Stelsels substitutie

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Substitutie

  1. \(\left\{\begin{matrix}-5x-6y=36\\5x-y=-29\end{matrix}\right.\)
  2. \(\left\{\begin{matrix}5x-5y=30\\-x=-4y-18\end{matrix}\right.\)
  3. \(\left\{\begin{matrix}4y=-8+4x\\6x+y=-37\end{matrix}\right.\)
  4. \(\left\{\begin{matrix}2x+4y=0\\x=3y-10\end{matrix}\right.\)
  5. \(\left\{\begin{matrix}y=-9-3x\\6x-6y=6\end{matrix}\right.\)
  6. \(\left\{\begin{matrix}2x+2y=6\\x=3y-37\end{matrix}\right.\)
  7. \(\left\{\begin{matrix}x+6y=-54\\2x=-3y-36\end{matrix}\right.\)
  8. \(\left\{\begin{matrix}-6y=51-5x\\-x-3y=-6\end{matrix}\right.\)
  9. \(\left\{\begin{matrix}-2x+y=21\\-2x+5y=41\end{matrix}\right.\)
  10. \(\left\{\begin{matrix}-5x-3y=31\\2x=-y-13\end{matrix}\right.\)
  11. \(\left\{\begin{matrix}-6x-2y=-10\\-x-6y=21\end{matrix}\right.\)
  12. \(\left\{\begin{matrix}4x-5y=61\\3x+y=22\end{matrix}\right.\)

