Stelsels combinatie

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Combinatie

1. \(\left\{\begin{matrix}6x-7y=11\\4x=-5y+75\end{matrix}\right.\)
2. \(\left\{\begin{matrix}3y=28+7x\\9x-9y=-72\end{matrix}\right.\)
3. \(\left\{\begin{matrix}4x+3y=1\\-5y-4x=-15\end{matrix}\right.\)
4. \(\left\{\begin{matrix}3x-10y=-37\\-10x=7y+83\end{matrix}\right.\)
5. \(\left\{\begin{matrix}5x+3y=25\\-4y-3x=-37\end{matrix}\right.\)
6. \(\left\{\begin{matrix}-5y+2x=-25\\4x-10y=-50\end{matrix}\right.\)
7. \(\left\{\begin{matrix}-10y=180+8x\\-10x-7y=170\end{matrix}\right.\)
8. \(\left\{\begin{matrix}-9x-10y=89\\-6y+6x=42\end{matrix}\right.\)
9. \(\left\{\begin{matrix}-2y-4x=12\\-3x-4y=-11\end{matrix}\right.\)
10. \(\left\{\begin{matrix}-9x-5y=-65\\-10x+10y=-10\end{matrix}\right.\)
11. \(\left\{\begin{matrix}7y-4x=-44\\-7x+2y=-36\end{matrix}\right.\)
12. \(\left\{\begin{matrix}6y=24-2x\\-2x+4y=6\end{matrix}\right.\)

