Stelsels combinatie

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Combinatie

1. \(\left\{\begin{matrix}-3y+7x=-13\\-6x+6y=18\end{matrix}\right.\)
2. \(\left\{\begin{matrix}-9x+8y=-13\\2x-10y=44\end{matrix}\right.\)
3. \(\left\{\begin{matrix}2y=9+9x\\-10x-9y=-91\end{matrix}\right.\)
4. \(\left\{\begin{matrix}-6x+3y=27\\4y-6x=34\end{matrix}\right.\)
5. \(\left\{\begin{matrix}-3x+5y=-54\\-2x+6y=-60\end{matrix}\right.\)
6. \(\left\{\begin{matrix}-4x-4y=-16\\7y+3x=32\end{matrix}\right.\)
7. \(\left\{\begin{matrix}-10x-9y=24\\-5x=-7y-57\end{matrix}\right.\)
8. \(\left\{\begin{matrix}2x+4y=-10\\3x=-3y-3\end{matrix}\right.\)
9. \(\left\{\begin{matrix}-5y+3x=46\\9x-8y=103\end{matrix}\right.\)
10. \(\left\{\begin{matrix}9y=-126-4x\\10x-2y=-70\end{matrix}\right.\)
11. \(\left\{\begin{matrix}10x+6y=106\\3x-2y=28\end{matrix}\right.\)
12. \(\left\{\begin{matrix}-10x+6y=-66\\-2y+3x=20\end{matrix}\right.\)

