Stelsels substitutie

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Substitutie

1. \(\left\{\begin{matrix}3x+y=0\\3x+3y=18\end{matrix}\right.\)
2. \(\left\{\begin{matrix}-y=53-6x\\-5x-3y=-48\end{matrix}\right.\)
3. \(\left\{\begin{matrix}x+3y=-14\\2x+5y=-22\end{matrix}\right.\)
4. \(\left\{\begin{matrix}6x-3y=-33\\2x=-y-5\end{matrix}\right.\)
5. \(\left\{\begin{matrix}-5x+6y=-48\\-5x=y-27\end{matrix}\right.\)
6. \(\left\{\begin{matrix}x-3y=-11\\-4x=-4y+28\end{matrix}\right.\)
7. \(\left\{\begin{matrix}6x+4y=-48\\-2x+y=-5\end{matrix}\right.\)
8. \(\left\{\begin{matrix}5x-4y=61\\-3x+y=-31\end{matrix}\right.\)
9. \(\left\{\begin{matrix}2x+2y=-10\\-5x=y+37\end{matrix}\right.\)
10. \(\left\{\begin{matrix}5x-y=-47\\4x=3y-31\end{matrix}\right.\)
11. \(\left\{\begin{matrix}-5x+3y=-18\\-6x+y=-19\end{matrix}\right.\)
12. \(\left\{\begin{matrix}-3x+6y=-33\\-4x+y=12\end{matrix}\right.\)

