Answer Key 5.4
- \(\begin{array}{rr}
\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\
\begin{array}{rrrrrrrrl}
\\
&a&-&b&+&2c&=&2& \\
+&a&+&b&+&c&=&3& \\
\hline
&&&2a&+&3c&=&5& \\ \\
&a&-&b&+&2c&=&2& \\
+&2a&+&b&-&c&=&2& \\
\hline
&&&(3a&+&c&=&4)&(-3) \\
&&&-9a&-&3c&=&-12& \\ \\
&&&-9a&-&3c&=&-12& \\
+&&&2a&+&3c&=&5& \\
\hline
&&&&&\dfrac{-7a}{-7}&=&\dfrac{-7}{-7}& \\ \\
&&&&&a&=&1&
\end{array}
&\hspace{0.25in}
\begin{array}{rrrrrrrl}
&&3a&+&c&=&4& \\
&&3(1)&+&c&=&4& \\
&&3&+&c&=&4& \\
&&-3&&&&-3& \\
\hline
&&&&c&=&1& \\ \\
a&+&b&+&c&=&3& \\
(1)&+&b&+&(1)&=&3& \\
&&b&+&2&=&3& \\
&&&-&2&&-2& \\
\hline
&&&&b&=&1&
\end{array}
\end{array}\) - \(\begin{array}{rr}
\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\
\begin{array}{rrrrrrrrl}
\\ \\
&2a&+&3b&-&c&=&12& \\
+&3a&+&4b&+&c&=&19& \\
\hline
&&&5a&+&7b&=&31& \\ \\
&2a&+&3b&-&c&=&12& \\
+&a&-&2b&+&c&=&-3& \\
\hline
&&&(3a&+&b&=&9)&(-7) \\
&&&-21a&-&7b&=&-63& \\ \\
&&&5a&+&7b&=&31& \\
+&&&-21a&-&7b&=&-63& \\
\hline
&&&&&\dfrac{-16a}{-16}&=&\dfrac{-32}{-16}& \\ \\
&&&&&a&=&2&
\end{array}
&\hspace{0.25in}
\begin{array}{rrrrrrr}
&&3a&+&b&=&9 \\
&&3(2)&+&b&=&9 \\
&&6&+&b&=&9 \\
&&-6&&&&-6 \\
\hline
&&&&b&=&3 \\ \\
a&-&2b&+&c&=&-3 \\
(2)&-&2(3)&+&c&=&-3 \\
2&-&6&+&c&=&-3 \\
&&-4&+&c&=&-3 \\
&&+4&&&&+4 \\
\hline
&&&&c&=&1
\end{array}
\end{array}\) - \(\begin{array}{rr}
\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\
\begin{array}{rrrrrrrrl}
&(3x&+&y&-&z&=&7)&(-1) \\
&-3x&-&y&+&z&=&-7& \\
+&x&+&3y&-&z&=&5& \\
\hline
&&&(-2x&+&2y&=&-2)&(\div 2) \\
&&&(-x&+&y&=&-1)&(7) \\
&&&-7x&+&7y&=&-7& \\ \\
&(3x&+&y&-&z&=&7)&(2) \\
&6x&+&2y&-&2z&=&14& \\
+&x&+&y&+&2z&=&3& \\
\hline
&&&7x&+&3y&=&17& \\
+&&&-7x&+&7y&=&-7& \\
\hline
&&&&&\dfrac{10y}{10}&=&\dfrac{10}{10}& \\ \\
&&&&&y&=&1& \\
\end{array}
&\hspace{0.25in}
\begin{array}{rrrrrrr}
\\ \\
&&-x&+&y&=&-1 \\
&&-x&+&(1)&=&-1 \\
&&-x&+&1&=&-1 \\
&&&-&1&&-1 \\
\hline
&&&&-x&=&-2 \\
&&&&x&=&2 \\ \\
