Stelsels combinatie

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Combinatie

1. \(\left\{\begin{matrix}5x-9y=58\\2y+6x=-20\end{matrix}\right.\)
2. \(\left\{\begin{matrix}8x-9y=-105\\4x+4y=24\end{matrix}\right.\)
3. \(\left\{\begin{matrix}-4y+4x=24\\8x-5y=21\end{matrix}\right.\)
4. \(\left\{\begin{matrix}-5x+10y=135\\-6y+9x=-135\end{matrix}\right.\)
5. \(\left\{\begin{matrix}-10y+3x=5\\7x+3y=-41\end{matrix}\right.\)
6. \(\left\{\begin{matrix}-6x+2y=22\\-7y-9x=13\end{matrix}\right.\)
7. \(\left\{\begin{matrix}5x+3y=-14\\4y-7x=-46\end{matrix}\right.\)
8. \(\left\{\begin{matrix}2y=-18-4x\\-7x-3y=34\end{matrix}\right.\)
9. \(\left\{\begin{matrix}8x-5y=17\\-5x+4y=-8\end{matrix}\right.\)
10. \(\left\{\begin{matrix}-8x+4y=-32\\10y-10x=-50\end{matrix}\right.\)
11. \(\left\{\begin{matrix}-3x+9y=108\\-9x+6y=114\end{matrix}\right.\)
12. \(\left\{\begin{matrix}-5x-3y=23\\-2x=-4y-22\end{matrix}\right.\)

