Stelsels combinatie

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Combinatie

1. \(\left\{\begin{matrix}-3x-7y=35\\3x+6y=-33\end{matrix}\right.\)
2. \(\left\{\begin{matrix}-2x-2y=6\\4x+8y=-4\end{matrix}\right.\)
3. \(\left\{\begin{matrix}8y=48-8x\\-10x-10y=-60\end{matrix}\right.\)
4. \(\left\{\begin{matrix}-4x-4y=-8\\5y-5x=80\end{matrix}\right.\)
5. \(\left\{\begin{matrix}-4x-3y=-22\\9y-4x=-46\end{matrix}\right.\)
6. \(\left\{\begin{matrix}7x+5y=42\\2x=7y-47\end{matrix}\right.\)
7. \(\left\{\begin{matrix}5x+2y=-11\\2x-5y=-45\end{matrix}\right.\)
8. \(\left\{\begin{matrix}-2x-5y=-19\\-7y-8x=-63\end{matrix}\right.\)
9. \(\left\{\begin{matrix}6x+7y=8\\3x=-8y+40\end{matrix}\right.\)
10. \(\left\{\begin{matrix}-2x-9y=21\\3x=10y-8\end{matrix}\right.\)
11. \(\left\{\begin{matrix}-5x-8y=18\\-10x+10y=10\end{matrix}\right.\)
12. \(\left\{\begin{matrix}7x+2y=-37\\2x+10y=-20\end{matrix}\right.\)

