Midterm 2: Version D Answer Key
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[latex]x-2y=-6[/latex] [latex]x[/latex] [latex]y[/latex] 0 3 −6 0 [latex]x+y=6[/latex] [latex]x[/latex] [latex]y[/latex] 0 6 6 0 
- [latex]\phantom{a}[/latex]
[latex]\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)[/latex] - [latex]\phantom{a}[/latex]
[latex]\begin{array}[t]{rrrrrr} &2x&-&3y&=&8 \\ +&-2x&+&3y&=&4 \\ \hline &&&0&=&12 \\ \end{array}[/latex]
[latex]\phantom{1}[/latex]
[latex]\therefore[/latex] No solution. Parallel lines. - [latex]\phantom{a}[/latex]
[latex]\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}[/latex]
[latex](1,0,3)[/latex] - [latex]36-\cancel{\{-2x-\left[6x-3(5-2x)\right]\}^0}1+3x^2[/latex]
[latex]36-1+3x^2[/latex]
[latex]35+3x^2[/latex] - [latex]3a^2(a^2-4a+4)[/latex]
[latex]3a^4-12a^3+12a^2[/latex] - [latex]\phantom{a}[/latex]
[latex]\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}[/latex] - [latex]\phantom{a}[/latex]
- [latex]x^2-3x+6x-18[/latex]
[latex]x(x-3)+6(x-3)[/latex]
[latex](x-3)(x+6)[/latex] - [latex]3x^2+xy+24xy+8y^2[/latex]
[latex]x(3x+y)+8y(3x+y)[/latex]
[latex](3x+y)(x+8y)[/latex] - [latex](5x)^3-y^3[/latex]
[latex](5x-y)(25x^2+5xy+y^2)[/latex] - [latex](9y^2-4x^2)(9y^2+4x^2)[/latex]
[latex](3y-2x)(3y+2x)(9y^2+4x^2)[/latex] - [latex]B+G=18\Rightarrow G=18-B[/latex]
[latex]\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}[/latex] - [latex]\phantom{a}[/latex]
[latex]\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}[/latex] - [latex]A+B=60\Rightarrow B=60-A[/latex]
[latex]\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}[/latex]
