# Midterm 2: Version E Answer Key

1. $x-y=-3$
$x$ $y$
0 3
−3 0
$x+2y=3$
$x$ $y$
3 0
0 $\dfrac{3}{2}$

2. $\phantom{a}$
$\begin{array}[t]{rrrrrl} &(2x&-&5y&=&-2)(-3) \\ &(3x&-&4y&=&\phantom{-}4)(2) \\ \\ &-6x&+&15y&=&6 \\ +&6x&-&8y&=&8 \\ \hline &&&7y&=&14 \\ &&&y&=&2 \\ \\ &2x&-&5(2)&=&-2 \\ &2x&-&10&=&-2 \\ &&+&10&&+10 \\ \hline &&&2x&=&8 \\ &&&x&=&4 \end{array}$
$(4,2)$
3. $\phantom{a}$
$\begin{array}[t]{rrrrrl} &(4x&+&3y&=&-29)(-2) \\ &(3x&+&2y&=&-21)(3) \\ \\ &-8x&-&6y&=&\phantom{-}58 \\ +&9x&+&6y&=&-63 \\ \hline &&&x&=&-5 \\ \\ &3(-5)&+&2y&=&-21 \\ &-15&+&2y&=&-21 \\ +&15&&&&+15 \\ \hline &&&2y&=&-6 \\ &&&y&=&-3 \end{array}$
$(-5,-3)$
4. $\phantom{a}$
$\begin{array}[t]{ll} \begin{array}[t]{rrrrrrrl} &(x&+&y&-&3z&=&0)(-2) \\ \\ &-2x&-&2y&+&6z&=&0 \\ +&2x&-&3y&&&=&16 \\ \hline &&&-5y&+&6z&=&16 \\ \\ &&&(2y&-&2z&=&-12)(3) \\ \\ &&&6y&-&6z&=&-36 \\ +&&&-5y&+&6z&=&\phantom{-}16 \\ \hline &&&&&y&=&-20 \end{array} & \hspace{0.25in} \begin{array}[t]{rrrrr} 2x&-&3y&=&16 \\ 2x&-&3(-20)&=&16 \\ 2x&+&60&=&16 \\ &&-60&&-60 \\ \hline &&2x&=&-44 \\ &&x&=&-22 \\ \\ 2y&-&2z&=&-12 \\ 2(-20)&-&2z&=&-12 \\ -40&-&2z&=&-12 \\ +40&&&&+40 \\ \hline &&-2z&=&28 \\ &&z&=&-14 \end{array} \end{array}$
$(-22,-20,-14)$
5. $5-4\left[2x-2\cancel{(6x-5)^0}1-(7-2x)\right]$
$5-4\left[2x-2(1)-7+2x\right]$
$5-4\left[4x-9\right]$
$5-16x+36$
$-16x+41$
6. $3ab^4(a^2-25)$
$3a^3b^4-75ab^4$
7. $\begin{array}[t]{rrrrrlrrrr} &x^2&+&3x&-&6&&&& \\ \times &x^2&+&3x&-&6&&&& \\ \hline &x^4&+&3x^3&-&6x^2&&&& \\ &&&3x^3&+&9x^2&-&18x&& \\ +&&&&-&6x^2&-&18x&+&36 \\ \hline &x^4&+&6x^3&-&3x^2&-&36x&+&36 \end{array}$
8. $\phantom{a}$
9. $x^2-3z+7x-21$
$x(x-3)+7(x-3)$
$(x-3)(x+7)$
10. $4x^2(x+1)-9(x+1)$
$(x+1)(4x^2-9)$
$(x+1)(2x-3)(2x+3)$
11. $(2x)^3-(3y)^3$
$(2x-3y)(4x^2+6xy+9y^2)$
12. $x^4-625x^2+x^2-625$
$x^2(x^2-625)+1(x^2-625)$
$(x^2+1)(x^2-625)$
$(x^2+1)(x-25)(x+25)$
13. $B+G=20\Rightarrow B=20-G$
$\begin{array}{rrrrrrrrrl} &G&-&4&=&2(B&-&4)&& \\ &G&-&4&=&2B&-&8&& \\ \\ &G&-&4&=&2(20&-&G)&-&8 \\ &G&-&4&=&40&-&2G&-&8 \\ &G&-&4&=&32&-&2G&& \\ +&2G&+&4&&4&+&2G&& \\ \hline &&&3G&=&36&&&& \\ &&&G&=&12&&&& \\ \\ &&&B&=&20&-&G&& \\ &&&B&=&20&-&12&=&8 \\ \end{array}$
14. $x=16\%\text{ solution}$
$\begin{array}{rrrrrrrr} &16x&+&6(20)&=&12(x&+&20) \\ &16x&+&120&=&12x&+&240 \\ -&12x&-&120&&-12x&-&120 \\ \hline &&&4x&=&120&& \\ \\ &&&x&=&\dfrac{120}{4}&=&30\text{ ml} \\ \end{array}$
15. $\phantom{a}$
$\begin{array}[t]{rrrrrl} &(C&+&R&=&60)(-3.40) \\ \\ &-3.40C&-&3.40R&=&-204 \\ +&3.40C&+&3.90R&=&\phantom{-}213\\ \hline &&&0.50R&=&\phantom{-}9 \\ \\ &&&R&=&\dfrac{9}{0.50}\text{ or 18 kg} \\ \\ &C&+&R&=&\phantom{-}60 \\ &C&+&18&=&\phantom{-}60 \\ &&-&18&&-18 \\ \hline &&&C&=&\phantom{-}42\text{ kg} \end{array}$