Stelsels combinatie

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Combinatie

1. \(\left\{\begin{matrix}-8x-4y=4\\-8x+3y=-45\end{matrix}\right.\)
2. \(\left\{\begin{matrix}7x-9y=107\\-10x=4y-18\end{matrix}\right.\)
3. \(\left\{\begin{matrix}-8x-5y=-55\\-2y+10x=44\end{matrix}\right.\)
4. \(\left\{\begin{matrix}7x-3y=70\\5x=10y+105\end{matrix}\right.\)
5. \(\left\{\begin{matrix}-10x+2y=84\\-2x=9y-2\end{matrix}\right.\)
6. \(\left\{\begin{matrix}4y=42+5x\\9x-5y=-80\end{matrix}\right.\)
7. \(\left\{\begin{matrix}4x-10y=-94\\-9y-8x=-73\end{matrix}\right.\)
8. \(\left\{\begin{matrix}3y=21+3x\\10x+9y=-32\end{matrix}\right.\)
9. \(\left\{\begin{matrix}-4x+2y=28\\2x=-6y-28\end{matrix}\right.\)
10. \(\left\{\begin{matrix}-5y=38-9x\\-7x+10y=1\end{matrix}\right.\)
11. \(\left\{\begin{matrix}10x-6y=-88\\-8x+3y=56\end{matrix}\right.\)
12. \(\left\{\begin{matrix}-4x+3y=-39\\2x-5y=51\end{matrix}\right.\)

