\hfill \hfill } %} \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}[theorem]{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}} \input{tcilatex} \geometry{left=0.6in,right=0.7in,top=0.7in,bottom=0.9in} \pagestyle{fancy} \lhead{\color{blue} } \lfoot{\small \copyright $\;$ copyright Hidegkuti, Powell, 2007} \cfoot{} \chead{\Large Solutions of Equations} \rhead{page \thepage\\\color{red}\large Solutions} \textwidth 7.0in \textheight 9in \setlength{\headheight}{35pt} \begin{document} {\Large Part 1. Sample Problems} \bigskip \begin{enumerate} \item Consider the equation $2x^{2}+x+34=21x-8$. \begin{enumerate} \item Is the number $1$ a solution of the equation?$~~~$% %TCIMACRO{\TeXButton{red}{\color{red}}}% %BeginExpansion \color{red}% %EndExpansion no, since $% %TCIMACRO{\TeXButton{red}{\color{red}}}% %BeginExpansion \color{red}% %EndExpansion 37\not=13$% %TCIMACRO{\TeXButton{black}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion Solution: \ We need to substitute $1$ for $x$ into both the left-hand side and right-hand side of the equation and evaluate those algebraic expressions to see whether the left-hand side equals to the right-hand side. The left-hand side:% \begin{eqnarray*} \text{LHS} &=&2x^{2}+x+34=2\left( 1\right) ^{2}+\left( 1\right) +34 \\ &=&2\cdot 1+1+34=2+1+34=3+34=37 \end{eqnarray*}% The right-hand side:% \begin{equation*} \text{RHS}=21x-8=21\left( 1\right) -8=21-8=13 \end{equation*}% Since the two expressions are not equal when $x=1$, \ $1$ is NOT a solution of the equation. \item Is $3$ a solution of the equation?$~~~$% %TCIMACRO{\TeXButton{red}{\color{red}}}% %BeginExpansion \color{red}% %EndExpansion yes, since $% %TCIMACRO{\TeXButton{red}{\color{red}}}% %BeginExpansion \color{red}% %EndExpansion 55=55$% %TCIMACRO{\TeXButton{black}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion Solution: \ We need to substitute $3$ for $x$ into both the left-hand side and right-hand side of the equation and evaluate those algebraic expressions to see whether the left-hand side equals to the right-hand side. The left-hand side:% \begin{eqnarray*} \text{LHS} &=&2x^{2}+x+34=2\left( 3\right) ^{2}+\left( 3\right) +34 \\ &=&2\cdot 9+3+34=18+3+34=21+34=55 \end{eqnarray*}% The right-hand side:% \begin{equation*} \text{RHS}=21x-8=21\left( 3\right) -8=63-8=55 \end{equation*}% Since the two expressions are equal when $x=3$, \ $3$ IS a solution of the equation. \item Is $x=4$ a solution of the equation?$~~~$% %TCIMACRO{\TeXButton{red}{\color{red}}}% %BeginExpansion \color{red}% %EndExpansion no, since $% %TCIMACRO{\TeXButton{red}{\color{red}}}% %BeginExpansion \color{red}% %EndExpansion 70\not=76$% %TCIMACRO{\TeXButton{black}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion Solution: \ We need to substitute $4$ for $x$ into both the left-hand side and right-hand side of the equation and evaluate those algebraic expressions to see whether the left-hand side equals to the right-hand side. The left-hand side:% \begin{eqnarray*} \text{LHS} &=&2x^{2}+x+34=2\left( 4\right) ^{2}+\left( 4\right) +34 \\ &=&2\cdot 16+4+34=32+4+34=36+34=70 \end{eqnarray*}% The right-hand side:% \begin{equation*} \text{RHS}=21x-8=21\left( 4\right) -8=84-8=76 \end{equation*}% Since the two expressions are not equal when $x=4$, \ $4$ is NOT a solution of the equation. \item Is $7$ a solution of the equation?