Midterm Two Review

1. [latex]x-2y=-4[/latex]

   [latex]x[/latex]	[latex]y[/latex]
   −4	0
   0	2
   −2	1

   [latex]x+y=5[/latex]

   [latex]x[/latex]	[latex]y[/latex]
   0	5
   5	0
   2	3

   

2. [latex]\phantom{a}[/latex]
   [latex]\begin{array}[t]{rrcrlrl} 2x&-&y&=&0&\Rightarrow &y=2x \\ 3x&+&4y&=&-22&& \\ \\ \therefore 3x&+&4(2x)&=&-22&& \\ 3x&+&8x&=&-22&& \\ &&11x&=&-22&& \\ &&x&=&-2&& \\ \\ &&y&=&2x&& \\ &&y&=&2(-2)&=&-4 \\ \end{array}[/latex]
   [latex](-2,-4)[/latex]
3. [latex]\phantom{a}[/latex]
   [latex]\begin{array}[t]{rrrrrl} &(2x&-&5y&=&15)(2) \\ &(3x&+&2y&=&13)(5) \\ \hline &4x&-&10y&=&30 \\ +&15x&+&10y&=&65 \\ \hline &&&19x&=&95 \\ &&&x&=&5 \\ \\ &\therefore 3(5)&+&2y&=&\phantom{-}13 \\ &15&+&2y&=&\phantom{-}13 \\ &-15&&&&-15 \\ \hline &&&2y&=&-2 \\ &&&y&=&-1 \end{array}[/latex]
   [latex](5,-1)[/latex]
4. [latex]\phantom{a}[/latex]
   [latex]\begin{array}[t]{rr} \begin{array}[t]{rrrrrrrl} &&&(5x&+&6z&=&-4)(-1) \\ \\ &5x&+&y&+&6z&=&-2 \\ +&-5x&&&-&6z&=&\phantom{-}4 \\ \hline &&&&&y&=&2 \\ \\ &&&\therefore 2y&-&3z&=&\phantom{-}3 \\ &&&2(2)&-&3z&=&\phantom{-}3 \\ &&&-4&&&&-4 \\ \hline &&&&&-3z&=&-1 \\ &&&&&z&=&\dfrac{1}{3} \\ \end{array} & \hspace{0.25in} \begin{array}[t]{rrrrr} 5x&+&6z&=&-4 \\ 5x&+&6\left(\dfrac{1}{3}\right)&=&-4 \\ 5x&+&2&=&-4 \\ &-&2&&-2 \\ \hline &&5x&=&-6 \\ &&x&=&-\dfrac{6}{5} \end{array} \end{array}[/latex]
   [latex]-\dfrac{6}{5}, 2, \dfrac{1}{3}[/latex]
5. [latex]\phantom{a}[/latex]
   [latex]\begin{array}[t]{rrrrrr} &4a^2&-&9a&+&2 \\ &-a^2&+&4a&+&5 \\ +&3a^2&-&a&+&9 \\ \hline &6a^2&-&6a&+&16 \end{array}[/latex]
6. [latex]8x^4-12x^2y^2-15x^2y^2-3x^4\Rightarrow 5x^4-27x^2y^2[/latex]
7. [latex]6-2\left[3x-20x+8-1\right][/latex]
   [latex]\begin{array}[t]{l} 6-2\left[-17x+7\right] \\ 6+34x-14 \\ 34x-8 \end{array}[/latex]
8. [latex]25a^{-10}b^6\text{ or } \dfrac{25b^6}{a^{10}}[/latex]
9. [latex]8a^2(a^2+10a+25)[/latex]
   [latex]8a^4+80a^3+200a^2[/latex]
10. [latex]4ab^2(a^2-4)[/latex]
    [latex]4a^3b^2-16ab^2[/latex]
11. [latex]\phantom{a}[/latex]
    [latex]\begin{array}[t]{rrrrrrrr} &x^2&-&4x&+&7\phantom{x}&& \\ \times &&&x&-&3\phantom{x}&& \\ \hline &x^3&-&4x^2&+&7x&& \\ +&&-&3x^2&+&12x&-&21 \\ \hline &x^3&-&7x^2&+&19x&-&21 \\ \end{array}[/latex]
12. [latex]\phantom{a}[/latex]
    [latex]\begin{array}[t]{rrrrrrrrrr} &2x^2&+&x&-&3\phantom{x^2}&&&& \\ \times &2x^2&+&x&-&3\phantom{x^2}&&&& \\ \hline &4x^4&+&2x^3&-&6x^2&&&& \\ &&&2x^3&+&x^2&-&3x&& \\ +&&&&&-6x^2&-&3x&+&9 \\ \hline &4x^4&+&4x^3&-&11x^2&-&6x&+&9 \end{array}[/latex]
13. [latex]\phantom{a}[/latex]
    [latex]\begin{array}[t]{rrrrrrrrrr} &x^2&+&5x&-&2\phantom{x^2}&&&& \\ \times &2x^2&-&x&+&3\phantom{x^2}&&&& \\ \hline &2x^4&+&10x^3&-&4x^2&&&& \\ &&&-x^3&-&5x^2&+&2x&& \\ +&&&&&3x^2&+&15x&-&6 \\ \hline &2x^4&+&9x^3&-&6x^2&+&17x&-&6 \end{array}[/latex]
14. [latex]\phantom{a}[/latex]
    [latex]\begin{array}[t]{rrrrrrrrrr} (x+4)(x+4)&\Rightarrow &&x^2&+&8x&+&16&& \\ &&\times&&&x&+&4&& \\ \hline &&&x^3&+&8x^2&+&16x&& \\ &&+&&&4x^2&+&32x&+&64 \\ \hline &&&x^3&+&12x^2&+&48x&+&64 \end{array}[/latex]
15. [latex]r^{-4-3}s^{9+9}\Rightarrow r^{-7}s^{18}\Rightarrow \dfrac{s^{18}}{r^7}[/latex]
    [latex]\dfrac{s^{18}}{r^7}[/latex]
16. [latex](x^{-2--2}y^{-3-4})^{-1}[/latex]
    [latex]\begin{array}[t]{l} (1\cancel{x^0}y^{-7})^{-1} \\ y^7 y^7 \end{array}[/latex]
17. [latex]\phantom{a}[/latex]
    
