Stelsels substitutie

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Substitutie

1. \(\left\{\begin{matrix}-5y=-27+3x\\-4x-y=-19\end{matrix}\right.\)
2. \(\left\{\begin{matrix}2y=-17+3x\\x-3y=22\end{matrix}\right.\)
3. \(\left\{\begin{matrix}5x-5y=-50\\2x=y-11\end{matrix}\right.\)
4. \(\left\{\begin{matrix}5x-3y=53\\-4x+y=-34\end{matrix}\right.\)
5. \(\left\{\begin{matrix}-x+3y=-10\\2x=6y+20\end{matrix}\right.\)
6. \(\left\{\begin{matrix}-6x+4y=30\\3x-y=-21\end{matrix}\right.\)
7. \(\left\{\begin{matrix}-2y=14+3x\\6x+y=-34\end{matrix}\right.\)
8. \(\left\{\begin{matrix}3x-y=-2\\-5x-4y=-42\end{matrix}\right.\)
9. \(\left\{\begin{matrix}-6x-2y=-64\\-4x=y-42\end{matrix}\right.\)
10. \(\left\{\begin{matrix}-2y=-7+x\\2x-6y=-56\end{matrix}\right.\)
11. \(\left\{\begin{matrix}-4x-y=-31\\6x-4y=74\end{matrix}\right.\)
12. \(\left\{\begin{matrix}6x+6y=0\\2x=-y+7\end{matrix}\right.\)

