Stelsels combinatie

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Combinatie

1. \(\left\{\begin{matrix}10x-8y=-24\\-6y+4x=-4\end{matrix}\right.\)
2. \(\left\{\begin{matrix}-9y+3x=-108\\-7x+5y=92\end{matrix}\right.\)
3. \(\left\{\begin{matrix}-6x-6y=36\\3x=2y+7\end{matrix}\right.\)
4. \(\left\{\begin{matrix}-9x-10y=136\\9x+7y=-106\end{matrix}\right.\)
5. \(\left\{\begin{matrix}-5y-3x=23\\-9x-8y=20\end{matrix}\right.\)
6. \(\left\{\begin{matrix}-5y-6x=-52\\8x+6y=64\end{matrix}\right.\)
7. \(\left\{\begin{matrix}2x+8y=18\\9x-10y=-11\end{matrix}\right.\)
8. \(\left\{\begin{matrix}-9x+3y=33\\8x=-10y-156\end{matrix}\right.\)
9. \(\left\{\begin{matrix}-2y+6x=22\\-6x+6y=-54\end{matrix}\right.\)
10. \(\left\{\begin{matrix}7x-3y=41\\-2x+2y=-6\end{matrix}\right.\)
11. \(\left\{\begin{matrix}10y=161+9x\\3x+6y=21\end{matrix}\right.\)
12. \(\left\{\begin{matrix}3y=40+7x\\2x-10y=16\end{matrix}\right.\)

