Stelsels combinatie

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Combinatie

1. \(\left\{\begin{matrix}9x+10y=41\\3x=-9y-9\end{matrix}\right.\)
2. \(\left\{\begin{matrix}-7x-7y=-14\\-6x=-5y-45\end{matrix}\right.\)
3. \(\left\{\begin{matrix}10y-4x=10\\-9x-3y=-54\end{matrix}\right.\)
4. \(\left\{\begin{matrix}3y=99-8x\\9x+8y=153\end{matrix}\right.\)
5. \(\left\{\begin{matrix}-9x-9y=117\\10x=4y-46\end{matrix}\right.\)
6. \(\left\{\begin{matrix}4x-5y=44\\-4y+7x=39\end{matrix}\right.\)
7. \(\left\{\begin{matrix}-10x-7y=-22\\-7x=-4y+38\end{matrix}\right.\)
8. \(\left\{\begin{matrix}-4y-6x=-70\\-3x+5y=14\end{matrix}\right.\)
9. \(\left\{\begin{matrix}-3y-6x=-63\\-9x+10y=7\end{matrix}\right.\)
10. \(\left\{\begin{matrix}5y=65+5x\\7x-8y=-95\end{matrix}\right.\)
11. \(\left\{\begin{matrix}-8x-9y=25\\10x-6y=124\end{matrix}\right.\)
12. \(\left\{\begin{matrix}5x+3y=-62\\9x=-6y-114\end{matrix}\right.\)