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Combinatie

Verbetersleutel

1. \(\left\{\begin{matrix}5x-9y=58\\2y+6x=-20\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}5x-9y=58\\6x+2y=-20\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{array}{c|c|c}5x-9y=58& \color{red}{6.} & \color{blue}{2.} \\6x+2y=-20& \color{red}{-5.} & \color{blue}{9.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{30x-30x}-54y-10y=348+100} \\ \color{blue}{10x+54x\underline{-18y+18y}=116-180} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-64y=448 \\64x=-64\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{448}{-64}=-7 \\x=\frac{-64}{64}=-1\end{matrix}\right.\\ \qquad V=\{(-1,-7)\}\)
2. \(\left\{\begin{matrix}8x-9y=-105\\4x+4y=24\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{array}{c|c|c}8x-9y=-105& \color{red}{1.} & \color{blue}{4.} \\4x+4y=24& \color{red}{-2.} & \color{blue}{9.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{8x-8x}-9y-8y=-105-48} \\ \color{blue}{32x+36x\underline{-36y+36y}=-420+216} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-17y=-153 \\68x=-204\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{-153}{-17}=9 \\x=\frac{-204}{68}=-3\end{matrix}\right.\\ \qquad V=\{(-3,9)\}\)
3. \(\left\{\begin{matrix}-4y+4x=24\\8x-5y=21\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}4x-4y=24\\8x-5y=21\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{array}{c|c|c}4x-4y=24& \color{red}{2.} & \color{blue}{5.} \\8x-5y=21& \color{red}{-1.} & \color{blue}{-4.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{8x-8x}-8y+5y=48-21} \\ \color{blue}{20x-32x\underline{-20y+20y}=120-84} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-3y=27 \\-12x=36\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{27}{-3}=-9 \\x=\frac{36}{-12}=-3\end{matrix}\right.\\ \qquad V=\{(-3,-9)\}\)
4. \(\left\{\begin{matrix}-5x+10y=135\\-6y+9x=-135\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-5x+10y=135\\9x-6y=-135\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{array}{c|c|c}-5x+10y=135& \color{red}{9.} & \color{blue}{3.} \\9x-6y=-135& \color{red}{5.} & \color{blue}{5.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{-45x+45x}+90y-30y=1215-675} \\ \color{blue}{-15x+45x\underline{+30y-30y}=405-675} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}60y=540 \\30x=-270\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{540}{60}=9 \\x=\frac{-270}{30}=-9\end{matrix}\right.\\ \qquad V=\{(-9,9)\}\)
5. \(\left\{\begin{matrix}-10y+3x=5\\7x+3y=-41\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}3x-10y=5\\7x+3y=-41\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{array}{c|c|c}3x-10y=5& \color{red}{7.} & \color{blue}{3.} \\7x+3y=-41& \color{red}{-3.} & \color{blue}{10.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{21x-21x}-70y-9y=35+123} \\ \color{blue}{9x+70x\underline{-30y+30y}=15-410} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-79y=158 \\79x=-395\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{158}{-79}=-2 \\x=\frac{-395}{79}=-5\end{matrix}\right.\\ \qquad V=\{(-5,-2)\}\)
6. \(\left\{\begin{matrix}-6x+2y=22\\-7y-9x=13\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-6x+2y=22\\-9x-7y=13\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{array}{c|c|c}-6x+2y=22& \color{red}{3.} & \color{blue}{7.} \\-9x-7y=13& \color{red}{-2.} & \color{blue}{2.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{-18x+18x}+6y+14y=66-26} \\ \color{blue}{-42x-18x\underline{+14y-14y}=154+26} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}20y=40 \\-60x=180\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{40}{20}=2 \\x=\frac{180}{-60}=-3\end{matrix}\right.\\ \qquad V=\{(-3,2)\}\)
7. \(\left\{\begin{matrix}5x+3y=-14\\4y-7x=-46\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}5x+3y=-14\\-7x+4y=-46\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{array}{c|c|c}5x+3y=-14& \color{red}{7.} & \color{blue}{4.} \\-7x+4y=-46& \color{red}{5.} & \color{blue}{-3.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{35x-35x}+21y+20y=-98-230} \\ \color{blue}{20x+21x\underline{+12y-12y}=-56+138} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}41y=-328 \\41x=82\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{-328}{41}=-8 \\x=\frac{82}{41}=2\end{matrix}\right.\\ \qquad V=\{(2,-8)\}\)
8. \(\left\{\begin{matrix}2y=-18-4x\\-7x-3y=34\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}4x+2y=-18\\-7x-3y=34\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{array}{c|c|c}4x+2y=-18& \color{red}{7.} & \color{blue}{3.} \\-7x-3y=34& \color{red}{4.} & \color{blue}{2.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{28x-28x}+14y-12y=-126+136} \\ \color{blue}{12x-14x\underline{+6y-6y}=-54+68} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}2y=10 \\-2x=14\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{10}{2}=5 \\x=\frac{14}{-2}=-7\end{matrix}\right.\\ \qquad V=\{(-7,5)\}\)
9. \(\left\{\begin{matrix}8x-5y=17\\-5x+4y=-8\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{array}{c|c|c}8x-5y=17& \color{red}{5.} & \color{blue}{4.} \\-5x+4y=-8& \color{red}{8.} & \color{blue}{5.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{40x-40x}-25y+32y=85-64} \\ \color{blue}{32x-25x\underline{-20y+20y}=68-40} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}7y=21 \\7x=28\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{21}{7}=3 \\x=\frac{28}{7}=4\end{matrix}\right.\\ \qquad V=\{(4,3)\}\)
10. \(\left\{\begin{matrix}-8x+4y=-32\\10y-10x=-50\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-8x+4y=-32\\-10x+10y=-50\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{array}{c|c|c}-8x+4y=-32& \color{red}{5.} & \color{blue}{5.} \\-10x+10y=-50& \color{red}{-4.} & \color{blue}{-2.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{-40x+40x}+20y-40y=-160+200} \\ \color{blue}{-40x+20x\underline{+20y-20y}=-160+100} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-20y=40 \\-20x=-60\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{40}{-20}=-2 \\x=\frac{-60}{-20}=3\end{matrix}\right.\\ \qquad V=\{(3,-2)\}\)
11. \(\left\{\begin{matrix}-3x+9y=108\\-9x+6y=114\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{array}{c|c|c}-3x+9y=108& \color{red}{3.} & \color{blue}{2.} \\-9x+6y=114& \color{red}{-1.} & \color{blue}{-3.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{-9x+9x}+27y-6y=324-114} \\ \color{blue}{-6x+27x\underline{+18y-18y}=216-342} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}21y=210 \\21x=-126\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{210}{21}=10 \\x=\frac{-126}{21}=-6\end{matrix}\right.\\ \qquad V=\{(-6,10)\}\)
12. \(\left\{\begin{matrix}-5x-3y=23\\-2x=-4y-22\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-5x-3y=23\\-2x+4y=-22\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{array}{c|c|c}-5x-3y=23& \color{red}{2.} & \color{blue}{4.} \\-2x+4y=-22& \color{red}{-5.} & \color{blue}{3.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{-10x+10x}-6y-20y=46+110} \\ \color{blue}{-20x-6x\underline{-12y+12y}=92-66} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-26y=156 \\-26x=26\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{156}{-26}=-6 \\x=\frac{26}{-26}=-1\end{matrix}\right.\\ \qquad V=\{(-1,-6)\}\)