Stelsels substitutie

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Substitutie

1. \(\left\{\begin{matrix}-3x-5y=-37\\x=2y-6\end{matrix}\right.\)
2. \(\left\{\begin{matrix}2x-4y=-8\\3x+y=30\end{matrix}\right.\)
3. \(\left\{\begin{matrix}3x+4y=27\\5x-y=-1\end{matrix}\right.\)
4. \(\left\{\begin{matrix}4y=-17+5x\\-x+5y=5\end{matrix}\right.\)
5. \(\left\{\begin{matrix}-2y=32-2x\\-6x+y=-61\end{matrix}\right.\)
6. \(\left\{\begin{matrix}6x+5y=-88\\-x-3y=32\end{matrix}\right.\)
7. \(\left\{\begin{matrix}4x+3y=-32\\x-2y=3\end{matrix}\right.\)
8. \(\left\{\begin{matrix}-3x-2y=33\\x-5y=6\end{matrix}\right.\)
9. \(\left\{\begin{matrix}2y=22-2x\\-3x+y=-17\end{matrix}\right.\)
10. \(\left\{\begin{matrix}-2x-6y=-54\\x=-2y+20\end{matrix}\right.\)
11. \(\left\{\begin{matrix}2y=27+3x\\-x+3y=30\end{matrix}\right.\)
12. \(\left\{\begin{matrix}5x-3y=9\\-x=-4y+5\end{matrix}\right.\)