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Combinatie

Verbetersleutel

1. \(\left\{\begin{matrix}-3y+7x=-13\\-6x+6y=18\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}7x-3y=-13\\-6x+6y=18\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{array}{c|c|c}7x-3y=-13& \color{red}{6.} & \color{blue}{2.} \\-6x+6y=18& \color{red}{7.} & \color{blue}{1.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{42x-42x}-18y+42y=-78+126} \\ \color{blue}{14x-6x\underline{-6y+6y}=-26+18} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}24y=48 \\8x=-8\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{48}{24}=2 \\x=\frac{-8}{8}=-1\end{matrix}\right.\\ \qquad V=\{(-1,2)\}\)
2. \(\left\{\begin{matrix}-9x+8y=-13\\2x-10y=44\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{array}{c|c|c}-9x+8y=-13& \color{red}{2.} & \color{blue}{5.} \\2x-10y=44& \color{red}{9.} & \color{blue}{4.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{-18x+18x}+16y-90y=-26+396} \\ \color{blue}{-45x+8x\underline{+40y-40y}=-65+176} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-74y=370 \\-37x=111\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{370}{-74}=-5 \\x=\frac{111}{-37}=-3\end{matrix}\right.\\ \qquad V=\{(-3,-5)\}\)
3. \(\left\{\begin{matrix}2y=9+9x\\-10x-9y=-91\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-9x+2y=9\\-10x-9y=-91\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{array}{c|c|c}-9x+2y=9& \color{red}{10.} & \color{blue}{9.} \\-10x-9y=-91& \color{red}{-9.} & \color{blue}{2.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{-90x+90x}+20y+81y=90+819} \\ \color{blue}{-81x-20x\underline{+18y-18y}=81-182} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}101y=909 \\-101x=-101\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{909}{101}=9 \\x=\frac{-101}{-101}=1\end{matrix}\right.\\ \qquad V=\{(1,9)\}\)
4. \(\left\{\begin{matrix}-6x+3y=27\\4y-6x=34\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-6x+3y=27\\-6x+4y=34\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{array}{c|c|c}-6x+3y=27& \color{red}{1.} & \color{blue}{4.} \\-6x+4y=34& \color{red}{-1.} & \color{blue}{-3.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{-6x+6x}+3y-4y=27-34} \\ \color{blue}{-24x+18x\underline{+12y-12y}=108-102} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-y=-7 \\-6x=6\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{-7}{-1}=7 \\x=\frac{6}{-6}=-1\end{matrix}\right.\\ \qquad V=\{(-1,7)\}\)
5. \(\left\{\begin{matrix}-3x+5y=-54\\-2x+6y=-60\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{array}{c|c|c}-3x+5y=-54& \color{red}{2.} & \color{blue}{6.} \\-2x+6y=-60& \color{red}{-3.} & \color{blue}{-5.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{-6x+6x}+10y-18y=-108+180} \\ \color{blue}{-18x+10x\underline{+30y-30y}=-324+300} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-8y=72 \\-8x=-24\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{72}{-8}=-9 \\x=\frac{-24}{-8}=3\end{matrix}\right.\\ \qquad V=\{(3,-9)\}\)
6. \(\left\{\begin{matrix}-4x-4y=-16\\7y+3x=32\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-4x-4y=-16\\3x+7y=32\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{array}{c|c|c}-4x-4y=-16& \color{red}{3.} & \color{blue}{7.} \\3x+7y=32& \color{red}{4.} & \color{blue}{4.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{-12x+12x}-12y+28y=-48+128} \\ \color{blue}{-28x+12x\underline{-28y+28y}=-112+128} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}16y=80 \\-16x=16\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{80}{16}=5 \\x=\frac{16}{-16}=-1\end{matrix}\right.\\ \qquad V=\{(-1,5)\}\)
7. \(\left\{\begin{matrix}-10x-9y=24\\-5x=-7y-57\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-10x-9y=24\\-5x+7y=-57\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{array}{c|c|c}-10x-9y=24& \color{red}{1.} & \color{blue}{7.} \\-5x+7y=-57& \color{red}{-2.} & \color{blue}{9.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{-10x+10x}-9y-14y=24+114} \\ \color{blue}{-70x-45x\underline{-63y+63y}=168-513} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-23y=138 \\-115x=-345\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{138}{-23}=-6 \\x=\frac{-345}{-115}=3\end{matrix}\right.\\ \qquad V=\{(3,-6)\}\)
8. \(\left\{\begin{matrix}2x+4y=-10\\3x=-3y-3\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}2x+4y=-10\\3x+3y=-3\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{array}{c|c|c}2x+4y=-10& \color{red}{3.} & \color{blue}{3.} \\3x+3y=-3& \color{red}{-2.} & \color{blue}{-4.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{6x-6x}+12y-6y=-30+6} \\ \color{blue}{6x-12x\underline{+12y-12y}=-30+12} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}6y=-24 \\-6x=-18\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{-24}{6}=-4 \\x=\frac{-18}{-6}=3\end{matrix}\right.\\ \qquad V=\{(3,-4)\}\)
9. \(\left\{\begin{matrix}-5y+3x=46\\9x-8y=103\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}3x-5y=46\\9x-8y=103\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{array}{c|c|c}3x-5y=46& \color{red}{3.} & \color{blue}{8.} \\9x-8y=103& \color{red}{-1.} & \color{blue}{-5.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{9x-9x}-15y+8y=138-103} \\ \color{blue}{24x-45x\underline{-40y+40y}=368-515} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-7y=35 \\-21x=-147\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{35}{-7}=-5 \\x=\frac{-147}{-21}=7\end{matrix}\right.\\ \qquad V=\{(7,-5)\}\)
10. \(\left\{\begin{matrix}9y=-126-4x\\10x-2y=-70\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}4x+9y=-126\\10x-2y=-70\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{array}{c|c|c}4x+9y=-126& \color{red}{5.} & \color{blue}{2.} \\10x-2y=-70& \color{red}{-2.} & \color{blue}{9.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{20x-20x}+45y+4y=-630+140} \\ \color{blue}{8x+90x\underline{+18y-18y}=-252-630} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}49y=-490 \\98x=-882\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{-490}{49}=-10 \\x=\frac{-882}{98}=-9\end{matrix}\right.\\ \qquad V=\{(-9,-10)\}\)
11. \(\left\{\begin{matrix}10x+6y=106\\3x-2y=28\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{array}{c|c|c}10x+6y=106& \color{red}{3.} & \color{blue}{1.} \\3x-2y=28& \color{red}{-10.} & \color{blue}{3.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{30x-30x}+18y+20y=318-280} \\ \color{blue}{10x+9x\underline{+6y-6y}=106+84} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}38y=38 \\19x=190\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{38}{38}=1 \\x=\frac{190}{19}=10\end{matrix}\right.\\ \qquad V=\{(10,1)\}\)
12. \(\left\{\begin{matrix}-10x+6y=-66\\-2y+3x=20\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-10x+6y=-66\\3x-2y=20\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{array}{c|c|c}-10x+6y=-66& \color{red}{3.} & \color{blue}{1.} \\3x-2y=20& \color{red}{10.} & \color{blue}{3.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{-30x+30x}+18y-20y=-198+200} \\ \color{blue}{-10x+9x\underline{+6y-6y}=-66+60} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-2y=2 \\-x=-6\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{2}{-2}=-1 \\x=\frac{-6}{-1}=6\end{matrix}\right.\\ \qquad V=\{(6,-1)\}\)