Stelsels combinatie

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Combinatie

1. \(\left\{\begin{matrix}-7x+8y=-51\\-3x+3y=-21\end{matrix}\right.\)
2. \(\left\{\begin{matrix}-6x+5y=14\\4y+9x=94\end{matrix}\right.\)
3. \(\left\{\begin{matrix}9y=-43+10x\\9x+4y=-46\end{matrix}\right.\)
4. \(\left\{\begin{matrix}-2y-4x=16\\3x-9y=-96\end{matrix}\right.\)
5. \(\left\{\begin{matrix}-7x-5y=51\\-9x-9y=63\end{matrix}\right.\)
6. \(\left\{\begin{matrix}-2x-9y=-108\\7y-5x=25\end{matrix}\right.\)
7. \(\left\{\begin{matrix}-3y=-19-2x\\-5x-8y=-30\end{matrix}\right.\)
8. \(\left\{\begin{matrix}-6x-5y=-3\\-3x=-2y+39\end{matrix}\right.\)
9. \(\left\{\begin{matrix}3y=-21-3x\\-5x+7y=-61\end{matrix}\right.\)
10. \(\left\{\begin{matrix}5x+8y=37\\-10x+5y=115\end{matrix}\right.\)
11. \(\left\{\begin{matrix}-5x-5y=35\\-8x-4y=32\end{matrix}\right.\)
12. \(\left\{\begin{matrix}-10y=55-5x\\3x-2y=17\end{matrix}\right.\)

