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Answer Key 5.3

  1. \(\begin{array}{rrrrrr}
    \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\
    &4x&+&2y&=&0 \\
    +&-4x&-&9y&=&-28 \\
    \hline
    &&&\dfrac{-7y}{-7}&=&\dfrac{-28}{-7} \\ \\
    &&&y&=&4 \\ \\
    &4x&+&2(4)&=&0 \\
    &4x&+&8&=&0 \\
    &&-&8&&-8 \\
    \hline
    &&&\dfrac{4x}{4}&=&\dfrac{-8}{4} \\ \\
    &&&x&=&-2 \\
    (-2,4)&&&&&
    \end{array}\)
  2. \(\begin{array}{rrrcrr}
    \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\
    &-7x&+&y&=&-10 \\
    +&-9x&-&y&=&-22 \\
    \hline
    &&&\dfrac{-16x}{-16}&=&\dfrac{-32}{-16} \\ \\
    &&&x&=&2 \\ \\
    &-7(2)&+&y&=&-10 \\
    &-14&+&y&=&-10 \\
    &+14&&&&+14 \\
    \hline
    &&&y&=&4 \\
    (2,4)&&&&&
    \end{array}\)
  3. \(\begin{array}{rrrrrr}
    \\ \\
    &-9x&+&5y&=&-22 \\
    +&9x&-&5y&=&13 \\
    \hline
    &&&0&=&-9
    \end{array}\)∴ \(\text{Two parallel lines. No solution}\)
  4. \(\begin{array}{rrrrrr}
    \\ \\
    &-x&-&2y&=&-7 \\
    +&x&+&2y&=&7 \\
    \hline
    &&&0&=&0
    \end{array}\)∴ \(\text{Two identical lines. Infinite solutions}\)
  5. \(\begin{array}{rrrrrr}
    \\ \\
    &-6x&+&9y&=&3 \\
    +&6x&-&9y&=&-9 \\
    \hline
    &&&0&=&-6
    \end{array}\)∴ \(\text{Two parallel lines. No solution}\)
  6. \(\begin{array}{rrrrrrr}
    \\ \\ \\ \\ \\
    &5x&-&5y&=&-15&(\div 5) \\
    &(x&-&y&=&-3)&(-1) \\ \\
    &x&-&y&=&-3& \\
    +&-x&+&y&=&3& \\
    \hline
    &&&0&=&0& \\
    \end{array}\)∴ \(\text{Two identical lines. Infinite solutions}\)
  7. \(\begin{array}{rrrrrr}
    \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\
    &4x&-&6y&=&-10 \\
    +&4x&+&6y&=&-14 \\
    \hline
    &&&\dfrac{8x}{8}&=&\dfrac{-24}{8} \\ \\
    &&&x&=&-3 \\ \\
    &4(-3)&-&6y&=&-10 \\
    &-12&-&6y&=&-10 \\
    &+12&&&&+12 \\
    \hline
    &&&\dfrac{-6y}{-6}&=&\dfrac{2}{-6} \\ \\
    &&&y&=&-\dfrac{1}{3} \\
    (-3, -\dfrac{1}{3})&&&&&
    \end{array}\)
  8. \(\begin{array}{rrrrrrrl}
    \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\
    &-3x&+&3y&=&-12&\div &(-3) \\
    &-3x&+&9y&=&-24&\div &(3) \\ \\
    & x&-&y&=&4&& \\
    +&-x&+&3y&=&-8&& \\
    \hline
    &&&\dfrac{2y}{2}&=&\dfrac{-4}{2}&& \\ \\
    &&&y&=&-2&& \\ \\
    &\therefore x&-&y&=&4&& \\
    &x&-&-2&=&4&& \\
    &x&+&2&=&4&& \\
    &&-&2&&-2&& \\
    \hline
    &&&x&=&2&& \\
    (2,-2)&&&&&&&
    \end{array}\)
  9. \(\begin{array}{rrrrrrr}
    \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\
    &(-x&-&5y&=&28)&(-1) \\ \\
    &x&+&5y&=&-28& \\
    +&-x&+&4y&=&-17& \\
    \hline
    &&&\dfrac{9y}{9}&=&\dfrac{-45}{9}& \\ \\
    &&&y&=&-5& \\ \\
    &x&+&5(-5)&=&-28& \\
    &x&-&25&=&-28& \\
    &&+&25&&+25& \\
    \hline
    &&&x&=&-3& \\
    (-3,-5)&&&&&&
    \end{array}\)
  10. \(\begin{array}{rrrrrrr}
    \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\
    &(-10x&-&5y&=&0)&(-1) \\ \\
    &10x&+&5y&=&0& \\
    +&-10x&-&10y&=&-30& \\
    \hline
    &&&\dfrac{-5y}{-5}&=&\dfrac{-30}{-5}& \\ \\
    &&&y&=&6& \\ \\
    &10x&+&5(6)&=&0& \\
    &10x&+&30&=&0& \\
    &&-&30&&-30& \\
    \hline
