Stelsels substitutie

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Substitutie

1. \(\left\{\begin{matrix}-6x-y=-67\\6x=6y+18\end{matrix}\right.\)
2. \(\left\{\begin{matrix}-4x+3y=-54\\-4x-y=-14\end{matrix}\right.\)
3. \(\left\{\begin{matrix}5y=0+4x\\-x-6y=58\end{matrix}\right.\)
4. \(\left\{\begin{matrix}-4y=8-3x\\-5x+y=32\end{matrix}\right.\)
5. \(\left\{\begin{matrix}6x+3y=-6\\2x=-y-2\end{matrix}\right.\)
6. \(\left\{\begin{matrix}-6x-3y=48\\-x=2y+5\end{matrix}\right.\)
7. \(\left\{\begin{matrix}-3x-6y=-6\\-3x-y=9\end{matrix}\right.\)
8. \(\left\{\begin{matrix}-6x-5y=-53\\-x-2y=-10\end{matrix}\right.\)
9. \(\left\{\begin{matrix}-x+2y=7\\2x=-4y+18\end{matrix}\right.\)
10. \(\left\{\begin{matrix}-4x-3y=33\\-6x-y=53\end{matrix}\right.\)
11. \(\left\{\begin{matrix}-3x+3y=0\\-4x=y-15\end{matrix}\right.\)
12. \(\left\{\begin{matrix}-6y=40-2x\\-3x-y=-20\end{matrix}\right.\)

