Stelsels combinatie

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Combinatie

1. \(\left\{\begin{matrix}3y=9-3x\\-2x+5y=43\end{matrix}\right.\)
2. \(\left\{\begin{matrix}-4x+3y=32\\8y-10x=84\end{matrix}\right.\)
3. \(\left\{\begin{matrix}7x+10y=-47\\-7x+5y=-13\end{matrix}\right.\)
4. \(\left\{\begin{matrix}-8y+10x=102\\6x-7y=70\end{matrix}\right.\)
5. \(\left\{\begin{matrix}2y=42+4x\\-3x-8y=-54\end{matrix}\right.\)
6. \(\left\{\begin{matrix}4x+3y=-20\\4x-7y=60\end{matrix}\right.\)
7. \(\left\{\begin{matrix}5x-9y=29\\-9x=-6y-42\end{matrix}\right.\)
8. \(\left\{\begin{matrix}6x-4y=68\\10y+7x=6\end{matrix}\right.\)
9. \(\left\{\begin{matrix}7y=61-6x\\-8x-2y=-30\end{matrix}\right.\)
10. \(\left\{\begin{matrix}-9x-8y=-31\\-9y-6x=-39\end{matrix}\right.\)
11. \(\left\{\begin{matrix}-2y-5x=39\\-4x+7y=57\end{matrix}\right.\)
12. \(\left\{\begin{matrix}5y+5x=45\\-5x-2y=-42\end{matrix}\right.\)

