Stelsels combinatie

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Combinatie

1. \(\left\{\begin{matrix}3y+3x=-24\\7x+8y=-61\end{matrix}\right.\)
2. \(\left\{\begin{matrix}-4y=40-3x\\-8x+9y=-95\end{matrix}\right.\)
3. \(\left\{\begin{matrix}9x+5y=104\\-6x=-4y+4\end{matrix}\right.\)
4. \(\left\{\begin{matrix}-8x+5y=-9\\-2x+6y=-26\end{matrix}\right.\)
5. \(\left\{\begin{matrix}2x+10y=-82\\-5x+9y=-33\end{matrix}\right.\)
6. \(\left\{\begin{matrix}-7x-3y=0\\-4x=-4y+40\end{matrix}\right.\)
7. \(\left\{\begin{matrix}-4y=52-2x\\8x+3y=18\end{matrix}\right.\)
8. \(\left\{\begin{matrix}2x+5y=36\\6y-3x=54\end{matrix}\right.\)
9. \(\left\{\begin{matrix}3x+10y=87\\8y+3x=69\end{matrix}\right.\)
10. \(\left\{\begin{matrix}3y-6x=-3\\-6x-7y=-53\end{matrix}\right.\)
11. \(\left\{\begin{matrix}-7x-2y=55\\10x-8y=30\end{matrix}\right.\)
12. \(\left\{\begin{matrix}-3x+8y=56\\-10x-6y=-140\end{matrix}\right.\)