x&+&y&+&2z&=&3 \\
(2)&+&(1)&+&2z&=&3 \\
&&2z&+&3&=&3 \\
&&&-&3&&-3 \\
\hline
&&&&2z&=&0 \\
&&&&z&=&0 \\
\end{array}
\end{array}\) - \(\begin{array}{rr}
\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\
\begin{array}{rrrrrrrrl}
&x&+&y&+&z&=&4&(-1) \\
&-x&-&y&-&z&=&-4& \\ \\
&-x&-&y&-&z&=&-4& \\
+&x&+&2y&+&3z&=&10& \\
\hline
&&&(y&+&2z&=&6)&(2) \\
&&&2y&+&4z&=&12& \\ \\
&-x&-&y&-&z&=&-4& \\
+&x&-&y&+&4z&=&20& \\
\hline
&&&-2y&+&3z&=&16& \\
+&&&2y&+&4z&=&12& \\
\hline
&&&&&\dfrac{7z}{7}&=&\dfrac{28}{7}& \\ \\
&&&&&z&=&4&
\end{array}
& \hspace{0.25in}
\begin{array}{rrrrrrr}
&&y&+&2z&=&6 \\
&&y&+&2(4)&=&6 \\
&&y&+&8&=&6 \\
&&&-&8&&-8 \\
\hline
&&&&y&=&-2 \\ \\
x&+&y&+&z&=&4 \\
x&+&(-2)&+&(4)&=&4 \\
&&x&+&2&=&4 \\
&&&-&2&&-2 \\
\hline
&&&&x&=&2
\end{array}
\end{array}\) - \(\begin{array}{rr}
\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\
\begin{array}{rrrrrrrrl}
&x&+&2y&-&z&=&0& \\
+&3x&-&2y&-&4z&=&-5& \\
\hline
&&&4x&-&5z&=&-5& \\ \\
&(2x&-&y&+&z&=&15)&(2) \\
&4x&-&2y&+&2z&=&30& \\
+&x&+&2y&-&z&=&0& \\
\hline
&&&(5x&+&z&=&30)&(5) \\
&&&25x&+&5z&=&150& \\ \\
&&&4x&-&5z&=&-5& \\
+&&&25x&+&5z&=&150& \\
\hline
&&&&&\dfrac{29x}{29}&=&\dfrac{145}{29}& \\ \\
&&&&&x&=&5& \\
\end{array}
& \hspace{0.25in}
\begin{array}{rrrrrrr}
&&5x&+&z&=&30 \\
&&5(5)&+&z&=&30 \\
&&25&+&z&=&30 \\
&&-25&&&&-25 \\
\hline
&&&&z&=&5 \\ \\
x&+&2y&-&z&=&0 \\
(5)&+&2y&-&(5)&=&0 \\
5&+&2y&-&5&=&0 \\
&&&&2y&=&0 \\
&&&&y&=&0 \\
\end{array}
\end{array}\) - \(\begin{array}{rr}
\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\
\begin{array}{rrrrrrrrl}
&(x&-&y&+&2z&=&-3)&(2) \\
&2x&-&2y&+&4z&=&-6& \\
+&x&+&2y&+&3z&=&4& \\
\hline
&&&(3x&+&7z&=&-2)&(-1) \\
&&&-3x&-&7z&=&2& \\ \\
&2x&+&y&+&z&=&-3& \\
+&x&-&y&+&2z&=&-3& \\
\hline
&&&3x&+&3z&=&-6& \\
+&&&-3x&-&7z&=&2& \\
\hline
&&&&&\dfrac{-4z}{-4}&=&\dfrac{-4}{-4}& \\ \\
&&&&&z&=&1& \\
\end{array}
&\hspace{0.25in}
\begin{array}{rrrrrrr}
&&3x&+&3z&=&-6 \\
&&3x&+&3(1)&=&-6 \\