    &&&\dfrac{10x}{10}&=&\dfrac{-30}{10}& \\ \\
    &&&x&=&-3& \\
    (-3,6)&&&&&&
    \end{array}\)
  11. \(\begin{array}{rrrrrrr}
    \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\
    &(2x&-&y&=&5)&(2) \\ \\
    &4x&-&2y&=&10& \\
    +&5x&+&2y&=&-28& \\
    \hline
    &&&\dfrac{9x}{9}&=&\dfrac{-18}{9}& \\ \\
    &&&x&=&-2& \\ \\
    &2(x)&-&y&=&5& \\
    &2(-2)&-&y&=&5& \\
    &-4&-&y&=&5& \\
    &+4&&&&+4& \\
    \hline
    &&&-y&=&9& \\ \\
    &&&y&=&-9& \\
    (-2,-9)&&&&&&
    \end{array}\)
  12. \(\begin{array}{rrrrrrr}
    \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\
    &-5x&+&6y&=&-17& \\
    &(x&-&2y&=&5)&(3) \\ \\
    &-5x&+&6y&=&-17& \\
    +&3x&-&6y&=&15& \\
    \hline
    &&&\dfrac{-2x}{-2}&=&\dfrac{-2}{-2}& \\ \\
    &&&x&=&1& \\ \\
    &x&-&2y&=&5& \\
    &1&-&2y&=&5& \\
    &-1&&&&-1& \\
    \hline
    &&&\dfrac{-2y}{-2}&=&\dfrac{4}{-2}& \\ \\
    &&&y&=&-2& \\
    (1,-2)&&&&&&
    \end{array}\)
  13. \(\begin{array}{rrrrrrr}
    \\ \\ \\ \\ \\ \\ \\ \\ \\ \\
    &(10x&+&6y&=&24)&(\div 2) \\
    &(-6x&+&y&=&4)&(-3) \\ \\
    &5x&+&3y&=&12& \\
    +&18x&-&3y&=&-12& \\
    \hline
    &&&23x&=&0& \\
    &&&x&=&0& \\ \\
    &-6(x)&+&y&=&4& \\
    &-6(0)&+&y&=&4& \\
    &&&y&=&4& \\
    (0,4)&&&&&&
    \end{array}\)
  14. \(\begin{array}{rrrrrrr}
    \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\
    &(10x&+&6y&=&-10)&(\div -2) \\ \\
    &x&+&3y&=&-1& \\
    +&-5x&-&3y&=&5& \\
    \hline
    &&&\dfrac{-4x}{-4}&=&\dfrac{4}{-4}& \\ \\
    &&&x&=&-1& \\ \\
    &-1&+&3y&=&-1& \\
    &+1&&&&+1& \\
    \hline
    &&&3y&=&0& \\ \\
    &&&y&=&0& \\
    (-1,0)&&&&&&
    \end{array}\)
  15. \(\begin{array}{rrrrrrl}
    \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\
    &(2x&+&4y&=&24)&(\div 2) \\
    &(4x&-&12y&=&8)&(\div -4) \\ \\
    &x&+&2y&=&12& \\
    +&-x&+&3y&=&-2& \\
    \hline
    &&&\dfrac{5y}{5}&=&\dfrac{10}{5}& \\ \\
    &&&y&=&2& \\ \\
    &x&+&2(y)&=&12& \\
    &x&+&2(2)&=&12& \\
    &x&+&4&=&12& \\
    &&-&4&&-4& \\
    \hline
    &&&x&=&8& \\
    (8,2)&&&&&&
    \end{array}\)
  16. \(\begin{array}{rrrrrrl}
    \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\
    &(-6x&+&4y&=&12)&(\div 2) \\
    &(12x&+&6y&=&18)&(\div -3) \\ \\
    &-3x&+&2y&=&6& \\
    +&-4x&-&2y&=&-6& \\
    \hline
    &&&-7x&=&0& \\
    &&&x&=&0& \\ \\
    &\dfrac{-3(x)}{2}&+&\dfrac{2y}{2}&=&\dfrac{6}{2}& \\ \\
    &\dfrac{-3(0)}{2}&+&\dfrac{2y}{2}&=&\dfrac{6}{2}& \\ \\
    &&&y&=&3& \\
    (0,3)&&&&&&
    \end{array}\)
  17. \(\begin{array}{rrrrrrr}
    \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\
    &(10x&-&8y&=&-8)&(\div 2) \\ \\
    &-7x&+&4y&=&-4& \\
    +&5x&-&4y&=&-4& \\
    \hline
    &&&\dfrac{-2x}{-2}&=&\dfrac{-8}{-2}& \\ \\
    &&&x&=&4& \\ \\
    &5(4)&-&4y&=&-4& \\
    &20&-&4y&=&-4& \\
    &-20&&&&-20& \\
    \hline
    &&&\dfrac{-4y}{-4}&=&\dfrac{-24}{-4}& \\ \\
    &&&y&=&6& \\
    (4,6)&&&&&&
    \end{array}\)
  18. \(\begin{array}{rrrrrrr}
    \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\
    &(-6x&+&4y&=&4)&(\div 2) \\ \\
    &-3x&+&2y&=&2& \\
    +&3x&-&y&=&26& \\
    \hline
