Stelsels substitutie

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Substitutie

1. \(\left\{\begin{matrix}2x+4y=8\\x=3y-21\end{matrix}\right.\)
2. \(\left\{\begin{matrix}-6y=-30+6x\\-x-2y=-13\end{matrix}\right.\)
3. \(\left\{\begin{matrix}4y=30-6x\\-2x+y=-24\end{matrix}\right.\)
4. \(\left\{\begin{matrix}y=-11+4x\\-3x-2y=11\end{matrix}\right.\)
5. \(\left\{\begin{matrix}-2y=9+x\\-3x+4y=-53\end{matrix}\right.\)
6. \(\left\{\begin{matrix}5x-3y=-10\\-4x=-y+1\end{matrix}\right.\)
7. \(\left\{\begin{matrix}-6y=-14+4x\\3x+y=14\end{matrix}\right.\)
8. \(\left\{\begin{matrix}-4y=44+6x\\-2x+y=10\end{matrix}\right.\)
9. \(\left\{\begin{matrix}-6x+5y=-44\\3x+y=-13\end{matrix}\right.\)
10. \(\left\{\begin{matrix}-2x-4y=-20\\3x-y=16\end{matrix}\right.\)
11. \(\left\{\begin{matrix}-x+4y=46\\3x=-3y+12\end{matrix}\right.\)
12. \(\left\{\begin{matrix}-4x-4y=-24\\6x+y=26\end{matrix}\right.\)

