Answer Key 10.7

$\begin{array}{rrrrrrrrrrl} \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ x&+&y&=&22&\Rightarrow &x&=&22&-&y \\ x&-&y&=&120&&&&&& \\ \\ (22&-&y)y&=&120&&&&&& \\ 22y&-&y^2&=&120&&&&&& \\ \\ &&0&=&y^2&-&22y&+&120&& \\ &&0&=&y^2&-&12y&-&10y&+&120 \\ \midrule &&0&=&y(y&-&12)&-&10(y&-&12) \\ &&0&=&(y&-&12)(y&-&10)&& \\ \\ &&y&=&12,&10&&&&& \end{array}$
$\therefore \text{ numbers are }10, 12$
$\begin{array}{rrrrccrrrrrrrrl} \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ &&&&x&-&y&=&4&\Rightarrow &x&=&y&+&4 \\ &&&&x&\cdot &y&=&140&&&&&& \\ \\ &&&&(y&+&4)y&=&140&&&&&& \\ &&&&y^2&+&4y&=&140&&&&&& \\ \\ &&y^2&+&4y&-&140&=&0&&&&&& \\ y^2&-&10y&+&14y&-&140&=&0&&&&&& \\ \midrule y(y&-&10)&+&14(y&-&10)&=&0&&&&&& \\ &&(y&-&10)(y&+&14)&=&0&&&&&& \\ \\ &&&&&&y&=&10,&-14&&&&& \\ \\ &&&&&&y&=&10,&x&=&10&+&4&= 14 \\ &&&&&&y&=&-14,&x&=&-14&+&4&= -10 \\ \end{array}$
$\therefore \text{ numbers are }10, 14\text{ and }-10, -14$
$\begin{array}{rrrrcrrrrrrrrrl} \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ &&&&A&-&B&=&8&\Rightarrow &A&=&B&+&8 \\ &&&&A^2&+&B^2&=&320&&&&&& \\ \\ &&(B&+&8)^2&+&B^2&=&320&&&&&& \\ B^2&+&16B&+&64&+&B^2&=&320&&&&&& \\ &&&-&320&&&&-320&&&&&& \\ \midrule &&2B^2&+&16B&-&256&=&0&&&&&& \\ &&2(B^2&+&8B&-&128)&=&0&&&&&& \\ &&2(B&+&16)(B&-&8)&=&0&&&&&& \\ \\ &&&&&&B&=&-16,&8&&&&& \\ \\ &&&&&&\therefore A&=&B&+&8&&&& \\ &&&&&&A&=&-8,&16&&&&& \end{array}$
$\therefore (-16, -8)\text{ and }(8,16)$
$\begin{array}{rrrrcrrrrrrrrr} \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ x, &x&+&2&&&&&&&&&& \\ \\ &&x^2&+&(x&+&2)^2&=&244&&&&& \\ x^2&+&x^2&+&4x&+&4&=&244&&&&& \\ &&&&&&-244&&-244&&&&& \\ \midrule &&2x^2&+&4x&-&240&=&0&&&&& \\ &&2(x^2&+&2x&-&120)&=&0&&&&& \\ &&2(x&-&10)(x&+&12)&=&0&&&&& \\ \\ &&&&&&x&=&10, &-12&&&& \\ \\ &&&&&&x&=&10, &x&+&2&=&12 \\ &&&&&&x&=&-12, &x&+&2&=&-10 \end{array}$
$\therefore \text{ numbers are }10, 12\text{ or }-12, -10$