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Combinatie

Verbetersleutel

1. \(\left\{\begin{matrix}6x-7y=11\\4x=-5y+75\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}6x-7y=11\\4x+5y=75\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{array}{c|c|c}6x-7y=11& \color{red}{2.} & \color{blue}{5.} \\4x+5y=75& \color{red}{-3.} & \color{blue}{7.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{12x-12x}-14y-15y=22-225} \\ \color{blue}{30x+28x\underline{-35y+35y}=55+525} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-29y=-203 \\58x=580\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{-203}{-29}=7 \\x=\frac{580}{58}=10\end{matrix}\right.\\ \qquad V=\{(10,7)\}\)
2. \(\left\{\begin{matrix}3y=28+7x\\9x-9y=-72\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-7x+3y=28\\9x-9y=-72\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{array}{c|c|c}-7x+3y=28& \color{red}{9.} & \color{blue}{3.} \\9x-9y=-72& \color{red}{7.} & \color{blue}{1.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{-63x+63x}+27y-63y=252-504} \\ \color{blue}{-21x+9x\underline{+9y-9y}=84-72} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-36y=-252 \\-12x=12\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{-252}{-36}=7 \\x=\frac{12}{-12}=-1\end{matrix}\right.\\ \qquad V=\{(-1,7)\}\)
3. \(\left\{\begin{matrix}4x+3y=1\\-5y-4x=-15\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}4x+3y=1\\-4x-5y=-15\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{array}{c|c|c}4x+3y=1& \color{red}{1.} & \color{blue}{5.} \\-4x-5y=-15& \color{red}{1.} & \color{blue}{3.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{4x-4x}+3y-5y=1-15} \\ \color{blue}{20x-12x\underline{+15y-15y}=5-45} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-2y=-14 \\8x=-40\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{-14}{-2}=7 \\x=\frac{-40}{8}=-5\end{matrix}\right.\\ \qquad V=\{(-5,7)\}\)
4. \(\left\{\begin{matrix}3x-10y=-37\\-10x=7y+83\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}3x-10y=-37\\-10x-7y=83\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{array}{c|c|c}3x-10y=-37& \color{red}{10.} & \color{blue}{7.} \\-10x-7y=83& \color{red}{3.} & \color{blue}{-10.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{30x-30x}-100y-21y=-370+249} \\ \color{blue}{21x+100x\underline{-70y+70y}=-259-830} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-121y=-121 \\121x=-1089\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{-121}{-121}=1 \\x=\frac{-1089}{121}=-9\end{matrix}\right.\\ \qquad V=\{(-9,1)\}\)
5. \(\left\{\begin{matrix}5x+3y=25\\-4y-3x=-37\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}5x+3y=25\\-3x-4y=-37\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{array}{c|c|c}5x+3y=25& \color{red}{3.} & \color{blue}{4.} \\-3x-4y=-37& \color{red}{5.} & \color{blue}{3.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{15x-15x}+9y-20y=75-185} \\ \color{blue}{20x-9x\underline{+12y-12y}=100-111} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-11y=-110 \\11x=-11\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{-110}{-11}=10 \\x=\frac{-11}{11}=-1\end{matrix}\right.\\ \qquad V=\{(-1,10)\}\)
6. \(\left\{\begin{matrix}-5y+2x=-25\\4x-10y=-50\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}2x-5y=-25\\4x-10y=-50\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{array}{c|c|c}2x-5y=-25& \color{red}{2.} & \color{blue}{2.} \\4x-10y=-50& \color{red}{-1.} & \color{blue}{-1.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{4x-4x}-10y+10y=-50+50} \\ \color{blue}{4x-4x\underline{-10y+10y}=-50+50} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}0y=0 \\0x=0\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{0}{0}=1 \\x=\frac{0}{0}=-10\end{matrix}\right.\\ \qquad V=\{(-10,1)\}\)
7. \(\left\{\begin{matrix}-10y=180+8x\\-10x-7y=170\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-8x-10y=180\\-10x-7y=170\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{array}{c|c|c}-8x-10y=180& \color{red}{5.} & \color{blue}{7.} \\-10x-7y=170& \color{red}{-4.} & \color{blue}{-10.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{-40x+40x}-50y+28y=900-680} \\ \color{blue}{-56x+100x\underline{-70y+70y}=1260-1700} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-22y=220 \\44x=-440\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{220}{-22}=-10 \\x=\frac{-440}{44}=-10\end{matrix}\right.\\ \qquad V=\{(-10,-10)\}\)
8. \(\left\{\begin{matrix}-9x-10y=89\\-6y+6x=42\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-9x-10y=89\\6x-6y=42\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{array}{c|c|c}-9x-10y=89& \color{red}{2.} & \color{blue}{3.} \\6x-6y=42& \color{red}{3.} & \color{blue}{-5.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{-18x+18x}-20y-18y=178+126} \\ \color{blue}{-27x-30x\underline{-30y+30y}=267-210} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-38y=304 \\-57x=57\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{304}{-38}=-8 \\x=\frac{57}{-57}=-1\end{matrix}\right.\\ \qquad V=\{(-1,-8)\}\)
9. \(\left\{\begin{matrix}-2y-4x=12\\-3x-4y=-11\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-4x-2y=12\\-3x-4y=-11\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{array}{c|c|c}-4x-2y=12& \color{red}{3.} & \color{blue}{2.} \\-3x-4y=-11& \color{red}{-4.} & \color{blue}{-1.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{-12x+12x}-6y+16y=36+44} \\ \color{blue}{-8x+3x\underline{-4y+4y}=24+11} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}10y=80 \\-5x=35\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{80}{10}=8 \\x=\frac{35}{-5}=-7\end{matrix}\right.\\ \qquad V=\{(-7,8)\}\)
10. \(\left\{\begin{matrix}-9x-5y=-65\\-10x+10y=-10\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{array}{c|c|c}-9x-5y=-65& \color{red}{10.} & \color{blue}{2.} \\-10x+10y=-10& \color{red}{-9.} & \color{blue}{1.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{-90x+90x}-50y-90y=-650+90} \\ \color{blue}{-18x-10x\underline{-10y+10y}=-130-10} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-140y=-560 \\-28x=-140\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{-560}{-140}=4 \\x=\frac{-140}{-28}=5\end{matrix}\right.\\ \qquad V=\{(5,4)\}\)
11. \(\left\{\begin{matrix}7y-4x=-44\\-7x+2y=-36\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-4x+7y=-44\\-7x+2y=-36\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{array}{c|c|c}-4x+7y=-44& \color{red}{7.} & \color{blue}{2.} \\-7x+2y=-36& \color{red}{-4.} & \color{blue}{-7.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{-28x+28x}+49y-8y=-308+144} \\ \color{blue}{-8x+49x\underline{+14y-14y}=-88+252} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}41y=-164 \\41x=164\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{-164}{41}=-4 \\x=\frac{164}{41}=4\end{matrix}\right.\\ \qquad V=\{(4,-4)\}\)
12. \(\left\{\begin{matrix}6y=24-2x\\-2x+4y=6\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}2x+6y=24\\-2x+4y=6\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{array}{c|c|c}2x+6y=24& \color{red}{1.} & \color{blue}{2.} \\-2x+4y=6& \color{red}{1.} & \color{blue}{-3.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{2x-2x}+6y+4y=24+6} \\ \color{blue}{4x+6x\underline{+12y-12y}=48-18} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}10y=30 \\10x=30\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{30}{10}=3 \\x=\frac{30}{10}=3\end{matrix}\right.\\ \qquad V=\{(3,3)\}\)