Stelsels substitutie

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Substitutie

1. \(\left\{\begin{matrix}-6x-6y=-24\\3x-y=16\end{matrix}\right.\)
2. \(\left\{\begin{matrix}-2x+3y=-40\\x=-5y-45\end{matrix}\right.\)
3. \(\left\{\begin{matrix}6x+3y=39\\-6x=y-25\end{matrix}\right.\)
4. \(\left\{\begin{matrix}3y=-54+6x\\4x+y=30\end{matrix}\right.\)
5. \(\left\{\begin{matrix}6x-6y=6\\-2x+y=8\end{matrix}\right.\)
6. \(\left\{\begin{matrix}-3x+2y=21\\4x=y-18\end{matrix}\right.\)
7. \(\left\{\begin{matrix}3x+4y=-63\\x=2y+9\end{matrix}\right.\)
8. \(\left\{\begin{matrix}-2y=35+5x\\x-4y=37\end{matrix}\right.\)
9. \(\left\{\begin{matrix}5x+5y=-85\\2x=y-4\end{matrix}\right.\)
10. \(\left\{\begin{matrix}3y=-41+5x\\2x+y=12\end{matrix}\right.\)
11. \(\left\{\begin{matrix}6x-y=-31\\-5x=4y+50\end{matrix}\right.\)
12. \(\left\{\begin{matrix}3x+3y=-9\\-x=-6y-46\end{matrix}\right.\)