&&3x&+&3&=&-6 \\
&&&-&3&&-3 \\
\hline
&&&&\dfrac{3x}{3}&=&\dfrac{-9}{3} \\ \\
&&&&x&=&-3 \\ \\
x&-&y&+&2z&=&-3 \\
(-3)&-&y&+&2(1)&=&-3 \\
-3&-&y&+&2&=&-3 \\
&&-y&-&1&=&-3 \\
&&&+&1&&+1 \\
\hline
&&&&-y&=&-2 \\
&&&&y&=&2
\end{array}
\end{array}\) - \(\begin{array}{rr}
\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\
\begin{array}{rrrrrrrrl}
\\
&x&+&y&+&z&=&6& \\
+&2x&-&y&-&z&=&-3& \\
\hline
&&&&&\dfrac{3x}{3}&=&\dfrac{3}{3}& \\ \\
&&&&&x&=&1& \\ \\
&x&-&2y&+&3z&=&6& \\
&(1)&-&2y&+&3z&=&6& \\
&1&-&2y&+&3z&=&6& \\
&-1&&&&&&-1& \\
\hline
&&&-2y&+&3z&=&5& \\ \\
&x&+&y&+&z&=&6& \\
&(1)&+&y&+&z&=&6& \\
&1&+&y&+&z&=&6& \\
&-1&&&&&&-1& \\
\hline
&&&(y&+&z&=&5)&(2) \\
&&&2y&+&2z&=&10&
\end{array}
& \hspace{0.25in}
\begin{array}{rrrrrrr}
\\ \\ \\ \\ \\
&&-2y&+&3z&=&5 \\
+&&2y&+&2z&=&10 \\
\hline
&&&&\dfrac{5z}{5}&=&\dfrac{15}{5} \\ \\
&&&&z&=&3 \\ \\
x&+&y&+&z&=&6 \\
(1)&+&y&+&(3)&=&6 \\
1&+&y&+&3&=&6 \\
&&y&+&4&=&6 \\
&&&-&4&&-4 \\
\hline
&&&&y&=&2 \\
\end{array}
\end{array}\) - \(\begin{array}{rr}
\\ \\ \\ \\ \\ \\ \\ \\
\begin{array}{rrrrrrrrl}
&x&+&y&-&z&=&0& \\
+&2x&+&y&+&z&=&0& \\
\hline
&&&3x&+&2y&=&0& \\ \\
&(x&+&y&-&z&=&0)&(-4) \\
&-4x&-&4y&+&4z&=&0& \\
+&x&+&2y&-&4z&=&0& \\
\hline
&&&-3x&-&2y&=&0& \\
\end{array}
& \hspace{0.25in}
\begin{array}{rrrrrr}
&-3x&-&2y&=&0 \\
+&3x&+&2y&=&0 \\
\hline
&&&0&=&0 \\ \\
&&\therefore &x&=&0 \\
&&&y&=&0 \\
&&&z&=&0 \\
\end{array}
\end{array}\) - \(\begin{array}{rr}
\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\
\begin{array}{rrrrrrrrl}
&x&+&y&+&z&=&2& \\
+&2x&-&y&+&3z&=&9& \\
\hline
&&&3x&+&4z&=&11& \\ \\
&2x&-&y&+&3z&=&9& \\
+&&&y&-&z&=&-3& \\
\hline
&&&(2x&+&2z&=&6)&(-2) \\
&&&-4x&-&4z&=&-12& \\ \\
&&&3x&+&4z&=&11& \\
+&&&-4x&-&4z&=&-12& \\
\hline
&&&&&-x&=&-1& \\
&&&&&x&=&1&
\end{array}
& \hspace{0.25in}
\begin{array}{rrrrrrr}
&&2x&+&2z&=&6 \\
&&2(1)&+&2z&=&6 \\
&&2&+&2z&=&6 \\
&&-2&&&&-2 \\
\hline
&&&&\dfrac{2z}{2}&=&\dfrac{4}{2} \\ \\
