Stelsels substitutie

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Substitutie

1. \(\left\{\begin{matrix}5y=41+6x\\4x-y=-11\end{matrix}\right.\)
2. \(\left\{\begin{matrix}-6y=18+2x\\3x-y=-7\end{matrix}\right.\)
3. \(\left\{\begin{matrix}2x+4y=-4\\2x-y=16\end{matrix}\right.\)
4. \(\left\{\begin{matrix}-5x-2y=64\\x=4y+18\end{matrix}\right.\)
5. \(\left\{\begin{matrix}y=-11+5x\\-4x+3y=-11\end{matrix}\right.\)
6. \(\left\{\begin{matrix}-6y=18-3x\\2x-y=0\end{matrix}\right.\)
7. \(\left\{\begin{matrix}-3y=-2-2x\\-4x-y=32\end{matrix}\right.\)
8. \(\left\{\begin{matrix}y=1+2x\\4x+3y=-47\end{matrix}\right.\)
9. \(\left\{\begin{matrix}-3x-6y=0\\-x=-2y+20\end{matrix}\right.\)
10. \(\left\{\begin{matrix}-6y=-90-6x\\x+5y=15\end{matrix}\right.\)
11. \(\left\{\begin{matrix}3y=51+6x\\x+4y=23\end{matrix}\right.\)
12. \(\left\{\begin{matrix}3x+6y=-33\\x-2y=29\end{matrix}\right.\)

