{"id":1855,"date":"2021-12-02T19:39:44","date_gmt":"2021-12-03T00:39:44","guid":{"rendered":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/back-matter\/answer-key-5-4\/"},"modified":"2022-11-02T10:37:55","modified_gmt":"2022-11-02T14:37:55","slug":"answer-key-5-4","status":"publish","type":"back-matter","link":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/back-matter\/answer-key-5-4\/","title":{"raw":"Answer Key 5.4","rendered":"Answer Key 5.4"},"content":{"raw":"
    \n \t
  1. [latex]\\phantom{a}[\/latex]\n[latex]\\begin{array}[t]{rr}\n\\begin{array}[t]{rrrrrrrrl}\n&a&-&b&+&2c&=&2& \\\\\n+&a&+&b&+&c&=&3& \\\\\n\\hline\n&&&2a&+&3c&=&5& \\\\ \\\\\n&a&-&b&+&2c&=&2& \\\\\n+&2a&+&b&-&c&=&2& \\\\\n\\hline\n&&&(3a&+&c&=&4)&(-3) \\\\\n&&&-9a&-&3c&=&-12& \\\\ \\\\\n&&&-9a&-&3c&=&-12& \\\\\n+&&&2a&+&3c&=&5& \\\\\n\\hline\n&&&&&\\dfrac{-7a}{-7}&=&\\dfrac{-7}{-7}& \\\\\n&&&&&a&=&1&\n\\end{array}\n&\\hspace{0.25in}\n\\begin{array}[t]{rrrrrrrl}\\\\\n&&3a&+&c&=&4& \\\\\n&&3(1)&+&c&=&4& \\\\\n&&3&+&c&=&4& \\\\\n&&-3&&&&-3& \\\\\n\\hline\n&&&&c&=&1& \\\\ \\\\\na&+&b&+&c&=&3& \\\\\n(1)&+&b&+&(1)&=&3& \\\\\n&&b&+&2&=&3& \\\\\n&&&-&2&&-2& \\\\\n\\hline\n&&&&b&=&1&\n\\end{array}\n\\end{array}[\/latex]<\/li>\n \t
  2. [latex]\\phantom{a}[\/latex]\n[latex]\\begin{array}[t]{rr}\n\\begin{array}[t]{rrrrrrrrl}\n&2a&+&3b&-&c&=&12& \\\\\n+&3a&+&4b&+&c&=&19& \\\\\n\\hline\n&&&5a&+&7b&=&31& \\\\ \\\\\n&2a&+&3b&-&c&=&12& \\\\\n+&a&-&2b&+&c&=&-3& \\\\\n\\hline\n&&&(3a&+&b&=&9)&(-7) \\\\\n&&&-21a&-&7b&=&-63& \\\\ \\\\\n&&&5a&+&7b&=&31& \\\\\n+&&&-21a&-&7b&=&-63& \\\\\n\\hline\n&&&&&\\dfrac{-16a}{-16}&=&\\dfrac{-32}{-16}& \\\\ \\\\\n&&&&&a&=&2&\n\\end{array}\n&\\hspace{0.25in}\n\\begin{array}[t]{rrrrrrr}\n&&3a&+&b&=&9 \\\\\n&&3(2)&+&b&=&9 \\\\\n&&6&+&b&=&9 \\\\\n&&-6&&&&-6 \\\\\n\\hline\n&&&&b&=&3 \\\\ \\\\\na&-&2b&+&c&=&-3 \\\\\n(2)&-&2(3)&+&c&=&-3 \\\\\n2&-&6&+&c&=&-3 \\\\\n&&-4&+&c&=&-3 \\\\\n&&+4&&&&+4 \\\\\n\\hline\n&&&&c&=&1\n\\end{array}\n\\end{array}[\/latex]<\/li>\n \t
  3. [latex]\\phantom{a}[\/latex]\n[latex]\\begin{array}[t]{rr}\n\\begin{array}[t]{rrrrrrrrl}\n&(3x&+&y&-&z&=&7)&(-1) \\\\\n&-3x&-&y&+&z&=&-7& \\\\\n+&x&+&3y&-&z&=&5& \\\\\n\\hline\n&&&(-2x&+&2y&=&-2)&(\\div 2) \\\\\n&&&(-x&+&y&=&-1)&(7) \\\\\n&&&-7x&+&7y&=&-7& \\\\ \\\\\n&(3x&+&y&-&z&=&7)&(2) \\\\\n&6x&+&2y&-&2z&=&14& \\\\\n+&x&+&y&+&2z&=&3& \\\\\n\\hline\n&&&7x&+&3y&=&17& \\\\\n+&&&-7x&+&7y&=&-7& \\\\\n\\hline\n&&&&&\\dfrac{10y}{10}&=&\\dfrac{10}{10}& \\\\ \\\\\n&&&&&y&=&1& \\\\\n\\end{array}\n&\\hspace{0.25in}\n\\begin{array}[t]{rrrrrrr}\n&&-x&+&y&=&-1 \\\\\n&&-x&+&(1)&=&-1 \\\\\n&&-x&+&1&=&-1 \\\\\n&&&-&1&&-1 \\\\\n\\hline\n&&&&-x&=&-2 \\\\\n&&&&x&=&2 \\\\ \\\\\nx&+&y&+&2z&=&3 \\\\\n(2)&+&(1)&+&2z&=&3 \\\\\n&&2z&+&3&=&3 \\\\\n&&&-&3&&-3 \\\\\n\\hline\n&&&&2z&=&0 \\\\\n&&&&z&=&0 \\\\\n\\end{array}\n\\end{array}[\/latex]<\/li>\n \t
  4. [latex]\\phantom{a}[\/latex]\n[latex]\\begin{array}[t]{rr}\n\\begin{array}[t]{rrrrrrrrl}\n&x&+&y&+&z&=&4&(-1) \\\\\n&-x&-&y&-&z&=&-4& \\\\ \\\\\n&-x&-&y&-&z&=&-4& \\\\\n+&x&+&2y&+&3z&=&10& \\\\\n\\hline\n&&&(y&+&2z&=&6)&(2) \\\\\n&&&2y&+&4z&=&12& \\\\ \\\\\n&-x&-&y&-&z&=&-4& \\\\\n+&x&-&y&+&4z&=&20& \\\\\n\\hline\n&&&-2y&+&3z&=&16& \\\\\n+&&&2y&+&4z&=&12& \\\\\n\\hline\n&&&&&\\dfrac{7z}{7}&=&\\dfrac{28}{7}& \\\\ \\\\\n&&&&&z&=&4&\n\\end{array}\n& \\hspace{0.25in}\n\\begin{array}[t]{rrrrrrr}\n&&y&+&2z&=&6 \\\\\n&&y&+&2(4)&=&6 \\\\\n&&y&+&8&=&6 \\\\\n&&&-&8&&-8 \\\\\n\\hline\n&&&&y&=&-2 \\\\ \\\\\nx&+&y&+&z&=&4 \\\\\nx&+&(-2)&+&(4)&=&4 \\\\\n&&x&+&2&=&4 \\\\\n&&&-&2&&-2 \\\\\n\\hline\n&&&&x&=&2\n\\end{array}\n\\end{array}[\/latex]<\/li>\n \t
  5. [latex]\\phantom{a}[\/latex]\n[latex]\\begin{array}[t]{rr}\n\\begin{array}[t]{rrrrrrrrl}\n&x&+&2y&-&z&=&0& \\\\\n+&3x&-&2y&-&4z&=&-5& \\\\\n\\hline\n&&&4x&-&5z&=&-5& \\\\ \\\\\n&(2x&-&y&+&z&=&15)&(2) \\\\\n&4x&-&2y&+&2z&=&30& \\\\\n+&x&+&2y&-&z&=&0& \\\\\n\\hline\n&&&(5x&+&z&=&30)&(5) \\\\\n&&&25x&+&5z&=&150& \\\\ \\\\\n&&&4x&-&5z&=&-5& \\\\\n+&&&25x&+&5z&=&150& \\\\\n\\hline\n&&&&&\\dfrac{29x}{29}&=&\\dfrac{145}{29}& \\\\ \\\\\n&&&&&x&=&5& \\\\\n\\end{array}\n& \\hspace{0.25in}\n\\begin{array}[t]{rrrrrrr}\\\\ \\\\ \\\\\n&&5x&+&z&=&30 \\\\\n&&5(5)&+&z&=&30 \\\\\n&&25&+&z&=&30 \\\\\n&&-25&&&&-25 \\\\\n\\hline\n&&&&z&=&5 \\\\ \\\\\nx&+&2y&-&z&=&0 \\\\\n(5)&+&2y&-&(5)&=&0 \\\\\n5&+&2y&-&5&=&0 \\\\\n&&&&2y&=&0 \\\\\n&&&&y&=&0 \\\\\n\\end{array}\n\\end{array}[\/latex]<\/li>\n \t
  6. [latex]\\phantom{a}[\/latex]\n[latex]\\begin{array}[t]{rr}\n\\begin{array}[t]{rrrrrrrrl}\n&(x&-&y&+&2z&=&-3)&(2) \\\\\n&2x&-&2y&+&4z&=&-6& \\\\\n+&x&+&2y&+&3z&=&4& \\\\\n\\hline\n&&&(3x&+&7z&=&-2)&(-1) \\\\\n&&&-3x&-&7z&=&2& \\\\ \\\\\n&2x&+&y&+&z&=&-3& \\\\\n+&x&-&y&+&2z&=&-3& \\\\\n\\hline\n&&&3x&+&3z&=&-6& \\\\\n+&&&-3x&-&7z&=&2& \\\\\n\\hline\n&&&&&\\dfrac{-4z}{-4}&=&\\dfrac{-4}{-4}& \\\\ \\\\\n&&&&&z&=&1& \\\\\n\\end{array}\n&\\hspace{0.25in}\n\\begin{array}[t]{rrrrrrr}\n&&3x&+&3z&=&-6 \\\\\n&&3x&+&3(1)&=&-6 \\\\\n&&3x&+&3&=&-6 \\\\\n&&&-&3&&-3 \\\\\n\\hline\n&&&&\\dfrac{3x}{3}&=&\\dfrac{-9}{3} \\\\ \\\\\n&&&&x&=&-3 \\\\ \\\\\nx&-&y&+&2z&=&-3 \\\\\n(-3)&-&y&+&2(1)&=&-3 \\\\\n-3&-&y&+&2&=&-3 \\\\\n&&-y&-&1&=&-3 \\\\\n&&&+&1&&+1 \\\\\n\\hline\n&&&&-y&=&-2 \\\\\n&&&&y&=&2\n\\end{array}\n\\end{array}[\/latex]<\/li>\n \t
  7. [latex]\\phantom{a}[\/latex]\n[latex]\\begin{array}[t]{rr}\n\\begin{array}[t]{rrrrrrrrl}\n&x&+&y&+&z&=&6& \\\\\n+&2x&-&y&-&z&=&-3& \\\\\n\\hline\n&&&&&\\dfrac{3x}{3}&=&\\dfrac{3}{3}& \\\\ \\\\\n&&&&&x&=&1& \\\\ \\\\\n&x&-&2y&+&3z&=&6& \\\\\n&(1)&-&2y&+&3z&=&6& \\\\\n&1&-&2y&+&3z&=&6& \\\\\n&-1&&&&&&-1& \\\\\n\\hline\n&&&-2y&+&3z&=&5& \\\\ \\\\\n&x&+&y&+&z&=&6& \\\\\n&(1)&+&y&+&z&=&6& \\\\\n&1&+&y&+&z&=&6& \\\\\n&-1&&&&&&-1& \\\\\n\\hline\n&&&(y&+&z&=&5)&(2) \\\\\n&&&2y&+&2z&=&10&\n\\end{array}\n& \\hspace{0.25in}\n\\begin{array}[t]{rrrrrrr}\n&&-2y&+&3z&=&5 \\\\\n+&&2y&+&2z&=&10 \\\\\n\\hline\n&&&&\\dfrac{5z}{5}&=&\\dfrac{15}{5} \\\\ \\\\\n&&&&z&=&3 \\\\ \\\\\nx&+&y&+&z&=&6 \\\\\n(1)&+&y&+&(3)&=&6 \\\\\n1&+&y&+&3&=&6 \\\\\n&&y&+&4&=&6 \\\\\n&&&-&4&&-4 \\\\\n\\hline\n&&&&y&=&2 \\\\\n\\end{array}\n\\end{array}[\/latex]<\/li>\n \t
  8. [latex]\\phantom{a}[\/latex]\n[latex]\\begin{array}[t]{rr}\n\\begin{array}[t]{rrrrrrrrl}\n&x&+&y&-&z&=&0& \\\\\n+&2x&+&y&+&z&=&0& \\\\\n\\hline\n&&&3x&+&2y&=&0& \\\\ \\\\\n&(x&+&y&-&z&=&0)&(-4) \\\\\n&-4x&-&4y&+&4z&=&0& \\\\\n+&x&+&2y&-&4z&=&0& \\\\\n\\hline\n&&&-3x&-&2y&=&0& \\\\\n\\end{array}\n& \\hspace{0.25in}\n\\begin{array}[t]{rrrrrr}\n&-3x&-&2y&=&0 \\\\\n+&3x&+&2y&=&0 \\\\\n\\hline\n&&&0&=&0 \\\\ \\\\\n&&\\therefore &x&=&0 \\\\\n&&&y&=&0 \\\\\n&&&z&=&0 \\\\\n\\end{array}\n\\end{array}[\/latex]<\/li>\n \t
  9. [latex]\\phantom{a}[\/latex]\n[latex]\\begin{array}[t]{rr}\n\\begin{array}[t]{rrrrrrrrl}\n&x&+&y&+&z&=&2& \\\\\n+&2x&-&y&+&3z&=&9& \\\\\n\\hline\n&&&3x&+&4z&=&11& \\\\ \\\\\n&2x&-&y&+&3z&=&9& \\\\\n+&&&y&-&z&=&-3& \\\\\n\\hline\n&&&(2x&+&2z&=&6)&(-2) \\\\\n&&&-4x&-&4z&=&-12& \\\\ \\\\\n&&&3x&+&4z&=&11& \\\\\n+&&&-4x&-&4z&=&-12& \\\\\n\\hline\n&&&&&-x&=&-1& \\\\\n&&&&&x&=&1&\n\\end{array}\n& \\hspace{0.25in}\n\\begin{array}[t]{rrrrrrr}\n&&2x&+&2z&=&6 \\\\\n&&2(1)&+&2z&=&6 \\\\\n&&2&+&2z&=&6 \\\\\n&&-2&&&&-2 \\\\\n\\hline\n&&&&\\dfrac{2z}{2}&=&\\dfrac{4}{2} \\\\ \\\\\n&&&&z&=&2 \\\\ \\\\\nx&+&y&+&z&=&2 \\\\\n(1)&+&y&+&(2)&=&2 \\\\\n&&y&+&3&=&2 \\\\\n&&&-&3&&-3 \\\\\n\\hline\n&&&&y&=&-1 \\\\\n\\end{array}\n\\end{array}[\/latex]<\/li>\n \t
