Midterm 2: Version C Answer Key
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[latex]2x-y-2=0[/latex] [latex]x[/latex] [latex]y[/latex] 0 −2 1 0 2 2 [latex]2x+3y+6=0[/latex] [latex]x[/latex] [latex]y[/latex] 0 −2 −3 0 −6 2 
- [latex]x+y=2\Rightarrow x=2-y[/latex]
[latex]\begin{array}{rrrrrrl} 3(2&-&y)&-&4y&=&13 \\ 6&-&3y&-&4y&=&13 \\ -6&&&&&&-6 \\ \hline &&&&-7y&=&\phantom{-}7 \\ &&&&y&=&-1 \\ \\ &&&&x&=&2--1 \\ &&&&x&=&3 \end{array}[/latex]
[latex](3,-1)[/latex] - [latex]\phantom{a}[/latex]
[latex]\begin{array}[t]{rrrrrr} &4x&-&3y&=&6 \\ +&4x&+&3y&=&2 \\ \hline &&&8x&=&8 \\ &&&x&=&1 \\ \\ &3y&+&4(1)&=&2 \\ &&-&4&&-4 \\ \hline &&&3y&=&-2 \\ \\ &&&y&=&-\dfrac{2}{3} \end{array}[/latex]
[latex]\left(1,-\dfrac{2}{3}\right)[/latex] - [latex]\phantom{a}[/latex]
[latex]\begin{array}[t]{ll} \begin{array}[t]{rrrrrrrl} \left[1\right]&(x&+&y&+&z&=&\phantom{-}6)(-2) \\ \left[2\right]&&&(-2x&+&z&=&-3)(-1) \\ \\ \left[1\right]&-2x&-&2y&-&2z&=&-12 \\ +&&&2y&+&4z&=&\phantom{-}10 \\ \hline &&&-2x&+&2z&=&-2 \\ +&&\left[2\right]&2x&-&z&=&\phantom{-}3 \\ \hline &&&&&z&=&1 \\ \end{array} & \hspace{0.25in} \begin{array}[t]{rrrrr} 2y&+&4z&=&10 \\ 2y&+&4(1)&=&10 \\ &-&4&&-4 \\ \hline &&2y&=&6 \\ &&y&=&3 \\ \\ -2x&+&z&=&-3 \\ -2x&+&1&=&-3 \\ &-&1&&-1 \\ \hline &&-2x&=&-4 \\ &&x&=&2 \\ \end{array} \end{array}[/latex]
[latex](2,3,1)[/latex] - [latex]36+\{\cancel{-2x-\left[6x-3(5-2x)\right]\}^0}1+6x^3[/latex]
[latex]36+1+6x^3\Rightarrow 6x^3+37[/latex] - [latex]6a^2b(a^2-9)[/latex]
[latex]6a^4b-54a^2b[/latex] - [latex]\phantom{a}[/latex]
[latex]\begin{array}[t]{rrrrrlrrrr} &x^2&+&3x&+&5&&&& \\ \times &x^2&+&3x&+&5&&&& \\ \hline &x^4&+&3x^3&+&5x^2&&&& \\ &&&3x^3&+&9x^2&+&15x&& \\ +&&&&&5x^2&+&15x&+&25 \\ \hline &x^4&+&6x^3&+&19x^2&+&30x&+&25 \end{array}[/latex] - [latex]\phantom{a}[/latex]
- [latex]x^2-x+18x-18[/latex]
[latex]x(x-1)+18(x-1)[/latex]
[latex](x-1)(x+18)[/latex] - [latex]2(a^2-2ab-15b^2)[/latex]
[latex]2(a^2-5ab+3ab-15b^2)[/latex]
[latex]2\left[a(a-5b)+3b(a-5b)\right][/latex]
[latex]2(a-5b)(a+3b)[/latex] - [latex](2x)^3-y^3[/latex]
[latex](2x-y)(4x^2+2xy+y^2)[/latex] - [latex](4y^2-x^2)(4y^2+x^2)[/latex]
[latex](2y-x)(2y+x)(4y^2+x^2)[/latex] - [latex]B+S=30\Rightarrow B=30-S[/latex]
[latex]\begin{array}{rrrrrrr} B&-&10&=&4(S&-&10) \\ 30-S&-&10&=&4S&-&40 \\ +S&+&40&&+S&+&40 \\ \hline &&60&=&5S&& \\ \\ &&S&=&\dfrac{60}{5}&=&12 \\ \\ &&\therefore B&=&30&-&S \\ &&B&=&30&-&12 \\ &&B&=&18&& \end{array}[/latex] - [latex]\phantom{a}[/latex]
[latex]\begin{array}[t]{rrrrrl} &(D&+&N&=&\phantom{1}18)(-1) \\ &(10D&+&5N&=&120)(\div 5) \\ \\ &-D&-&N&=&-18 \\ +&2D&+&N&=&\phantom{-}24 \\ \hline &&&D&=&6 \\ \\ \therefore &D&+&N&=&18 \\ &6&+&N&=&18 \\ &-6&&&&-6 \\ \hline &&&N&=&12 \end{array}[/latex] - [latex]\text{if }x=5\%, \text{ then }10-x=30\%[/latex]
[latex]\begin{array}{rrrrcrl} 5x&+&30(10&-&x)&=&25(10) \\ 5x&+&300&-&30x&=&\phantom{-}250 \\ &-&300&&&&-300 \\ \hline &&&&-25x&=&-50 \\ \\ &&&&x&=&\dfrac{-50}{-25}\text{ or 2 L of 5%} \\ \\ &&10&-&x&=&\text{8 L of 30%} \end{array}[/latex]
