x+3$?%
\begin{eqnarray*}
\text{LHS} &=&2\left( 5\right) -3=10-3=7 \\
\text{RHS} &=&4+3=7
\end{eqnarray*}%
Since LHS$~=~$RHS, \ the point $C\left( 4,5\right) $ \ is on the graph of $%
2y-3=x+3$.

\item Is it possible that the graph is that of \ $2y-3=x+3$? \ 

Based on the points $A,$ $B$, and $C$, the answer is yes, since $A,$ $B,$
and $C$ are all on the graph of the equation $2y-3=x+3$.
\end{enumerate}

\item Consider the equation \ $6\left( 1-y\right) =\left( x-1\right) \left(
x^{2}-x-20\right) $.

\begin{enumerate}
\item Is the point $A\left( -4,1\right) $ \ on the graph of \ $6\left(
1-y\right) =\left( x-1\right) \left( x^{2}-x-20\right) $?

Solution: \ We need to determine whether the coordinates of $A$ are a
solution of the equation or not. \ Let $x=-4$ \ and $y=1$. \ Is this pair a
solution of $\ 6\left( 1-y\right) =\left( x-1\right) \left(
x^{2}-x-20\right) $?%
\begin{eqnarray*}
\text{LHS} &=&6\left( 1-1\right) =6\left( 0\right) =0 \\
\text{RHS} &=&\left( \left( -4\right) -1\right) \left( \left( -4\right)
^{2}-\left( -4\right) -20\right) =-5\left( 16+4-20\right) =-5\left(
20-20\right) \\
&=&-5\cdot 0=0
\end{eqnarray*}%
Since LHS$~=~$RHS$,$ \ the point $A\left( -4,1\right) $ \ is on the graph of 
$6\left( 1-y\right) =\left( x-1\right) \left( x^{2}-x-20\right) $.

\item Is the point $A\left( 2,4\right) $ \ on the graph of \ $6\left(
1-y\right) =\left( x-1\right) \left( x^{2}-x-20\right) $?

Solution: \ We need to determine whether the coordinates of $A$ are a
solution of the equation or not. \ Let $x=2$ \ and $y=4$. \ Is this pair a
solution of $6\left( 1-y\right) =\left( x-1\right) \left( x^{2}-x-20\right) $%
?%
\begin{eqnarray*}
\text{LHS} &=&6\left( 1-4\right) =6\left( -3\right) =-18 \\
\text{RHS} &=&\left( 2-1\right) \left( 2^{2}-2-20\right) =1\left(
4-2-20\right) =1\left( 2-20\right) \\
&=&1\left( -18\right) =-18
\end{eqnarray*}%
Since LHS$~=~$RHS$,$ \ the point $B\left( 2,4\right) $ \ is on the graph of $%
6\left( 1-y\right) =\left( x-1\right) \left( x^{2}-x-20\right) $.

\item Is the point $C\left( 4,5\right) $ \ on the graph of \ $6\left(
1-y\right) =\left( x-1\right) \left( x^{2}-x-20\right) $?

Solution: \ We need to determine whether the coordinates of $A$ are a
solution of the equation or not. \ Let $x=4$ \ and $y=5$. \ Is this pair a
solution of $6\left( 1-y\right) =\left( x-1\right) \left( x^{2}-x-20\right) $%
?%
\begin{eqnarray*}
\text{LHS} &=&6\left( 1-5\right) =6\left( -4\right) =-24 \\
\text{RHS} &=&\left( 4-1\right) \left( 4^{2}-4-20\right) =3\left(
16-4-20\right) =3\left( 12-20\right) \\
&=&3\left( -8\right) =-24
\end{eqnarray*}%
Since LHS$~=~$RHS, \ the point $C\left( 4,5\right) $ \ is on the graph of $%
6\left( 1-y\right) =\left( x-1\right) \left( x^{2}-x-20\right) $.

\item Is it possible that the graph is that of \ $6\left( 1-y\right) =\left(
x-1\right) \left( x^{2}-x-20\right) $? \ 

Based on the points $A,$ $B$, and $C$, the answer is yes, since $A,$ $B,$
and $C$ are all on the graph of the equation $6\left( 1-y\right) =\left(
x-1\right) \left( x^{2}-x-20\right) $.
\end{enumerate}
\end{enumerate}

We need to rule out one of the equations from b) and c). \ Let us test these
equations using point $D\left( -1,-5\right) $.

For part b):

Is the point $D\left( -1,-5\right) $ \ on the graph of \ $2y-3=x+3$?

Solution: \ We need to determine whether the coordinates of $A$ are a
solution of the equation or not. \ Let $x=-1$ \ and $y=-5$. \ Is this pair a
solution of $2y-3=x+3$?%
\begin{eqnarray*}
\text{LHS} &=&2\left( -5\right) -3=-10-3=-13 \\
\text{RHS} &=&-1+3=2
\end{eqnarray*}%
Since LHS$~\not=~$RHS, \ the point $D\left( -1,-5\right) $ \ is NOT on the
graph of $2y-3=x+3$. \ Consequently, \ the graph is NOT that of $2y-3=x+3$.

For part c):

Is the point $D\left( -1,-5\right) $ \ on the graph of \ $6\left( 1-y\right)
=\left( x-1\right) \left( x^{2}-x-20\right) $?

Solution: \ We need to determine whether the coordinates of $A$ are a
solution of the equation or not. \ Let $x=-1$ \ and $y=-5$. \ Is this pair a
solution of $6\left( 1-y\right) =\left( x-1\right) \left( x^{2}-x-20\right) $%
?%
\begin{eqnarray*}
\text{LHS} &=&6\left( 1-\left( -5\right) \right) =6\cdot 6=36 \\
\text{RHS} &=&\left( -1-1\right) \left( \left( -1\right) ^{2}-\left(
-1\right) -20\right) =-2\left( 1+1-20\right) =-2\left( 2-20\right) \\
&=&-2\left( -18\right) =36
\end{eqnarray*}%
Since LHS$~=~$RHS, \ the point $D\left( -1,-5\right) $ \ is on the graph of $%
6\left( 1-y\right) =\left( x-1\right) \left( x^{2}-x-20\right) $. \
Consequently, \ it is possible that the graph is that of $6\left( 1-y\right)
=\left( x-1\right) \left( x^{2}-x-20\right) $.
\end{enumerate}

\end{document}