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Combinatie

Verbetersleutel

1. \(\left\{\begin{matrix}10x-8y=-24\\-6y+4x=-4\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}10x-8y=-24\\4x-6y=-4\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{array}{c|c|c}10x-8y=-24& \color{red}{2.} & \color{blue}{3.} \\4x-6y=-4& \color{red}{-5.} & \color{blue}{-4.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{20x-20x}-16y+30y=-48+20} \\ \color{blue}{30x-16x\underline{-24y+24y}=-72+16} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}14y=-28 \\14x=-56\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{-28}{14}=-2 \\x=\frac{-56}{14}=-4\end{matrix}\right.\\ \qquad V=\{(-4,-2)\}\)
2. \(\left\{\begin{matrix}-9y+3x=-108\\-7x+5y=92\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}3x-9y=-108\\-7x+5y=92\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{array}{c|c|c}3x-9y=-108& \color{red}{7.} & \color{blue}{5.} \\-7x+5y=92& \color{red}{3.} & \color{blue}{9.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{21x-21x}-63y+15y=-756+276} \\ \color{blue}{15x-63x\underline{-45y+45y}=-540+828} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-48y=-480 \\-48x=288\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{-480}{-48}=10 \\x=\frac{288}{-48}=-6\end{matrix}\right.\\ \qquad V=\{(-6,10)\}\)
3. \(\left\{\begin{matrix}-6x-6y=36\\3x=2y+7\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-6x-6y=36\\3x-2y=7\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{array}{c|c|c}-6x-6y=36& \color{red}{1.} & \color{blue}{1.} \\3x-2y=7& \color{red}{2.} & \color{blue}{-3.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{-6x+6x}-6y-4y=36+14} \\ \color{blue}{-6x-9x\underline{-6y+6y}=36-21} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-10y=50 \\-15x=15\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{50}{-10}=-5 \\x=\frac{15}{-15}=-1\end{matrix}\right.\\ \qquad V=\{(-1,-5)\}\)
4. \(\left\{\begin{matrix}-9x-10y=136\\9x+7y=-106\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{array}{c|c|c}-9x-10y=136& \color{red}{1.} & \color{blue}{7.} \\9x+7y=-106& \color{red}{1.} & \color{blue}{10.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{-9x+9x}-10y+7y=136-106} \\ \color{blue}{-63x+90x\underline{-70y+70y}=952-1060} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-3y=30 \\27x=-108\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{30}{-3}=-10 \\x=\frac{-108}{27}=-4\end{matrix}\right.\\ \qquad V=\{(-4,-10)\}\)
5. \(\left\{\begin{matrix}-5y-3x=23\\-9x-8y=20\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-3x-5y=23\\-9x-8y=20\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{array}{c|c|c}-3x-5y=23& \color{red}{3.} & \color{blue}{8.} \\-9x-8y=20& \color{red}{-1.} & \color{blue}{-5.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{-9x+9x}-15y+8y=69-20} \\ \color{blue}{-24x+45x\underline{-40y+40y}=184-100} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-7y=49 \\21x=84\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{49}{-7}=-7 \\x=\frac{84}{21}=4\end{matrix}\right.\\ \qquad V=\{(4,-7)\}\)
6. \(\left\{\begin{matrix}-5y-6x=-52\\8x+6y=64\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-6x-5y=-52\\8x+6y=64\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{array}{c|c|c}-6x-5y=-52& \color{red}{4.} & \color{blue}{6.} \\8x+6y=64& \color{red}{3.} & \color{blue}{5.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{-24x+24x}-20y+18y=-208+192} \\ \color{blue}{-36x+40x\underline{-30y+30y}=-312+320} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-2y=-16 \\4x=8\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{-16}{-2}=8 \\x=\frac{8}{4}=2\end{matrix}\right.\\ \qquad V=\{(2,8)\}\)
7. \(\left\{\begin{matrix}2x+8y=18\\9x-10y=-11\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{array}{c|c|c}2x+8y=18& \color{red}{9.} & \color{blue}{5.} \\9x-10y=-11& \color{red}{-2.} & \color{blue}{4.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{18x-18x}+72y+20y=162+22} \\ \color{blue}{10x+36x\underline{+40y-40y}=90-44} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}92y=184 \\46x=46\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{184}{92}=2 \\x=\frac{46}{46}=1\end{matrix}\right.\\ \qquad V=\{(1,2)\}\)
8. \(\left\{\begin{matrix}-9x+3y=33\\8x=-10y-156\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-9x+3y=33\\8x+10y=-156\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{array}{c|c|c}-9x+3y=33& \color{red}{8.} & \color{blue}{10.} \\8x+10y=-156& \color{red}{9.} & \color{blue}{-3.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{-72x+72x}+24y+90y=264-1404} \\ \color{blue}{-90x-24x\underline{+30y-30y}=330+468} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}114y=-1140 \\-114x=798\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{-1140}{114}=-10 \\x=\frac{798}{-114}=-7\end{matrix}\right.\\ \qquad V=\{(-7,-10)\}\)
9. \(\left\{\begin{matrix}-2y+6x=22\\-6x+6y=-54\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}6x-2y=22\\-6x+6y=-54\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{array}{c|c|c}6x-2y=22& \color{red}{1.} & \color{blue}{3.} \\-6x+6y=-54& \color{red}{1.} & \color{blue}{1.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{6x-6x}-2y+6y=22-54} \\ \color{blue}{18x-6x\underline{-6y+6y}=66-54} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}4y=-32 \\12x=12\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{-32}{4}=-8 \\x=\frac{12}{12}=1\end{matrix}\right.\\ \qquad V=\{(1,-8)\}\)
10. \(\left\{\begin{matrix}7x-3y=41\\-2x+2y=-6\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{array}{c|c|c}7x-3y=41& \color{red}{2.} & \color{blue}{2.} \\-2x+2y=-6& \color{red}{7.} & \color{blue}{3.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{14x-14x}-6y+14y=82-42} \\ \color{blue}{14x-6x\underline{-6y+6y}=82-18} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}8y=40 \\8x=64\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{40}{8}=5 \\x=\frac{64}{8}=8\end{matrix}\right.\\ \qquad V=\{(8,5)\}\)
11. \(\left\{\begin{matrix}10y=161+9x\\3x+6y=21\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-9x+10y=161\\3x+6y=21\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{array}{c|c|c}-9x+10y=161& \color{red}{1.} & \color{blue}{3.} \\3x+6y=21& \color{red}{3.} & \color{blue}{-5.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{-9x+9x}+10y+18y=161+63} \\ \color{blue}{-27x-15x\underline{+30y-30y}=483-105} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}28y=224 \\-42x=378\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{224}{28}=8 \\x=\frac{378}{-42}=-9\end{matrix}\right.\\ \qquad V=\{(-9,8)\}\)
12. \(\left\{\begin{matrix}3y=40+7x\\2x-10y=16\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-7x+3y=40\\2x-10y=16\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{array}{c|c|c}-7x+3y=40& \color{red}{2.} & \color{blue}{10.} \\2x-10y=16& \color{red}{7.} & \color{blue}{3.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{-14x+14x}+6y-70y=80+112} \\ \color{blue}{-70x+6x\underline{+30y-30y}=400+48} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-64y=192 \\-64x=448\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{192}{-64}=-3 \\x=\frac{448}{-64}=-7\end{matrix}\right.\\ \qquad V=\{(-7,-3)\}\)