Stelsels substitutie

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Substitutie

  1. \(\left\{\begin{matrix}6x+4y=62\\6x=y+37\end{matrix}\right.\)
  2. \(\left\{\begin{matrix}3x-6y=-30\\-x+5y=37\end{matrix}\right.\)
  3. \(\left\{\begin{matrix}-5y=43-2x\\-3x-y=-22\end{matrix}\right.\)
  4. \(\left\{\begin{matrix}-3x+4y=38\\x+5y=38\end{matrix}\right.\)
  5. \(\left\{\begin{matrix}2y=-29+5x\\-x+3y=2\end{matrix}\right.\)
  6. \(\left\{\begin{matrix}5y=-44+6x\\x-3y=3\end{matrix}\right.\)
  7. \(\left\{\begin{matrix}3x+6y=-51\\-x-4y=33\end{matrix}\right.\)
  8. \(\left\{\begin{matrix}x+3y=21\\3x=4y-67\end{matrix}\right.\)
  9. \(\left\{\begin{matrix}-6x+2y=38\\-5x=-y+29\end{matrix}\right.\)
  10. \(\left\{\begin{matrix}-6x-6y=54\\-x+3y=1\end{matrix}\right.\)
  11. \(\left\{\begin{matrix}-3x+3y=-24\\-x-4y=-18\end{matrix}\right.\)
  12. \(\left\{\begin{matrix}-y=-9+x\\6x-4y=44\end{matrix}\right.\)

