\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 Evaluating Algebraic Expressions\\ with Natural Numbers} \rhead{page \thepage\\\color{red}\large Solutions} \textwidth 7.0in \textheight 9in \setlength{\headheight}{35pt} \begin{document} {\Large Part 1. Sample Problems} \bigskip Evaluate each of the algebraic expressions when $p=7$ \ and \ $q=3$. \begin{enumerate} \item $15-p=~~% %TCIMACRO{\TeXButton{red}{\color{red}}}% %BeginExpansion \color{red}% %EndExpansion 8% %TCIMACRO{\TeXButton{black}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion $ Solution: Step 1. We re-write the expression with one modification: we replace each variable by an empty pair of parenthses. Step 2. We insert the values into the parentheses. \ Now the problem becomes an order of operations problem. Step 3. We drop the unnecessary parentheses and work out the order of operations problem. \ (It may appear awkward to create these parentheses but they will later become extremely helpful.)% \begin{eqnarray*} \text{Step 1. \ \ \ \ \ \ \ \ \ \ }15-p &=&15-\left( ~~\right) \\ \text{Step 2. \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ } &=&15-\left( 7\right) \\ \text{Step 3. \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ } &=&15-7 \\ &=&8 \end{eqnarray*} \item $pq=~~% %TCIMACRO{\TeXButton{red}{\color{red}}}% %BeginExpansion \color{red}% %EndExpansion 21% %TCIMACRO{\TeXButton{black}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion $ Solution:% \begin{eqnarray*} \text{Step 1. \ \ \ \ \ \ \ \ \ \ \ \ \ \ }pq &=&\left( ~~\right) \left( ~~\right) \\ \text{Step 2. \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ } &=&\left( 7\right) \left( 3\right) \\ \text{Step 3. \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ } &=&21 \end{eqnarray*} \item $4p-q=~~% %TCIMACRO{\TeXButton{red}{\color{red}}}% %BeginExpansion \color{red}% %EndExpansion 25% %TCIMACRO{\TeXButton{black}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion $ Solution:% \begin{eqnarray*} \text{Step 1. \ \ \ \ \ \ \ \ \ \ \ \ \ \ }4p-q &=&4\left( ~~\right) -\left( ~~\right) \\ \text{Step 2. \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ } &=&4\left( 7\right) -\left( 3\right) \\ \text{Step 3.\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ } &=&4\cdot 7-3% \text{ \ \ \ \ \ \ \ \ \ \ \ \ multiplication} \\ &=&28-3\text{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ subtraction} \\ &=&25 \end{eqnarray*} \item $p-2q=~~% %TCIMACRO{\TeXButton{red}{\color{red}}}% %BeginExpansion \color{red}% %EndExpansion 1% %TCIMACRO{\TeXButton{black}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion $ Solution:% \begin{eqnarray*} \text{Step 1. \ \ \ \ \ }p-2q &=&\left( ~~\right) -2\left( ~~\right) \\ \text{Step 2.\ \ \ \ \ \ \ \ \ \ \ \ \ \ } &=&\left( 7\right) -2\left( 3\right) \\ \text{Step 3.\ \ \ \ \ \ \ \ \ \ \ \ \ \ } &=&7-2\cdot 3\text{ \ \ \ \ \ \ multiplication} \\ &=&7-6\text{ \ \ \ \ \ \ \ \ \ \ \ subtraction} \\ &=&1 \end{eqnarray*} \item $p^{2}-q^{2}=~~% %TCIMACRO{\TeXButton{red}{\color{red}}}% %BeginExpansion \color{red}% %EndExpansion 40% %TCIMACRO{\TeXButton{black}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion $ Solution: \begin{eqnarray*} p^{2}-q^{2} &=&\left( ~~\right) ^{2}-\left( ~~\right) ^{2} \\ &=&\left( 7\right) ^{2}-\left( 3\right) ^{2} \\ &=&7^{2}-3^{2}\text{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ exponents,} \\ &=&49-3^{2}\text{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ left to right} \\ &=&49-9\text{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ subtraction} \\ &=&40 \end{eqnarray*} \item $\left( p-q\right) ^{2}=~~% %TCIMACRO{\TeXButton{red}{\color{red}}}% %BeginExpansion \color{red}% %EndExpansion 16% %TCIMACRO{\TeXButton{black}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion $ Solution: \begin{eqnarray*} \left( p-q\right) ^{2} &=&\left[ \left( ~~\right) -\left( ~~\right) \right] ^{2} \\ &=&\left[ \left( 7\right) -\left( 3\right) \right] ^{2} \\ &=&\left( 7-3\right) ^{2}\text{ \ \ \ \ \ \ \ \ \ subtraction in parentheses} \\ &=&4^{2}\text{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ exponentiation} \\ &=&16 \end{eqnarray*} \item $2q^{2}=~~% %TCIMACRO{\TeXButton{red}{\color{red}}}% %BeginExpansion \color{red}% %EndExpansion 18% %TCIMACRO{\TeXButton{black}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion $ Solution: \begin{eqnarray*} 2q^{2} &=&2\left( ~~\right) ^{2} \\ &=&2\left( 3\right) ^{2} \\ &=&2\cdot 3^{2}\text{ \ \ \ \ \ \ \ \ exponentiation} \\ &=&2\cdot 9\text{ \ \ \ \ \ \ \ \ \ \ multiplication} \\ &=&18 \end{eqnarray*} \item $\left( 2q\right) ^{2}=~~% %TCIMACRO{\TeXButton{red}{\color{red}}}% %BeginExpansion \color{red}% %EndExpansion 36% %TCIMACRO{\TeXButton{black}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion $ Solution: \begin{eqnarray*} \left( 2q\right) ^{2} &=&\left[ 2\left( ~~\right) \right] ^{2} \\ &=&\left[ 2\left( 3\right) \right] ^{2} \\ &=&\left( 2\cdot 3\right) ^{2}\text{ \ \ \ \ \ \ \ \ \ \ multiplication in parentheses} \\ &=&6^{2}\text{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ exponents} \\ &=&36 \end{eqnarray*}% \pagebreak \item $15-\dfrac{p+q}{5}=~~% %TCIMACRO{\TeXButton{red}{\color{red}}}% %BeginExpansion \color{red}% %EndExpansion 13% %TCIMACRO{\TeXButton{black}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion $ Solution: \ From here on, we show computations \textbf{in the form they should appear}. \ Once you wrote down the expression with little parentheses instead of the letters, you can insert the values into it. \ (In this problem, we skipped the line $15-\dfrac{\left( ~~\right) +\left( ~~\right) }{% 5}$) \begin{eqnarray*} 15-\dfrac{p+q}{5} &=&15-\dfrac{\left( 7\right) +\left( 3\right) }{5} \\ &=&15-\dfrac{7+3}{5}\text{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ invisible parentheses!% } \\ &=&15-\dfrac{10}{5}\text{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ division} \\ &=&15-2\text{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ subtraction} \\ &=&13 \end{eqnarray*} \item $\left( p+q\right) ^{2}-\left( 5q-2p\right) ^{4}=~~% %TCIMACRO{\TeXButton{red}{\color{red}}}% %BeginExpansion \color{red}% %EndExpansion 99% %TCIMACRO{\TeXButton{black}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion $ Solution:% \begin{equation*} \left( p+q\right) ^{2}-\left( 5q-2p\right) ^{4}=\left[ \left( 7\right) +\left( 3\right) \right] ^{2}-\left[ 5\left( 3\right) -2\left( 7\right) % \right] ^{4}\text{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ } \end{equation*}% \begin{eqnarray*} &=&\left( 7+3\right) ^{2}-\left( 5\cdot 3-2\cdot 7\right) ^{4}\text{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ addition in parentheses} \\ &=&10^{2}-\left( 5\cdot 3-2\cdot 7\right) ^{4}\text{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ multiplications in parentheses,} \\ &=&10^{2}-\left( 15-2\cdot 7\right) ^{4}\text{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ left to right} \\ &=&10^{2}-\left( 15-14\right) ^{4}\text{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ subtraction in parentheses} \\ &=&10^{2}-1^{4}\text{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ exponents, left to right} \\ &=&100-1^{4}\text{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ careful! \ }1^{4}\not=4 \\ &=&100-1 \\ &=&99 \end{eqnarray*}% \pagebreak \end{enumerate} {\Large Part 2. Practice Problems}\bigskip \begin{enumerate} \item Evaluate each of the algebraic expressions when $x=6$ \ and \ $y=8$. \begin{enumerate} \item $19-y+x=~~% %TCIMACRO{\TeXButton{red}{\color{red}}}% %BeginExpansion \color{red}% %EndExpansion 17$% %TCIMACRO{\TeXButton{black}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion \item $19-\left( y+x\right) =~~% %TCIMACRO{\TeXButton{red}{\color{red}}}% %BeginExpansion \color{red}% %EndExpansion 5$% %TCIMACRO{\TeXButton{black}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion \item $2x^{2}-5y+3=~~% %TCIMACRO{\TeXButton{red}{\color{red}}}% %BeginExpansion \color{red}% %EndExpansion 35$% %TCIMACRO{\TeXButton{black}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion \item $x^{2}+y^{2}=~~% %TCIMACRO{\TeXButton{red}{\color{red}}}% %BeginExpansion \color{red}% %EndExpansion 100$% %TCIMACRO{\TeXButton{black}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion \item $\left( x+y\right) ^{2}=~~% %TCIMACRO{\TeXButton{red}{\color{red}}}% %BeginExpansion \color{red}% %EndExpansion 196$% %TCIMACRO{\TeXButton{black}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion \item $3y-x=~~% %TCIMACRO{\TeXButton{red}{\color{red}}}% %BeginExpansion \color{red}% %EndExpansion 18$% %TCIMACRO{\TeXButton{black}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion \item $3\left( y-x\right) =~~% %TCIMACRO{\TeXButton{red}{\color{red}}}% %BeginExpansion \color{red}% %EndExpansion 6$% %TCIMACRO{\TeXButton{black}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion \item $\dfrac{5x-y}{2}=~~% %TCIMACRO{\TeXButton{red}{\color{red}}}% %BeginExpansion \color{red}% %EndExpansion 11$% %TCIMACRO{\TeXButton{black}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion \item $5x-\dfrac{y}{2}=~~% %TCIMACRO{\TeXButton{red}{\color{red}}}% %BeginExpansion \color{red}% %EndExpansion 26$% %TCIMACRO{\TeXButton{black}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion \item $\dfrac{x^{2}-5x+4}{y-3}=~~% %TCIMACRO{\TeXButton{red}{\color{red}}}% %BeginExpansion \color{red}% %EndExpansion 2$% %TCIMACRO{\TeXButton{black}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion \end{enumerate} \item Consider the expression $\dfrac{6a-3b-ab+2a^{2}}{2a-b}-3$. \begin{enumerate} \item Evaluate the expression if \ $a=5$ \ and \ $b=1$.~$~~% %TCIMACRO{\TeXButton{red}{\color{red}}}% %BeginExpansion \color{red}% %EndExpansion 5$% %TCIMACRO{\TeXButton{black}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion \item Evaluate the expression if \ $a=5$ \ and \ $b=2$.~$~~% %TCIMACRO{\TeXButton{red}{\color{red}}}% %BeginExpansion \color{red}% %EndExpansion 5$% %TCIMACRO{\TeXButton{black}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion \item Evaluate the expression if \ $a=5$ \ and \ $b=3$.~$~~% %TCIMACRO{\TeXButton{red}{\color{red}}}% %BeginExpansion \color{red}% %EndExpansion 5$% %TCIMACRO{\TeXButton{black}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion \item Evaluate the expression if \ $a=4$ \ and \ $b=1$.~$~~% %TCIMACRO{\TeXButton{red}{\color{red}}}% %BeginExpansion \color{red}% %EndExpansion 4$% %TCIMACRO{\TeXButton{black}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion \end{enumerate} \end{enumerate} \end{document}