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Substitutie

Verbetersleutel

1. \(\left\{\begin{matrix}-6x-y=-67\\6x=6y+18\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-6x-y=-67\\6x-6y=18\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}-6x+67=y\\6x-6y=18\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}y=-6x+67\\ 6x-6\left(-6x+67\right)=18\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}y=-6x+67\\ 6x+36x-402=18\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}y=-6x+67\\ 42x=18+402=420\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=-6x+67\\ x=\frac{420}{42}=10\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=-6.(10)+67=7\\ x=10\end{matrix}\right.\\ \qquad V=\{(10,7)\}\)
2. \(\left\{\begin{matrix}-4x+3y=-54\\-4x-y=-14\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-4x+3y=-54\\ -4x+14=y\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-4x+3\left(-4x+14\right)=-54\\y=-4x+14\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-4x-12x+42=-54\\y=-4x+14\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-16x=-54-42=-96\\y=-4x+14\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{-96}{-16} = 6 \\ y=-4x+14\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = 6 \\ y=-4.(6)+14=-10\end{matrix}\right.\\ \qquad V=\{(6,-10)\}\)
3. \(\left\{\begin{matrix}5y=0+4x\\-x-6y=58\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-4x+5y=0\\-x-6y=58\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}-4x+5y=0\\ -6y-58=x\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-4\left(-6y-58\right)+5y=0\\x=-6y-58\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}24y+232+5y=0\\x=-6y-58\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}29y=0-232=-232\\x=-6y-58\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{-232}{29} = -8 \\ x=-6y-58\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = -8 \\ x=-6.(-8)-58=-10\end{matrix}\right.\\ \qquad V=\{(-10,-8)\}\)
4. \(\left\{\begin{matrix}-4y=8-3x\\-5x+y=32\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}3x-4y=8\\-5x+y=32\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}3x-4y=8\\ y=5x+32\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}3x-4\left(5x+32\right)=8\\y=5x+32\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}3x-20x-128=8\\y=5x+32\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-17x=8+128=136\\y=5x+32\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{136}{-17} = -8 \\ y=5x+32\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = -8 \\ y=5.(-8)+32=-8\end{matrix}\right.\\ \qquad V=\{(-8,-8)\}\)
5. \(\left\{\begin{matrix}6x+3y=-6\\2x=-y-2\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}6x+3y=-6\\2x+y=-2\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}6x+3y=-6\\ y=-2x-2\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}6x+3\left(-2x-2\right)=-6\\y=-2x-2\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}6x-6x-6=-6\\y=-2x-2\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}0x=-6+6=0\\y=-2x-2\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{0}{0} = 2 \\ y=-2x-2\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = 2 \\ y=-2.(2)-2=-6\end{matrix}\right.\\ \qquad V=\{(2,-6)\}\)
6. \(\left\{\begin{matrix}-6x-3y=48\\-x=2y+5\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-6x-3y=48\\-x-2y=5\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}-6x-3y=48\\ -2y-5=x\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-6\left(-2y-5\right)-3y=48\\x=-2y-5\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}12y+30-3y=48\\x=-2y-5\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}9y=48-30=18\\x=-2y-5\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{18}{9} = 2 \\ x=-2y-5\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = 2 \\ x=-2.(2)-5=-9\end{matrix}\right.\\ \qquad V=\{(-9,2)\}\)
7. \(\left\{\begin{matrix}-3x-6y=-6\\-3x-y=9\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-3x-6y=-6\\ -3x-9=y\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-3x-6\left(-3x-9\right)=-6\\y=-3x-9\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-3x+18x+54=-6\\y=-3x-9\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}15x=-6-54=-60\\y=-3x-9\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{-60}{15} = -4 \\ y=-3x-9\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = -4 \\ y=-3.(-4)-9=3\end{matrix}\right.\\ \qquad V=\{(-4,3)\}\)
8. \(\left\{\begin{matrix}-6x-5y=-53\\-x-2y=-10\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-6x-5y=-53\\ -2y+10=x\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-6\left(-2y+10\right)-5y=-53\\x=-2y+10\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}12y-60-5y=-53\\x=-2y+10\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}7y=-53+60=7\\x=-2y+10\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{7}{7} = 1 \\ x=-2y+10\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = 1 \\ x=-2.(1)+10=8\end{matrix}\right.\\ \qquad V=\{(8,1)\}\)
9. \(\left\{\begin{matrix}-x+2y=7\\2x=-4y+18\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-x+2y=7\\2x+4y=18\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}2y-7=x\\2x+4y=18\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}x=2y-7\\ 2.\left(2y-7\right)+4y=18\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}x=2y-7\\ 4y-14+4y=18\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}x=2y-7\\ 8y=18+14=32\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=2y-7\\ y=\frac{32}{8}=4\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=2.(4)-7=1\\ y=4\end{matrix}\right.\\ \qquad V=\{(1,4)\}\)
10. \(\left\{\begin{matrix}-4x-3y=33\\-6x-y=53\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-4x-3y=33\\ -6x-53=y\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-4x-3\left(-6x-53\right)=33\\y=-6x-53\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-4x+18x+159=33\\y=-6x-53\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}14x=33-159=-126\\y=-6x-53\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{-126}{14} = -9 \\ y=-6x-53\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = -9 \\ y=-6.(-9)-53=1\end{matrix}\right.\\ \qquad V=\{(-9,1)\}\)
11. \(\left\{\begin{matrix}-3x+3y=0\\-4x=y-15\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-3x+3y=0\\-4x-y=-15\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}-3x+3y=0\\ -4x+15=y\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-3x+3\left(-4x+15\right)=0\\y=-4x+15\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-3x-12x+45=0\\y=-4x+15\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-15x=0-45=-45\\y=-4x+15\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{-45}{-15} = 3 \\ y=-4x+15\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = 3 \\ y=-4.(3)+15=3\end{matrix}\right.\\ \qquad V=\{(3,3)\}\)
12. \(\left\{\begin{matrix}-6y=40-2x\\-3x-y=-20\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}2x-6y=40\\-3x-y=-20\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}2x-6y=40\\ -3x+20=y\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}2x-6\left(-3x+20\right)=40\\y=-3x+20\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}2x+18x-120=40\\y=-3x+20\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}20x=40+120=160\\y=-3x+20\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{160}{20} = 8 \\ y=-3x+20\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = 8 \\ y=-3.(8)+20=-4\end{matrix}\right.\\ \qquad V=\{(8,-4)\}\)