    &&&y&=&28& \\ \\
    &3x&-&28&=&26& \\
    &&+&28&&+28& \\
    \hline
    &&&\dfrac{3x}{3}&=&\dfrac{54}{3}& \\ \\
    &&&x&=&18& \\
    (18,28)&&&&&&
    \end{array}\)
  19. \(\begin{array}{rrrrrrr}
    \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\
    &(-6x&-&5y&=&-3)&(2) \\ \\
    &5x&+&10y&=&20& \\
    +&-12x&-&10y&=&-6& \\
    \hline
    &&&\dfrac{-7x}{-7}&=&\dfrac{14}{-7}& \\ \\
    &&&x&=&-2& \\ \\
    &5(-2)&+&10y&=&20& \\
    &-10&+&10y&=&20& \\
    &+10&&&&+10& \\
    \hline
    &&&\dfrac{10y}{10}&=&\dfrac{30}{10}& \\ \\
    &&&y&=&3& \\
    (-2,3)&&&&&&
    \end{array}\)
  20. \(\begin{array}{rrrrrrr}
    \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\
    &(3x&-&7y&=&-11)&(3) \\ \\
    &-9x&-&5y&=&-19& \\
    +&9x&-&21y&=&-33& \\
    \hline
    &&&\dfrac{-26y}{-26}&=&\dfrac{-52}{-26}& \\ \\
    &&&y&=&2& \\ \\
    &3x&-&7(2)&=&-11& \\
    &3x&-&14&=&-11& \\
    &&+&14&&+14& \\
    \hline
    &&&\dfrac{3x}{3}&=&\dfrac{3}{3}& \\ \\
    &&&x&=&1& \\
    (1,2)&&&&&&
    \end{array}\)
  21. \(\begin{array}{rrrrrrl}
    \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\
    &(-7x&+&5y&=&-8)&(3) \\
    &(-3x&-&3y&=&12)&(5) \\ \\
    &-21x&+&15y&=&-24& \\
    +&-15x&-&15y&=&60& \\
    \hline
    &&&\dfrac{-36x}{-36}&=&\dfrac{36}{-36}& \\ \\
    &&&x&=&-1& \\ \\
    &-3(-1)&-&3y&=&12& \\
    &3&-&3y&=&12& \\
    &-3&&&&-3& \\
    \hline
    &&&\dfrac{-3y}{-3}&=&\dfrac{9}{-3}& \\ \\
    &&&y&=&-3& \\
    (-1,-3)&&&&&&
    \end{array}\)
  22. \(\begin{array}{rrrrrrcll}
    \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\
    &&(6x&+&3y&=&-18)&\div &-3 \\ \\
    &&-2x&-&y&=&6&& \\
    &&+2x&&&&+2x&& \\
    \hline
    &&&&-y&=&2x&+&6 \\
    &&&&y&=&-2x&-&6 \\ \\
    8x&+&7(-2x&-&6)&=&-24&& \\
    8x&-&14x&-&42&=&-24&& \\
    &&&+&42&&+42&& \\
    \hline
    &&&&\dfrac{-6x}{-6}&=&\dfrac{18}{-6}&& \\ \\
    &&&&x&=&-3&& \\ \\
    &&&&y&=&-2(-3)&-&6 \\
    &&&&y&=&6&-&6 \\
    &&&&y&=&0&& \\
    (-3,0)&&&&&&&& \\
    \end{array}\)
  23. \(\begin{array}{rrrrrrrrr}
    \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\
    &&(-8x&-&8y&=&-8)&\div &(-8) \\ \\
    &&x&+&y&=&1&& \\
    &&-x&&&&-x&& \\
    \hline
    &&&&y&=&1&-&x \\ \\
    10x&+&9(1&-&x)&=&1&& \\
    10x&+&9&-&9x&=&1&& \\
    &-&9&&&&-9&& \\
    \hline
    &&&&x&=&-8&& \\ \\
    &&&&y&=&1&-&-8 \\
    &&&&y&=&9&& \\
    (-8,9)&&&&&&&&
    \end{array}\)
  24. \(\begin{array}{rrrrrrl}
    \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\
    &(-7x&+&10y&=&13)&(4) \\
    &(4x&+&9y&=&22)&(7) \\ \\
    &-28x&+&40y&=&52& \\
    +&28x&+&63y&=&154& \\
    \hline
    &&&\dfrac{103y}{103}&=&\dfrac{206}{103}& \\ \\
    &&&y&=&2& \\ \\
    &4x&+&9(2)&=&22& \\
    &4x&+&18&=&22& \\
    &&-&18&&-18& \\
    \hline
    &&&\dfrac{4x}{4}&=&\dfrac{4}{4}& \\ \\
    &&&x&=&1& \\
    (1,2)&&&&&&
    \end{array}\)

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Intermediate Algebra Copyright © 2020 by Terrance Berg is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License, except where otherwise noted.

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