&&&&z&=&2 \\ \\
x&+&y&+&z&=&2 \\
(1)&+&y&+&(2)&=&2 \\
&&y&+&3&=&2 \\
&&&-&3&&-3 \\
\hline
&&&&y&=&-1 \\
\end{array}
\end{array}\) - \(\begin{array}{rr}
\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\
\begin{array}{rrrrrrrrl}
&(4x&&&+&z&=&3)&(2) \\
&8x&&&+&2z&=&6& \\
+&6x&-&y&-&2z&=&-1& \\
\hline
&&&(14x&-&y&=&5)&(3) \\
&&&42x&-&3y&=&15& \\
+&&&-2x&+&3y&=&5& \\
\hline
&&&&&\dfrac{40x}{40}&=&\dfrac{20}{40}& \\ \\
&&&&&x&=&\dfrac{1}{2}&
\end{array}
& \hspace{0.25in}
\begin{array}{rrrrrrr}
\\ \\ \\ \\ \\
&&-2x&+&3y&=&5 \\
&&-2\left(\dfrac{1}{2}\right)&+&3y&=&5 \\
&&-1&+&3y&=&5 \\
&&+1&&&&+1 \\
\hline
&&&&\dfrac{3y}{3}&=&\dfrac{6}{3} \\ \\
&&&&y&=&2 \\ \\
&&4x&+&z&=&3 \\
&&4\left(\dfrac{1}{2}\right)&+&z&=&3 \\
&&2&+&z&=&3 \\
&&-2&&&&-2 \\
\hline
&&&&z&=&1 \\
\end{array}
\end{array}\) - \(\begin{array}{rr}
\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\
\begin{array}{rrrrrrrrl}
&&&x&-&z&=&-2& \\
+&&&y&+&z&=&5& \\
\hline
&&&x&+&y&=&3& \\ \\
&2x&-&3y&+&z&=&-1& \\
+&x&&&-&z&=&-2& \\
\hline
&&&(3x&-&3y&=&-3)&(\div 3) \\
&&&x&-&y&=&-1& \\
+&&&x&+&y&=&3& \\
\hline
&&&&&\dfrac{2x}{2}&=&\dfrac{2}{2}& \\ \\
&&&&&x&=&1&
\end{array}
& \hspace{0.25in}
\begin{array}{rrrrr}
x&-&z&=&-2 \\
(1)&-&z&=&-2 \\
1&-&z&=&-2 \\
-1&&&&-1 \\
\hline
&&-z&=&-3 \\
&&z&=&3 \\ \\
y&+&z&=&5 \\
y&+&(3)&=&5 \\
y&+&3&=&5 \\
&-&3&&-3 \\
\hline
&&y&=&2 \\
\end{array}
\end{array}\) - \(\begin{array}{rr}
\\ \\ \\ \\ \\ \\ \\ \\
\begin{array}{rrrrrrrrl}
&(3x&+&4y&-&z&=&11)&(2) \\
&6x&+&8y&-&2z&=&22& \\
+&&&y&+&2z&=&-4& \\
\hline
&&&(6x&+&9y&=&18)&(\div 3) \\
&&&2x&+&3y&=&6& \\
+&&&-2x&+&y&=&-6& \\
\hline
&&&&&4y&=&0& \\
&&&&&y&=&0& \\ \\
\end{array}
& \hspace{0.25in}
\begin{array}{rrrrr}
\\ \\ \\
-2x&+&y&=&-6 \\
-2x&+&0&=&-6 \\
&&\dfrac{-2x}{-2}&=&\dfrac{-6}{-2} \\ \\
&&x&=&3 \\ \\
y&+&2z&=&-4 \\
0&+&2z&=&-4 \\
&&\dfrac{2z}{2}&=&\dfrac{-4}{2} \\ \\
&&z&=&-2
\end{array}
\end{array}\) - \(\begin{array}{rr}
\\ \\ \\ \\ \\ \\ \\ \\ \\
