Stelsels combinatie

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Combinatie

1. \(\left\{\begin{matrix}-5x-8y=-22\\-8x+3y=107\end{matrix}\right.\)
2. \(\left\{\begin{matrix}7y-2x=-38\\-10x+6y=42\end{matrix}\right.\)
3. \(\left\{\begin{matrix}5y=45+6x\\-8x+6y=58\end{matrix}\right.\)
4. \(\left\{\begin{matrix}8x+7y=31\\-5x-5y=-25\end{matrix}\right.\)
5. \(\left\{\begin{matrix}-7x+2y=-65\\-9x-6y=-75\end{matrix}\right.\)
6. \(\left\{\begin{matrix}8x-3y=-21\\4x=10y-2\end{matrix}\right.\)
7. \(\left\{\begin{matrix}-7y+2x=0\\3x+10y=41\end{matrix}\right.\)
8. \(\left\{\begin{matrix}-7x-9y=-125\\-6y-10x=-110\end{matrix}\right.\)
9. \(\left\{\begin{matrix}8x-8y=-32\\4x=6y-8\end{matrix}\right.\)
10. \(\left\{\begin{matrix}9x-10y=-87\\-10x+8y=78\end{matrix}\right.\)
11. \(\left\{\begin{matrix}-3y+7x=91\\10x-6y=142\end{matrix}\right.\)
12. \(\left\{\begin{matrix}8y=-98-6x\\6x+2y=-38\end{matrix}\right.\)