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Substitutie

Verbetersleutel

1. \(\left\{\begin{matrix}-5y=-27+3x\\-4x-y=-19\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-3x-5y=-27\\-4x-y=-19\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}-3x-5y=-27\\ -4x+19=y\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-3x-5\left(-4x+19\right)=-27\\y=-4x+19\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-3x+20x-95=-27\\y=-4x+19\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}17x=-27+95=68\\y=-4x+19\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{68}{17} = 4 \\ y=-4x+19\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = 4 \\ y=-4.(4)+19=3\end{matrix}\right.\\ \qquad V=\{(4,3)\}\)
2. \(\left\{\begin{matrix}2y=-17+3x\\x-3y=22\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-3x+2y=-17\\x-3y=22\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}-3x+2y=-17\\ x=3y+22\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-3\left(3y+22\right)+2y=-17\\x=3y+22\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-9y-66+2y=-17\\x=3y+22\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-7y=-17+66=49\\x=3y+22\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{49}{-7} = -7 \\ x=3y+22\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = -7 \\ x=3.(-7)+22=1\end{matrix}\right.\\ \qquad V=\{(1,-7)\}\)
3. \(\left\{\begin{matrix}5x-5y=-50\\2x=y-11\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}5x-5y=-50\\2x-y=-11\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}5x-5y=-50\\ 2x+11=y\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}5x-5\left(2x+11\right)=-50\\y=2x+11\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}5x-10x-55=-50\\y=2x+11\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-5x=-50+55=5\\y=2x+11\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{5}{-5} = -1 \\ y=2x+11\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = -1 \\ y=2.(-1)+11=9\end{matrix}\right.\\ \qquad V=\{(-1,9)\}\)
4. \(\left\{\begin{matrix}5x-3y=53\\-4x+y=-34\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}5x-3y=53\\ y=4x-34\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}5x-3\left(4x-34\right)=53\\y=4x-34\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}5x-12x+102=53\\y=4x-34\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-7x=53-102=-49\\y=4x-34\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{-49}{-7} = 7 \\ y=4x-34\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = 7 \\ y=4.(7)-34=-6\end{matrix}\right.\\ \qquad V=\{(7,-6)\}\)
5. \(\left\{\begin{matrix}-x+3y=-10\\2x=6y+20\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-x+3y=-10\\2x-6y=20\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}3y+10=x\\2x-6y=20\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}x=3y+10\\ 2.\left(3y+10\right)-6y=20\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}x=3y+10\\ 6y+20-6y=20\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}x=3y+10\\ 0y=20-20=0\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=3y+10\\ y=\frac{0}{0}=-3\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=3.(-3)+10=1\\ y=-3\end{matrix}\right.\\ \qquad V=\{(1,-3)\}\)
6. \(\left\{\begin{matrix}-6x+4y=30\\3x-y=-21\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-6x+4y=30\\ 3x+21=y\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-6x+4\left(3x+21\right)=30\\y=3x+21\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-6x+12x+84=30\\y=3x+21\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}6x=30-84=-54\\y=3x+21\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{-54}{6} = -9 \\ y=3x+21\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = -9 \\ y=3.(-9)+21=-6\end{matrix}\right.\\ \qquad V=\{(-9,-6)\}\)
7. \(\left\{\begin{matrix}-2y=14+3x\\6x+y=-34\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-3x-2y=14\\6x+y=-34\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}-3x-2y=14\\ y=-6x-34\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-3x-2\left(-6x-34\right)=14\\y=-6x-34\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-3x+12x+68=14\\y=-6x-34\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}9x=14-68=-54\\y=-6x-34\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{-54}{9} = -6 \\ y=-6x-34\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = -6 \\ y=-6.(-6)-34=2\end{matrix}\right.\\ \qquad V=\{(-6,2)\}\)
8. \(\left\{\begin{matrix}3x-y=-2\\-5x-4y=-42\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}3x+2=y\\-5x-4y=-42\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}y=3x+2\\ -5x-4\left(3x+2\right)=-42\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}y=3x+2\\ -5x-12x-8=-42\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}y=3x+2\\ -17x=-42+8=-34\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=3x+2\\ x=\frac{-34}{-17}=2\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=3.(2)+2=8\\ x=2\end{matrix}\right.\\ \qquad V=\{(2,8)\}\)
9. \(\left\{\begin{matrix}-6x-2y=-64\\-4x=y-42\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-6x-2y=-64\\-4x-y=-42\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}-6x-2y=-64\\ -4x+42=y\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-6x-2\left(-4x+42\right)=-64\\y=-4x+42\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-6x+8x-84=-64\\y=-4x+42\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}2x=-64+84=20\\y=-4x+42\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{20}{2} = 10 \\ y=-4x+42\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = 10 \\ y=-4.(10)+42=2\end{matrix}\right.\\ \qquad V=\{(10,2)\}\)
10. \(\left\{\begin{matrix}-2y=-7+x\\2x-6y=-56\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-x-2y=-7\\2x-6y=-56\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}-2y+7=x\\2x-6y=-56\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}x=-2y+7\\ 2.\left(-2y+7\right)-6y=-56\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}x=-2y+7\\ -4y+14-6y=-56\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}x=-2y+7\\ -10y=-56-14=-70\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=-2y+7\\ y=\frac{-70}{-10}=7\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=-2.(7)+7=-7\\ y=7\end{matrix}\right.\\ \qquad V=\{(-7,7)\}\)
11. \(\left\{\begin{matrix}-4x-y=-31\\6x-4y=74\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-4x+31=y\\6x-4y=74\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}y=-4x+31\\ 6x-4\left(-4x+31\right)=74\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}y=-4x+31\\ 6x+16x-124=74\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}y=-4x+31\\ 22x=74+124=198\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=-4x+31\\ x=\frac{198}{22}=9\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=-4.(9)+31=-5\\ x=9\end{matrix}\right.\\ \qquad V=\{(9,-5)\}\)
12. \(\left\{\begin{matrix}6x+6y=0\\2x=-y+7\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}6x+6y=0\\2x+y=7\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}6x+6y=0\\ y=-2x+7\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}6x+6\left(-2x+7\right)=0\\y=-2x+7\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}6x-12x+42=0\\y=-2x+7\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-6x=0-42=-42\\y=-2x+7\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{-42}{-6} = 7 \\ y=-2x+7\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = 7 \\ y=-2.(7)+7=-7\end{matrix}\right.\\ \qquad V=\{(7,-7)\}\)