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Combinatie

Verbetersleutel

1. \(\left\{\begin{matrix}-3x-7y=35\\3x+6y=-33\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{array}{c|c|c}-3x-7y=35& \color{red}{1.} & \color{blue}{6.} \\3x+6y=-33& \color{red}{1.} & \color{blue}{7.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{-3x+3x}-7y+6y=35-33} \\ \color{blue}{-18x+21x\underline{-42y+42y}=210-231} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-y=2 \\3x=-21\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{2}{-1}=-2 \\x=\frac{-21}{3}=-7\end{matrix}\right.\\ \qquad V=\{(-7,-2)\}\)
2. \(\left\{\begin{matrix}-2x-2y=6\\4x+8y=-4\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{array}{c|c|c}-2x-2y=6& \color{red}{2.} & \color{blue}{4.} \\4x+8y=-4& \color{red}{1.} & \color{blue}{1.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{-4x+4x}-4y+8y=12-4} \\ \color{blue}{-8x+4x\underline{-8y+8y}=24-4} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}4y=8 \\-4x=20\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{8}{4}=2 \\x=\frac{20}{-4}=-5\end{matrix}\right.\\ \qquad V=\{(-5,2)\}\)
3. \(\left\{\begin{matrix}8y=48-8x\\-10x-10y=-60\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}8x+8y=48\\-10x-10y=-60\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{array}{c|c|c}8x+8y=48& \color{red}{5.} & \color{blue}{5.} \\-10x-10y=-60& \color{red}{4.} & \color{blue}{4.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{40x-40x}+40y-40y=240-240} \\ \color{blue}{40x-40x\underline{+40y-40y}=240-240} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}0y=0 \\0x=0\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{0}{0}=8 \\x=\frac{0}{0}=-2\end{matrix}\right.\\ \qquad V=\{(-2,8)\}\)
4. \(\left\{\begin{matrix}-4x-4y=-8\\5y-5x=80\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-4x-4y=-8\\-5x+5y=80\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{array}{c|c|c}-4x-4y=-8& \color{red}{5.} & \color{blue}{5.} \\-5x+5y=80& \color{red}{-4.} & \color{blue}{4.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{-20x+20x}-20y-20y=-40-320} \\ \color{blue}{-20x-20x\underline{-20y+20y}=-40+320} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-40y=-360 \\-40x=280\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{-360}{-40}=9 \\x=\frac{280}{-40}=-7\end{matrix}\right.\\ \qquad V=\{(-7,9)\}\)
5. \(\left\{\begin{matrix}-4x-3y=-22\\9y-4x=-46\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-4x-3y=-22\\-4x+9y=-46\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{array}{c|c|c}-4x-3y=-22& \color{red}{1.} & \color{blue}{3.} \\-4x+9y=-46& \color{red}{-1.} & \color{blue}{1.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{-4x+4x}-3y-9y=-22+46} \\ \color{blue}{-12x-4x\underline{-9y+9y}=-66-46} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-12y=24 \\-16x=-112\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{24}{-12}=-2 \\x=\frac{-112}{-16}=7\end{matrix}\right.\\ \qquad V=\{(7,-2)\}\)
6. \(\left\{\begin{matrix}7x+5y=42\\2x=7y-47\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}7x+5y=42\\2x-7y=-47\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{array}{c|c|c}7x+5y=42& \color{red}{2.} & \color{blue}{7.} \\2x-7y=-47& \color{red}{-7.} & \color{blue}{5.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{14x-14x}+10y+49y=84+329} \\ \color{blue}{49x+10x\underline{+35y-35y}=294-235} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}59y=413 \\59x=59\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{413}{59}=7 \\x=\frac{59}{59}=1\end{matrix}\right.\\ \qquad V=\{(1,7)\}\)
7. \(\left\{\begin{matrix}5x+2y=-11\\2x-5y=-45\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{array}{c|c|c}5x+2y=-11& \color{red}{2.} & \color{blue}{5.} \\2x-5y=-45& \color{red}{-5.} & \color{blue}{2.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{10x-10x}+4y+25y=-22+225} \\ \color{blue}{25x+4x\underline{+10y-10y}=-55-90} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}29y=203 \\29x=-145\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{203}{29}=7 \\x=\frac{-145}{29}=-5\end{matrix}\right.\\ \qquad V=\{(-5,7)\}\)
8. \(\left\{\begin{matrix}-2x-5y=-19\\-7y-8x=-63\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-2x-5y=-19\\-8x-7y=-63\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{array}{c|c|c}-2x-5y=-19& \color{red}{4.} & \color{blue}{7.} \\-8x-7y=-63& \color{red}{-1.} & \color{blue}{-5.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{-8x+8x}-20y+7y=-76+63} \\ \color{blue}{-14x+40x\underline{-35y+35y}=-133+315} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-13y=-13 \\26x=182\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{-13}{-13}=1 \\x=\frac{182}{26}=7\end{matrix}\right.\\ \qquad V=\{(7,1)\}\)
9. \(\left\{\begin{matrix}6x+7y=8\\3x=-8y+40\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}6x+7y=8\\3x+8y=40\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{array}{c|c|c}6x+7y=8& \color{red}{1.} & \color{blue}{8.} \\3x+8y=40& \color{red}{-2.} & \color{blue}{-7.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{6x-6x}+7y-16y=8-80} \\ \color{blue}{48x-21x\underline{+56y-56y}=64-280} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-9y=-72 \\27x=-216\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{-72}{-9}=8 \\x=\frac{-216}{27}=-8\end{matrix}\right.\\ \qquad V=\{(-8,8)\}\)
10. \(\left\{\begin{matrix}-2x-9y=21\\3x=10y-8\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-2x-9y=21\\3x-10y=-8\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{array}{c|c|c}-2x-9y=21& \color{red}{3.} & \color{blue}{10.} \\3x-10y=-8& \color{red}{2.} & \color{blue}{-9.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{-6x+6x}-27y-20y=63-16} \\ \color{blue}{-20x-27x\underline{-90y+90y}=210+72} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-47y=47 \\-47x=282\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{47}{-47}=-1 \\x=\frac{282}{-47}=-6\end{matrix}\right.\\ \qquad V=\{(-6,-1)\}\)
11. \(\left\{\begin{matrix}-5x-8y=18\\-10x+10y=10\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{array}{c|c|c}-5x-8y=18& \color{red}{2.} & \color{blue}{5.} \\-10x+10y=10& \color{red}{-1.} & \color{blue}{4.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{-10x+10x}-16y-10y=36-10} \\ \color{blue}{-25x-40x\underline{-40y+40y}=90+40} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-26y=26 \\-65x=130\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{26}{-26}=-1 \\x=\frac{130}{-65}=-2\end{matrix}\right.\\ \qquad V=\{(-2,-1)\}\)
12. \(\left\{\begin{matrix}7x+2y=-37\\2x+10y=-20\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{array}{c|c|c}7x+2y=-37& \color{red}{2.} & \color{blue}{5.} \\2x+10y=-20& \color{red}{-7.} & \color{blue}{-1.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{14x-14x}+4y-70y=-74+140} \\ \color{blue}{35x-2x\underline{+10y-10y}=-185+20} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-66y=66 \\33x=-165\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{66}{-66}=-1 \\x=\frac{-165}{33}=-5\end{matrix}\right.\\ \qquad V=\{(-5,-1)\}\)