$\begin{array}{rrrrrrrrrrrr} \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ x,&x&+&2&&&&&&&& \\ \\ &&x^2&-&(x&+&2)^2&=&60&&& \\ x^2&-&(x^2&+&4x&+&4)&=&60&&& \\ x^2&-&x^2&-&4x&-&4&=&60&&& \\ &&&&&+&4&&+4&&& \\ \midrule &&&&&&\dfrac{-4x}{-4}&=&\dfrac{64}{-4}&&& \\ \\ &&&&&&x&=&-16&&& \\ \\ &&&&x&+&2&\Rightarrow &-16&+&2& \\ &&&&&&&\Rightarrow &-14&&& \\ \end{array}$
$-16, -14$
$\begin{array}{rrrrcrrrl} \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ x,&x&+&2&&&&& \\ \\ &&x^2&+&(x&+&2)^2&=&\phantom{-}452 \\ x^2&+&x^2&+&4x&+&4&=&\phantom{-}452 \\ &&&&&-&452&&-452 \\ \midrule &&2x^2&+&4x&-&448&=&0 \\ &&2(x^2&+&2x&-&224)&=&0 \\ &&2(x&-&14)(x&+&16)&=&0 \\ \\ &&&&&&x&=&14, -16 \\ \\ &&&&&&x&=&14 \\ &&&&x&+&2&=&16 \\ \\ &&&&&&x&=&-16 \\ &&&&x&+&2&=&-14 \end{array}$
$14,16\text{ and }-16,-14$
$\begin{array}{rrcrrrrrrrr} \\ \\ \\ \\ \\ \\ \\ x,&x&+&2,&x&+&4&&&& \\ \\ &&x(x&+&2)&=&38&+&x&+&4 \\ x^2&+&2x&&&=&42&+&x&& \\ &-&x&-&42&&-42&-&x&& \\ \midrule x^2&+&x&-&42&=&0&&&& \\ (x&+&7)(x&-&6)&=&0&&&& \\ &&&&x&=&\cancel{-7},&6&&& \\ \end{array}$
$\therefore \text{ numbers are }6,8,10$
$x, x+2, x+4$ $\begin{array}{rrrrrrrrrrr} \\ \\ \\ \\ \\ \\ \\ \\ \\ &&(x)(x&+&2)&=&52&+&x&+&4 \\ \\ x^2&+&2x&&&=&56&+&x&& \\ &-&x&-&56&&-56&-&x&& \\ \midrule x^2&+&x&-&56&=&0&&&& \\ (x&+&8)(x&-&7)&=&0&&&& \\ \\ &&&&x&=&\cancel{-8}, 7&&&& \end{array}$
$\therefore \text{ numbers are }7,9,11$
$\begin{array}{rrrrrrrrlrrr} \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ &&A&=&T&+&4&&&&& \\ \\ A&\cdot &T&=&80&+&(A&-&4)&(T&-&4) \\ (T&+&4)T&=&80&+&(T&+&\cancel{4-4})&(T&-&4) \\ \\ T^2&+&4T&=&80&+&T^2&-&4T&&& \\ -T^2&+&4T&&&-&T^2&+&4T&&& \\ \midrule &&\dfrac{8T}{8}&=&\dfrac{80}{8}&&&&&&& \\ \\ &&T&=&10&&&&&&& \\ \\ &&\therefore A&=&T&+&4&&&&& \\ &&A&=&10&+&4&=&14&&& \end{array}$
$\begin{array}{rrcrrrcrcrrrr} \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ &&C&=&K&+&3&&&&&& \\ &&CK&=&(C&+&5)&&(K&+&5)&-&130 \\ \\ (K&+&3)K&=&(K&+&3&+&5)(K&+&5)&-&130 \\ K^2&+&3K&=&K^2&+&13K&+&40&-&130&& \\ -K^2&-&13K&&-K^2&-&13K&&&&&& \\ \midrule &&\dfrac{-10K}{-10}&=&\dfrac{-90}{-10}&&&&&&&& \\ \\ &&K&=&9&&&&&&&& \\ \\ &&\therefore C&=&9&+&3&=&12&&&& \end{array}$