  10. [latex]\\phantom{a}[\/latex]\n[latex]\\begin{array}[t]{rr}\n\\begin{array}[t]{rrrrrrrrl}\n&(4x&&&+&z&=&3)&(2) \\\\\n&8x&&&+&2z&=&6& \\\\\n+&6x&-&y&-&2z&=&-1& \\\\\n\\hline\n&&&(14x&-&y&=&5)&(3) \\\\\n&&&42x&-&3y&=&15& \\\\\n+&&&-2x&+&3y&=&5& \\\\\n\\hline\n&&&&&\\dfrac{40x}{40}&=&\\dfrac{20}{40}& \\\\ \\\\\n&&&&&x&=&\\dfrac{1}{2}&\n\\end{array}\n& \\hspace{0.25in}\n\\begin{array}[t]{rrrrrrr}\n&&-2x&+&3y&=&5 \\\\\n&&-2\\left(\\dfrac{1}{2}\\right)&+&3y&=&5 \\\\\n&&-1&+&3y&=&5 \\\\\n&&+1&&&&+1 \\\\\n\\hline\n&&&&\\dfrac{3y}{3}&=&\\dfrac{6}{3} \\\\ \\\\\n&&&&y&=&2 \\\\ \\\\\n&&4x&+&z&=&3 \\\\\n&&4\\left(\\dfrac{1}{2}\\right)&+&z&=&3 \\\\\n&&2&+&z&=&3 \\\\\n&&-2&&&&-2 \\\\\n\\hline\n&&&&z&=&1 \\\\\n\\end{array}\n\\end{array}[\/latex]<\/li>\n \t
  11. [latex]\\phantom{a}[\/latex]\n[latex]\\begin{array}[t]{rr}\n\\begin{array}[t]{rrrrrrrrl}\n&&&x&-&z&=&-2& \\\\\n+&&&y&+&z&=&5& \\\\\n\\hline\n&&&x&+&y&=&3& \\\\ \\\\\n&2x&-&3y&+&z&=&-1& \\\\\n+&x&&&-&z&=&-2& \\\\\n\\hline\n&&&(3x&-&3y&=&-3)&(\\div 3) \\\\\n&&&x&-&y&=&-1& \\\\\n+&&&x&+&y&=&3& \\\\\n\\hline\n&&&&&\\dfrac{2x}{2}&=&\\dfrac{2}{2}& \\\\ \\\\\n&&&&&x&=&1&\n\\end{array}\n& \\hspace{0.25in}\n\\begin{array}[t]{rrrrr}\nx&-&z&=&-2 \\\\\n(1)&-&z&=&-2 \\\\\n1&-&z&=&-2 \\\\\n-1&&&&-1 \\\\\n\\hline\n&&-z&=&-3 \\\\\n&&z&=&3 \\\\ \\\\\ny&+&z&=&5 \\\\\ny&+&(3)&=&5 \\\\\ny&+&3&=&5 \\\\\n&-&3&&-3 \\\\\n\\hline\n&&y&=&2 \\\\\n\\end{array}\n\\end{array}[\/latex]<\/li>\n \t
  12. [latex]\\phantom{a}[\/latex]\n[latex]\\begin{array}[t]{rr}\n\\begin{array}[t]{rrrrrrrrl}\n&(3x&+&4y&-&z&=&11)&(2) \\\\\n&6x&+&8y&-&2z&=&22& \\\\\n+&&&y&+&2z&=&-4& \\\\\n\\hline\n&&&(6x&+&9y&=&18)&(\\div 3) \\\\\n&&&2x&+&3y&=&6& \\\\\n+&&&-2x&+&y&=&-6& \\\\\n\\hline\n&&&&&4y&=&0& \\\\\n&&&&&y&=&0& \\\\ \\\\\n\\end{array}\n& \\hspace{0.25in}\n\\begin{array}[t]{rrrrr}\n-2x&+&y&=&-6 \\\\\n-2x&+&0&=&-6 \\\\\n&&\\dfrac{-2x}{-2}&=&\\dfrac{-6}{-2} \\\\ \\\\\n&&x&=&3 \\\\ \\\\\ny&+&2z&=&-4 \\\\\n0&+&2z&=&-4 \\\\\n&&\\dfrac{2z}{2}&=&\\dfrac{-4}{2} \\\\ \\\\\n&&z&=&-2\n\\end{array}\n\\end{array}[\/latex]<\/li>\n \t
