Stelsels combinatie

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Combinatie

1. \(\left\{\begin{matrix}-4x-6y=16\\-5x-4y=-8\end{matrix}\right.\)
2. \(\left\{\begin{matrix}-9x-4y=-36\\5y+2x=-29\end{matrix}\right.\)
3. \(\left\{\begin{matrix}10x-7y=135\\-3x=5y-5\end{matrix}\right.\)
4. \(\left\{\begin{matrix}3y-5x=44\\4x+8y=-4\end{matrix}\right.\)
5. \(\left\{\begin{matrix}-10x+2y=-62\\5x=-3y+27\end{matrix}\right.\)
6. \(\left\{\begin{matrix}-6x+4y=22\\-10x+8y=34\end{matrix}\right.\)
7. \(\left\{\begin{matrix}-7y=-34-6x\\5x+4y=70\end{matrix}\right.\)
8. \(\left\{\begin{matrix}-10x-2y=-46\\-8x+8y=-56\end{matrix}\right.\)
9. \(\left\{\begin{matrix}-4y-4x=-20\\10x-6y=-94\end{matrix}\right.\)
10. \(\left\{\begin{matrix}-8x-10y=-94\\-8x=4y-76\end{matrix}\right.\)
11. \(\left\{\begin{matrix}6y-2x=30\\3x-4y=-10\end{matrix}\right.\)
12. \(\left\{\begin{matrix}3y=14+2x\\5x-8y=-39\end{matrix}\right.\)

