Stelsels substitutie

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Substitutie

  1. \(\left\{\begin{matrix}-4x+2y=12\\-4x=y-18\end{matrix}\right.\)
  2. \(\left\{\begin{matrix}y=-13+x\\-5x+3y=-49\end{matrix}\right.\)
  3. \(\left\{\begin{matrix}-3y=31+5x\\4x-y=-35\end{matrix}\right.\)
  4. \(\left\{\begin{matrix}-x+5y=48\\5x=4y-51\end{matrix}\right.\)
  5. \(\left\{\begin{matrix}4x-3y=11\\2x-y=5\end{matrix}\right.\)
  6. \(\left\{\begin{matrix}-2y=16-4x\\-6x+y=-40\end{matrix}\right.\)
  7. \(\left\{\begin{matrix}-2x+4y=0\\5x=-y+33\end{matrix}\right.\)
  8. \(\left\{\begin{matrix}-3x-3y=15\\x+2y=-12\end{matrix}\right.\)
  9. \(\left\{\begin{matrix}-x+3y=-4\\5x+5y=-60\end{matrix}\right.\)
  10. \(\left\{\begin{matrix}6x+3y=75\\-x+2y=0\end{matrix}\right.\)
  11. \(\left\{\begin{matrix}-y=-21-5x\\-6x+3y=18\end{matrix}\right.\)
  12. \(\left\{\begin{matrix}6x-5y=21\\-2x-y=1\end{matrix}\right.\)