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Combinatie

Verbetersleutel

1. \(\left\{\begin{matrix}9x+10y=41\\3x=-9y-9\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}9x+10y=41\\3x+9y=-9\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{array}{c|c|c}9x+10y=41& \color{red}{1.} & \color{blue}{9.} \\3x+9y=-9& \color{red}{-3.} & \color{blue}{-10.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{9x-9x}+10y-27y=41+27} \\ \color{blue}{81x-30x\underline{+90y-90y}=369+90} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-17y=68 \\51x=459\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{68}{-17}=-4 \\x=\frac{459}{51}=9\end{matrix}\right.\\ \qquad V=\{(9,-4)\}\)
2. \(\left\{\begin{matrix}-7x-7y=-14\\-6x=-5y-45\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-7x-7y=-14\\-6x+5y=-45\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{array}{c|c|c}-7x-7y=-14& \color{red}{6.} & \color{blue}{5.} \\-6x+5y=-45& \color{red}{-7.} & \color{blue}{7.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{-42x+42x}-42y-35y=-84+315} \\ \color{blue}{-35x-42x\underline{-35y+35y}=-70-315} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-77y=231 \\-77x=-385\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{231}{-77}=-3 \\x=\frac{-385}{-77}=5\end{matrix}\right.\\ \qquad V=\{(5,-3)\}\)
3. \(\left\{\begin{matrix}10y-4x=10\\-9x-3y=-54\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-4x+10y=10\\-9x-3y=-54\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{array}{c|c|c}-4x+10y=10& \color{red}{9.} & \color{blue}{3.} \\-9x-3y=-54& \color{red}{-4.} & \color{blue}{10.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{-36x+36x}+90y+12y=90+216} \\ \color{blue}{-12x-90x\underline{+30y-30y}=30-540} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}102y=306 \\-102x=-510\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{306}{102}=3 \\x=\frac{-510}{-102}=5\end{matrix}\right.\\ \qquad V=\{(5,3)\}\)
4. \(\left\{\begin{matrix}3y=99-8x\\9x+8y=153\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}8x+3y=99\\9x+8y=153\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{array}{c|c|c}8x+3y=99& \color{red}{9.} & \color{blue}{8.} \\9x+8y=153& \color{red}{-8.} & \color{blue}{-3.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{72x-72x}+27y-64y=891-1224} \\ \color{blue}{64x-27x\underline{+24y-24y}=792-459} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-37y=-333 \\37x=333\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{-333}{-37}=9 \\x=\frac{333}{37}=9\end{matrix}\right.\\ \qquad V=\{(9,9)\}\)
5. \(\left\{\begin{matrix}-9x-9y=117\\10x=4y-46\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-9x-9y=117\\10x-4y=-46\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{array}{c|c|c}-9x-9y=117& \color{red}{10.} & \color{blue}{4.} \\10x-4y=-46& \color{red}{9.} & \color{blue}{-9.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{-90x+90x}-90y-36y=1170-414} \\ \color{blue}{-36x-90x\underline{-36y+36y}=468+414} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-126y=756 \\-126x=882\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{756}{-126}=-6 \\x=\frac{882}{-126}=-7\end{matrix}\right.\\ \qquad V=\{(-7,-6)\}\)
6. \(\left\{\begin{matrix}4x-5y=44\\-4y+7x=39\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}4x-5y=44\\7x-4y=39\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{array}{c|c|c}4x-5y=44& \color{red}{7.} & \color{blue}{4.} \\7x-4y=39& \color{red}{-4.} & \color{blue}{-5.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{28x-28x}-35y+16y=308-156} \\ \color{blue}{16x-35x\underline{-20y+20y}=176-195} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-19y=152 \\-19x=-19\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{152}{-19}=-8 \\x=\frac{-19}{-19}=1\end{matrix}\right.\\ \qquad V=\{(1,-8)\}\)
7. \(\left\{\begin{matrix}-10x-7y=-22\\-7x=-4y+38\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-10x-7y=-22\\-7x+4y=38\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{array}{c|c|c}-10x-7y=-22& \color{red}{7.} & \color{blue}{4.} \\-7x+4y=38& \color{red}{-10.} & \color{blue}{7.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{-70x+70x}-49y-40y=-154-380} \\ \color{blue}{-40x-49x\underline{-28y+28y}=-88+266} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-89y=-534 \\-89x=178\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{-534}{-89}=6 \\x=\frac{178}{-89}=-2\end{matrix}\right.\\ \qquad V=\{(-2,6)\}\)
8. \(\left\{\begin{matrix}-4y-6x=-70\\-3x+5y=14\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-6x-4y=-70\\-3x+5y=14\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{array}{c|c|c}-6x-4y=-70& \color{red}{1.} & \color{blue}{5.} \\-3x+5y=14& \color{red}{-2.} & \color{blue}{4.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{-6x+6x}-4y-10y=-70-28} \\ \color{blue}{-30x-12x\underline{-20y+20y}=-350+56} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-14y=-98 \\-42x=-294\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{-98}{-14}=7 \\x=\frac{-294}{-42}=7\end{matrix}\right.\\ \qquad V=\{(7,7)\}\)
9. \(\left\{\begin{matrix}-3y-6x=-63\\-9x+10y=7\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-6x-3y=-63\\-9x+10y=7\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{array}{c|c|c}-6x-3y=-63& \color{red}{3.} & \color{blue}{10.} \\-9x+10y=7& \color{red}{-2.} & \color{blue}{3.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{-18x+18x}-9y-20y=-189-14} \\ \color{blue}{-60x-27x\underline{-30y+30y}=-630+21} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-29y=-203 \\-87x=-609\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{-203}{-29}=7 \\x=\frac{-609}{-87}=7\end{matrix}\right.\\ \qquad V=\{(7,7)\}\)
10. \(\left\{\begin{matrix}5y=65+5x\\7x-8y=-95\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-5x+5y=65\\7x-8y=-95\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{array}{c|c|c}-5x+5y=65& \color{red}{7.} & \color{blue}{8.} \\7x-8y=-95& \color{red}{5.} & \color{blue}{5.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{-35x+35x}+35y-40y=455-475} \\ \color{blue}{-40x+35x\underline{+40y-40y}=520-475} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-5y=-20 \\-5x=45\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{-20}{-5}=4 \\x=\frac{45}{-5}=-9\end{matrix}\right.\\ \qquad V=\{(-9,4)\}\)
11. \(\left\{\begin{matrix}-8x-9y=25\\10x-6y=124\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{array}{c|c|c}-8x-9y=25& \color{red}{5.} & \color{blue}{2.} \\10x-6y=124& \color{red}{4.} & \color{blue}{-3.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{-40x+40x}-45y-24y=125+496} \\ \color{blue}{-16x-30x\underline{-18y+18y}=50-372} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-69y=621 \\-46x=-322\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{621}{-69}=-9 \\x=\frac{-322}{-46}=7\end{matrix}\right.\\ \qquad V=\{(7,-9)\}\)
12. \(\left\{\begin{matrix}5x+3y=-62\\9x=-6y-114\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}5x+3y=-62\\9x+6y=-114\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{array}{c|c|c}5x+3y=-62& \color{red}{9.} & \color{blue}{2.} \\9x+6y=-114& \color{red}{-5.} & \color{blue}{-1.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{45x-45x}+27y-30y=-558+570} \\ \color{blue}{10x-9x\underline{+6y-6y}=-124+114} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-3y=12 \\x=-10\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{12}{-3}=-4 \\x=\frac{-10}{1}=-10\end{matrix}\right.\\ \qquad V=\{(-10,-4)\}\)