18. [latex]2^3\cdot 11[/latex]
19. [latex]2^5\cdot 3\cdot 7 \left\{ \begin{array}{l} 84=2^2\cdot 3\cdot 7 \\ 96=2^5\cdot 3 \end{array}\right.[/latex]
20. [latex]x(5y+6z)-3(5y+6z)[/latex]
    [latex](5y+6z)(x-3)[/latex]
21. [latex]-12=4\times -3[/latex]
    [latex]1=4+-3[/latex]
    [latex]x^2+4x-3x-12[/latex]
    [latex]x(x+4)-3(x+4)[/latex]
    [latex](x+4)(x-3)[/latex]
22. [latex]x^2(x+1)-4(x+1)[/latex]
    [latex](x+1)(x^2-4)[/latex]
    [latex]x+1)(x-2)(x+2)[/latex]
23. [latex]x^3-(3y)^3[/latex]
    [latex](x-3y)(x^2+3xy+9y^2)[/latex]
24. [latex](x^2-36)(x^2+1)[/latex]
    [latex](x-6)(x+6)(x^2+1)[/latex]
25. [latex]\phantom{a}[/latex]
    [latex]\begin{array}[t]{lll} \begin{array}[t]{rrrrl} (A&+&B&=&70)(-4) \\ 4A&+&7B&=&430 \end{array} & \Rightarrow \hspace{0.25in} \begin{array}[t]{rrrrrl} &-4A&-&4B&=&-280 \\ +&4A&+&7B&=&\phantom{-}430 \\ \hline &&&3B&=&\phantom{-}150 \\ \\ &&&B&=&\dfrac{150}{3}\text{ or }50 \end{array} & \hspace{0.25in} \begin{array}[t]{rrrrr} \therefore A&+&B&=&70 \\ \\ A&+&50&=&70 \\ &&-50&&-50 \\ \hline &&A&=&20 \end{array} \end{array}[/latex]
26. [latex]\phantom{a}[/latex]
    [latex]\begin{array}[t]{rrcrrrl} 5x&+&21(2)&=&11(x&+&2) \\ \\ 5x&+&42&=&11x&+&22 \\ -5x&-&22&&-5x&-&22 \\ \hline &&20&=&6x&& \\ \\ &&x&=&\dfrac{20}{6}&=&3\dfrac{1}{3}\text{ litres} \\ \end{array}[/latex]
27. [latex]B+G=16\Rightarrow B=16-G\text{ or }G=16-B[/latex]
    [latex]\begin{array}[t]{ll} \begin{array}[t]{rrrrrrr} G&-&4&=&3(B&-&4) \\ 16-B&-&4&=&3B&-&12 \\ +B&+&12&&+B&+&12 \\ \hline &&24&=&4B&& \\ \\ &&B&=&\dfrac{24}{4}&=&6 \end{array} & \hspace{0.25in} \begin{array}[t]{rrrrr} \therefore G&=&16&-&B \\ G&=&16&-&6 \\ G&=&10&& \end{array} \end{array}[/latex]