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Substitutie

Verbetersleutel

1. \(\left\{\begin{matrix}3x+y=0\\3x+3y=18\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=-3x+0\\ 3x+3y=18\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}y=-3x+0\\ 3x+3\left(-3x+0\right)=18\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}y=-3x+0\\ 3x-9x+0=18\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}y=-3x+0\\ -6x=18+0=18\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=-3x+0\\ x=\frac{18}{-6}=-3\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=-3.(-3)+0=9\\ x=-3\end{matrix}\right.\\ \qquad V=\{(-3,9)\}\)
2. \(\left\{\begin{matrix}-y=53-6x\\-5x-3y=-48\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}6x-y=53\\-5x-3y=-48\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}6x-53=y\\-5x-3y=-48\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}y=6x-53\\ -5x-3\left(6x-53\right)=-48\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}y=6x-53\\ -5x-18x+159=-48\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}y=6x-53\\ -23x=-48-159=-207\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=6x-53\\ x=\frac{-207}{-23}=9\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=6.(9)-53=1\\ x=9\end{matrix}\right.\\ \qquad V=\{(9,1)\}\)
3. \(\left\{\begin{matrix}x+3y=-14\\2x+5y=-22\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=-3y-14\\ 2x+5y=-22\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}x=-3y-14\\ 2.\left(-3y-14\right)+5y=-22\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}x=-3y-14\\ -6y-28+5y=-22\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}x=-3y-14\\ -y=-22+28=6\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=-3y-14\\ y=\frac{6}{-1}=-6\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=-3.(-6)-14=4\\ y=-6\end{matrix}\right.\\ \qquad V=\{(4,-6)\}\)
4. \(\left\{\begin{matrix}6x-3y=-33\\2x=-y-5\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}6x-3y=-33\\2x+y=-5\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}6x-3y=-33\\ y=-2x-5\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}6x-3\left(-2x-5\right)=-33\\y=-2x-5\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}6x+6x+15=-33\\y=-2x-5\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}12x=-33-15=-48\\y=-2x-5\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{-48}{12} = -4 \\ y=-2x-5\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = -4 \\ y=-2.(-4)-5=3\end{matrix}\right.\\ \qquad V=\{(-4,3)\}\)
5. \(\left\{\begin{matrix}-5x+6y=-48\\-5x=y-27\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-5x+6y=-48\\-5x-y=-27\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}-5x+6y=-48\\ -5x+27=y\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-5x+6\left(-5x+27\right)=-48\\y=-5x+27\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-5x-30x+162=-48\\y=-5x+27\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-35x=-48-162=-210\\y=-5x+27\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{-210}{-35} = 6 \\ y=-5x+27\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = 6 \\ y=-5.(6)+27=-3\end{matrix}\right.\\ \qquad V=\{(6,-3)\}\)
6. \(\left\{\begin{matrix}x-3y=-11\\-4x=-4y+28\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x-3y=-11\\-4x+4y=28\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}x=3y-11\\ -4x+4y=28\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}x=3y-11\\ -4.\left(3y-11\right)+4y=28\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}x=3y-11\\ -12y+44+4y=28\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}x=3y-11\\ -8y=28-44=-16\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=3y-11\\ y=\frac{-16}{-8}=2\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=3.(2)-11=-5\\ y=2\end{matrix}\right.\\ \qquad V=\{(-5,2)\}\)
7. \(\left\{\begin{matrix}6x+4y=-48\\-2x+y=-5\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}6x+4y=-48\\ y=2x-5\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}6x+4\left(2x-5\right)=-48\\y=2x-5\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}6x+8x-20=-48\\y=2x-5\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}14x=-48+20=-28\\y=2x-5\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{-28}{14} = -2 \\ y=2x-5\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = -2 \\ y=2.(-2)-5=-9\end{matrix}\right.\\ \qquad V=\{(-2,-9)\}\)
8. \(\left\{\begin{matrix}5x-4y=61\\-3x+y=-31\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}5x-4y=61\\ y=3x-31\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}5x-4\left(3x-31\right)=61\\y=3x-31\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}5x-12x+124=61\\y=3x-31\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-7x=61-124=-63\\y=3x-31\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{-63}{-7} = 9 \\ y=3x-31\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = 9 \\ y=3.(9)-31=-4\end{matrix}\right.\\ \qquad V=\{(9,-4)\}\)
9. \(\left\{\begin{matrix}2x+2y=-10\\-5x=y+37\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}2x+2y=-10\\-5x-y=37\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}2x+2y=-10\\ -5x-37=y\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}2x+2\left(-5x-37\right)=-10\\y=-5x-37\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}2x-10x-74=-10\\y=-5x-37\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-8x=-10+74=64\\y=-5x-37\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{64}{-8} = -8 \\ y=-5x-37\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = -8 \\ y=-5.(-8)-37=3\end{matrix}\right.\\ \qquad V=\{(-8,3)\}\)
10. \(\left\{\begin{matrix}5x-y=-47\\4x=3y-31\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}5x-y=-47\\4x-3y=-31\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}5x+47=y\\4x-3y=-31\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}y=5x+47\\ 4x-3\left(5x+47\right)=-31\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}y=5x+47\\ 4x-15x-141=-31\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}y=5x+47\\ -11x=-31+141=110\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=5x+47\\ x=\frac{110}{-11}=-10\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=5.(-10)+47=-3\\ x=-10\end{matrix}\right.\\ \qquad V=\{(-10,-3)\}\)
11. \(\left\{\begin{matrix}-5x+3y=-18\\-6x+y=-19\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-5x+3y=-18\\ y=6x-19\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-5x+3\left(6x-19\right)=-18\\y=6x-19\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-5x+18x-57=-18\\y=6x-19\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}13x=-18+57=39\\y=6x-19\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{39}{13} = 3 \\ y=6x-19\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = 3 \\ y=6.(3)-19=-1\end{matrix}\right.\\ \qquad V=\{(3,-1)\}\)
12. \(\left\{\begin{matrix}-3x+6y=-33\\-4x+y=12\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-3x+6y=-33\\ y=4x+12\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-3x+6\left(4x+12\right)=-33\\y=4x+12\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-3x+24x+72=-33\\y=4x+12\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}21x=-33-72=-105\\y=4x+12\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{-105}{21} = -5 \\ y=4x+12\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = -5 \\ y=4.(-5)+12=-8\end{matrix}\right.\\ \qquad V=\{(-5,-8)\}\)