\hfill \thepage} %} \newenvironment{proof}[1][Proof]{\noindent\textbf{#1.} }{\ \rule{0.5em}{0.5em}} \input{tcilatex} \geometry{left=0.5in,right=0.5in,top=0.5in,bottom=0.5in} \pagestyle{fancy} \lhead{\Large \color{blue}Lecture Notes} \lfoot{} \cfoot{} \chead{\LARGE Combining Like Terms} \rhead{\large page \thepage} \lfoot{\small \copyright \; Hidegkuti, Powell, 2012} \rfoot{\small Last revised: June 26, 2012} \textwidth 7.6in \textheight 9.6in \setlength{\headheight}{28pt} \setlength{\parindent}{0in} \begin{document} \begin{center} {\LARGE Sample Problems\bigskip }\bigskip \end{center} Simplify each of the following by combining like terms.\bigskip %TCIMACRO{\TeXButton{2 col begin}{\begin{multicols}{2}}}% %BeginExpansion \begin{multicols}{2}% %EndExpansion \begin{enumerate} \item $3x+8x\medskip $ \item $4a-7a\medskip $ \item $-p+3+3p\medskip $ \item $2b-3+5b+10-b\medskip $ \item $2m-1-9m+1\medskip $ \item $3a+b-2-3a-10b-2\medskip $ \item $x+1-2x+3-3x-4+4x\medskip $ \item $3p-q-3p-q\medskip $ \item $3x^{2}-5x+2-x^{2}+x-2\medskip $ \item $\dfrac{2}{3}x-1-x+\dfrac{2}{5}\medskip $ \item $ab-2a+5b-a+b-ab+3a$ \item $-x+4-x+6+5x$ %TCIMACRO{\TeXButton{multcol end}{\end{multicols}}}% %BeginExpansion \end{multicols}% %EndExpansion {\LARGE \bigskip \bigskip \bigskip } \end{enumerate} \begin{center} {\LARGE Practice Problems\bigskip } \end{center} Simplify each of the following by combining like terms.\bigskip %TCIMACRO{\TeXButton{2 col begin}{\begin{multicols}{2}}}% %BeginExpansion \begin{multicols}{2}% %EndExpansion \begin{enumerate} \item $3a-5-7a+1+a\medskip $ \item $3x-5+5x-3-8x+8\medskip $ \item $m^{2}-m+1-m^{2}+5m\medskip $ \item $\dfrac{1}{2}x-\dfrac{1}{3}x+1\medskip $ \item $-1+p+q-p+2q+3\medskip $ \item $5a-2+b-ab+a-b\medskip $ \item $\dfrac{1}{2}x+\dfrac{1}{2}y-\dfrac{1}{2}x+\dfrac{1}{2}y\medskip $ \item $2r-R+r+2R-1\medskip $ \item $-x+3-5x+1\medskip $ \item $x-y+2z+3x-2y-z\medskip $ \item $3y^{2}-y-2+y^{2}+y+2-5y^{2}\medskip $ \end{enumerate} %TCIMACRO{\TeXButton{multcol end}{\end{multicols}}}% %BeginExpansion \end{multicols}% %EndExpansion \pagebreak \begin{center} {\LARGE Sample Problems - Answers\bigskip } \end{center} 1.) $\ 11x$ \ \ \ \ 2.) $\ -3a$ \ \ \ \ \ 3.) $\ 2p+3$ \ \ \ \ \ 4.) $\ 6b+7$ \ \ \ \ \ 5.) $\ -7m$ \ \ \ \ \ \ 6.) $\ -9b-4$ \ \ \ \ \ \ 7.) $\ 0$ \ \ \ \ 8.) $\ -2q\medskip \medskip $ 9.) $\ 2x^{2}-4x$ \ \ \ \ \ \ 10.) $\ -\dfrac{1}{3}x-\dfrac{3}{5}$ \ \ \ \ \ 11.) $\ 6b$ \ \ \ \ \ \ \ 12.) \ $3x+10$\ $\medskip \medskip \medskip \medskip $ \begin{center} {\LARGE Practice Problems - Answers\bigskip } \end{center} 1.) $\ -3a-4$ \ \ \ \ \ 2.) $\ 0$ \ \ \ \ 3.) $\ 4m+1$ \ \ \ \ 4.) $\ \dfrac{% 1}{6}x+1$ \ \ \ \ \ 5.) $\ 3q+2$ \ \ \ \ \ 6.) $\ 6a-ab-2$ \ \ \ \ \ \ 7.) $% \ y\medskip $ 8.) $\ R+3r-1\medskip $ \ \ \ \ \ \ 9.) $\ -6x+4\medskip $ \ \ \ \ \ \ 10.) $% \ 4x-3y+z\medskip $ \ \ \ \ \ 11.) $\ -y^{2}${\LARGE \bigskip \bigskip } \begin{center} {\LARGE Sample Problems - Solutions\bigskip } \end{center} \begin{enumerate} \item $3x+8x\smallskip $ Solution: \ $3x$ and $8x$ are like terms. \ We add the coefficients, $3$ and $8$ and that is how many $x$ we have: \ $3x+8x=\left( 3+8\right) x=11x$. \ We usually just write% \begin{equation*} 3x+8x=11x \end{equation*} \item $4a-7a\smallskip $ Solution: \ We can re-write $4a-7a$ as an addition: $4a-7a=4a+\left( -7\right) a$. \ In other words, the coefficints are $4$ and $-7$. \ $4a$ and $-7a$ are like terms. \ We add the coefficients, $4$ and $-7$ and that is how many $a$ we have: \ The rigorous notation that follows the definitions is% \begin{equation*} 4a-7a=4a+\left( -7\right) a=\left( 4+\left( -7\right) \right) a=-3a \end{equation*}% but we usually just write \begin{equation*} 4a-7a=-3a \end{equation*} \item $-p+3+3p\smallskip $ Solution: \ We can re-write $-p$ and $3p$ are like terms and can be combined. \ However, the number $3$ can not be combined with the other two terms. \ The first term, $-p$ has coefficient $-1$. \ Recall that addition is commutative and so we can rearrange the terms so that like terms are grouped together.