  13. [latex]\\phantom{a}[\/latex]\n[latex]\\begin{array}[t]{rr}\n\\begin{array}[t]{rrrrrrrrl}\n&&&(-2y&+&z&=&-6)&(3) \\\\\n&&&-6y&+&3z&=&-18& \\\\\n+&x&+&6y&+&3z&=&30& \\\\\n\\hline\n&&&(x&+&6z&=&12)&(-2) \\\\\n&&&-2x&-&12z&=&-24& \\\\\n+&&&2x&+&2z&=&4& \\\\\n\\hline\n&&&&&\\dfrac{-10z}{-10}&=&\\dfrac{-20}{-10}& \\\\ \\\\\n&&&&&z&=&2&\n\\end{array}\n& \\hspace{0.25in}\n\\begin{array}[t]{rrrrr}\n2x&+&2z&=&4 \\\\\n2x&+&2(2)&=&4 \\\\\n2x&+&4&=&4 \\\\\n&-&4&&-4 \\\\\n\\hline\n&&2x&=&0 \\\\\n&&x&=&0 \\\\ \\\\\n-2y&+&z&=&-6 \\\\\n-2y&+&2&=&-6 \\\\\n&-&2&&-2 \\\\\n\\hline\n&&\\dfrac{-2y}{-2}&=&\\dfrac{-8}{-2} \\\\ \\\\\n&&y&=&4\n\\end{array}\n\\end{array}[\/latex]<\/li>\n \t
  14. [latex]\\phantom{a}[\/latex]\n[latex]\\begin{array}[t]{rr}\n\\begin{array}[t]{rrrrrrrrl}\n&(x&-&y&+&2z&=&0)&(2) \\\\\n&2x&-&2y&+&4z&=&0& \\\\\n+&x&+&2y&&&=&1& \\\\\n\\hline\n&&&3x&+&4z&=&1& \\\\ \\\\\n&&&(2x&+&z&=&4)&(-4) \\\\\n&&&-8x&-&4z&=&-16& \\\\\n+&&&3x&+&4z&=&1& \\\\\n\\hline\n&&&&&\\dfrac{-5x}{-5}&=&\\dfrac{-15}{-5}& \\\\ \\\\\n&&&&&x&=&3&\n\\end{array}\n& \\hspace{0.25in}\n\\begin{array}[t]{rrrrr}\nx&+&2y&=&1 \\\\\n3&+&2y&=&1 \\\\\n-3&&&&-3 \\\\\n\\hline\n&&\\dfrac{2y}{2}&=&\\dfrac{-2}{2} \\\\ \\\\\n&&y&=&-1 \\\\ \\\\\n2x&+&z&=&4 \\\\\n2(3)&+&z&=&4 \\\\\n6&+&z&=&4 \\\\\n-6&&&&-6 \\\\\n\\hline\n&&z&=&-2\n\\end{array}\n\\end{array}[\/latex]<\/li>\n \t
  15. [latex]\\phantom{a}[\/latex]\n[latex]\\begin{array}[t]{rr}\n\\begin{array}[t]{rrrrrrrr}\n&x&+&y&+&z&=&4 \\\\\n+&&-&y&-&z&=&-4 \\\\\n\\hline\n&&&&&x&=&0\n\\end{array}\n& \\hspace{0.25in}\n\\begin{array}[t]{rrrrr}\nx&-&2y&=&0 \\\\\n0&-&2y&=&0 \\\\\n&&-2y&=&0 \\\\\n&&y&=&0 \\\\ \\\\\n-y&-&z&=&-4 \\\\\n0&-&z&=&-4 \\\\\n&&-z&=&-4 \\\\\n&&z&=&4\n\\end{array}\n\\end{array}[\/latex]<\/li>\n \t
  16. [latex]\\phantom{a}[\/latex]\n[latex]\\begin{array}[t]{rr}\n\\begin{array}[t]{rrrrrrrrl}\n&(x&+&y&-&z&=&2)&(-2) \\\\\n&-2x&-&2y&+&2z&=&-4& \\\\\n+&2x&&&+&z&=&6& \\\\\n\\hline\n&&&-2y&+&3z&=&2& \\\\\n+&&&2y&-&4z&=&-4& \\\\\n\\hline\n&&&&&-z&=&-2& \\\\\n&&&&&z&=&2&\n\\end{array}\n& \\hspace{0.25in}\n\\begin{array}[t]{rrrrr}\n2x&+&z&=&6 \\\\\n2x&+&2&=&6 \\\\\n&-&2&&-2 \\\\\n\\hline\n&&\\dfrac{2x}{2}&=&\\dfrac{4}{2} \\\\ \\\\\n&&x&=&2 \\\\ \\\\\n2y&-&4z&=&-4 \\\\\n2y&-&4(2)&=&-4 \\\\\n2y&-&8&=&-4 \\\\\n&+&8&&+8 \\\\\n\\hline\n&&\\dfrac{2y}{2}&=&\\dfrac{4}{2} \\\\ \\\\\n&&y&=&2\n\\end{array}\n\\end{array}[\/latex]<\/li>\n<\/ol>","rendered":"
      \n
    1. [latex]\\phantom{a}[\/latex]
      \n[latex]\\begin{array}[t]{rr} \\begin{array}[t]{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}[t]{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]<\/li>\n
    2. [latex]\\phantom{a}[\/latex]
      \n[latex]\\begin{array}[t]{rr} \\begin{array}[t]{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}[t]{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]<\/li>\n