$\begin{array}{rrrrcrrrrrrrr} \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ &&&&&&J&=&S&+&1&& \\ \\ &&(J&+&5)(S&+&5)&=&230&+&J&\cdot &S \\ (S&+&1&+&5)(S&+&5)&=&230&+&(S&+&1)S \\ &&(S&+&6)(S&+&5)&=&230&+&S^2&+&S \\ \\ &&S^2&+&11S&+&30&=&S^2&+&S&+&230 \\ &&-S^2&-&S&-&30&&-S^2&-&S&-&30 \\ \midrule &&&&&&\dfrac{10S}{10}&=&\dfrac{200}{10}&&&& \\ \\ &&&&&&S&=&20&&&& \\ &&&&&&J&=&21&&&& \end{array}$
$\begin{array}{rrcrcrrrrrrrr} \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ &&&&&&J&=&S&+&2&& \\ &&(S&+&2)(J&+&2)&=&48&+&S&\cdot &J \\ \\ (S&+&2)(S&+&2&+&2)&=&48&+&S(S&+&2) \\ &&(S&+&2)(S&+&4)&=&48&+&S^2&+&25 \\ \\ &&S^2&+&6S&+&8&=&48&+&S^2&+&25 \\ &&-S^2&-&2S&-&8&&-8&-&S^2&-&25 \\ \midrule &&&&&&\dfrac{4S}{4}&=&\dfrac{40}{4}&&&& \\ \\ &&&&&&S&=&10&&&& \\ &&&&&&J&=&12&&&& \end{array}$
$\begin{array}{ll} \\ \\ \\ \begin{array}{rrrrrrrrl} \\ \\ \\ \\ \\ \\ \\ &&&&&&d&=&r\cdot t \\ \\ &&&&r&\cdot &t&=&240\ \\ &&&&&&\therefore r&=&\dfrac{240}{t} \\ \\ &&(r&+&20)(t&-&1)&=&240 \\ &&(\dfrac{240}{t}&+&20)(t&-&1)&=&240 \\ \\ \cancel{240}&+&20t&-&\dfrac{240}{t}&-&20&=&\cancel{240} \\ \\ &&(20t&-&\dfrac{240}{t}&-&20&=&0)(t) \\ \\ &&(20t^2&-&240&-&20t&=&0)(\div 20) \end{array} &\hspace{0.25in} \begin{array}{rrcrcrl} t^2&-&12&-&t&=&0 \\ (t&-&4)(t&+&3)&=&0 \\ \\ &&&&t&=&4, \cancel{-3} \\ \\ &&&&r&=&\dfrac{240}{4}\text{ or }60\text{ km/h} \\ \\ &&&&\text{faster}&=&80\text{ km/h} \end{array} \end{array}$
$\begin{array}{ll} \begin{array}{rrrrrrrrl} \\ \\ \\ \\ \\ \\ \\ &&&&&&d&=&r\cdot t \\ &&&&r&\cdot &t&=&100 \\ &&&&&&r&=&\dfrac{100}{t} \\ \\ &&(r&+&20)(t&-&0.5)&=&120 \\ &&(\dfrac{100}{t}&+&20)(t&-&0.5)&=&120 \\ 100&+&20t&-&\dfrac{50}{t}&-&10&=&120 \\ &&&&&-&120&&-120 \\ \midrule &&20t&-&30&-&\dfrac{50}{t}&=&0 \\ \\ &&(20t&-&30&-&\dfrac{50}{t}&=&0)(t) \\ \\ &&(20t^2&-&30t&-&50&=&0)(\div 10) \end{array} &\hspace{0.25in} \begin{array}{rrrrrrl} 2t^2&-&3t&-&5&=&0 \\ &&&&t&=&\dfrac{-(-3)\pm \sqrt{(-3)^2-4(2)(-5)}}{2(2)} \\ \\ &&&&t&=&\dfrac{3\pm 7}{4}=\dfrac{10}{4}\text{ or }\cancel{\dfrac{-4}{4}} \\ \\ &&&&t&=&2.5\text{ h} \end{array} \end{array}$
