Answer Key 5.4
- [latex]\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}[/latex]
- [latex]\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}[/latex]
- [latex]\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}[/latex]
- [latex]\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}[/latex]
- [latex]\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}[/latex]
- [latex]\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}[/latex]
- [latex]\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}[/latex]
- [latex]\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}[/latex]
- [latex]\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}[/latex]
- [latex]\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}[/latex]
- [latex]\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}[/latex]
- [latex]\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}[/latex]
- [latex]\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}[/latex]
- [latex]\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}[/latex]
- [latex]\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}[/latex]
- [latex]\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}[/latex]