    3. [latex]\\phantom{a}[\/latex]
      \n[latex]\\begin{array}[t]{rr} \\begin{array}[t]{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}[t]{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]<\/li>\n
    4. [latex]\\phantom{a}[\/latex]
      \n[latex]\\begin{array}[t]{rr} \\begin{array}[t]{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}[t]{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]<\/li>\n
    5. [latex]\\phantom{a}[\/latex]
      \n[latex]\\begin{array}[t]{rr} \\begin{array}[t]{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}[t]{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]<\/li>\n
    6. [latex]\\phantom{a}[\/latex]
      \n[latex]\\begin{array}[t]{rr} \\begin{array}[t]{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}[t]{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]<\/li>\n
    7. [latex]\\phantom{a}[\/latex]
      \n[latex]\\begin{array}[t]{rr} \\begin{array}[t]{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}[t]{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]<\/li>\n
    8. [latex]\\phantom{a}[\/latex]
      \n[latex]\\begin{array}[t]{rr} \\begin{array}[t]{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}[t]{rrrrrr} &-3x&-&2y&=&0 \\\\ +&3x&+&2y&=&0 \\\\ \\hline &&&0&=&0 \\\\ \\\\ &&\\therefore &x&=&0 \\\\ &&&y&=&0 \\\\ &&&z&=&0 \\\\ \\end{array} \\end{array}[\/latex]<\/li>\n
    9. [latex]\\phantom{a}[\/latex]
      \n[latex]\\begin{array}[t]{rr} \\begin{array}[t]{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}[t]{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]<\/li>\n
    10. [latex]\\phantom{a}[\/latex]
      \n[latex]\\begin{array}[t]{rr} \\begin{array}[t]{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}[t]{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]<\/li>\n
    11. [latex]\\phantom{a}[\/latex]
      \n[latex]\\begin{array}[t]{rr} \\begin{array}[t]{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}[t]{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]<\/li>\n
    12. [latex]\\phantom{a}[\/latex]
      \n[latex]\\begin{array}[t]{rr} \\begin{array}[t]{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}[t]{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]<\/li>\n
    13. [latex]\\phantom{a}[\/latex]