Substitutie

Verbetersleutel

  1. \(\left\{\begin{matrix}-5x-6y=36\\5x-y=-29\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-5x-6y=36\\ 5x+29=y\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-5x-6\left(5x+29\right)=36\\y=5x+29\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-5x-30x-174=36\\y=5x+29\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-35x=36+174=210\\y=5x+29\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{210}{-35} = -6 \\ y=5x+29\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = -6 \\ y=5.(-6)+29=-1\end{matrix}\right.\\ \qquad V=\{(-6,-1)\}\)
  2. \(\left\{\begin{matrix}5x-5y=30\\-x=-4y-18\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}5x-5y=30\\-x+4y=-18\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}5x-5y=30\\ 4y+18=x\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}5\left(4y+18\right)-5y=30\\x=4y+18\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}20y+90-5y=30\\x=4y+18\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}15y=30-90=-60\\x=4y+18\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{-60}{15} = -4 \\ x=4y+18\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = -4 \\ x=4.(-4)+18=2\end{matrix}\right.\\ \qquad V=\{(2,-4)\}\)
  3. \(\left\{\begin{matrix}4y=-8+4x\\6x+y=-37\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-4x+4y=-8\\6x+y=-37\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}-4x+4y=-8\\ y=-6x-37\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-4x+4\left(-6x-37\right)=-8\\y=-6x-37\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-4x-24x-148=-8\\y=-6x-37\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-28x=-8+148=140\\y=-6x-37\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{140}{-28} = -5 \\ y=-6x-37\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = -5 \\ y=-6.(-5)-37=-7\end{matrix}\right.\\ \qquad V=\{(-5,-7)\}\)
  4. \(\left\{\begin{matrix}2x+4y=0\\x=3y-10\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}2x+4y=0\\x-3y=-10\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}2x+4y=0\\ x=3y-10\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}2\left(3y-10\right)+4y=0\\x=3y-10\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}6y-20+4y=0\\x=3y-10\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}10y=0+20=20\\x=3y-10\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{20}{10} = 2 \\ x=3y-10\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = 2 \\ x=3.(2)-10=-4\end{matrix}\right.\\ \qquad V=\{(-4,2)\}\)
  5. \(\left\{\begin{matrix}y=-9-3x\\6x-6y=6\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}3x+y=-9\\6x-6y=6\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}y=-3x-9\\ 6x-6y=6\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}y=-3x-9\\ 6x-6\left(-3x-9\right)=6\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}y=-3x-9\\ 6x+18x+54=6\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}y=-3x-9\\ 24x=6-54=-48\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=-3x-9\\ x=\frac{-48}{24}=-2\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=-3.(-2)-9=-3\\ x=-2\end{matrix}\right.\\ \qquad V=\{(-2,-3)\}\)
  6. \(\left\{\begin{matrix}2x+2y=6\\x=3y-37\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}2x+2y=6\\x-3y=-37\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}2x+2y=6\\ x=3y-37\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}2\left(3y-37\right)+2y=6\\x=3y-37\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}6y-74+2y=6\\x=3y-37\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}8y=6+74=80\\x=3y-37\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{80}{8} = 10 \\ x=3y-37\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = 10 \\ x=3.(10)-37=-7\end{matrix}\right.\\ \qquad V=\{(-7,10)\}\)
  7. \(\left\{\begin{matrix}x+6y=-54\\2x=-3y-36\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x+6y=-54\\2x+3y=-36\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}x=-6y-54\\ 2x+3y=-36\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}x=-6y-54\\ 2.\left(-6y-54\right)+3y=-36\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}x=-6y-54\\ -12y-108+3y=-36\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}x=-6y-54\\ -9y=-36+108=72\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=-6y-54\\ y=\frac{72}{-9}=-8\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=-6.(-8)-54=-6\\ y=-8\end{matrix}\right.\\ \qquad V=\{(-6,-8)\}\)
  8. \(\left\{\begin{matrix}-6y=51-5x\\-x-3y=-6\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}5x-6y=51\\-x-3y=-6\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}5x-6y=51\\ -3y+6=x\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}5\left(-3y+6\right)-6y=51\\x=-3y+6\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-15y+30-6y=51\\x=-3y+6\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-21y=51-30=21\\x=-3y+6\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{21}{-21} = -1 \\ x=-3y+6\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = -1 \\ x=-3.(-1)+6=9\end{matrix}\right.\\ \qquad V=\{(9,-1)\}\)
  9. \(\left\{\begin{matrix}-2x+y=21\\-2x+5y=41\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=2x+21\\ -2x+5y=41\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}y=2x+21\\ -2x+5\left(2x+21\right)=41\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}y=2x+21\\ -2x+10x+105=41\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}y=2x+21\\ 8x=41-105=-64\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=2x+21\\ x=\frac{-64}{8}=-8\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=2.(-8)+21=5\\ x=-8\end{matrix}\right.\\ \qquad V=\{(-8,5)\}\)
  10. \(\left\{\begin{matrix}-5x-3y=31\\2x=-y-13\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-5x-3y=31\\2x+y=-13\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}-5x-3y=31\\ y=-2x-13\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-5x-3\left(-2x-13\right)=31\\y=-2x-13\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-5x+6x+39=31\\y=-2x-13\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}x=31-39=-8\\y=-2x-13\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{-8}{1} = -8 \\ y=-2x-13\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = -8 \\ y=-2.(-8)-13=3\end{matrix}\right.\\ \qquad V=\{(-8,3)\}\)
  11. \(\left\{\begin{matrix}-6x-2y=-10\\-x-6y=21\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-6x-2y=-10\\ -6y-21=x\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-6\left(-6y-21\right)-2y=-10\\x=-6y-21\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}36y+126-2y=-10\\x=-6y-21\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}34y=-10-126=-136\\x=-6y-21\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{-136}{34} = -4 \\ x=-6y-21\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = -4 \\ x=-6.(-4)-21=3\end{matrix}\right.\\ \qquad V=\{(3,-4)\}\)
  12. \(\left\{\begin{matrix}4x-5y=61\\3x+y=22\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}4x-5y=61\\ y=-3x+22\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}4x-5\left(-3x+22\right)=61\\y=-3x+22\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}4x+15x-110=61\\y=-3x+22\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}19x=61+110=171\\y=-3x+22\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{171}{19} = 9 \\ y=-3x+22\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = 9 \\ y=-3.(9)+22=-5\end{matrix}\right.\\ \qquad V=\{(9,-5)\}\)
Oefeningengenerator vanhoeckes.be/wiskunde 2024-04-26 12:51:06