\begin{array}{rrrrrrrrl}
&&&(-2y&+&z&=&-6)&(3) \\
&&&-6y&+&3z&=&-18& \\
+&x&+&6y&+&3z&=&30& \\
\hline
&&&(x&+&6z&=&12)&(-2) \\
&&&-2x&-&12z&=&-24& \\
+&&&2x&+&2z&=&4& \\
\hline
&&&&&\dfrac{-10z}{-10}&=&\dfrac{-20}{-10}& \\ \\
&&&&&z&=&2&
\end{array}
& \hspace{0.25in}
\begin{array}{rrrrr}
\\ \\ \\
2x&+&2z&=&4 \\
2x&+&2(2)&=&4 \\
2x&+&4&=&4 \\
&-&4&&-4 \\
\hline
&&2x&=&0 \\
&&x&=&0 \\ \\
-2y&+&z&=&-6 \\
-2y&+&2&=&-6 \\
&-&2&&-2 \\
\hline
&&\dfrac{-2y}{-2}&=&\dfrac{-8}{-2} \\ \\
&&y&=&4
\end{array}
\end{array}\) - \(\begin{array}{rr}
\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\
\begin{array}{rrrrrrrrl}
&(x&-&y&+&2z&=&0)&(2) \\
&2x&-&2y&+&4z&=&0& \\
+&x&+&2y&&&=&1& \\
\hline
&&&3x&+&4z&=&1& \\ \\
&&&(2x&+&z&=&4)&(-4) \\
&&&-8x&-&4z&=&-16& \\
+&&&3x&+&4z&=&1& \\
\hline
&&&&&\dfrac{-5x}{-5}&=&\dfrac{-15}{-5}& \\ \\
&&&&&x&=&3&
\end{array}
& \hspace{0.25in}
\begin{array}{rrrrr}
x&+&2y&=&1 \\
3&+&2y&=&1 \\
-3&&&&-3 \\
\hline
&&\dfrac{2y}{2}&=&\dfrac{-2}{2} \\ \\
&&y&=&-1 \\ \\
2x&+&z&=&4 \\
2(3)&+&z&=&4 \\
6&+&z&=&4 \\
-6&&&&-6 \\
\hline
&&z&=&-2
\end{array}
\end{array}\) - \(\begin{array}{rr}
\\ \\
\begin{array}{rrrrrrrr}
&x&+&y&+&z&=&4 \\
+&&-&y&-&z&=&-4 \\
\hline
&&&&&x&=&0
\end{array}
& \hspace{0.25in}
\begin{array}{rrrrr}
\\ \\ \\ \\ \\ \\
x&-&2y&=&0 \\
0&-&2y&=&0 \\
&&-2y&=&0 \\
&&y&=&0 \\ \\
-y&-&z&=&-4 \\
0&-&z&=&-4 \\
&&-z&=&-4 \\
&&z&=&4
\end{array}
\end{array}\) - \(\begin{array}{rr}
\\ \\ \\ \\ \\ \\ \\
\begin{array}{rrrrrrrrl}
&(x&+&y&-&z&=&2)&(-2) \\
&-2x&-&2y&+&2z&=&-4& \\
+&2x&&&+&z&=&6& \\
\hline
&&&-2y&+&3z&=&2& \\
+&&&2y&-&4z&=&-4& \\
\hline
&&&&&-z&=&-2& \\
&&&&&z&=&2&
\end{array}
& \hspace{0.25in}
\begin{array}{rrrrr}
\\ \\ \\ \\ \\ \\ \\ \\
2x&+&z&=&6 \\
2x&+&2&=&6 \\
&-&2&&-2 \\
\hline
&&\dfrac{2x}{2}&=&\dfrac{4}{2} \\ \\
&&x&=&2 \\ \\
2y&-&4z&=&-4 \\
2y&-&4(2)&=&-4 \\
2y&-&8&=&-4 \\
&+&8&&+8 \\
\hline
&&\dfrac{2y}{2}&=&\dfrac{4}{2} \\ \\
&&y&=&2
\end{array}
\end{array}\)