% \begin{equation*} -p+3+3p=-1p+3p+3=\left( -1+3\right) p+3=2p+3 \end{equation*}% We usually just write% \begin{equation*} -p+3+3p=2p+3 \end{equation*} \item $2b-3+5b+10-b\smallskip $ Solution: \ Recall that addition is commutative and so we can rearrange the terms so that like terms are grouped together. \ Also, we can re-write subtractions as additions with negative coefficients. \ Then we add the coefficients in like terms.% \begin{eqnarray*} 2b-3+5b+10-b &=&2b+5b+\left( -1\right) b+\left( -3\right) +10 \\ &=&\left( 2+5+\left( -1\right) \right) b+\left( -3+10\right) \\ &=&6b+7 \end{eqnarray*}% We do not have to re-write sustractions as above. \ The following computation is also perfectly acceptable:% \begin{eqnarray*} 2b-3+5b+10-b &=&2b+5b-b-3+10 \\ &=&6b+7 \end{eqnarray*} \item $2m-1-9m+1\smallskip $ Solution: \ Recall that addition is commutative and so we can rearrange the terms so that like terms are grouped together. \ Also, we can re-write subtractions as additions with negative coefficients. \ Then we add the coefficients in like terms.% \begin{eqnarray*} 2m-1-9m+1 &=&2m-9m-1+1 \\ &=&-7m+0 \\ &=&-7m \end{eqnarray*} \item $3a+b-2-3a-10b-2\smallskip $ Solution: \ This problem contains three groups of unlike terms: \ $3a$ and $% -3a$ are like terms, $-b$ and $-10b$ ae like terms, and $-2$ and $-2$ are like terms. \ When we combine $3a$ and $-3a$, we get $0a$ which is $0$.% \begin{eqnarray*} 3a+b-2-3a-10b-2 &=&3a-3a+b-10b-2-2 \\ &=&0a-9b-4 \\ &=&-9b-4 \end{eqnarray*} \item $x+1-2x+3-3x-4+4x\smallskip $ Solution: \ \begin{eqnarray*} x+1-2x+3-3x-4+4x &=&x-2x-3x+4x+1+3-4 \\ &=&\left( 1-2-3+4\right) x+\left( 1+3-4\right) \\ &=&0x+0=0 \end{eqnarray*} \item $3p-q-3p-q\smallskip $ Solution: \ $3p$ and $-3p$ are like terms and $-q$ and $-q$ are like terms.% \begin{equation*} 3p-q-3p-q=3p-3p-q-q=0p-2q=-2q \end{equation*} \item $3x^{2}-5x+2-x^{2}+x-2\smallskip $ Solution: \ $x^{2}$ and $x$ and $1$ are all unlike terms. \ \begin{eqnarray*} 3x^{2}-5x+2-x^{2}+x-2 &=&3x^{2}-x^{2}-5x+x+2-2 \\ &=&2x^{2}-4x+0=2x^{2}-4x \end{eqnarray*} \item $\dfrac{2}{3}x-1-x+\dfrac{2}{5}\smallskip $ Solution: \ We add the coefficients. $\ $The operations are $\dfrac{2}{3}-1$ \ and $-1+\dfrac{2}{5}\smallskip $ $\dfrac{2}{3}-1=\dfrac{2}{3}-\dfrac{3}{3}=\dfrac{2-3}{3}=\dfrac{-1}{3}=-% \dfrac{1}{3}$ \ and $-1+\dfrac{2}{5}=\dfrac{-5}{5}+\dfrac{2}{5}=\dfrac{-5+2}{% 5}=\dfrac{-3}{5}=-\dfrac{3}{5}\smallskip $% \begin{eqnarray*} \dfrac{2}{3}x-1-x+\dfrac{2}{5} &=&\dfrac{2}{3}x-x-1+\dfrac{2}{5} \\ &=&\left( \dfrac{2}{3}-1\right) x+\left( -1+\dfrac{2}{5}\right) \\ &=&-\dfrac{1}{3}x-\dfrac{3}{5} \end{eqnarray*} \item $ab-2a+5b-a+b-ab+3a\smallskip $ Solution: \ $a,$ $b$, and $ab$ are all unlike terms. We will have three groups of like terms.% \begin{eqnarray*} ab-2a+5b-a+b-ab+3a &=&ab-ab-2a-a+3a+5b+b \\ &=&\left( 1-1\right) ab+\left( -2-1+3\right) a+\left( 5+1\right) b \\ &=&0ab+0a+6b=6b \end{eqnarray*} \item $-x+4-x+6+5x\smallskip $ Solution: \ \begin{equation*} -x+4-x+6+5x=-x-x+5x+4+6=3x+10 \end{equation*} \end{enumerate} \end{document}