$\text{Answer: }\dfrac{100\text{ km}}{2.5\text{ h}}=\dfrac{40\text{ km}}{\text{h}}, \dfrac{120\text{ km}}{2\text{ h}}=\dfrac{60\text{ km}}{\text{h}}$
$\begin{array}{ll} \\ \\ \\ \\ \begin{array}{rrrrrrrrl} \\ \\ \\ \\ \\ \\ \\ \\ &&&&&&d&=&r\cdot t \\ &&&&r&\cdot &t&=&150 \\ &&&&&&r&=&\dfrac{150}{t} \\ \\ &&(r&+&5)(t&-&1.5)&=&150 \\ &&(\dfrac{150}{t}&+&5)(t&-&1.5)&=&150 \\ \\ \cancel{150}&+&5t&-&\dfrac{225}{t}&-&7.5&=&\cancel{150} \\ \\ &&(5t&-&\dfrac{225}{t}&-&7.5&=&0)(t) \\ \\ &&(5t^2&-&7.5t&-&225&=&0)(2) \\ &&(10t^2&-&15t&-&450&=&0)(\div 5) \end{array} &\hspace{0.25in} \begin{array}{rrrrrrl} \\ \\ \\ \\ \\ \\ 2t^2&-&3t&-&90&=&0 \\ (t&+&6)(2t&-&15)&=&0 \\ &&&&t&=&\cancel{-6}, \dfrac{15}{2} \\ \\ &&&&r&=&\dfrac{150}{t} \\ \\ &&&&r&=&\dfrac{150}{\dfrac{15}{2}} \\ \\ &&&&r&=&\dfrac{150}{1}\cdot \dfrac{2}{15} \\ \\ &&&&r&=&20\text{ km/h} \end{array} \end{array}$
$\begin{array}{rrrrrrrrl} \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ &&&&&&d&=&r\cdot t \\ &&&&r&\cdot &t&=&180\Rightarrow r=\dfrac{180}{t} \\ \\ &&(r&+&15)(t&-&1)&=&180 \\ &&(\dfrac{180}{t}&+&15)(t&-&1)&=&180 \\ \\ \cancel{180}&+&15t&-&\dfrac{180}{t}&-&15&=&\cancel{180} \\ &&(15t&-&15&-&\dfrac{180}{t}&=&0)(t) \\ &&(15t^2&-&15t&-&180&=&0)(\div 15) \\ \\ &&t^2&-&t&-&12&=&0 \\ &&(t&-&4)(t&+&3)&=&0 \\ &&&&&&t&=&4, \cancel{-3} \\ \\ &&&&&&r&=&\dfrac{180}{4}=45 \end{array}$
$\begin{array}{rrrrcrrrl} \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ &&&&r&\cdot &t&=&72\Rightarrow r=\dfrac{72}{t} \\ \\ &&(r&+&12)(9&-&t)&=&72 \\ \\ &&(\dfrac{72}{t}&+&12)(9&-&t)&=&72 \\ \dfrac{648}{t}&+&108&-&72&-&12t&=&72 \\ &&&-&72&&&&-72 \\ \midrule &&(-12t&-&36&+&\dfrac{648}{t}&=&0)(t) \\ \\ &&(-12t^2&-&36t&+&648&=&0)(\div -12) \\ \\ &&t^2&+&3t&-&54&=&0 \\ &&(t&+&9)(t&-&6)&=&0 \\ &&&&&&t&=&\cancel{-9}, 6 \\ \\ &&&&&&r&=&\dfrac{72}{6}=12\text{ (there)} \\ \\ &&&&&&r&=&24\text{ (return)} \end{array}$