Substitutie

Verbetersleutel

  1. \(\left\{\begin{matrix}6x+4y=62\\6x=y+37\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}6x+4y=62\\6x-y=37\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}6x+4y=62\\ 6x-37=y\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}6x+4\left(6x-37\right)=62\\y=6x-37\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}6x+24x-148=62\\y=6x-37\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}30x=62+148=210\\y=6x-37\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{210}{30} = 7 \\ y=6x-37\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = 7 \\ y=6.(7)-37=5\end{matrix}\right.\\ \qquad V=\{(7,5)\}\)
  2. \(\left\{\begin{matrix}3x-6y=-30\\-x+5y=37\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}3x-6y=-30\\ 5y-37=x\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}3\left(5y-37\right)-6y=-30\\x=5y-37\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}15y-111-6y=-30\\x=5y-37\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}9y=-30+111=81\\x=5y-37\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{81}{9} = 9 \\ x=5y-37\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = 9 \\ x=5.(9)-37=8\end{matrix}\right.\\ \qquad V=\{(8,9)\}\)
  3. \(\left\{\begin{matrix}-5y=43-2x\\-3x-y=-22\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}2x-5y=43\\-3x-y=-22\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}2x-5y=43\\ -3x+22=y\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}2x-5\left(-3x+22\right)=43\\y=-3x+22\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}2x+15x-110=43\\y=-3x+22\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}17x=43+110=153\\y=-3x+22\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{153}{17} = 9 \\ y=-3x+22\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = 9 \\ y=-3.(9)+22=-5\end{matrix}\right.\\ \qquad V=\{(9,-5)\}\)
  4. \(\left\{\begin{matrix}-3x+4y=38\\x+5y=38\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-3x+4y=38\\ x=-5y+38\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-3\left(-5y+38\right)+4y=38\\x=-5y+38\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}15y-114+4y=38\\x=-5y+38\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}19y=38+114=152\\x=-5y+38\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{152}{19} = 8 \\ x=-5y+38\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = 8 \\ x=-5.(8)+38=-2\end{matrix}\right.\\ \qquad V=\{(-2,8)\}\)
  5. \(\left\{\begin{matrix}2y=-29+5x\\-x+3y=2\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-5x+2y=-29\\-x+3y=2\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}-5x+2y=-29\\ 3y-2=x\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-5\left(3y-2\right)+2y=-29\\x=3y-2\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-15y+10+2y=-29\\x=3y-2\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-13y=-29-10=-39\\x=3y-2\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{-39}{-13} = 3 \\ x=3y-2\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = 3 \\ x=3.(3)-2=7\end{matrix}\right.\\ \qquad V=\{(7,3)\}\)
  6. \(\left\{\begin{matrix}5y=-44+6x\\x-3y=3\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-6x+5y=-44\\x-3y=3\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}-6x+5y=-44\\ x=3y+3\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-6\left(3y+3\right)+5y=-44\\x=3y+3\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-18y-18+5y=-44\\x=3y+3\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-13y=-44+18=-26\\x=3y+3\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{-26}{-13} = 2 \\ x=3y+3\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = 2 \\ x=3.(2)+3=9\end{matrix}\right.\\ \qquad V=\{(9,2)\}\)
  7. \(\left\{\begin{matrix}3x+6y=-51\\-x-4y=33\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}3x+6y=-51\\ -4y-33=x\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}3\left(-4y-33\right)+6y=-51\\x=-4y-33\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-12y-99+6y=-51\\x=-4y-33\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-6y=-51+99=48\\x=-4y-33\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{48}{-6} = -8 \\ x=-4y-33\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = -8 \\ x=-4.(-8)-33=-1\end{matrix}\right.\\ \qquad V=\{(-1,-8)\}\)
  8. \(\left\{\begin{matrix}x+3y=21\\3x=4y-67\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x+3y=21\\3x-4y=-67\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}x=-3y+21\\ 3x-4y=-67\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}x=-3y+21\\ 3.\left(-3y+21\right)-4y=-67\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}x=-3y+21\\ -9y+63-4y=-67\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}x=-3y+21\\ -13y=-67-63=-130\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=-3y+21\\ y=\frac{-130}{-13}=10\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=-3.(10)+21=-9\\ y=10\end{matrix}\right.\\ \qquad V=\{(-9,10)\}\)
  9. \(\left\{\begin{matrix}-6x+2y=38\\-5x=-y+29\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-6x+2y=38\\-5x+y=29\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}-6x+2y=38\\ y=5x+29\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-6x+2\left(5x+29\right)=38\\y=5x+29\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-6x+10x+58=38\\y=5x+29\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}4x=38-58=-20\\y=5x+29\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{-20}{4} = -5 \\ y=5x+29\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = -5 \\ y=5.(-5)+29=4\end{matrix}\right.\\ \qquad V=\{(-5,4)\}\)
  10. \(\left\{\begin{matrix}-6x-6y=54\\-x+3y=1\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-6x-6y=54\\ 3y-1=x\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-6\left(3y-1\right)-6y=54\\x=3y-1\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-18y+6-6y=54\\x=3y-1\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-24y=54-6=48\\x=3y-1\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{48}{-24} = -2 \\ x=3y-1\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = -2 \\ x=3.(-2)-1=-7\end{matrix}\right.\\ \qquad V=\{(-7,-2)\}\)
  11. \(\left\{\begin{matrix}-3x+3y=-24\\-x-4y=-18\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-3x+3y=-24\\ -4y+18=x\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-3\left(-4y+18\right)+3y=-24\\x=-4y+18\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}12y-54+3y=-24\\x=-4y+18\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}15y=-24+54=30\\x=-4y+18\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{30}{15} = 2 \\ x=-4y+18\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = 2 \\ x=-4.(2)+18=10\end{matrix}\right.\\ \qquad V=\{(10,2)\}\)
  12. \(\left\{\begin{matrix}-y=-9+x\\6x-4y=44\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-x-y=-9\\6x-4y=44\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}-y+9=x\\6x-4y=44\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}x=-y+9\\ 6.\left(-y+9\right)-4y=44\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}x=-y+9\\ -6y+54-4y=44\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}x=-y+9\\ -10y=44-54=-10\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=-y+9\\ y=\frac{-10}{-10}=1\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=-.(1)+9=8\\ y=1\end{matrix}\right.\\ \qquad V=\{(8,1)\}\)
Oefeningengenerator vanhoeckes.be/wiskunde 2024-12-03 18:12:03