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Substitutie

Verbetersleutel

1. \(\left\{\begin{matrix}2x+4y=8\\x=3y-21\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}2x+4y=8\\x-3y=-21\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}2x+4y=8\\ x=3y-21\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}2\left(3y-21\right)+4y=8\\x=3y-21\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}6y-42+4y=8\\x=3y-21\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}10y=8+42=50\\x=3y-21\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{50}{10} = 5 \\ x=3y-21\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = 5 \\ x=3.(5)-21=-6\end{matrix}\right.\\ \qquad V=\{(-6,5)\}\)
2. \(\left\{\begin{matrix}-6y=-30+6x\\-x-2y=-13\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-6x-6y=-30\\-x-2y=-13\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}-6x-6y=-30\\ -2y+13=x\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-6\left(-2y+13\right)-6y=-30\\x=-2y+13\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}12y-78-6y=-30\\x=-2y+13\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}6y=-30+78=48\\x=-2y+13\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{48}{6} = 8 \\ x=-2y+13\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = 8 \\ x=-2.(8)+13=-3\end{matrix}\right.\\ \qquad V=\{(-3,8)\}\)
3. \(\left\{\begin{matrix}4y=30-6x\\-2x+y=-24\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}6x+4y=30\\-2x+y=-24\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}6x+4y=30\\ y=2x-24\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}6x+4\left(2x-24\right)=30\\y=2x-24\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}6x+8x-96=30\\y=2x-24\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}14x=30+96=126\\y=2x-24\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{126}{14} = 9 \\ y=2x-24\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = 9 \\ y=2.(9)-24=-6\end{matrix}\right.\\ \qquad V=\{(9,-6)\}\)
4. \(\left\{\begin{matrix}y=-11+4x\\-3x-2y=11\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-4x+y=-11\\-3x-2y=11\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}y=4x-11\\ -3x-2y=11\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}y=4x-11\\ -3x-2\left(4x-11\right)=11\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}y=4x-11\\ -3x-8x+22=11\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}y=4x-11\\ -11x=11-22=-11\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=4x-11\\ x=\frac{-11}{-11}=1\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=4.(1)-11=-7\\ x=1\end{matrix}\right.\\ \qquad V=\{(1,-7)\}\)
5. \(\left\{\begin{matrix}-2y=9+x\\-3x+4y=-53\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-x-2y=9\\-3x+4y=-53\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}-2y-9=x\\-3x+4y=-53\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}x=-2y-9\\ -3.\left(-2y-9\right)+4y=-53\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}x=-2y-9\\ 6y+27+4y=-53\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}x=-2y-9\\ 10y=-53-27=-80\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=-2y-9\\ y=\frac{-80}{10}=-8\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=-2.(-8)-9=7\\ y=-8\end{matrix}\right.\\ \qquad V=\{(7,-8)\}\)
6. \(\left\{\begin{matrix}5x-3y=-10\\-4x=-y+1\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}5x-3y=-10\\-4x+y=1\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}5x-3y=-10\\ y=4x+1\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}5x-3\left(4x+1\right)=-10\\y=4x+1\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}5x-12x-3=-10\\y=4x+1\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-7x=-10+3=-7\\y=4x+1\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{-7}{-7} = 1 \\ y=4x+1\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = 1 \\ y=4.(1)+1=5\end{matrix}\right.\\ \qquad V=\{(1,5)\}\)
7. \(\left\{\begin{matrix}-6y=-14+4x\\3x+y=14\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-4x-6y=-14\\3x+y=14\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}-4x-6y=-14\\ y=-3x+14\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-4x-6\left(-3x+14\right)=-14\\y=-3x+14\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-4x+18x-84=-14\\y=-3x+14\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}14x=-14+84=70\\y=-3x+14\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{70}{14} = 5 \\ y=-3x+14\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = 5 \\ y=-3.(5)+14=-1\end{matrix}\right.\\ \qquad V=\{(5,-1)\}\)
8. \(\left\{\begin{matrix}-4y=44+6x\\-2x+y=10\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-6x-4y=44\\-2x+y=10\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}-6x-4y=44\\ y=2x+10\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-6x-4\left(2x+10\right)=44\\y=2x+10\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-6x-8x-40=44\\y=2x+10\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-14x=44+40=84\\y=2x+10\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{84}{-14} = -6 \\ y=2x+10\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = -6 \\ y=2.(-6)+10=-2\end{matrix}\right.\\ \qquad V=\{(-6,-2)\}\)
9. \(\left\{\begin{matrix}-6x+5y=-44\\3x+y=-13\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-6x+5y=-44\\ y=-3x-13\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-6x+5\left(-3x-13\right)=-44\\y=-3x-13\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-6x-15x-65=-44\\y=-3x-13\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-21x=-44+65=21\\y=-3x-13\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{21}{-21} = -1 \\ y=-3x-13\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = -1 \\ y=-3.(-1)-13=-10\end{matrix}\right.\\ \qquad V=\{(-1,-10)\}\)
10. \(\left\{\begin{matrix}-2x-4y=-20\\3x-y=16\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-2x-4y=-20\\ 3x-16=y\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-2x-4\left(3x-16\right)=-20\\y=3x-16\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-2x-12x+64=-20\\y=3x-16\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-14x=-20-64=-84\\y=3x-16\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{-84}{-14} = 6 \\ y=3x-16\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = 6 \\ y=3.(6)-16=2\end{matrix}\right.\\ \qquad V=\{(6,2)\}\)
11. \(\left\{\begin{matrix}-x+4y=46\\3x=-3y+12\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-x+4y=46\\3x+3y=12\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}4y-46=x\\3x+3y=12\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}x=4y-46\\ 3.\left(4y-46\right)+3y=12\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}x=4y-46\\ 12y-138+3y=12\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}x=4y-46\\ 15y=12+138=150\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=4y-46\\ y=\frac{150}{15}=10\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=4.(10)-46=-6\\ y=10\end{matrix}\right.\\ \qquad V=\{(-6,10)\}\)
12. \(\left\{\begin{matrix}-4x-4y=-24\\6x+y=26\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-4x-4y=-24\\ y=-6x+26\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-4x-4\left(-6x+26\right)=-24\\y=-6x+26\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-4x+24x-104=-24\\y=-6x+26\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}20x=-24+104=80\\y=-6x+26\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{80}{20} = 4 \\ y=-6x+26\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = 4 \\ y=-6.(4)+26=2\end{matrix}\right.\\ \qquad V=\{(4,2)\}\)