$~~~$% %TCIMACRO{\TeXButton{red}{\color{red}}}% %BeginExpansion \color{red}% %EndExpansion yes, since $% %TCIMACRO{\TeXButton{red}{\color{red}}}% %BeginExpansion \color{red}% %EndExpansion 139=139$% %TCIMACRO{\TeXButton{black}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion Solution: \ We need to substitute $7$ for $x$ into both the left-hand side and right-hand side of the equation and evaluate those algebraic expressions to see whether the left-hand side equals to the right-hand side. The left-hand side:% \begin{eqnarray*} \text{LHS} &=&2x^{2}+x+34=2\left( 7\right) ^{2}+\left( 7\right) +34 \\ &=&2\cdot 49+7+34=98+7+34=105+34=139 \end{eqnarray*}% The right-hand side:% \begin{equation*} \text{RHS}=21x-8=21\left( 7\right) -8=147-8=139 \end{equation*}% Since the two expressions are equal when $x=7$, \ \ $7$ \ IS a solution of the equation. \end{enumerate} \item Consider the equation \ $3a-2b-1=\left( a-b\right) ^{2}+4$. \begin{enumerate} \item Is the pair of numbers $a=8$ and $b=5$ a solution of the equation?$~~~$% %TCIMACRO{\TeXButton{red}{\color{red}}}% %BeginExpansion \color{red}% %EndExpansion yes, since $% %TCIMACRO{\TeXButton{red}{\color{red}}}% %BeginExpansion \color{red}% %EndExpansion 13=13$% %TCIMACRO{\TeXButton{black}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion Solution: \ We need to substitute $a=8$ \ \ and $b=5$ into both the left-hand side and right-hand side of the equation and evaluate those algebraic expressions to see whether the left-hand side equals to the right-hand side. The left-hand side:% \begin{eqnarray*} \text{LHS} &=&3a-2b-1=3\left( 8\right) -2\left( 5\right) -1=3\cdot 8-2\cdot 5-1 \\ &=&24-2\cdot 5-1=24-10-1=14-1=13 \end{eqnarray*}% The right-hand side:% \begin{equation*} \text{RHS}=\left( a-b\right) ^{2}+4=\left( \left( 8\right) -\left( 5\right) \right) ^{2}+4=\left( 8-5\right) ^{2}+4=3^{2}+4=9+4=13 \end{equation*}% Since the two expressions are equal when $a=8$ \ \ and $b=5$, \ this pair IS a solution of the equation. \item Is the pair of numbers $a=10$ and $b=7$ a solution of the equation?$% ~~~ $% %TCIMACRO{\TeXButton{red}{\color{red}}}% %BeginExpansion \color{red}% %EndExpansion no, since $% %TCIMACRO{\TeXButton{red}{\color{red}}}% %BeginExpansion \color{red}% %EndExpansion 15\not=13$% %TCIMACRO{\TeXButton{black}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion Solution: \ We need to substitute $a=10$ \ \ and $b=7$ into both the left-hand side and right-hand side of the equation and evaluate those algebraic expressions to see whether the left-hand side equals to the right-hand side. The left-hand side:% \begin{eqnarray*} \text{LHS} &=&3a-2b-1=3\left( 10\right) -2\left( 7\right) -1=3\cdot 10-2\cdot 7-1 \\ &=&30-2\cdot 7-1=30-14-1=16-1=15 \end{eqnarray*}% The right-hand side:% \begin{equation*} \text{RHS}=\left( a-b\right) ^{2}+4=\left( \left( 10\right) -\left( 7\right) \right) ^{2}+4=\left( 10-7\right) ^{2}+4=3^{2}+4=9+4=13 \end{equation*}% Since the two expressions are not equal when $a=8$ \ \ and $b=5$, \ this pair is NOT a solution of the equation.\bigskip \bigskip \end{enumerate} \end{enumerate} {\Large Part 2. Practice Problems}\bigskip \begin{enumerate} \item Consider the equation $\dfrac{2x^{2}-11x-21}{2x+3}=\allowbreak 3x-\left( 2x+7\right) $. \begin{enumerate} \item Is the number $8$ a solution of the equation?