Answer Key 7.9

[latexpage]

1. \(\begin{array}{rrl}
   \\
   R&=&J+10 \\
   R+4&=&2(J+4)
   \end{array}\)
2. \(\begin{array}{rrl}
   \\
   F&=&4S \\
   F+20&=&2(S+20)
   \end{array}\)
3. \(\begin{array}{rrl}
   \\
   P&=&J+20 \\
   P+2&=&2(J+2)
   \end{array}\)
4. \(\begin{array}{rrl}
   \\
   D&=&23+A \\
   D+6&=&2(A+6)
   \end{array}\)
5. \(\begin{array}{rrl}
   \\
   F&=&B+4 \\
   (F-5)+(B-5)&=&48
   \end{array}\)
6. \(\begin{array}{rrl}
   \\
   J&=&4M \\
   (J-5)+(M-5)&=&50
   \end{array}\)
7. \(\begin{array}{rrl}
   \\
   T&=&5+J \\
   (T+6)+(J+6)&=&79
   \end{array}\)
8. \(\begin{array}{rrl}
   \\
   J&=&2L \\
   (J+3)+(L+3)&=&54
   \end{array}\)
9. \(\begin{array}{rrrrrrrrr}
   \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\
   &&J&+&m&=&32&& \\
   &&&-&m&&-m&& \\
   \midrule
   &&&&J&=&32&-&m \\ \\
   &&J&-&4&=&2(m&-&4) \\
   (32&-&m)&-&4&=&2m&-&8 \\
   32&-&m&-&4&=&2m&-&8 \\
   &&28&-&m&=&2m&-&8 \\
   &&+8&+&m&&+m&+&8 \\
   \midrule
   &&&&\dfrac{36}{3}&=&\dfrac{3m}{3}&& \\ \\
   &&&&m&=&12&& \\ \\
   &&&&\therefore J&=&32&-&m \\
   &&&&J&=&32&-&12 \\
   &&&&J&=&20&&
   \end{array}\)
10. \(\begin{array}{rrrrrrrrr}
    \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\
    &&F&+&S&=&56&& \\
    &&&-&S&=&&-&S \\
    \midrule
    &&&&F&=&56&-&S \\ \\
    &&F&-&4&=&3(S&-&4) \\
    56&-&S&-&4&=&3S&-&12 \\
    &+&S&+&12&&+S&+&12 \\
    \midrule
    &&&&\dfrac{64}{4}&=&\dfrac{4S}{4}&& \\ \\
    &&&&S&=&16&& \\ \\
    &&&&\therefore F&=&56&-&S \\
    &&&&F&=&56&-&16 \\
    &&&&F&=&40&&
    \end{array}\)
11. \(\begin{array}{rr}
    \\ \\ \\ \\ \\
    \begin{array}{rrrrrrrrrl}
    \\
    &&w&+&B&=&20&&& \\
    &&-w&&&&&-&w& \\
    \midrule
    &&&&B&=&20&-&w& \\ \\
    &&B&-&4&=&\dfrac{1}{2}(w&-&4)& \\ \\
    20&-&w&-&4&=&\dfrac{1}{2}(w&-&4)& \\ \\
    &&[16&-&w&=&\dfrac{1}{2}(w&-&4)]&(2) \\
    \end{array}
    &
    \begin{array}{rrrrrrr}
    32&-&2w&=&w&-&4 \\
    +4&+&2w&&+2w&+&4 \\
    \midrule
    &&\dfrac{36}{3}&=&\dfrac{3w}{3}&& \\ \\
    &&w&=&12&& \\ \\
    &&B&=&20&-&w \\
    &&B&=&20&-&12 \\
    &&B&=&8&&
    \end{array}
    \end{array}\)
12. \(\begin{array}{rrrrrrl}
    \\ \\ \\ \\ \\ \\ \\ \\ \\ \\
    &&m&=&36&& \\
    &&D&=&3&& \\ \\
    m&+&x&=&4(D&+&x) \\
    36&+&x&=&4(3&+&x) \\
    36&+&x&=&12&+&4x \\
    -12&-&x&&-12&-&x \\
    \midrule
    &&\dfrac{24}{3}&=&\dfrac{3x}{3}&& \\ \\
    &&x&=&8&&\text{ years} \\
    \end{array}\)
13. \(\begin{array}{rrrrlrr}
    \\ \\ \\ \\ \\ \\ \\ \\ \\ \\
    &&B_\text{o}&=&2B_\text{y}&& \\ \\
    B_\text{o}&-&5&=&\phantom{-}3(B_\text{y}&-&5) \\
    2B_\text{y}&-&5&=&\phantom{-}3B_\text{y}&-&15 \\
    -3B_\text{y}&+&5&&-3B_\text{y}&+&5 \\
    \midrule
    &&-B_\text{y}&=&-10&& \\
    &&B_\text{y}&=&\phantom{-}10&& \\ \\
