Stelsels combinatie

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Combinatie

1. \(\left\{\begin{matrix}-3y=1-4x\\3x+4y=57\end{matrix}\right.\)
2. \(\left\{\begin{matrix}4y+8x=68\\-6x+8y=48\end{matrix}\right.\)
3. \(\left\{\begin{matrix}6x+9y=-42\\-5y+8x=46\end{matrix}\right.\)
4. \(\left\{\begin{matrix}6y=-63+5x\\-4x-10y=68\end{matrix}\right.\)
5. \(\left\{\begin{matrix}-7y=-143+10x\\7x-8y=-16\end{matrix}\right.\)
6. \(\left\{\begin{matrix}2y+9x=25\\3x-7y=16\end{matrix}\right.\)
7. \(\left\{\begin{matrix}4x-7y=90\\-6y+4x=80\end{matrix}\right.\)
8. \(\left\{\begin{matrix}6y=28-10x\\-9x+2y=-77\end{matrix}\right.\)
9. \(\left\{\begin{matrix}10x-7y=-11\\8y-2x=-44\end{matrix}\right.\)
10. \(\left\{\begin{matrix}-6y+2x=-20\\4x+5y=11\end{matrix}\right.\)
11. \(\left\{\begin{matrix}9y+8x=27\\-6x-10y=-4\end{matrix}\right.\)
12. \(\left\{\begin{matrix}-6y+3x=27\\-9x-3y=24\end{matrix}\right.\)

