# Midterm 2: Version D Answer Key

1. $x-2y=-6$
$x$ $y$
0 3
−6 0
$x+y=6$
$x$ $y$
0 6
6 0

2. $\phantom{a}$
$\begin{array}[t]{rrrrrl} &(3x&-&2y&=&0)(5) \\ &(2x&+&5y&=&0)(2) \\ \\ &15x&-&10y&=&0 \\ +&4x&+&10y&=&0 \\ \hline &&&19x&=&0 \\ &&&x&=&0 \\ \\ &\therefore \cancel{2x}0&+&5y&=&0 \\ &&&5y&=&0 \\ &&&y&=&0\\ \end{array}\\ (0,0)$
3. $\phantom{a}$
$\begin{array}[t]{rrrrrr} &2x&-&3y&=&8 \\ +&-2x&+&3y&=&4 \\ \hline &&&0&=&12 \\ \end{array}$
$\phantom{1}$
$\therefore$ No solution. Parallel lines.
4. $\phantom{a}$
$\begin{array}[t]{ll} \begin{array}[t]{rrrrrrrl} &(2x&+&y&-&3z&=&-7)(2) \\ \\ &4x&+&2y&-&6z&=&-14 \\ +&&-&2y&+&3z&=&\phantom{-1}9 \\ \hline &&&4x&-&3z&=&-5 \\ \\ &&&(3x&+&z&=&6)(3) \\ \\ &&&4x&-&3z&=&-5 \\ +&&&9x&+&3z&=&18 \\ \hline &&&&&13x&=&13 \\ &&&&&x&=&1 \end{array} & \hspace{0.25in} \begin{array}[t]{rrrrr} 3x&+&z&=&6 \\ 3(1)&+&z&=&6 \\ -3&&&&-3 \\ \hline &&z&=&3 \\ \\ -2y&+&3z&=&9 \\ -2y&+&3(3)&=&9 \\ &&-9&&-9 \\ \hline &&-2y&=&0 \\ &&y&=&0 \end{array} \end{array}$
$(1,0,3)$
5. $36-\cancel{\{-2x-\left[6x-3(5-2x)\right]\}^0}1+3x^2$
$36-1+3x^2$
$35+3x^2$
6. $3a^2(a^2-4a+4)$
$3a^4-12a^3+12a^2$
7. $\phantom{a}$
$\begin{array}[t]{rrrrrlrrrr} &x^2&+&2x&-&4&&&& \\ \times &x^2&+&2x&-&4&&&& \\ \hline &x^4&+&2x^3&-&4x^2&&&& \\ &&&2x^3&+&4x^2&-&8x&& \\ +&&&&-&4x^2&-&8x&+&16 \\ \hline &x^4&+&4x^3&-&4x^2&-&16x&+&16 \end{array}$
8. $\phantom{a}$
9. $x^2-3x+6x-18$
$x(x-3)+6(x-3)$
$(x-3)(x+6)$
10. $3x^2+xy+24xy+8y^2$
$x(3x+y)+8y(3x+y)$
$(3x+y)(x+8y)$
11. $(5x)^3-y^3$
$(5x-y)(25x^2+5xy+y^2)$
12. $(9y^2-4x^2)(9y^2+4x^2)$
$(3y-2x)(3y+2x)(9y^2+4x^2)$
13. $B+G=18\Rightarrow G=18-B$
$\begin{array}{rrrcrrrr} &G&-&4&=&4(B&-&4) \\ &18-B&-&4&=&4B&-&16 \\ +&16+B&&&&+B&+&16 \\ \hline &&&30&=&5B&& \\ \\ &&&B&=&\dfrac{30}{5}&=&6 \\ \\ &&&\therefore G&=&18&-&B \\ &&&\phantom{\therefore}G&=&18&-&6 \\ &&&\phantom{\therefore}G&=&12&& \end{array}$
14. $\phantom{a}$
$\begin{array}[t]{rrrrrl} &(D&+&Q&=&20)(-10) \\ &10D&+&25Q&=&350 \\ \\ &-10D&-&10Q&=&-200 \\ +&10D&+&25Q&=&\phantom{-}350 \\ \hline &&&15Q&=&150 \\ \\ &&&Q&=&\dfrac{150}{15}\text{ or }10 \\ \\ \therefore &D&+&Q&=&\phantom{-}20 \\ &D&+&10&=&\phantom{-}20 \\ &&-&10&&-10 \\ \hline &&&D&=&10 \end{array}$
15. $A+B=60\Rightarrow B=60-A$
$\begin{array}[t]{llclrll} &&3.80A&+&3.55B&=&\phantom{-}218.50 \\ 3.80A&+&3.55(60&-&A)&=&\phantom{-}218.50 \\ 3.80A&+&213&-&3.55A&=&\phantom{-}218.50 \\ &-&213&&&=&-213 \\ \hline &&&&0.25A&=&5.50 \\ \\ &&&&A&=&\dfrac{5.50}{0.25}\text{ or 22 kg} \\ \\ &&&&B&=&60-A \\ &&&&B&=&60-22 \\ &&&&B&=&38\text{ kg} \end{array}$