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Combinatie

Verbetersleutel

1. \(\left\{\begin{matrix}-5x-8y=-22\\-8x+3y=107\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{array}{c|c|c}-5x-8y=-22& \color{red}{8.} & \color{blue}{3.} \\-8x+3y=107& \color{red}{-5.} & \color{blue}{8.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{-40x+40x}-64y-15y=-176-535} \\ \color{blue}{-15x-64x\underline{-24y+24y}=-66+856} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-79y=-711 \\-79x=790\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{-711}{-79}=9 \\x=\frac{790}{-79}=-10\end{matrix}\right.\\ \qquad V=\{(-10,9)\}\)
2. \(\left\{\begin{matrix}7y-2x=-38\\-10x+6y=42\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-2x+7y=-38\\-10x+6y=42\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{array}{c|c|c}-2x+7y=-38& \color{red}{5.} & \color{blue}{6.} \\-10x+6y=42& \color{red}{-1.} & \color{blue}{-7.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{-10x+10x}+35y-6y=-190-42} \\ \color{blue}{-12x+70x\underline{+42y-42y}=-228-294} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}29y=-232 \\58x=-522\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{-232}{29}=-8 \\x=\frac{-522}{58}=-9\end{matrix}\right.\\ \qquad V=\{(-9,-8)\}\)
3. \(\left\{\begin{matrix}5y=45+6x\\-8x+6y=58\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-6x+5y=45\\-8x+6y=58\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{array}{c|c|c}-6x+5y=45& \color{red}{4.} & \color{blue}{6.} \\-8x+6y=58& \color{red}{-3.} & \color{blue}{-5.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{-24x+24x}+20y-18y=180-174} \\ \color{blue}{-36x+40x\underline{+30y-30y}=270-290} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}2y=6 \\4x=-20\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{6}{2}=3 \\x=\frac{-20}{4}=-5\end{matrix}\right.\\ \qquad V=\{(-5,3)\}\)
4. \(\left\{\begin{matrix}8x+7y=31\\-5x-5y=-25\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{array}{c|c|c}8x+7y=31& \color{red}{5.} & \color{blue}{5.} \\-5x-5y=-25& \color{red}{8.} & \color{blue}{7.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{40x-40x}+35y-40y=155-200} \\ \color{blue}{40x-35x\underline{+35y-35y}=155-175} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-5y=-45 \\5x=-20\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{-45}{-5}=9 \\x=\frac{-20}{5}=-4\end{matrix}\right.\\ \qquad V=\{(-4,9)\}\)
5. \(\left\{\begin{matrix}-7x+2y=-65\\-9x-6y=-75\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{array}{c|c|c}-7x+2y=-65& \color{red}{9.} & \color{blue}{3.} \\-9x-6y=-75& \color{red}{-7.} & \color{blue}{1.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{-63x+63x}+18y+42y=-585+525} \\ \color{blue}{-21x-9x\underline{+6y-6y}=-195-75} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}60y=-60 \\-30x=-270\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{-60}{60}=-1 \\x=\frac{-270}{-30}=9\end{matrix}\right.\\ \qquad V=\{(9,-1)\}\)
6. \(\left\{\begin{matrix}8x-3y=-21\\4x=10y-2\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}8x-3y=-21\\4x-10y=-2\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{array}{c|c|c}8x-3y=-21& \color{red}{1.} & \color{blue}{10.} \\4x-10y=-2& \color{red}{-2.} & \color{blue}{-3.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{8x-8x}-3y+20y=-21+4} \\ \color{blue}{80x-12x\underline{-30y+30y}=-210+6} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}17y=-17 \\68x=-204\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{-17}{17}=-1 \\x=\frac{-204}{68}=-3\end{matrix}\right.\\ \qquad V=\{(-3,-1)\}\)
7. \(\left\{\begin{matrix}-7y+2x=0\\3x+10y=41\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}2x-7y=0\\3x+10y=41\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{array}{c|c|c}2x-7y=0& \color{red}{3.} & \color{blue}{10.} \\3x+10y=41& \color{red}{-2.} & \color{blue}{7.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{6x-6x}-21y-20y=0-82} \\ \color{blue}{20x+21x\underline{-70y+70y}=0+287} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-41y=-82 \\41x=287\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{-82}{-41}=2 \\x=\frac{287}{41}=7\end{matrix}\right.\\ \qquad V=\{(7,2)\}\)
8. \(\left\{\begin{matrix}-7x-9y=-125\\-6y-10x=-110\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-7x-9y=-125\\-10x-6y=-110\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{array}{c|c|c}-7x-9y=-125& \color{red}{10.} & \color{blue}{2.} \\-10x-6y=-110& \color{red}{-7.} & \color{blue}{-3.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{-70x+70x}-90y+42y=-1250+770} \\ \color{blue}{-14x+30x\underline{-18y+18y}=-250+330} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-48y=-480 \\16x=80\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{-480}{-48}=10 \\x=\frac{80}{16}=5\end{matrix}\right.\\ \qquad V=\{(5,10)\}\)
9. \(\left\{\begin{matrix}8x-8y=-32\\4x=6y-8\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}8x-8y=-32\\4x-6y=-8\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{array}{c|c|c}8x-8y=-32& \color{red}{1.} & \color{blue}{3.} \\4x-6y=-8& \color{red}{-2.} & \color{blue}{-4.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{8x-8x}-8y+12y=-32+16} \\ \color{blue}{24x-16x\underline{-24y+24y}=-96+32} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}4y=-16 \\8x=-64\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{-16}{4}=-4 \\x=\frac{-64}{8}=-8\end{matrix}\right.\\ \qquad V=\{(-8,-4)\}\)
10. \(\left\{\begin{matrix}9x-10y=-87\\-10x+8y=78\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{array}{c|c|c}9x-10y=-87& \color{red}{10.} & \color{blue}{4.} \\-10x+8y=78& \color{red}{9.} & \color{blue}{5.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{90x-90x}-100y+72y=-870+702} \\ \color{blue}{36x-50x\underline{-40y+40y}=-348+390} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-28y=-168 \\-14x=42\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{-168}{-28}=6 \\x=\frac{42}{-14}=-3\end{matrix}\right.\\ \qquad V=\{(-3,6)\}\)
11. \(\left\{\begin{matrix}-3y+7x=91\\10x-6y=142\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}7x-3y=91\\10x-6y=142\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{array}{c|c|c}7x-3y=91& \color{red}{10.} & \color{blue}{2.} \\10x-6y=142& \color{red}{-7.} & \color{blue}{-1.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{70x-70x}-30y+42y=910-994} \\ \color{blue}{14x-10x\underline{-6y+6y}=182-142} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}12y=-84 \\4x=40\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{-84}{12}=-7 \\x=\frac{40}{4}=10\end{matrix}\right.\\ \qquad V=\{(10,-7)\}\)
12. \(\left\{\begin{matrix}8y=-98-6x\\6x+2y=-38\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}6x+8y=-98\\6x+2y=-38\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{array}{c|c|c}6x+8y=-98& \color{red}{1.} & \color{blue}{1.} \\6x+2y=-38& \color{red}{-1.} & \color{blue}{-4.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{6x-6x}+8y-2y=-98+38} \\ \color{blue}{6x-24x\underline{+8y-8y}=-98+152} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}6y=-60 \\-18x=54\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{-60}{6}=-10 \\x=\frac{54}{-18}=-3\end{matrix}\right.\\ \qquad V=\{(-3,-10)\}\)