$\begin{array}{rrrrcrrrl} \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ &&&&r&\cdot &t&=&120\Rightarrow r=\dfrac{120}{t} \\ &&(r&+&10)(7&-&t)&=&120 \\ \\ &&(\dfrac{120}{t}&+&10)(7&-&t)&=&120 \\ \dfrac{840}{t}&+&70&-&120&-&10t&=&120 \\ &&&-&120&&&&-120 \\ \midrule &&(-10t&-&170&+&\dfrac{840}{t}&=&0)(t) \\ &&(-10t^2&-&170t&+&840&=&0)(\div -10) \\ \\ &&t^2&+&17t&-&84&=&0 \\ &&(t&+&21)(t&-&4)&=&0 \\ &&&&&&t&=&\cancel{-21}, 4 \\ \\ &&&&&&r&=&\dfrac{120}{4}\text{ or }30\text{ km/h} \\ \\ &&&&r&+&10&=&40\text{ km/h} \\ \end{array}$
$\begin{array}{rrrrcrrrl} \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ &&&&r&\cdot &t&=&240\Rightarrow r=\dfrac{240}{t} \\ \\ &&(r&+&20)(t&-&1)&=&240 \\ &&(\dfrac{240}{t}&+&20)(t&-&1)&=&240 \\ \cancel{240}&+&20t&-&\dfrac{240}{t}&-&20&=&\cancel{240} \\ \\ &&(20t&-&20&-&\dfrac{240}{t}&=&0)(t) \\ &&(20t^2&-&20t&-&240&=&0)(\div 20) \\ \\ &&t^2&-&t&-&12&=&0 \\ &&(t&-&4)(t&+&3)&=&0 \\ &&&&&&t&=&4, \cancel{-3} \\ \\ &&&&&&r&=&\dfrac{240}{4}\text{ or }60\text{ km/h} \end{array}$
$\begin{array}{rrrrcrrrl} \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ &&&&r&\cdot &t&=&600\Rightarrow r=\dfrac{600}{t} \\ \\ &&(r&-&50)(7&-&t)&=&600 \\ &&(\dfrac{600}{t}&-&50)(7&-&t)&=&600 \\ \dfrac{4200}{t}&-&350&-&600&+&50t&=&600 \\ &&&-&600&&&&-600 \\ \midrule &&(50t&-&1550&+&\dfrac{4200}{t}&=&0)(t) \\ &&(50t^2&-&1550t&+&4200&=&0)(\div 50) \\ \\ &&t^2&-&31t&+&84&=&0 \\ &&(t&-&3)(t&-&28)&=&0 \\ &&&&&&t&=&3, \cancel{28} \\ \\ &&&&&&r&=&\dfrac{600}{3}\text{ or }200\text{ km/h} \end{array}$
$\begin{array}{rrrrlrrrl} \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ L&=&4&+&W&&&& \\ \text{Area}&=&L&\cdot &W&&&& \\ \\ 60&=&(4&+&W)W&&&& \\ 60&=&4W&+&W^2&&&& \\ \\ 0&=&W^2&+&4W&-&60&& \\ 0&=&W^2&+&10W&-&6W&-&60 \\ \midrule 0&=&W(W&+&10)&-&6(W&+&10) \\ 0&=&(W&+&10)(W&-&6)&& \\ \\ W&=&\cancel{-10},&6&&&&& \\ L&=&6&+&4&=&10&& \end{array}$
$\begin{array}{rrrrcrrrr} \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ W&=&L&-&10&&&& \\ \text{Area}&=&L&\cdot &W&&&& \\ \\ 200&=&L(L&-&10)&&&& \\ 200&=&L^2&-&10L&&&& \\ \\ 0&=&L^2&-&10L&-&200&& \\ 0&=&L^2&+&10L&-&20L&-&200 \\ \midrule 0&=&L(L&+&10)&-&20(L&+&10) \\ 0&=&(L&+&10)(L&-&20)&& \\ \\ L&=&\cancel{-10},&20&&&&& \\ W&=&20&-&10&=&10&& \end{array}$
