Stelsels combinatie

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Combinatie

1. \(\left\{\begin{matrix}-9x-6y=99\\-8y-9x=111\end{matrix}\right.\)
2. \(\left\{\begin{matrix}10y-3x=-118\\-10x+2y=-80\end{matrix}\right.\)
3. \(\left\{\begin{matrix}9x+8y=-53\\-6y-3x=21\end{matrix}\right.\)
4. \(\left\{\begin{matrix}5x+2y=-37\\-5x+3y=32\end{matrix}\right.\)
5. \(\left\{\begin{matrix}9x+9y=0\\-7x-6y=9\end{matrix}\right.\)
6. \(\left\{\begin{matrix}10y+10x=-150\\7x+4y=-90\end{matrix}\right.\)
7. \(\left\{\begin{matrix}6x-6y=114\\-3x-5y=23\end{matrix}\right.\)
8. \(\left\{\begin{matrix}-5x+5y=45\\-3x=6y+18\end{matrix}\right.\)
9. \(\left\{\begin{matrix}-8x-6y=6\\10x=-2y-24\end{matrix}\right.\)
10. \(\left\{\begin{matrix}-2x-7y=54\\4x=4y+72\end{matrix}\right.\)
11. \(\left\{\begin{matrix}-9x-7y=21\\-6y-10x=34\end{matrix}\right.\)
12. \(\left\{\begin{matrix}-9x-2y=65\\-4x=5y+70\end{matrix}\right.\)