      \n[latex]\\begin{array}[t]{rr} \\begin{array}[t]{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}[t]{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]<\/li>\n
    14. [latex]\\phantom{a}[\/latex]
      \n[latex]\\begin{array}[t]{rr} \\begin{array}[t]{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}[t]{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]<\/li>\n
    15. [latex]\\phantom{a}[\/latex]
      \n[latex]\\begin{array}[t]{rr} \\begin{array}[t]{rrrrrrrr} &x&+&y&+&z&=&4 \\\\ +&&-&y&-&z&=&-4 \\\\ \\hline &&&&&x&=&0 \\end{array} & \\hspace{0.25in} \\begin{array}[t]{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]<\/li>\n
    16. [latex]\\phantom{a}[\/latex]
      \n[latex]\\begin{array}[t]{rr} \\begin{array}[t]{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}[t]{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]<\/li>\n<\/ol>\n","protected":false},"author":90,"menu_order":45,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":"cc-by-nc-sa"},"back-matter-type":[],"contributor":[],"license":[56],"class_list":["post-1855","back-matter","type-back-matter","status-publish","hentry","license-cc-by-nc-sa"],"_links":{"self":[{"href":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-json\/pressbooks\/v2\/back-matter\/1855","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-json\/pressbooks\/v2\/back-matter"}],"about":[{"href":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-json\/wp\/v2\/types\/back-matter"}],"author":[{"embeddable":true,"href":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-json\/wp\/v2\/users\/90"}],"version-history":[{"count":1,"href":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-json\/pressbooks\/v2\/back-matter\/1855\/revisions"}],"predecessor-version":[{"id":1856,"href":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-json\/pressbooks\/v2\/back-matter\/1855\/revisions\/1856"}],"metadata":[{"href":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-json\/pressbooks\/v2\/back-matter\/1855\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-json\/wp\/v2\/media?parent=1855"}],"wp:term":[{"taxonomy":"back-matter-type","embeddable":true,"href":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-json\/pressbooks\/v2\/back-matter-type?post=1855"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-json\/wp\/v2\/contributor?post=1855"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-json\/wp\/v2\/license?post=1855"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}