$\begin{array}{rrrrcrcrcrl} \\ \\ \\ \\ \\ \\ \\ \\ &&&&&&\text{Area}_{\text{large}}&-&\text{Area}_{\text{small}}&=&2800\text{ m}^2 \\ \\ &&(150&+&2x)(120&+&2x)&-&(150)(120)&=&2800 \\ \cancel{18000}&+&240x&+&300x&+&4x^2&-&\cancel{18000}&=&2800 \\ &&&&&&&-&2800&&-2800 \\ \midrule &&&&4x^2&+&540x&-&2800&=&0 \\ &&&&x^2&+&135x&-&700&=&0 \\ &&&&(x&-&5)(x&+&140)&=&0 \\ &&&&&&&&x&=&5, \cancel{-140} \end{array}$
$\text{walkway}=5\text{ m}$
$\begin{array}{rrrrcrcrcrl} \\ \\ \\ \\ \\ \\ \\ \\ \\ &&&&&&\text{Area}_{\text{large}}&-&\text{Area}_{\text{small}}&=&74\text{ m}^2 \\ \\ &&(25&+&2x)(10&+&2x)&-&(25)(10)&=&74 \\ \cancel{250}&+&20x&+&50x&+&4x^2&-&\cancel{250}&=&74 \\ &&&&&&&-&74&&-74 \\ \midrule &&&&4x^2&+&70x&-&74&=&0 \\ &&&&2x^2&+&35x&-&37&=&0 \\ &&&&(x&-&1)(2x&+&37)&=&0 \\ &&&&&&&&x&=&1, \cancel{-\dfrac{37}{2}} \\ \end{array}$
$\text{the overlap}=1\text{ m}$
$\begin{array}{rrrrrrrrl} \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ &&&&L&=&W&+&4 \\ &&L&\cdot &W&=&60&& \\ \\ &&(W&+&4)W&=&60&& \\ W^2&+&4W&&&=&60&& \\ &&&-&60&&-60&& \\ \midrule W^2&+&4W&-&60&=&0&& \\ (W&-&6)(W&+&10)&=&0&& \\ \\ &&&&W&=&6,&\cancel{-10}& \\ &&&&L&=&6&+&4=10 \end{array}$
$\begin{array}{rrrrrrrrcrr} \\ \\ \\ \\ \\ \\ \\ &&(x&+&5)^2&=&4(x)^2&&&& \\ \\ x^2&+&10x&+&25&=&4x^2&&&& \\ -x^2&-&10x&-&25&&-x^2&-&10x&-&25 \\ \midrule &&&&0&=&3x^2&-&10x&-&25 \\ &&&&0&=&(x&-&5)(3x&+&5) \\ &&&&x&=&5, &\cancel{-\dfrac{5}{3}}&&& \end{array}$
$\begin{array}{rrrrrrlll} \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ &&&&L&=&20&+&W \\ &&L&\cdot &W&=&2400&& \\ \\ &&(20&+&W)W&=&2400&& \\ W^2&+&20W&&&=&2400&& \\ &&&-&2400&&-2400&& \\ \midrule W^2&+&20W&-&2400&=&0&& \\ (W&+&60)(W&-&40)&=&0&& \\ \\ &&&&W&=&\cancel{-60},&40& \\ &&&&L&=&20&+&40=60 \end{array}$
$\begin{array}{rrrrcrrrrrrrr} \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ &&&&&&L&=&W&+&8&& \\ &&(L&+&2)(W&+&2)&=&L&\cdot &W&+&60 \\ \\ (W&+&8&+&2)(W&+&2)&=&(W&+&8)W&+&60 \\ &&W^2&+&12W&+&20&=&W^2&+&8W&+&60 \\ &&-W^2&-&8W&-&20&&-W^2&-&8W&-&20 \\ \midrule &&&&&&\dfrac{4W}{4}&=&\dfrac{40}{4}&&&& \\ \\ &&&&&&W&=&10&&&& \\ &&&&&&L&=&10&+&8&=&18 \end{array}$

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