$~~~$% %TCIMACRO{\TeXButton{red}{\color{red}}}% %BeginExpansion \color{red}% %EndExpansion yes, since $% %TCIMACRO{\TeXButton{red}{\color{red}}}% %BeginExpansion \color{red}% %EndExpansion 1=1$% %TCIMACRO{\TeXButton{black}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion \item Is $13$ a solution of the equation?$~~~$% %TCIMACRO{\TeXButton{red}{\color{red}}}% %BeginExpansion \color{red}% %EndExpansion yes, since $% %TCIMACRO{\TeXButton{red}{\color{red}}}% %BeginExpansion \color{red}% %EndExpansion 6=6$% %TCIMACRO{\TeXButton{black}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion \item Is $x=10$ a solution of the equation?$~~~$% %TCIMACRO{\TeXButton{red}{\color{red}}}% %BeginExpansion \color{red}% %EndExpansion yes, since $% %TCIMACRO{\TeXButton{red}{\color{red}}}% %BeginExpansion \color{red}% %EndExpansion 3=3$% %TCIMACRO{\TeXButton{black}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion \end{enumerate} \item Consider the equation $y=\dfrac{5x-3}{2}$. \begin{enumerate} \item Is the pair of numbers $x=1$ and $y=1$ a solution of the equation?$~~~$% %TCIMACRO{\TeXButton{red}{\color{red}}}% %BeginExpansion \color{red}% %EndExpansion yes, since $% %TCIMACRO{\TeXButton{red}{\color{red}}}% %BeginExpansion \color{red}% %EndExpansion 1=1$% %TCIMACRO{\TeXButton{black}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion \item Is the pair of numbers $x=9$ and $y=4$ a solution of the equation?$~~~$% %TCIMACRO{\TeXButton{red}{\color{red}}}% %BeginExpansion \color{red}% %EndExpansion no, since $% %TCIMACRO{\TeXButton{red}{\color{red}}}% %BeginExpansion \color{red}% %EndExpansion 21\not=4$% %TCIMACRO{\TeXButton{black}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion \item Is the pair of numbers $x=3$ and $y=6$ a solution of the equation?$~~~$% %TCIMACRO{\TeXButton{red}{\color{red}}}% %BeginExpansion \color{red}% %EndExpansion yes, since $% %TCIMACRO{\TeXButton{red}{\color{red}}}% %BeginExpansion \color{red}% %EndExpansion 6=6$% %TCIMACRO{\TeXButton{black}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion \item Is the pair of numbers $x=17$ and $y=41$ a solution of the equation?$% ~~~$% %TCIMACRO{\TeXButton{red}{\color{red}}}% %BeginExpansion \color{red}% %EndExpansion yes, since $% %TCIMACRO{\TeXButton{red}{\color{red}}}% %BeginExpansion \color{red}% %EndExpansion 41=41$% %TCIMACRO{\TeXButton{black}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion \end{enumerate} \item Consider the equation \ $\left( p-q\right) ^{2}+\dfrac{3p-1}{6-q}% =4\left( p+1\right) $. \begin{enumerate} \item Is the pair of numbers $p=8$ and $q=5$ a solution of the equation?$~~~$% %TCIMACRO{\TeXButton{red}{\color{red}}}% %BeginExpansion \color{red}% %EndExpansion no, since $% %TCIMACRO{\TeXButton{red}{\color{red}}}% %BeginExpansion \color{red}% %EndExpansion 32\not=36$% %TCIMACRO{\TeXButton{black}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion \item Is the pair of numbers $p=7$ and $q=1$ a solution of the equation?$~~~$% %TCIMACRO{\TeXButton{red}{\color{red}}}% %BeginExpansion \color{red}% %EndExpansion yes, since $% %TCIMACRO{\TeXButton{red}{\color{red}}}% %BeginExpansion \color{red}% %EndExpansion 40=40$% %TCIMACRO{\TeXButton{black}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion \end{enumerate} \end{enumerate} \end{document}