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