Answer Key 2.3
- [latex]\phantom{a}[/latex]
[latex]\begin{array}[t]{rrrrrr} 2&-(-3a&-&8)&=&1 \\ -2&&&&&-2 \\ \hline &-(-3a&-&8)&=&-1 \\ &3a&+&8&=&-1 \\ &&-&8&&-8 \\ \hline &&&3a&=&-9 \\ &&&a&=&-3 \end{array}[/latex] - [latex]\phantom{a}[/latex]
[latex]\begin{array}[t]{rrcrc} 2(-3n&+&8)&=&-20\\ \\ \dfrac{2}{2}(-3n&+&8)&=&\dfrac{-20}{2} \\ \\ -3n&+&8&=&-10 \\ &-&8&&-8 \\ \hline &&\dfrac{-3n}{-3}&=&\dfrac{-18}{-3} \\ \\ &&n&=&6 \end{array}[/latex] - [latex]\phantom{a}[/latex]
[latex]\begin{array}[t]{rrcrc} -5(-4&+&2v)&=&-50\\ \\ \dfrac{-5}{-5}(-4&+&2v)&=&\dfrac{-50}{-5}\\ \\ -4&+&2v&=&10 \\ +4&&&&+4 \\ \hline &&\dfrac{2v}{2}&=&\dfrac{14}{2}\\ \\ &&v&=&7 \end{array}[/latex] - [latex]\phantom{a}[/latex]
[latex]\begin{array}[t]{rrcrrrl} 2&-&8(-4&+&3x)&=&34 \\ -2&&&&&&-2 \\ \hline &&\dfrac{-8}{-8}(-4&+&3x)&=&\dfrac{32}{-8} \\ \\ &&-4&+&3x&=&-4 \\ &&+4&&&&+4 \\ \hline &&&&3x&=&0 \\ &&&&x&=&0 \end{array}[/latex] - [latex]\phantom{a}[/latex]
[latex]\begin{array}[t]{rrrrr} \dfrac{66}{6}&=&\dfrac{6}{6}(6&+&5x) \\ \\ 11&=&6&+&5x \\ -6&&-6&& \\ \hline \dfrac{5}{5}&=&\dfrac{5x}{5}&& \\ \\ x&=&1&& \end{array}[/latex] - [latex]\phantom{a}[/latex]
[latex]\begin{array}[t]{rrcrcrr} 32&=&\phantom{-}2&-&5(-4n&+&6) \\ -2&&-2&&&& \\ \hline \dfrac{30}{-5}&=&\dfrac{-5}{-5}(-4n&+&6)&& \\ \\ -6&=&-4n&+&6&& \\ -6&&&-&6&& \\ \hline \dfrac{-12}{-4}&=&\dfrac{-4n}{-4}&&&& \\ \\ n&=&3&&&& \end{array}[/latex] - [latex]\phantom{a}[/latex]
[latex]\begin{array}[t]{rrrrrrr} -2&+&2(8x&-&9)&=&-16 \\ +2&&&&&&+2 \\ \hline &&\dfrac{2}{2}(8x&-&9)&=&\dfrac{-14}{2} \\ \\ &&8x&-&9&=&-7 \\ &&&+&9&&+9 \\ \hline &&&&8x&=&2 \\ \\ &&&&x&=&\dfrac{1}{4} \end{array}[/latex] - [latex]\phantom{a}[/latex]
[latex]\begin{array}[t]{rrrrl} -3&+&5n&=&12 \\ +3&&&&+3 \\ \hline &&\dfrac{5n}{5}&=&\dfrac{15}{5} \\ \\ &&n&=&3 \end{array}[/latex] - [latex]\phantom{a}[/latex]
[latex]\begin{array}[t]{rrrrlll} -1&-&7m&=&-8m&+&7 \\ -7&+&7m&&+7m&-&7 \\ \hline &&(-8&=&-m)(-1)&& \\ &&m&=&8&& \end{array}[/latex] - [latex]\phantom{a}[/latex]
[latex]\begin{array}[t]{rrrrrrr} 56p&-&48&=&6p&+&2 \\ -6p&+&48&&-6p&+&48 \\ \hline &&\dfrac{50p}{50}&=&\dfrac{50}{50}&& \\ \\ &&p&=&1&& \end{array}[/latex] - [latex]\phantom{a}[/latex]
[latex]\begin{array}[t]{rrrrrrr} 1&-&12r&=&29&-&8r \\ -1&+&8r&&-1&+&8r \\ \hline &&\dfrac{-4r}{-4}&=&\dfrac{28}{-4}&& \\ \\ &&r&=&-7&& \end{array}[/latex] - [latex]\phantom{a}[/latex]
[latex]\begin{array}[t]{rrrrrrr} 4&+&3x&=&-12x&+&4 \\ -4&+&12x&&+12x&-&4 \\ \hline &&15x&=&0&& \\ &&x&=&0&& \end{array}[/latex] - [latex]\phantom{a}[/latex]
[latex]\begin{array}[t]{rrrrrrr} 20&-&7b&=&-12b&+&30 \\ -20&+&12b&&+12b&-&20 \\ \hline &&\dfrac{5b}{5}&=&\dfrac{10}{5}&& \\ \\ &&b&=&2&& \end{array}[/latex] - [latex]\phantom{a}[/latex]