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Combinatie

Verbetersleutel

1. \(\left\{\begin{matrix}3y=9-3x\\-2x+5y=43\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}3x+3y=9\\-2x+5y=43\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{array}{c|c|c}3x+3y=9& \color{red}{2.} & \color{blue}{5.} \\-2x+5y=43& \color{red}{3.} & \color{blue}{-3.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{6x-6x}+6y+15y=18+129} \\ \color{blue}{15x+6x\underline{+15y-15y}=45-129} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}21y=147 \\21x=-84\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{147}{21}=7 \\x=\frac{-84}{21}=-4\end{matrix}\right.\\ \qquad V=\{(-4,7)\}\)
2. \(\left\{\begin{matrix}-4x+3y=32\\8y-10x=84\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-4x+3y=32\\-10x+8y=84\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{array}{c|c|c}-4x+3y=32& \color{red}{5.} & \color{blue}{8.} \\-10x+8y=84& \color{red}{-2.} & \color{blue}{-3.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{-20x+20x}+15y-16y=160-168} \\ \color{blue}{-32x+30x\underline{+24y-24y}=256-252} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-y=-8 \\-2x=4\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{-8}{-1}=8 \\x=\frac{4}{-2}=-2\end{matrix}\right.\\ \qquad V=\{(-2,8)\}\)
3. \(\left\{\begin{matrix}7x+10y=-47\\-7x+5y=-13\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{array}{c|c|c}7x+10y=-47& \color{red}{1.} & \color{blue}{1.} \\-7x+5y=-13& \color{red}{1.} & \color{blue}{-2.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{7x-7x}+10y+5y=-47-13} \\ \color{blue}{7x+14x\underline{+10y-10y}=-47+26} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}15y=-60 \\21x=-21\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{-60}{15}=-4 \\x=\frac{-21}{21}=-1\end{matrix}\right.\\ \qquad V=\{(-1,-4)\}\)
4. \(\left\{\begin{matrix}-8y+10x=102\\6x-7y=70\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}10x-8y=102\\6x-7y=70\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{array}{c|c|c}10x-8y=102& \color{red}{3.} & \color{blue}{7.} \\6x-7y=70& \color{red}{-5.} & \color{blue}{-8.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{30x-30x}-24y+35y=306-350} \\ \color{blue}{70x-48x\underline{-56y+56y}=714-560} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}11y=-44 \\22x=154\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{-44}{11}=-4 \\x=\frac{154}{22}=7\end{matrix}\right.\\ \qquad V=\{(7,-4)\}\)
5. \(\left\{\begin{matrix}2y=42+4x\\-3x-8y=-54\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-4x+2y=42\\-3x-8y=-54\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{array}{c|c|c}-4x+2y=42& \color{red}{3.} & \color{blue}{4.} \\-3x-8y=-54& \color{red}{-4.} & \color{blue}{1.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{-12x+12x}+6y+32y=126+216} \\ \color{blue}{-16x-3x\underline{+8y-8y}=168-54} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}38y=342 \\-19x=114\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{342}{38}=9 \\x=\frac{114}{-19}=-6\end{matrix}\right.\\ \qquad V=\{(-6,9)\}\)
6. \(\left\{\begin{matrix}4x+3y=-20\\4x-7y=60\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{array}{c|c|c}4x+3y=-20& \color{red}{1.} & \color{blue}{7.} \\4x-7y=60& \color{red}{-1.} & \color{blue}{3.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{4x-4x}+3y+7y=-20-60} \\ \color{blue}{28x+12x\underline{+21y-21y}=-140+180} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}10y=-80 \\40x=40\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{-80}{10}=-8 \\x=\frac{40}{40}=1\end{matrix}\right.\\ \qquad V=\{(1,-8)\}\)
7. \(\left\{\begin{matrix}5x-9y=29\\-9x=-6y-42\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}5x-9y=29\\-9x+6y=-42\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{array}{c|c|c}5x-9y=29& \color{red}{9.} & \color{blue}{2.} \\-9x+6y=-42& \color{red}{5.} & \color{blue}{3.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{45x-45x}-81y+30y=261-210} \\ \color{blue}{10x-27x\underline{-18y+18y}=58-126} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-51y=51 \\-17x=-68\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{51}{-51}=-1 \\x=\frac{-68}{-17}=4\end{matrix}\right.\\ \qquad V=\{(4,-1)\}\)
8. \(\left\{\begin{matrix}6x-4y=68\\10y+7x=6\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}6x-4y=68\\7x+10y=6\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{array}{c|c|c}6x-4y=68& \color{red}{7.} & \color{blue}{5.} \\7x+10y=6& \color{red}{-6.} & \color{blue}{2.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{42x-42x}-28y-60y=476-36} \\ \color{blue}{30x+14x\underline{-20y+20y}=340+12} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-88y=440 \\44x=352\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{440}{-88}=-5 \\x=\frac{352}{44}=8\end{matrix}\right.\\ \qquad V=\{(8,-5)\}\)
9. \(\left\{\begin{matrix}7y=61-6x\\-8x-2y=-30\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}6x+7y=61\\-8x-2y=-30\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{array}{c|c|c}6x+7y=61& \color{red}{4.} & \color{blue}{2.} \\-8x-2y=-30& \color{red}{3.} & \color{blue}{7.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{24x-24x}+28y-6y=244-90} \\ \color{blue}{12x-56x\underline{+14y-14y}=122-210} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}22y=154 \\-44x=-88\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{154}{22}=7 \\x=\frac{-88}{-44}=2\end{matrix}\right.\\ \qquad V=\{(2,7)\}\)
10. \(\left\{\begin{matrix}-9x-8y=-31\\-9y-6x=-39\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-9x-8y=-31\\-6x-9y=-39\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{array}{c|c|c}-9x-8y=-31& \color{red}{2.} & \color{blue}{9.} \\-6x-9y=-39& \color{red}{-3.} & \color{blue}{-8.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{-18x+18x}-16y+27y=-62+117} \\ \color{blue}{-81x+48x\underline{-72y+72y}=-279+312} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}11y=55 \\-33x=33\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{55}{11}=5 \\x=\frac{33}{-33}=-1\end{matrix}\right.\\ \qquad V=\{(-1,5)\}\)
11. \(\left\{\begin{matrix}-2y-5x=39\\-4x+7y=57\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-5x-2y=39\\-4x+7y=57\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{array}{c|c|c}-5x-2y=39& \color{red}{4.} & \color{blue}{7.} \\-4x+7y=57& \color{red}{-5.} & \color{blue}{2.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{-20x+20x}-8y-35y=156-285} \\ \color{blue}{-35x-8x\underline{-14y+14y}=273+114} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-43y=-129 \\-43x=387\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{-129}{-43}=3 \\x=\frac{387}{-43}=-9\end{matrix}\right.\\ \qquad V=\{(-9,3)\}\)
12. \(\left\{\begin{matrix}5y+5x=45\\-5x-2y=-42\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}5x+5y=45\\-5x-2y=-42\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{array}{c|c|c}5x+5y=45& \color{red}{1.} & \color{blue}{2.} \\-5x-2y=-42& \color{red}{1.} & \color{blue}{5.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{5x-5x}+5y-2y=45-42} \\ \color{blue}{10x-25x\underline{+10y-10y}=90-210} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}3y=3 \\-15x=-120\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{3}{3}=1 \\x=\frac{-120}{-15}=8\end{matrix}\right.\\ \qquad V=\{(8,1)\}\)