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Substitutie

Verbetersleutel

1. \(\left\{\begin{matrix}-3x-5y=-37\\x=2y-6\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-3x-5y=-37\\x-2y=-6\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}-3x-5y=-37\\ x=2y-6\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-3\left(2y-6\right)-5y=-37\\x=2y-6\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-6y+18-5y=-37\\x=2y-6\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-11y=-37-18=-55\\x=2y-6\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{-55}{-11} = 5 \\ x=2y-6\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = 5 \\ x=2.(5)-6=4\end{matrix}\right.\\ \qquad V=\{(4,5)\}\)
2. \(\left\{\begin{matrix}2x-4y=-8\\3x+y=30\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}2x-4y=-8\\ y=-3x+30\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}2x-4\left(-3x+30\right)=-8\\y=-3x+30\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}2x+12x-120=-8\\y=-3x+30\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}14x=-8+120=112\\y=-3x+30\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{112}{14} = 8 \\ y=-3x+30\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = 8 \\ y=-3.(8)+30=6\end{matrix}\right.\\ \qquad V=\{(8,6)\}\)
3. \(\left\{\begin{matrix}3x+4y=27\\5x-y=-1\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}3x+4y=27\\ 5x+1=y\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}3x+4\left(5x+1\right)=27\\y=5x+1\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}3x+20x+4=27\\y=5x+1\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}23x=27-4=23\\y=5x+1\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{23}{23} = 1 \\ y=5x+1\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = 1 \\ y=5.(1)+1=6\end{matrix}\right.\\ \qquad V=\{(1,6)\}\)
4. \(\left\{\begin{matrix}4y=-17+5x\\-x+5y=5\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-5x+4y=-17\\-x+5y=5\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}-5x+4y=-17\\ 5y-5=x\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-5\left(5y-5\right)+4y=-17\\x=5y-5\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-25y+25+4y=-17\\x=5y-5\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-21y=-17-25=-42\\x=5y-5\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{-42}{-21} = 2 \\ x=5y-5\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = 2 \\ x=5.(2)-5=5\end{matrix}\right.\\ \qquad V=\{(5,2)\}\)
5. \(\left\{\begin{matrix}-2y=32-2x\\-6x+y=-61\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}2x-2y=32\\-6x+y=-61\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}2x-2y=32\\ y=6x-61\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}2x-2\left(6x-61\right)=32\\y=6x-61\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}2x-12x+122=32\\y=6x-61\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-10x=32-122=-90\\y=6x-61\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{-90}{-10} = 9 \\ y=6x-61\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = 9 \\ y=6.(9)-61=-7\end{matrix}\right.\\ \qquad V=\{(9,-7)\}\)
6. \(\left\{\begin{matrix}6x+5y=-88\\-x-3y=32\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}6x+5y=-88\\ -3y-32=x\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}6\left(-3y-32\right)+5y=-88\\x=-3y-32\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-18y-192+5y=-88\\x=-3y-32\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-13y=-88+192=104\\x=-3y-32\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{104}{-13} = -8 \\ x=-3y-32\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = -8 \\ x=-3.(-8)-32=-8\end{matrix}\right.\\ \qquad V=\{(-8,-8)\}\)
7. \(\left\{\begin{matrix}4x+3y=-32\\x-2y=3\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}4x+3y=-32\\ x=2y+3\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}4\left(2y+3\right)+3y=-32\\x=2y+3\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}8y+12+3y=-32\\x=2y+3\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}11y=-32-12=-44\\x=2y+3\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{-44}{11} = -4 \\ x=2y+3\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = -4 \\ x=2.(-4)+3=-5\end{matrix}\right.\\ \qquad V=\{(-5,-4)\}\)
8. \(\left\{\begin{matrix}-3x-2y=33\\x-5y=6\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-3x-2y=33\\ x=5y+6\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-3\left(5y+6\right)-2y=33\\x=5y+6\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-15y-18-2y=33\\x=5y+6\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-17y=33+18=51\\x=5y+6\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{51}{-17} = -3 \\ x=5y+6\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = -3 \\ x=5.(-3)+6=-9\end{matrix}\right.\\ \qquad V=\{(-9,-3)\}\)
9. \(\left\{\begin{matrix}2y=22-2x\\-3x+y=-17\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}2x+2y=22\\-3x+y=-17\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}2x+2y=22\\ y=3x-17\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}2x+2\left(3x-17\right)=22\\y=3x-17\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}2x+6x-34=22\\y=3x-17\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}8x=22+34=56\\y=3x-17\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{56}{8} = 7 \\ y=3x-17\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = 7 \\ y=3.(7)-17=4\end{matrix}\right.\\ \qquad V=\{(7,4)\}\)
10. \(\left\{\begin{matrix}-2x-6y=-54\\x=-2y+20\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-2x-6y=-54\\x+2y=20\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}-2x-6y=-54\\ x=-2y+20\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-2\left(-2y+20\right)-6y=-54\\x=-2y+20\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}4y-40-6y=-54\\x=-2y+20\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-2y=-54+40=-14\\x=-2y+20\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{-14}{-2} = 7 \\ x=-2y+20\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = 7 \\ x=-2.(7)+20=6\end{matrix}\right.\\ \qquad V=\{(6,7)\}\)
11. \(\left\{\begin{matrix}2y=27+3x\\-x+3y=30\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-3x+2y=27\\-x+3y=30\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}-3x+2y=27\\ 3y-30=x\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-3\left(3y-30\right)+2y=27\\x=3y-30\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-9y+90+2y=27\\x=3y-30\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-7y=27-90=-63\\x=3y-30\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{-63}{-7} = 9 \\ x=3y-30\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = 9 \\ x=3.(9)-30=-3\end{matrix}\right.\\ \qquad V=\{(-3,9)\}\)
12. \(\left\{\begin{matrix}5x-3y=9\\-x=-4y+5\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}5x-3y=9\\-x+4y=5\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}5x-3y=9\\ 4y-5=x\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}5\left(4y-5\right)-3y=9\\x=4y-5\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}20y-25-3y=9\\x=4y-5\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}17y=9+25=34\\x=4y-5\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{34}{17} = 2 \\ x=4y-5\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = 2 \\ x=4.(2)-5=3\end{matrix}\right.\\ \qquad V=\{(3,2)\}\)