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Combinatie

Verbetersleutel

1. \(\left\{\begin{matrix}-9x-6y=99\\-8y-9x=111\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-9x-6y=99\\-9x-8y=111\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{array}{c|c|c}-9x-6y=99& \color{red}{1.} & \color{blue}{4.} \\-9x-8y=111& \color{red}{-1.} & \color{blue}{-3.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{-9x+9x}-6y+8y=99-111} \\ \color{blue}{-36x+27x\underline{-24y+24y}=396-333} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}2y=-12 \\-9x=63\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{-12}{2}=-6 \\x=\frac{63}{-9}=-7\end{matrix}\right.\\ \qquad V=\{(-7,-6)\}\)
2. \(\left\{\begin{matrix}10y-3x=-118\\-10x+2y=-80\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-3x+10y=-118\\-10x+2y=-80\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{array}{c|c|c}-3x+10y=-118& \color{red}{10.} & \color{blue}{1.} \\-10x+2y=-80& \color{red}{-3.} & \color{blue}{-5.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{-30x+30x}+100y-6y=-1180+240} \\ \color{blue}{-3x+50x\underline{+10y-10y}=-118+400} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}94y=-940 \\47x=282\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{-940}{94}=-10 \\x=\frac{282}{47}=6\end{matrix}\right.\\ \qquad V=\{(6,-10)\}\)
3. \(\left\{\begin{matrix}9x+8y=-53\\-6y-3x=21\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}9x+8y=-53\\-3x-6y=21\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{array}{c|c|c}9x+8y=-53& \color{red}{1.} & \color{blue}{3.} \\-3x-6y=21& \color{red}{3.} & \color{blue}{4.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{9x-9x}+8y-18y=-53+63} \\ \color{blue}{27x-12x\underline{+24y-24y}=-159+84} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-10y=10 \\15x=-75\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{10}{-10}=-1 \\x=\frac{-75}{15}=-5\end{matrix}\right.\\ \qquad V=\{(-5,-1)\}\)
4. \(\left\{\begin{matrix}5x+2y=-37\\-5x+3y=32\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{array}{c|c|c}5x+2y=-37& \color{red}{1.} & \color{blue}{3.} \\-5x+3y=32& \color{red}{1.} & \color{blue}{-2.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{5x-5x}+2y+3y=-37+32} \\ \color{blue}{15x+10x\underline{+6y-6y}=-111-64} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}5y=-5 \\25x=-175\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{-5}{5}=-1 \\x=\frac{-175}{25}=-7\end{matrix}\right.\\ \qquad V=\{(-7,-1)\}\)
5. \(\left\{\begin{matrix}9x+9y=0\\-7x-6y=9\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{array}{c|c|c}9x+9y=0& \color{red}{7.} & \color{blue}{2.} \\-7x-6y=9& \color{red}{9.} & \color{blue}{3.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{63x-63x}+63y-54y=0+81} \\ \color{blue}{18x-21x\underline{+18y-18y}=0+27} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}9y=81 \\-3x=27\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{81}{9}=9 \\x=\frac{27}{-3}=-9\end{matrix}\right.\\ \qquad V=\{(-9,9)\}\)
6. \(\left\{\begin{matrix}10y+10x=-150\\7x+4y=-90\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}10x+10y=-150\\7x+4y=-90\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{array}{c|c|c}10x+10y=-150& \color{red}{7.} & \color{blue}{2.} \\7x+4y=-90& \color{red}{-10.} & \color{blue}{-5.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{70x-70x}+70y-40y=-1050+900} \\ \color{blue}{20x-35x\underline{+20y-20y}=-300+450} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}30y=-150 \\-15x=150\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{-150}{30}=-5 \\x=\frac{150}{-15}=-10\end{matrix}\right.\\ \qquad V=\{(-10,-5)\}\)
7. \(\left\{\begin{matrix}6x-6y=114\\-3x-5y=23\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{array}{c|c|c}6x-6y=114& \color{red}{1.} & \color{blue}{5.} \\-3x-5y=23& \color{red}{2.} & \color{blue}{-6.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{6x-6x}-6y-10y=114+46} \\ \color{blue}{30x+18x\underline{-30y+30y}=570-138} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-16y=160 \\48x=432\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{160}{-16}=-10 \\x=\frac{432}{48}=9\end{matrix}\right.\\ \qquad V=\{(9,-10)\}\)
8. \(\left\{\begin{matrix}-5x+5y=45\\-3x=6y+18\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-5x+5y=45\\-3x-6y=18\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{array}{c|c|c}-5x+5y=45& \color{red}{3.} & \color{blue}{6.} \\-3x-6y=18& \color{red}{-5.} & \color{blue}{5.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{-15x+15x}+15y+30y=135-90} \\ \color{blue}{-30x-15x\underline{+30y-30y}=270+90} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}45y=45 \\-45x=360\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{45}{45}=1 \\x=\frac{360}{-45}=-8\end{matrix}\right.\\ \qquad V=\{(-8,1)\}\)
9. \(\left\{\begin{matrix}-8x-6y=6\\10x=-2y-24\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-8x-6y=6\\10x+2y=-24\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{array}{c|c|c}-8x-6y=6& \color{red}{5.} & \color{blue}{1.} \\10x+2y=-24& \color{red}{4.} & \color{blue}{3.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{-40x+40x}-30y+8y=30-96} \\ \color{blue}{-8x+30x\underline{-6y+6y}=6-72} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-22y=-66 \\22x=-66\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{-66}{-22}=3 \\x=\frac{-66}{22}=-3\end{matrix}\right.\\ \qquad V=\{(-3,3)\}\)
10. \(\left\{\begin{matrix}-2x-7y=54\\4x=4y+72\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-2x-7y=54\\4x-4y=72\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{array}{c|c|c}-2x-7y=54& \color{red}{2.} & \color{blue}{4.} \\4x-4y=72& \color{red}{1.} & \color{blue}{-7.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{-4x+4x}-14y-4y=108+72} \\ \color{blue}{-8x-28x\underline{-28y+28y}=216-504} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-18y=180 \\-36x=-288\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{180}{-18}=-10 \\x=\frac{-288}{-36}=8\end{matrix}\right.\\ \qquad V=\{(8,-10)\}\)
11. \(\left\{\begin{matrix}-9x-7y=21\\-6y-10x=34\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-9x-7y=21\\-10x-6y=34\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{array}{c|c|c}-9x-7y=21& \color{red}{10.} & \color{blue}{6.} \\-10x-6y=34& \color{red}{-9.} & \color{blue}{-7.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{-90x+90x}-70y+54y=210-306} \\ \color{blue}{-54x+70x\underline{-42y+42y}=126-238} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-16y=-96 \\16x=-112\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{-96}{-16}=6 \\x=\frac{-112}{16}=-7\end{matrix}\right.\\ \qquad V=\{(-7,6)\}\)
12. \(\left\{\begin{matrix}-9x-2y=65\\-4x=5y+70\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-9x-2y=65\\-4x-5y=70\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{array}{c|c|c}-9x-2y=65& \color{red}{4.} & \color{blue}{5.} \\-4x-5y=70& \color{red}{-9.} & \color{blue}{-2.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{-36x+36x}-8y+45y=260-630} \\ \color{blue}{-45x+8x\underline{-10y+10y}=325-140} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}37y=-370 \\-37x=185\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{-370}{37}=-10 \\x=\frac{185}{-37}=-5\end{matrix}\right.\\ \qquad V=\{(-5,-10)\}\)