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Substitutie

Verbetersleutel

1. \(\left\{\begin{matrix}-6x-6y=-24\\3x-y=16\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-6x-6y=-24\\ 3x-16=y\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-6x-6\left(3x-16\right)=-24\\y=3x-16\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-6x-18x+96=-24\\y=3x-16\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-24x=-24-96=-120\\y=3x-16\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{-120}{-24} = 5 \\ y=3x-16\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = 5 \\ y=3.(5)-16=-1\end{matrix}\right.\\ \qquad V=\{(5,-1)\}\)
2. \(\left\{\begin{matrix}-2x+3y=-40\\x=-5y-45\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-2x+3y=-40\\x+5y=-45\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}-2x+3y=-40\\ x=-5y-45\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-2\left(-5y-45\right)+3y=-40\\x=-5y-45\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}10y+90+3y=-40\\x=-5y-45\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}13y=-40-90=-130\\x=-5y-45\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{-130}{13} = -10 \\ x=-5y-45\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = -10 \\ x=-5.(-10)-45=5\end{matrix}\right.\\ \qquad V=\{(5,-10)\}\)
3. \(\left\{\begin{matrix}6x+3y=39\\-6x=y-25\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}6x+3y=39\\-6x-y=-25\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}6x+3y=39\\ -6x+25=y\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}6x+3\left(-6x+25\right)=39\\y=-6x+25\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}6x-18x+75=39\\y=-6x+25\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-12x=39-75=-36\\y=-6x+25\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{-36}{-12} = 3 \\ y=-6x+25\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = 3 \\ y=-6.(3)+25=7\end{matrix}\right.\\ \qquad V=\{(3,7)\}\)
4. \(\left\{\begin{matrix}3y=-54+6x\\4x+y=30\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-6x+3y=-54\\4x+y=30\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}-6x+3y=-54\\ y=-4x+30\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-6x+3\left(-4x+30\right)=-54\\y=-4x+30\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-6x-12x+90=-54\\y=-4x+30\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-18x=-54-90=-144\\y=-4x+30\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{-144}{-18} = 8 \\ y=-4x+30\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = 8 \\ y=-4.(8)+30=-2\end{matrix}\right.\\ \qquad V=\{(8,-2)\}\)
5. \(\left\{\begin{matrix}6x-6y=6\\-2x+y=8\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}6x-6y=6\\ y=2x+8\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}6x-6\left(2x+8\right)=6\\y=2x+8\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}6x-12x-48=6\\y=2x+8\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-6x=6+48=54\\y=2x+8\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{54}{-6} = -9 \\ y=2x+8\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = -9 \\ y=2.(-9)+8=-10\end{matrix}\right.\\ \qquad V=\{(-9,-10)\}\)
6. \(\left\{\begin{matrix}-3x+2y=21\\4x=y-18\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-3x+2y=21\\4x-y=-18\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}-3x+2y=21\\ 4x+18=y\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-3x+2\left(4x+18\right)=21\\y=4x+18\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-3x+8x+36=21\\y=4x+18\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}5x=21-36=-15\\y=4x+18\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{-15}{5} = -3 \\ y=4x+18\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = -3 \\ y=4.(-3)+18=6\end{matrix}\right.\\ \qquad V=\{(-3,6)\}\)
7. \(\left\{\begin{matrix}3x+4y=-63\\x=2y+9\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}3x+4y=-63\\x-2y=9\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}3x+4y=-63\\ x=2y+9\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}3\left(2y+9\right)+4y=-63\\x=2y+9\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}6y+27+4y=-63\\x=2y+9\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}10y=-63-27=-90\\x=2y+9\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{-90}{10} = -9 \\ x=2y+9\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = -9 \\ x=2.(-9)+9=-9\end{matrix}\right.\\ \qquad V=\{(-9,-9)\}\)
8. \(\left\{\begin{matrix}-2y=35+5x\\x-4y=37\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-5x-2y=35\\x-4y=37\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}-5x-2y=35\\ x=4y+37\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-5\left(4y+37\right)-2y=35\\x=4y+37\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-20y-185-2y=35\\x=4y+37\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-22y=35+185=220\\x=4y+37\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{220}{-22} = -10 \\ x=4y+37\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = -10 \\ x=4.(-10)+37=-3\end{matrix}\right.\\ \qquad V=\{(-3,-10)\}\)
9. \(\left\{\begin{matrix}5x+5y=-85\\2x=y-4\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}5x+5y=-85\\2x-y=-4\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}5x+5y=-85\\ 2x+4=y\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}5x+5\left(2x+4\right)=-85\\y=2x+4\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}5x+10x+20=-85\\y=2x+4\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}15x=-85-20=-105\\y=2x+4\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{-105}{15} = -7 \\ y=2x+4\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = -7 \\ y=2.(-7)+4=-10\end{matrix}\right.\\ \qquad V=\{(-7,-10)\}\)
10. \(\left\{\begin{matrix}3y=-41+5x\\2x+y=12\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-5x+3y=-41\\2x+y=12\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}-5x+3y=-41\\ y=-2x+12\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-5x+3\left(-2x+12\right)=-41\\y=-2x+12\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-5x-6x+36=-41\\y=-2x+12\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-11x=-41-36=-77\\y=-2x+12\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{-77}{-11} = 7 \\ y=-2x+12\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = 7 \\ y=-2.(7)+12=-2\end{matrix}\right.\\ \qquad V=\{(7,-2)\}\)
11. \(\left\{\begin{matrix}6x-y=-31\\-5x=4y+50\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}6x-y=-31\\-5x-4y=50\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}6x+31=y\\-5x-4y=50\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}y=6x+31\\ -5x-4\left(6x+31\right)=50\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}y=6x+31\\ -5x-24x-124=50\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}y=6x+31\\ -29x=50+124=174\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=6x+31\\ x=\frac{174}{-29}=-6\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=6.(-6)+31=-5\\ x=-6\end{matrix}\right.\\ \qquad V=\{(-6,-5)\}\)
12. \(\left\{\begin{matrix}3x+3y=-9\\-x=-6y-46\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}3x+3y=-9\\-x+6y=-46\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}3x+3y=-9\\ 6y+46=x\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}3\left(6y+46\right)+3y=-9\\x=6y+46\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}18y+138+3y=-9\\x=6y+46\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}21y=-9-138=-147\\x=6y+46\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{-147}{21} = -7 \\ x=6y+46\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = -7 \\ x=6.(-7)+46=4\end{matrix}\right.\\ \qquad V=\{(4,-7)\}\)