    &&\therefore B_\text{o}&=&2B_\text{y}&& \\
    &&B_\text{o}&=&2(10)&& \\
    &&B_\text{o}&=&20&&
    \end{array}\)
14. \(\begin{array}{rrrrrrr}
    \\ \\ \\ \\ \\ \\ \\ \\
    &&P&=&30&& \\
    &&V&=&22&& \\ \\
    P&-&x&=&2(V&-&x) \\
    30&-&x&=&2(22&-&x) \\
    30&-&x&=&44&-&2x \\
    -44&+&x&&-44&+&x \\
    \midrule
    &&-14&=&-x&& \\
    &&x&=&14&&
    \end{array}\)
15. \(\begin{array}{rrrrrrrrr}
    \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\
    &&&&&&m&=&2c \\ \\
    (m&-&7)&+&(c&-&7)&=&13 \\
    &&m&+&c&-&14&=&13 \\
    &&2c&+&c&-&14&=&13 \\
    &&&&&+&14&&+14 \\
    \midrule
    &&&&&&\dfrac{3c}{3}&=&\dfrac{27}{3} \\ \\
    &&&&&&c&=&9 \\ \\
    &&&&&&\therefore m&=&2c \\
    &&&&&&m&=&2(9) \\
    &&&&&&m&=&18
    \end{array}\)
16. \(\begin{array}{rrrrrrrrr}
    \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\
    &&J&+&m&=&35&& \\
    &&&-&m&&&-&m \\
    \midrule
    &&&&J&=&35&-&m \\ \\
    &&J&-&10&=&2(m&-&10) \\
    35&-&m&-&10&=&2m&-&20 \\
    &&25&-&m&=&2m&-&20 \\
    &&-25&-&2m&&-2m&-&25 \\
    \midrule
    &&&&-3m&=&-45&& \\ \\
    &&&&m&=&\dfrac{-45}{-3}&\text{or}&15 \\ \\
    &&&&\therefore J&=&35&-&m \\
    &&&&J&=&35&-&15 \\
    &&&&J&=&20&&
    \end{array}\)
17. \(\begin{array}{rrrrrrrrr}
    \\ \\ \\ \\ \\ \\ \\ \\ \\ \\
    &&&&S&=&28&+&B \\ \\
    &&S&+&6&=&2(B&+&6) \\
    28&+&B&+&6&=&2B&+&12 \\
    &&B&+&34&=&2B&+&12 \\
    &&-B&-&12&=&-B&-&12 \\
    \midrule
    &&&&22&=&B&& \\ \\
    &&&&S&=&28&+&B \\
    &&&&S&=&28&+&22 \\
    &&&&S&=&50&&
    \end{array}\)
18. \(\begin{array}{rrrrrrrrr}
    \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\
    &&c&+&w&=&64&& \\
    &&-c&&&&&-&c \\
    \midrule
    &&&&w&=&64&-&c \\
    &&&&w&=&64&-&14 \\
    &&&&\therefore w&=&50&& \\ \\
    &&w&+&4&=&3(c&+&4) \\
    64&-&c&+&4&=&3c&+&12 \\
    &+&c&-&12&&+c&-&12 \\
    \midrule
    &&&&\dfrac{56}{4}&=&\dfrac{4c}{4}&& \\ \\
    &&&&c&=&14&&
    \end{array}\)
19. \(\begin{array}{rrrrrrr}
    \\ \\ \\ \\ \\ \\ \\
    &&S&=&12&& \\
    &&T&=&36&& \\ \\
    T&+&x&=&2(S&+&x) \\
    36&+&x&=&2(12&+&x) \\
    36&+&x&=&24&+&2x \\
    -24&-&x&&-24&-&x \\
    \midrule
    &&x&=&12&&
    \end{array}\)
20. \(\begin{array}{rrrrrrrrrrrrrrl}
    \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\
    &&&&&&&&&&F&=&3S&& \\
    &&&&&&&&&&D&=&S&-&3 \\ \\
    F&-&3&+&D&-&3&+&S&-&3&=&63&& \\
    F&&&+&D&&&+&S&-&9&=&63&& \\
    3S&&&+&S&-&3&+&S&-&9&=&63&& \\
    &&&&&&&&5S&-&12&=&63&& \\
    &&&&&&&&&+&12&&+12&& \\
    \midrule
    &&&&&&&&&&\dfrac{5S}{5}&=&\dfrac{75}{5}&& \\ \\
    &&&&&&&&&&S&=&15&& \\ \\
    &&&&&&&&&&F&=&3S&& \\
    &&&&&&&&&&F&=&3(15)&\text{or}&45 \\ \\
    &&&&&&&&&&D&=&S&-&3 \\
    &&&&&&&&&&D&=&15&-&3\text{ or }12
    \end{array}\)