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Combinatie

Verbetersleutel

1. \(\left\{\begin{matrix}-4x-6y=16\\-5x-4y=-8\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{array}{c|c|c}-4x-6y=16& \color{red}{5.} & \color{blue}{2.} \\-5x-4y=-8& \color{red}{-4.} & \color{blue}{-3.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{-20x+20x}-30y+16y=80+32} \\ \color{blue}{-8x+15x\underline{-12y+12y}=32+24} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-14y=112 \\7x=56\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{112}{-14}=-8 \\x=\frac{56}{7}=8\end{matrix}\right.\\ \qquad V=\{(8,-8)\}\)
2. \(\left\{\begin{matrix}-9x-4y=-36\\5y+2x=-29\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-9x-4y=-36\\2x+5y=-29\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{array}{c|c|c}-9x-4y=-36& \color{red}{2.} & \color{blue}{5.} \\2x+5y=-29& \color{red}{9.} & \color{blue}{4.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{-18x+18x}-8y+45y=-72-261} \\ \color{blue}{-45x+8x\underline{-20y+20y}=-180-116} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}37y=-333 \\-37x=-296\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{-333}{37}=-9 \\x=\frac{-296}{-37}=8\end{matrix}\right.\\ \qquad V=\{(8,-9)\}\)
3. \(\left\{\begin{matrix}10x-7y=135\\-3x=5y-5\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}10x-7y=135\\-3x-5y=-5\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{array}{c|c|c}10x-7y=135& \color{red}{3.} & \color{blue}{5.} \\-3x-5y=-5& \color{red}{10.} & \color{blue}{-7.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{30x-30x}-21y-50y=405-50} \\ \color{blue}{50x+21x\underline{-35y+35y}=675+35} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-71y=355 \\71x=710\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{355}{-71}=-5 \\x=\frac{710}{71}=10\end{matrix}\right.\\ \qquad V=\{(10,-5)\}\)
4. \(\left\{\begin{matrix}3y-5x=44\\4x+8y=-4\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-5x+3y=44\\4x+8y=-4\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{array}{c|c|c}-5x+3y=44& \color{red}{4.} & \color{blue}{8.} \\4x+8y=-4& \color{red}{5.} & \color{blue}{-3.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{-20x+20x}+12y+40y=176-20} \\ \color{blue}{-40x-12x\underline{+24y-24y}=352+12} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}52y=156 \\-52x=364\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{156}{52}=3 \\x=\frac{364}{-52}=-7\end{matrix}\right.\\ \qquad V=\{(-7,3)\}\)
5. \(\left\{\begin{matrix}-10x+2y=-62\\5x=-3y+27\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-10x+2y=-62\\5x+3y=27\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{array}{c|c|c}-10x+2y=-62& \color{red}{1.} & \color{blue}{3.} \\5x+3y=27& \color{red}{2.} & \color{blue}{-2.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{-10x+10x}+2y+6y=-62+54} \\ \color{blue}{-30x-10x\underline{+6y-6y}=-186-54} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}8y=-8 \\-40x=-240\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{-8}{8}=-1 \\x=\frac{-240}{-40}=6\end{matrix}\right.\\ \qquad V=\{(6,-1)\}\)
6. \(\left\{\begin{matrix}-6x+4y=22\\-10x+8y=34\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{array}{c|c|c}-6x+4y=22& \color{red}{5.} & \color{blue}{2.} \\-10x+8y=34& \color{red}{-3.} & \color{blue}{-1.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{-30x+30x}+20y-24y=110-102} \\ \color{blue}{-12x+10x\underline{+8y-8y}=44-34} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-4y=8 \\-2x=10\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{8}{-4}=-2 \\x=\frac{10}{-2}=-5\end{matrix}\right.\\ \qquad V=\{(-5,-2)\}\)
7. \(\left\{\begin{matrix}-7y=-34-6x\\5x+4y=70\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}6x-7y=-34\\5x+4y=70\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{array}{c|c|c}6x-7y=-34& \color{red}{5.} & \color{blue}{4.} \\5x+4y=70& \color{red}{-6.} & \color{blue}{7.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{30x-30x}-35y-24y=-170-420} \\ \color{blue}{24x+35x\underline{-28y+28y}=-136+490} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-59y=-590 \\59x=354\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{-590}{-59}=10 \\x=\frac{354}{59}=6\end{matrix}\right.\\ \qquad V=\{(6,10)\}\)
8. \(\left\{\begin{matrix}-10x-2y=-46\\-8x+8y=-56\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{array}{c|c|c}-10x-2y=-46& \color{red}{4.} & \color{blue}{4.} \\-8x+8y=-56& \color{red}{-5.} & \color{blue}{1.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{-40x+40x}-8y-40y=-184+280} \\ \color{blue}{-40x-8x\underline{-8y+8y}=-184-56} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-48y=96 \\-48x=-240\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{96}{-48}=-2 \\x=\frac{-240}{-48}=5\end{matrix}\right.\\ \qquad V=\{(5,-2)\}\)
9. \(\left\{\begin{matrix}-4y-4x=-20\\10x-6y=-94\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-4x-4y=-20\\10x-6y=-94\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{array}{c|c|c}-4x-4y=-20& \color{red}{5.} & \color{blue}{3.} \\10x-6y=-94& \color{red}{2.} & \color{blue}{-2.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{-20x+20x}-20y-12y=-100-188} \\ \color{blue}{-12x-20x\underline{-12y+12y}=-60+188} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-32y=-288 \\-32x=128\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{-288}{-32}=9 \\x=\frac{128}{-32}=-4\end{matrix}\right.\\ \qquad V=\{(-4,9)\}\)
10. \(\left\{\begin{matrix}-8x-10y=-94\\-8x=4y-76\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-8x-10y=-94\\-8x-4y=-76\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{array}{c|c|c}-8x-10y=-94& \color{red}{1.} & \color{blue}{2.} \\-8x-4y=-76& \color{red}{-1.} & \color{blue}{-5.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{-8x+8x}-10y+4y=-94+76} \\ \color{blue}{-16x+40x\underline{-20y+20y}=-188+380} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-6y=-18 \\24x=192\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{-18}{-6}=3 \\x=\frac{192}{24}=8\end{matrix}\right.\\ \qquad V=\{(8,3)\}\)
11. \(\left\{\begin{matrix}6y-2x=30\\3x-4y=-10\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-2x+6y=30\\3x-4y=-10\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{array}{c|c|c}-2x+6y=30& \color{red}{3.} & \color{blue}{2.} \\3x-4y=-10& \color{red}{2.} & \color{blue}{3.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{-6x+6x}+18y-8y=90-20} \\ \color{blue}{-4x+9x\underline{+12y-12y}=60-30} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}10y=70 \\5x=30\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{70}{10}=7 \\x=\frac{30}{5}=6\end{matrix}\right.\\ \qquad V=\{(6,7)\}\)
12. \(\left\{\begin{matrix}3y=14+2x\\5x-8y=-39\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-2x+3y=14\\5x-8y=-39\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{array}{c|c|c}-2x+3y=14& \color{red}{5.} & \color{blue}{8.} \\5x-8y=-39& \color{red}{2.} & \color{blue}{3.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{-10x+10x}+15y-16y=70-78} \\ \color{blue}{-16x+15x\underline{+24y-24y}=112-117} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-y=-8 \\-x=-5\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{-8}{-1}=8 \\x=\frac{-5}{-1}=5\end{matrix}\right.\\ \qquad V=\{(5,8)\}\)