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