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Combinatie

Verbetersleutel

1. \(\left\{\begin{matrix}-3y=1-4x\\3x+4y=57\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}4x-3y=1\\3x+4y=57\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{array}{c|c|c}4x-3y=1& \color{red}{3.} & \color{blue}{4.} \\3x+4y=57& \color{red}{-4.} & \color{blue}{3.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{12x-12x}-9y-16y=3-228} \\ \color{blue}{16x+9x\underline{-12y+12y}=4+171} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-25y=-225 \\25x=175\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{-225}{-25}=9 \\x=\frac{175}{25}=7\end{matrix}\right.\\ \qquad V=\{(7,9)\}\)
2. \(\left\{\begin{matrix}4y+8x=68\\-6x+8y=48\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}8x+4y=68\\-6x+8y=48\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{array}{c|c|c}8x+4y=68& \color{red}{3.} & \color{blue}{2.} \\-6x+8y=48& \color{red}{4.} & \color{blue}{-1.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{24x-24x}+12y+32y=204+192} \\ \color{blue}{16x+6x\underline{+8y-8y}=136-48} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}44y=396 \\22x=88\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{396}{44}=9 \\x=\frac{88}{22}=4\end{matrix}\right.\\ \qquad V=\{(4,9)\}\)
3. \(\left\{\begin{matrix}6x+9y=-42\\-5y+8x=46\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}6x+9y=-42\\8x-5y=46\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{array}{c|c|c}6x+9y=-42& \color{red}{4.} & \color{blue}{5.} \\8x-5y=46& \color{red}{-3.} & \color{blue}{9.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{24x-24x}+36y+15y=-168-138} \\ \color{blue}{30x+72x\underline{+45y-45y}=-210+414} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}51y=-306 \\102x=204\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{-306}{51}=-6 \\x=\frac{204}{102}=2\end{matrix}\right.\\ \qquad V=\{(2,-6)\}\)
4. \(\left\{\begin{matrix}6y=-63+5x\\-4x-10y=68\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-5x+6y=-63\\-4x-10y=68\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{array}{c|c|c}-5x+6y=-63& \color{red}{4.} & \color{blue}{5.} \\-4x-10y=68& \color{red}{-5.} & \color{blue}{3.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{-20x+20x}+24y+50y=-252-340} \\ \color{blue}{-25x-12x\underline{+30y-30y}=-315+204} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}74y=-592 \\-37x=-111\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{-592}{74}=-8 \\x=\frac{-111}{-37}=3\end{matrix}\right.\\ \qquad V=\{(3,-8)\}\)
5. \(\left\{\begin{matrix}-7y=-143+10x\\7x-8y=-16\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-10x-7y=-143\\7x-8y=-16\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{array}{c|c|c}-10x-7y=-143& \color{red}{7.} & \color{blue}{8.} \\7x-8y=-16& \color{red}{10.} & \color{blue}{-7.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{-70x+70x}-49y-80y=-1001-160} \\ \color{blue}{-80x-49x\underline{-56y+56y}=-1144+112} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-129y=-1161 \\-129x=-1032\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{-1161}{-129}=9 \\x=\frac{-1032}{-129}=8\end{matrix}\right.\\ \qquad V=\{(8,9)\}\)
6. \(\left\{\begin{matrix}2y+9x=25\\3x-7y=16\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}9x+2y=25\\3x-7y=16\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{array}{c|c|c}9x+2y=25& \color{red}{1.} & \color{blue}{7.} \\3x-7y=16& \color{red}{-3.} & \color{blue}{2.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{9x-9x}+2y+21y=25-48} \\ \color{blue}{63x+6x\underline{+14y-14y}=175+32} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}23y=-23 \\69x=207\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{-23}{23}=-1 \\x=\frac{207}{69}=3\end{matrix}\right.\\ \qquad V=\{(3,-1)\}\)
7. \(\left\{\begin{matrix}4x-7y=90\\-6y+4x=80\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}4x-7y=90\\4x-6y=80\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{array}{c|c|c}4x-7y=90& \color{red}{1.} & \color{blue}{6.} \\4x-6y=80& \color{red}{-1.} & \color{blue}{-7.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{4x-4x}-7y+6y=90-80} \\ \color{blue}{24x-28x\underline{-42y+42y}=540-560} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-y=10 \\-4x=-20\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{10}{-1}=-10 \\x=\frac{-20}{-4}=5\end{matrix}\right.\\ \qquad V=\{(5,-10)\}\)
8. \(\left\{\begin{matrix}6y=28-10x\\-9x+2y=-77\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}10x+6y=28\\-9x+2y=-77\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{array}{c|c|c}10x+6y=28& \color{red}{9.} & \color{blue}{1.} \\-9x+2y=-77& \color{red}{10.} & \color{blue}{-3.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{90x-90x}+54y+20y=252-770} \\ \color{blue}{10x+27x\underline{+6y-6y}=28+231} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}74y=-518 \\37x=259\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{-518}{74}=-7 \\x=\frac{259}{37}=7\end{matrix}\right.\\ \qquad V=\{(7,-7)\}\)
9. \(\left\{\begin{matrix}10x-7y=-11\\8y-2x=-44\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}10x-7y=-11\\-2x+8y=-44\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{array}{c|c|c}10x-7y=-11& \color{red}{1.} & \color{blue}{8.} \\-2x+8y=-44& \color{red}{5.} & \color{blue}{7.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{10x-10x}-7y+40y=-11-220} \\ \color{blue}{80x-14x\underline{-56y+56y}=-88-308} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}33y=-231 \\66x=-396\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{-231}{33}=-7 \\x=\frac{-396}{66}=-6\end{matrix}\right.\\ \qquad V=\{(-6,-7)\}\)
10. \(\left\{\begin{matrix}-6y+2x=-20\\4x+5y=11\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}2x-6y=-20\\4x+5y=11\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{array}{c|c|c}2x-6y=-20& \color{red}{2.} & \color{blue}{5.} \\4x+5y=11& \color{red}{-1.} & \color{blue}{6.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{4x-4x}-12y-5y=-40-11} \\ \color{blue}{10x+24x\underline{-30y+30y}=-100+66} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-17y=-51 \\34x=-34\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{-51}{-17}=3 \\x=\frac{-34}{34}=-1\end{matrix}\right.\\ \qquad V=\{(-1,3)\}\)
11. \(\left\{\begin{matrix}9y+8x=27\\-6x-10y=-4\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}8x+9y=27\\-6x-10y=-4\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{array}{c|c|c}8x+9y=27& \color{red}{3.} & \color{blue}{10.} \\-6x-10y=-4& \color{red}{4.} & \color{blue}{9.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{24x-24x}+27y-40y=81-16} \\ \color{blue}{80x-54x\underline{+90y-90y}=270-36} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-13y=65 \\26x=234\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{65}{-13}=-5 \\x=\frac{234}{26}=9\end{matrix}\right.\\ \qquad V=\{(9,-5)\}\)
12. \(\left\{\begin{matrix}-6y+3x=27\\-9x-3y=24\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}3x-6y=27\\-9x-3y=24\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{array}{c|c|c}3x-6y=27& \color{red}{3.} & \color{blue}{1.} \\-9x-3y=24& \color{red}{1.} & \color{blue}{-2.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{9x-9x}-18y-3y=81+24} \\ \color{blue}{3x+18x\underline{-6y+6y}=27-48} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-21y=105 \\21x=-21\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{105}{-21}=-5 \\x=\frac{-21}{21}=-1\end{matrix}\right.\\ \qquad V=\{(-1,-5)\}\)