[latex]\begin{array}[t]{rrrrrrr} -16n&+&12&=&39&-&7n \\ +7n&-&12&&-12&+&7n \\ \hline &&\dfrac{-9n}{-9}&=&\dfrac{27}{-9}&& \\ \\ &&n&=&-3&& \end{array}[/latex] - [latex]\phantom{a}[/latex]
[latex]\begin{array}[t]{rrrrrrrrr} -2&-&5(2&-&4m)&=&33&+&5m \\ +2&&&&&&+2&& \\ \hline &&-10&+&20m&=&35&+&5m \\ &&+10&-&5m&&+10&-&5m \\ \hline &&&&\dfrac{15m}{15}&=&\dfrac{45}{15}&& \\ \\ &&&&m&=&3&& \end{array}[/latex] - [latex]\phantom{a}[/latex]
[latex]\begin{array}[t]{rrrrrrr} -25&-&7x&=&12x&-&6 \\ +25&-&12x&&-12x&+&25 \\ \hline &&\dfrac{-19x}{-19}&=&\dfrac{19}{-19}&& \\ \\ &&x&=&-1&& \end{array}[/latex] - [latex]\phantom{a}[/latex]
[latex]\begin{array}[t]{rrrrrrrrr} -4n&+&11&=&2&-&16n&+&3n \\ +16n&-&11&&-11&+&16n&-&3n \\ -3n&&&&&&&& \\ \hline &&\dfrac{9n}{9}&=&\dfrac{-9}{9}&&&& \\ \\ &&n&=&-1&&&& \end{array}[/latex] - [latex]\phantom{a}[/latex]
[latex]\begin{array}[t]{rrrrrrr} -7&-&7b&=&-5&-&5b \\ +7&+&5b&&+7&+&5b \\ \hline &&\dfrac{-2b}{-2}&=&\dfrac{2}{-2}&& \\ \\ &&b&=&-1&& \end{array}[/latex] - [latex]\phantom{a}[/latex]
[latex]\begin{array}[t]{rrrrlrrrr} -6v&-&29&=&-4v&-&5v&-&5 \\ +4v&+&29&&+4v&+&5v&+&29 \\ +5v&&&&&&&& \\ \hline &&\dfrac{3v}{3}&=&\dfrac{24}{3}&&&& \\ \\ &&v&=&8&&&& \end{array}[/latex] - [latex]\phantom{a}[/latex]
[latex]\begin{array}[t]{rrcrrrr} -64r&+&16&=&3r&+&16 \\ -3r&-&16&&-3r&-&16 \\ \hline &&-67r&=&0&& \\ &&r&=&0&& \end{array}[/latex] - [latex]\phantom{a}[/latex]
[latex]\begin{array}[t]{rrcrrrr} 8x&-&8&=&-20&-&4x \\ +4x&+&8&&+8&+&4x \\ \hline &&\dfrac{12x}{12}&&\dfrac{-12}{12}&& \\ \\ &&x&=&-1&& \end{array}[/latex] - [latex]\phantom{a}[/latex]
[latex]\begin{array}[t]{rrrrlrrrr} -8n&-&19&=&-16n&+&6&+&3n \\ +16n&+&19&&+16n&+&19&-&3n \\ -3n&&&&&&&& \\ \hline &&\dfrac{5n}{5}&=&\dfrac{25}{5}&&&& \\ \\ &&n&=&5&&&& \end{array}[/latex] - [latex]\phantom{a}[/latex]
[latex]\begin{array}[t]{rrrrrrrrr} -2m&+&4&+&7m&-&56&=&-67 \\ &-&4&&&+&56&&+56 \\ &&&&&&&&-4 \\ \hline &&&&&&\dfrac{5m}{5}&=&\dfrac{-15}{5} \\ \\ &&&&&&m&=&-3 \end{array}[/latex] - [latex]\phantom{a}[/latex]
[latex]\begin{array}[t]{rrlrrrrrr} 7&=&4n&-&28&+&35n&+&35 \\ +28&&&+&28&&&-&35 \\ -35&&&&&&&& \\ \hline 0&=&39n&&&&&& \\ n&=&0&&&&&& \end{array}[/latex] - [latex]\phantom{a}[/latex]
[latex]\begin{array}[t]{rrlrrrrrr} 50&=&\phantom{-}56&+&56r&-&4r&-&6 \\ -56&&-56&&&&&+&6 \\ +6&&&&&&&& \\ \hline 0&=&52r&&&&&& \\ r&=&0&&&&&& \end{array}[/latex] - [latex]\phantom{a}[/latex]
[latex]\begin{array}[t]{rrrrrrrrr} -48&-&48x&-&12&+&24x&=&-12 \\ +48&&&+&12&&&&+12 \\ &&&&&&&&+48 \\ \hline &&&&&&\dfrac{-24x}{-24}&=&\dfrac{48}{-24} \\ \\ &&&&&&x&=&-2 \end{array}[/latex]