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Combinatie

Verbetersleutel

1. \(\left\{\begin{matrix}3y+3x=-24\\7x+8y=-61\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}3x+3y=-24\\7x+8y=-61\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{array}{c|c|c}3x+3y=-24& \color{red}{7.} & \color{blue}{8.} \\7x+8y=-61& \color{red}{-3.} & \color{blue}{-3.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{21x-21x}+21y-24y=-168+183} \\ \color{blue}{24x-21x\underline{+24y-24y}=-192+183} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-3y=15 \\3x=-9\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{15}{-3}=-5 \\x=\frac{-9}{3}=-3\end{matrix}\right.\\ \qquad V=\{(-3,-5)\}\)
2. \(\left\{\begin{matrix}-4y=40-3x\\-8x+9y=-95\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}3x-4y=40\\-8x+9y=-95\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{array}{c|c|c}3x-4y=40& \color{red}{8.} & \color{blue}{9.} \\-8x+9y=-95& \color{red}{3.} & \color{blue}{4.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{24x-24x}-32y+27y=320-285} \\ \color{blue}{27x-32x\underline{-36y+36y}=360-380} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-5y=35 \\-5x=-20\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{35}{-5}=-7 \\x=\frac{-20}{-5}=4\end{matrix}\right.\\ \qquad V=\{(4,-7)\}\)
3. \(\left\{\begin{matrix}9x+5y=104\\-6x=-4y+4\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}9x+5y=104\\-6x+4y=4\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{array}{c|c|c}9x+5y=104& \color{red}{2.} & \color{blue}{4.} \\-6x+4y=4& \color{red}{3.} & \color{blue}{-5.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{18x-18x}+10y+12y=208+12} \\ \color{blue}{36x+30x\underline{+20y-20y}=416-20} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}22y=220 \\66x=396\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{220}{22}=10 \\x=\frac{396}{66}=6\end{matrix}\right.\\ \qquad V=\{(6,10)\}\)
4. \(\left\{\begin{matrix}-8x+5y=-9\\-2x+6y=-26\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{array}{c|c|c}-8x+5y=-9& \color{red}{1.} & \color{blue}{6.} \\-2x+6y=-26& \color{red}{-4.} & \color{blue}{-5.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{-8x+8x}+5y-24y=-9+104} \\ \color{blue}{-48x+10x\underline{+30y-30y}=-54+130} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-19y=95 \\-38x=76\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{95}{-19}=-5 \\x=\frac{76}{-38}=-2\end{matrix}\right.\\ \qquad V=\{(-2,-5)\}\)
5. \(\left\{\begin{matrix}2x+10y=-82\\-5x+9y=-33\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{array}{c|c|c}2x+10y=-82& \color{red}{5.} & \color{blue}{9.} \\-5x+9y=-33& \color{red}{2.} & \color{blue}{-10.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{10x-10x}+50y+18y=-410-66} \\ \color{blue}{18x+50x\underline{+90y-90y}=-738+330} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}68y=-476 \\68x=-408\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{-476}{68}=-7 \\x=\frac{-408}{68}=-6\end{matrix}\right.\\ \qquad V=\{(-6,-7)\}\)
6. \(\left\{\begin{matrix}-7x-3y=0\\-4x=-4y+40\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-7x-3y=0\\-4x+4y=40\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{array}{c|c|c}-7x-3y=0& \color{red}{4.} & \color{blue}{4.} \\-4x+4y=40& \color{red}{-7.} & \color{blue}{3.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{-28x+28x}-12y-28y=0-280} \\ \color{blue}{-28x-12x\underline{-12y+12y}=0+120} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-40y=-280 \\-40x=120\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{-280}{-40}=7 \\x=\frac{120}{-40}=-3\end{matrix}\right.\\ \qquad V=\{(-3,7)\}\)
7. \(\left\{\begin{matrix}-4y=52-2x\\8x+3y=18\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}2x-4y=52\\8x+3y=18\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{array}{c|c|c}2x-4y=52& \color{red}{4.} & \color{blue}{3.} \\8x+3y=18& \color{red}{-1.} & \color{blue}{4.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{8x-8x}-16y-3y=208-18} \\ \color{blue}{6x+32x\underline{-12y+12y}=156+72} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-19y=190 \\38x=228\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{190}{-19}=-10 \\x=\frac{228}{38}=6\end{matrix}\right.\\ \qquad V=\{(6,-10)\}\)
8. \(\left\{\begin{matrix}2x+5y=36\\6y-3x=54\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}2x+5y=36\\-3x+6y=54\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{array}{c|c|c}2x+5y=36& \color{red}{3.} & \color{blue}{6.} \\-3x+6y=54& \color{red}{2.} & \color{blue}{-5.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{6x-6x}+15y+12y=108+108} \\ \color{blue}{12x+15x\underline{+30y-30y}=216-270} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}27y=216 \\27x=-54\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{216}{27}=8 \\x=\frac{-54}{27}=-2\end{matrix}\right.\\ \qquad V=\{(-2,8)\}\)
9. \(\left\{\begin{matrix}3x+10y=87\\8y+3x=69\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}3x+10y=87\\3x+8y=69\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{array}{c|c|c}3x+10y=87& \color{red}{1.} & \color{blue}{4.} \\3x+8y=69& \color{red}{-1.} & \color{blue}{-5.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{3x-3x}+10y-8y=87-69} \\ \color{blue}{12x-15x\underline{+40y-40y}=348-345} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}2y=18 \\-3x=3\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{18}{2}=9 \\x=\frac{3}{-3}=-1\end{matrix}\right.\\ \qquad V=\{(-1,9)\}\)
10. \(\left\{\begin{matrix}3y-6x=-3\\-6x-7y=-53\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-6x+3y=-3\\-6x-7y=-53\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{array}{c|c|c}-6x+3y=-3& \color{red}{1.} & \color{blue}{7.} \\-6x-7y=-53& \color{red}{-1.} & \color{blue}{3.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{-6x+6x}+3y+7y=-3+53} \\ \color{blue}{-42x-18x\underline{+21y-21y}=-21-159} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}10y=50 \\-60x=-180\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{50}{10}=5 \\x=\frac{-180}{-60}=3\end{matrix}\right.\\ \qquad V=\{(3,5)\}\)
11. \(\left\{\begin{matrix}-7x-2y=55\\10x-8y=30\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{array}{c|c|c}-7x-2y=55& \color{red}{10.} & \color{blue}{4.} \\10x-8y=30& \color{red}{7.} & \color{blue}{-1.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{-70x+70x}-20y-56y=550+210} \\ \color{blue}{-28x-10x\underline{-8y+8y}=220-30} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-76y=760 \\-38x=190\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{760}{-76}=-10 \\x=\frac{190}{-38}=-5\end{matrix}\right.\\ \qquad V=\{(-5,-10)\}\)
12. \(\left\{\begin{matrix}-3x+8y=56\\-10x-6y=-140\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{array}{c|c|c}-3x+8y=56& \color{red}{10.} & \color{blue}{3.} \\-10x-6y=-140& \color{red}{-3.} & \color{blue}{4.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{-30x+30x}+80y+18y=560+420} \\ \color{blue}{-9x-40x\underline{+24y-24y}=168-560} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}98y=980 \\-49x=-392\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{980}{98}=10 \\x=\frac{-392}{-49}=8\end{matrix}\right.\\ \qquad V=\{(8,10)\}\)