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Combinatie

Verbetersleutel

1. \(\left\{\begin{matrix}-7x+8y=-51\\-3x+3y=-21\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{array}{c|c|c}-7x+8y=-51& \color{red}{3.} & \color{blue}{3.} \\-3x+3y=-21& \color{red}{-7.} & \color{blue}{-8.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{-21x+21x}+24y-21y=-153+147} \\ \color{blue}{-21x+24x\underline{+24y-24y}=-153+168} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}3y=-6 \\3x=15\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{-6}{3}=-2 \\x=\frac{15}{3}=5\end{matrix}\right.\\ \qquad V=\{(5,-2)\}\)
2. \(\left\{\begin{matrix}-6x+5y=14\\4y+9x=94\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-6x+5y=14\\9x+4y=94\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{array}{c|c|c}-6x+5y=14& \color{red}{3.} & \color{blue}{4.} \\9x+4y=94& \color{red}{2.} & \color{blue}{-5.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{-18x+18x}+15y+8y=42+188} \\ \color{blue}{-24x-45x\underline{+20y-20y}=56-470} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}23y=230 \\-69x=-414\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{230}{23}=10 \\x=\frac{-414}{-69}=6\end{matrix}\right.\\ \qquad V=\{(6,10)\}\)
3. \(\left\{\begin{matrix}9y=-43+10x\\9x+4y=-46\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-10x+9y=-43\\9x+4y=-46\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{array}{c|c|c}-10x+9y=-43& \color{red}{9.} & \color{blue}{4.} \\9x+4y=-46& \color{red}{10.} & \color{blue}{-9.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{-90x+90x}+81y+40y=-387-460} \\ \color{blue}{-40x-81x\underline{+36y-36y}=-172+414} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}121y=-847 \\-121x=242\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{-847}{121}=-7 \\x=\frac{242}{-121}=-2\end{matrix}\right.\\ \qquad V=\{(-2,-7)\}\)
4. \(\left\{\begin{matrix}-2y-4x=16\\3x-9y=-96\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-4x-2y=16\\3x-9y=-96\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{array}{c|c|c}-4x-2y=16& \color{red}{3.} & \color{blue}{9.} \\3x-9y=-96& \color{red}{4.} & \color{blue}{-2.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{-12x+12x}-6y-36y=48-384} \\ \color{blue}{-36x-6x\underline{-18y+18y}=144+192} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-42y=-336 \\-42x=336\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{-336}{-42}=8 \\x=\frac{336}{-42}=-8\end{matrix}\right.\\ \qquad V=\{(-8,8)\}\)
5. \(\left\{\begin{matrix}-7x-5y=51\\-9x-9y=63\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{array}{c|c|c}-7x-5y=51& \color{red}{9.} & \color{blue}{9.} \\-9x-9y=63& \color{red}{-7.} & \color{blue}{-5.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{-63x+63x}-45y+63y=459-441} \\ \color{blue}{-63x+45x\underline{-45y+45y}=459-315} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}18y=18 \\-18x=144\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{18}{18}=1 \\x=\frac{144}{-18}=-8\end{matrix}\right.\\ \qquad V=\{(-8,1)\}\)
6. \(\left\{\begin{matrix}-2x-9y=-108\\7y-5x=25\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-2x-9y=-108\\-5x+7y=25\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{array}{c|c|c}-2x-9y=-108& \color{red}{5.} & \color{blue}{7.} \\-5x+7y=25& \color{red}{-2.} & \color{blue}{9.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{-10x+10x}-45y-14y=-540-50} \\ \color{blue}{-14x-45x\underline{-63y+63y}=-756+225} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-59y=-590 \\-59x=-531\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{-590}{-59}=10 \\x=\frac{-531}{-59}=9\end{matrix}\right.\\ \qquad V=\{(9,10)\}\)
7. \(\left\{\begin{matrix}-3y=-19-2x\\-5x-8y=-30\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}2x-3y=-19\\-5x-8y=-30\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{array}{c|c|c}2x-3y=-19& \color{red}{5.} & \color{blue}{8.} \\-5x-8y=-30& \color{red}{2.} & \color{blue}{-3.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{10x-10x}-15y-16y=-95-60} \\ \color{blue}{16x+15x\underline{-24y+24y}=-152+90} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-31y=-155 \\31x=-62\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{-155}{-31}=5 \\x=\frac{-62}{31}=-2\end{matrix}\right.\\ \qquad V=\{(-2,5)\}\)
8. \(\left\{\begin{matrix}-6x-5y=-3\\-3x=-2y+39\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-6x-5y=-3\\-3x+2y=39\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{array}{c|c|c}-6x-5y=-3& \color{red}{1.} & \color{blue}{2.} \\-3x+2y=39& \color{red}{-2.} & \color{blue}{5.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{-6x+6x}-5y-4y=-3-78} \\ \color{blue}{-12x-15x\underline{-10y+10y}=-6+195} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-9y=-81 \\-27x=189\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{-81}{-9}=9 \\x=\frac{189}{-27}=-7\end{matrix}\right.\\ \qquad V=\{(-7,9)\}\)
9. \(\left\{\begin{matrix}3y=-21-3x\\-5x+7y=-61\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}3x+3y=-21\\-5x+7y=-61\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{array}{c|c|c}3x+3y=-21& \color{red}{5.} & \color{blue}{7.} \\-5x+7y=-61& \color{red}{3.} & \color{blue}{-3.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{15x-15x}+15y+21y=-105-183} \\ \color{blue}{21x+15x\underline{+21y-21y}=-147+183} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}36y=-288 \\36x=36\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{-288}{36}=-8 \\x=\frac{36}{36}=1\end{matrix}\right.\\ \qquad V=\{(1,-8)\}\)
10. \(\left\{\begin{matrix}5x+8y=37\\-10x+5y=115\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{array}{c|c|c}5x+8y=37& \color{red}{2.} & \color{blue}{5.} \\-10x+5y=115& \color{red}{1.} & \color{blue}{-8.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{10x-10x}+16y+5y=74+115} \\ \color{blue}{25x+80x\underline{+40y-40y}=185-920} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}21y=189 \\105x=-735\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{189}{21}=9 \\x=\frac{-735}{105}=-7\end{matrix}\right.\\ \qquad V=\{(-7,9)\}\)
11. \(\left\{\begin{matrix}-5x-5y=35\\-8x-4y=32\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{array}{c|c|c}-5x-5y=35& \color{red}{8.} & \color{blue}{4.} \\-8x-4y=32& \color{red}{-5.} & \color{blue}{-5.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{-40x+40x}-40y+20y=280-160} \\ \color{blue}{-20x+40x\underline{-20y+20y}=140-160} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-20y=120 \\20x=-20\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{120}{-20}=-6 \\x=\frac{-20}{20}=-1\end{matrix}\right.\\ \qquad V=\{(-1,-6)\}\)
12. \(\left\{\begin{matrix}-10y=55-5x\\3x-2y=17\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}5x-10y=55\\3x-2y=17\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{array}{c|c|c}5x-10y=55& \color{red}{3.} & \color{blue}{1.} \\3x-2y=17& \color{red}{-5.} & \color{blue}{-5.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{15x-15x}-30y+10y=165-85} \\ \color{blue}{5x-15x\underline{-10y+10y}=55-85} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-20y=80 \\-10x=-30\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{80}{-20}=-4 \\x=\frac{-30}{-10}=3\end{matrix}\right.\\ \qquad V=\{(3,-4)\}\)