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Substitutie

Verbetersleutel

1. \(\left\{\begin{matrix}5y=41+6x\\4x-y=-11\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-6x+5y=41\\4x-y=-11\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}-6x+5y=41\\ 4x+11=y\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-6x+5\left(4x+11\right)=41\\y=4x+11\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-6x+20x+55=41\\y=4x+11\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}14x=41-55=-14\\y=4x+11\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{-14}{14} = -1 \\ y=4x+11\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = -1 \\ y=4.(-1)+11=7\end{matrix}\right.\\ \qquad V=\{(-1,7)\}\)
2. \(\left\{\begin{matrix}-6y=18+2x\\3x-y=-7\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-2x-6y=18\\3x-y=-7\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}-2x-6y=18\\ 3x+7=y\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-2x-6\left(3x+7\right)=18\\y=3x+7\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-2x-18x-42=18\\y=3x+7\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-20x=18+42=60\\y=3x+7\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{60}{-20} = -3 \\ y=3x+7\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = -3 \\ y=3.(-3)+7=-2\end{matrix}\right.\\ \qquad V=\{(-3,-2)\}\)
3. \(\left\{\begin{matrix}2x+4y=-4\\2x-y=16\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}2x+4y=-4\\ 2x-16=y\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}2x+4\left(2x-16\right)=-4\\y=2x-16\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}2x+8x-64=-4\\y=2x-16\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}10x=-4+64=60\\y=2x-16\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{60}{10} = 6 \\ y=2x-16\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = 6 \\ y=2.(6)-16=-4\end{matrix}\right.\\ \qquad V=\{(6,-4)\}\)
4. \(\left\{\begin{matrix}-5x-2y=64\\x=4y+18\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-5x-2y=64\\x-4y=18\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}-5x-2y=64\\ x=4y+18\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-5\left(4y+18\right)-2y=64\\x=4y+18\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-20y-90-2y=64\\x=4y+18\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-22y=64+90=154\\x=4y+18\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{154}{-22} = -7 \\ x=4y+18\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = -7 \\ x=4.(-7)+18=-10\end{matrix}\right.\\ \qquad V=\{(-10,-7)\}\)
5. \(\left\{\begin{matrix}y=-11+5x\\-4x+3y=-11\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-5x+y=-11\\-4x+3y=-11\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}y=5x-11\\ -4x+3y=-11\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}y=5x-11\\ -4x+3\left(5x-11\right)=-11\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}y=5x-11\\ -4x+15x-33=-11\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}y=5x-11\\ 11x=-11+33=22\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=5x-11\\ x=\frac{22}{11}=2\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=5.(2)-11=-1\\ x=2\end{matrix}\right.\\ \qquad V=\{(2,-1)\}\)
6. \(\left\{\begin{matrix}-6y=18-3x\\2x-y=0\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}3x-6y=18\\2x-y=0\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}3x-6y=18\\ 2x+0=y\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}3x-6\left(2x+0\right)=18\\y=2x+0\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}3x-12x+0=18\\y=2x+0\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-9x=18+0=18\\y=2x+0\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{18}{-9} = -2 \\ y=2x+0\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = -2 \\ y=2.(-2)+0=-4\end{matrix}\right.\\ \qquad V=\{(-2,-4)\}\)
7. \(\left\{\begin{matrix}-3y=-2-2x\\-4x-y=32\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}2x-3y=-2\\-4x-y=32\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}2x-3y=-2\\ -4x-32=y\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}2x-3\left(-4x-32\right)=-2\\y=-4x-32\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}2x+12x+96=-2\\y=-4x-32\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}14x=-2-96=-98\\y=-4x-32\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{-98}{14} = -7 \\ y=-4x-32\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = -7 \\ y=-4.(-7)-32=-4\end{matrix}\right.\\ \qquad V=\{(-7,-4)\}\)
8. \(\left\{\begin{matrix}y=1+2x\\4x+3y=-47\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-2x+y=1\\4x+3y=-47\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}y=2x+1\\ 4x+3y=-47\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}y=2x+1\\ 4x+3\left(2x+1\right)=-47\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}y=2x+1\\ 4x+6x+3=-47\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}y=2x+1\\ 10x=-47-3=-50\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=2x+1\\ x=\frac{-50}{10}=-5\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=2.(-5)+1=-9\\ x=-5\end{matrix}\right.\\ \qquad V=\{(-5,-9)\}\)
9. \(\left\{\begin{matrix}-3x-6y=0\\-x=-2y+20\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-3x-6y=0\\-x+2y=20\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}-3x-6y=0\\ 2y-20=x\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-3\left(2y-20\right)-6y=0\\x=2y-20\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-6y+60-6y=0\\x=2y-20\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-12y=0-60=-60\\x=2y-20\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{-60}{-12} = 5 \\ x=2y-20\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = 5 \\ x=2.(5)-20=-10\end{matrix}\right.\\ \qquad V=\{(-10,5)\}\)
10. \(\left\{\begin{matrix}-6y=-90-6x\\x+5y=15\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}6x-6y=-90\\x+5y=15\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}6x-6y=-90\\ x=-5y+15\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}6\left(-5y+15\right)-6y=-90\\x=-5y+15\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-30y+90-6y=-90\\x=-5y+15\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-36y=-90-90=-180\\x=-5y+15\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{-180}{-36} = 5 \\ x=-5y+15\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = 5 \\ x=-5.(5)+15=-10\end{matrix}\right.\\ \qquad V=\{(-10,5)\}\)
11. \(\left\{\begin{matrix}3y=51+6x\\x+4y=23\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-6x+3y=51\\x+4y=23\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}-6x+3y=51\\ x=-4y+23\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-6\left(-4y+23\right)+3y=51\\x=-4y+23\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}24y-138+3y=51\\x=-4y+23\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}27y=51+138=189\\x=-4y+23\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{189}{27} = 7 \\ x=-4y+23\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = 7 \\ x=-4.(7)+23=-5\end{matrix}\right.\\ \qquad V=\{(-5,7)\}\)
12. \(\left\{\begin{matrix}3x+6y=-33\\x-2y=29\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}3x+6y=-33\\ x=2y+29\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}3\left(2y+29\right)+6y=-33\\x=2y+29\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}6y+87+6y=-33\\x=2y+29\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}12y=-33-87=-120\\x=2y+29\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{-120}{12} = -10 \\ x=2y+29\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = -10 \\ x=2.(-10)+29=9\end{matrix}\right.\\ \qquad V=\{(9,-10)\}\)