Substitutie

Verbetersleutel

  1. \(\left\{\begin{matrix}-4x+2y=12\\-4x=y-18\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-4x+2y=12\\-4x-y=-18\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}-4x+2y=12\\ -4x+18=y\end{matrix}\right.\text{(Afzonderen onbekende met coƫff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-4x+2\left(-4x+18\right)=12\\y=-4x+18\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-4x-8x+36=12\\y=-4x+18\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-12x=12-36=-24\\y=-4x+18\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{-24}{-12} = 2 \\ y=-4x+18\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = 2 \\ y=-4.(2)+18=10\end{matrix}\right.\\ \qquad V=\{(2,10)\}\)
  2. \(\left\{\begin{matrix}y=-13+x\\-5x+3y=-49\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-x+y=-13\\-5x+3y=-49\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}y+13=x\\-5x+3y=-49\end{matrix}\right.\text{(Afzonderen onbekende met coƫff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}x=y+13\\ -5.\left(y+13\right)+3y=-49\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}x=y+13\\ -5y-65+3y=-49\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}x=y+13\\ -2y=-49+65=16\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=y+13\\ y=\frac{16}{-2}=-8\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=.(-8)+13=5\\ y=-8\end{matrix}\right.\\ \qquad V=\{(5,-8)\}\)
  3. \(\left\{\begin{matrix}-3y=31+5x\\4x-y=-35\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-5x-3y=31\\4x-y=-35\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}-5x-3y=31\\ 4x+35=y\end{matrix}\right.\text{(Afzonderen onbekende met coƫff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-5x-3\left(4x+35\right)=31\\y=4x+35\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-5x-12x-105=31\\y=4x+35\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-17x=31+105=136\\y=4x+35\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{136}{-17} = -8 \\ y=4x+35\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = -8 \\ y=4.(-8)+35=3\end{matrix}\right.\\ \qquad V=\{(-8,3)\}\)
  4. \(\left\{\begin{matrix}-x+5y=48\\5x=4y-51\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-x+5y=48\\5x-4y=-51\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}5y-48=x\\5x-4y=-51\end{matrix}\right.\text{(Afzonderen onbekende met coƫff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}x=5y-48\\ 5.\left(5y-48\right)-4y=-51\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}x=5y-48\\ 25y-240-4y=-51\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}x=5y-48\\ 21y=-51+240=189\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=5y-48\\ y=\frac{189}{21}=9\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=5.(9)-48=-3\\ y=9\end{matrix}\right.\\ \qquad V=\{(-3,9)\}\)
  5. \(\left\{\begin{matrix}4x-3y=11\\2x-y=5\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}4x-3y=11\\ 2x-5=y\end{matrix}\right.\text{(Afzonderen onbekende met coƫff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}4x-3\left(2x-5\right)=11\\y=2x-5\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}4x-6x+15=11\\y=2x-5\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-2x=11-15=-4\\y=2x-5\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{-4}{-2} = 2 \\ y=2x-5\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = 2 \\ y=2.(2)-5=-1\end{matrix}\right.\\ \qquad V=\{(2,-1)\}\)
  6. \(\left\{\begin{matrix}-2y=16-4x\\-6x+y=-40\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}4x-2y=16\\-6x+y=-40\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}4x-2y=16\\ y=6x-40\end{matrix}\right.\text{(Afzonderen onbekende met coƫff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}4x-2\left(6x-40\right)=16\\y=6x-40\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}4x-12x+80=16\\y=6x-40\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-8x=16-80=-64\\y=6x-40\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{-64}{-8} = 8 \\ y=6x-40\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = 8 \\ y=6.(8)-40=8\end{matrix}\right.\\ \qquad V=\{(8,8)\}\)
  7. \(\left\{\begin{matrix}-2x+4y=0\\5x=-y+33\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-2x+4y=0\\5x+y=33\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}-2x+4y=0\\ y=-5x+33\end{matrix}\right.\text{(Afzonderen onbekende met coƫff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-2x+4\left(-5x+33\right)=0\\y=-5x+33\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-2x-20x+132=0\\y=-5x+33\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-22x=0-132=-132\\y=-5x+33\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{-132}{-22} = 6 \\ y=-5x+33\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = 6 \\ y=-5.(6)+33=3\end{matrix}\right.\\ \qquad V=\{(6,3)\}\)
  8. \(\left\{\begin{matrix}-3x-3y=15\\x+2y=-12\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-3x-3y=15\\ x=-2y-12\end{matrix}\right.\text{(Afzonderen onbekende met coƫff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-3\left(-2y-12\right)-3y=15\\x=-2y-12\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}6y+36-3y=15\\x=-2y-12\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}3y=15-36=-21\\x=-2y-12\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{-21}{3} = -7 \\ x=-2y-12\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = -7 \\ x=-2.(-7)-12=2\end{matrix}\right.\\ \qquad V=\{(2,-7)\}\)
  9. \(\left\{\begin{matrix}-x+3y=-4\\5x+5y=-60\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}3y+4=x\\5x+5y=-60\end{matrix}\right.\text{(Afzonderen onbekende met coƫff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}x=3y+4\\ 5.\left(3y+4\right)+5y=-60\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}x=3y+4\\ 15y+20+5y=-60\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}x=3y+4\\ 20y=-60-20=-80\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=3y+4\\ y=\frac{-80}{20}=-4\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=3.(-4)+4=-8\\ y=-4\end{matrix}\right.\\ \qquad V=\{(-8,-4)\}\)
  10. \(\left\{\begin{matrix}6x+3y=75\\-x+2y=0\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}6x+3y=75\\ 2y+0=x\end{matrix}\right.\text{(Afzonderen onbekende met coƫff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}6\left(2y+0\right)+3y=75\\x=2y+0\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}12y+0+3y=75\\x=2y+0\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}15y=75+0=75\\x=2y+0\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{75}{15} = 5 \\ x=2y+0\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = 5 \\ x=2.(5)+0=10\end{matrix}\right.\\ \qquad V=\{(10,5)\}\)
  11. \(\left\{\begin{matrix}-y=-21-5x\\-6x+3y=18\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}5x-y=-21\\-6x+3y=18\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}5x+21=y\\-6x+3y=18\end{matrix}\right.\text{(Afzonderen onbekende met coƫff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}y=5x+21\\ -6x+3\left(5x+21\right)=18\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}y=5x+21\\ -6x+15x+63=18\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}y=5x+21\\ 9x=18-63=-45\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=5x+21\\ x=\frac{-45}{9}=-5\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=5.(-5)+21=-4\\ x=-5\end{matrix}\right.\\ \qquad V=\{(-5,-4)\}\)
  12. \(\left\{\begin{matrix}6x-5y=21\\-2x-y=1\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}6x-5y=21\\ -2x-1=y\end{matrix}\right.\text{(Afzonderen onbekende met coƫff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}6x-5\left(-2x-1\right)=21\\y=-2x-1\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}6x+10x+5=21\\y=-2x-1\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}16x=21-5=16\\y=-2x-1\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{16}{16} = 1 \\ y=-2x-1\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = 1 \\ y=-2.(1)-1=-3\end{matrix}\right.\\ \qquad V=\{(1,-3)\}\)
Oefeningengenerator vanhoeckes.be/wiskunde 2025-04-03 16:48:26