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Combinatie

Verbetersleutel

1. \(\left\{\begin{matrix}-8x-4y=4\\-8x+3y=-45\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{array}{c|c|c}-8x-4y=4& \color{red}{1.} & \color{blue}{3.} \\-8x+3y=-45& \color{red}{-1.} & \color{blue}{4.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{-8x+8x}-4y-3y=4+45} \\ \color{blue}{-24x-32x\underline{-12y+12y}=12-180} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-7y=49 \\-56x=-168\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{49}{-7}=-7 \\x=\frac{-168}{-56}=3\end{matrix}\right.\\ \qquad V=\{(3,-7)\}\)
2. \(\left\{\begin{matrix}7x-9y=107\\-10x=4y-18\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}7x-9y=107\\-10x-4y=-18\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{array}{c|c|c}7x-9y=107& \color{red}{10.} & \color{blue}{4.} \\-10x-4y=-18& \color{red}{7.} & \color{blue}{-9.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{70x-70x}-90y-28y=1070-126} \\ \color{blue}{28x+90x\underline{-36y+36y}=428+162} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-118y=944 \\118x=590\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{944}{-118}=-8 \\x=\frac{590}{118}=5\end{matrix}\right.\\ \qquad V=\{(5,-8)\}\)
3. \(\left\{\begin{matrix}-8x-5y=-55\\-2y+10x=44\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-8x-5y=-55\\10x-2y=44\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{array}{c|c|c}-8x-5y=-55& \color{red}{5.} & \color{blue}{2.} \\10x-2y=44& \color{red}{4.} & \color{blue}{-5.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{-40x+40x}-25y-8y=-275+176} \\ \color{blue}{-16x-50x\underline{-10y+10y}=-110-220} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-33y=-99 \\-66x=-330\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{-99}{-33}=3 \\x=\frac{-330}{-66}=5\end{matrix}\right.\\ \qquad V=\{(5,3)\}\)
4. \(\left\{\begin{matrix}7x-3y=70\\5x=10y+105\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}7x-3y=70\\5x-10y=105\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{array}{c|c|c}7x-3y=70& \color{red}{5.} & \color{blue}{10.} \\5x-10y=105& \color{red}{-7.} & \color{blue}{-3.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{35x-35x}-15y+70y=350-735} \\ \color{blue}{70x-15x\underline{-30y+30y}=700-315} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}55y=-385 \\55x=385\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{-385}{55}=-7 \\x=\frac{385}{55}=7\end{matrix}\right.\\ \qquad V=\{(7,-7)\}\)
5. \(\left\{\begin{matrix}-10x+2y=84\\-2x=9y-2\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-10x+2y=84\\-2x-9y=-2\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{array}{c|c|c}-10x+2y=84& \color{red}{1.} & \color{blue}{9.} \\-2x-9y=-2& \color{red}{-5.} & \color{blue}{2.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{-10x+10x}+2y+45y=84+10} \\ \color{blue}{-90x-4x\underline{+18y-18y}=756-4} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}47y=94 \\-94x=752\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{94}{47}=2 \\x=\frac{752}{-94}=-8\end{matrix}\right.\\ \qquad V=\{(-8,2)\}\)
6. \(\left\{\begin{matrix}4y=42+5x\\9x-5y=-80\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-5x+4y=42\\9x-5y=-80\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{array}{c|c|c}-5x+4y=42& \color{red}{9.} & \color{blue}{5.} \\9x-5y=-80& \color{red}{5.} & \color{blue}{4.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{-45x+45x}+36y-25y=378-400} \\ \color{blue}{-25x+36x\underline{+20y-20y}=210-320} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}11y=-22 \\11x=-110\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{-22}{11}=-2 \\x=\frac{-110}{11}=-10\end{matrix}\right.\\ \qquad V=\{(-10,-2)\}\)
7. \(\left\{\begin{matrix}4x-10y=-94\\-9y-8x=-73\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}4x-10y=-94\\-8x-9y=-73\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{array}{c|c|c}4x-10y=-94& \color{red}{2.} & \color{blue}{9.} \\-8x-9y=-73& \color{red}{1.} & \color{blue}{-10.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{8x-8x}-20y-9y=-188-73} \\ \color{blue}{36x+80x\underline{-90y+90y}=-846+730} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-29y=-261 \\116x=-116\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{-261}{-29}=9 \\x=\frac{-116}{116}=-1\end{matrix}\right.\\ \qquad V=\{(-1,9)\}\)
8. \(\left\{\begin{matrix}3y=21+3x\\10x+9y=-32\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-3x+3y=21\\10x+9y=-32\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{array}{c|c|c}-3x+3y=21& \color{red}{10.} & \color{blue}{3.} \\10x+9y=-32& \color{red}{3.} & \color{blue}{-1.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{-30x+30x}+30y+27y=210-96} \\ \color{blue}{-9x-10x\underline{+9y-9y}=63+32} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}57y=114 \\-19x=95\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{114}{57}=2 \\x=\frac{95}{-19}=-5\end{matrix}\right.\\ \qquad V=\{(-5,2)\}\)
9. \(\left\{\begin{matrix}-4x+2y=28\\2x=-6y-28\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-4x+2y=28\\2x+6y=-28\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{array}{c|c|c}-4x+2y=28& \color{red}{1.} & \color{blue}{3.} \\2x+6y=-28& \color{red}{2.} & \color{blue}{-1.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{-4x+4x}+2y+12y=28-56} \\ \color{blue}{-12x-2x\underline{+6y-6y}=84+28} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}14y=-28 \\-14x=112\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{-28}{14}=-2 \\x=\frac{112}{-14}=-8\end{matrix}\right.\\ \qquad V=\{(-8,-2)\}\)
10. \(\left\{\begin{matrix}-5y=38-9x\\-7x+10y=1\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}9x-5y=38\\-7x+10y=1\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{array}{c|c|c}9x-5y=38& \color{red}{7.} & \color{blue}{2.} \\-7x+10y=1& \color{red}{9.} & \color{blue}{1.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{63x-63x}-35y+90y=266+9} \\ \color{blue}{18x-7x\underline{-10y+10y}=76+1} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}55y=275 \\11x=77\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{275}{55}=5 \\x=\frac{77}{11}=7\end{matrix}\right.\\ \qquad V=\{(7,5)\}\)
11. \(\left\{\begin{matrix}10x-6y=-88\\-8x+3y=56\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{array}{c|c|c}10x-6y=-88& \color{red}{4.} & \color{blue}{1.} \\-8x+3y=56& \color{red}{5.} & \color{blue}{2.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{40x-40x}-24y+15y=-352+280} \\ \color{blue}{10x-16x\underline{-6y+6y}=-88+112} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-9y=-72 \\-6x=24\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{-72}{-9}=8 \\x=\frac{24}{-6}=-4\end{matrix}\right.\\ \qquad V=\{(-4,8)\}\)
12. \(\left\{\begin{matrix}-4x+3y=-39\\2x-5y=51\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{array}{c|c|c}-4x+3y=-39& \color{red}{1.} & \color{blue}{5.} \\2x-5y=51& \color{red}{2.} & \color{blue}{3.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{-4x+4x}+3y-10y=-39+102} \\ \color{blue}{-20x+6x\underline{+15y-15y}=-195+153} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-7y=63 \\-14x=-42\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{63}{-7}=-9 \\x=\frac{-42}{-14}=3\end{matrix}\right.\\ \qquad V=\{(3,-9)\}\)