Stelsels substitutie

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Substitutie

  1. \(\left\{\begin{matrix}y=-3-5x\\5x+6y=32\end{matrix}\right.\)
  2. \(\left\{\begin{matrix}y=-14+6x\\-2x+6y=18\end{matrix}\right.\)
  3. \(\left\{\begin{matrix}x+6y=-14\\3x+3y=-12\end{matrix}\right.\)
  4. \(\left\{\begin{matrix}-4y=-17+3x\\-x-2y=-5\end{matrix}\right.\)
  5. \(\left\{\begin{matrix}3y=3-6x\\-x+2y=-8\end{matrix}\right.\)
  6. \(\left\{\begin{matrix}x-5y=35\\2x=-3y-34\end{matrix}\right.\)
  7. \(\left\{\begin{matrix}3y=-13+x\\6x-6y=18\end{matrix}\right.\)
  8. \(\left\{\begin{matrix}-4x-5y=12\\x-6y=26\end{matrix}\right.\)
  9. \(\left\{\begin{matrix}-3x+4y=54\\2x=-y-3\end{matrix}\right.\)
  10. \(\left\{\begin{matrix}6x+5y=-85\\-x=3y+25\end{matrix}\right.\)
  11. \(\left\{\begin{matrix}-6y=36-6x\\5x+y=-18\end{matrix}\right.\)
  12. \(\left\{\begin{matrix}4y=34+x\\2x-3y=-18\end{matrix}\right.\)

Substitutie

Verbetersleutel

  1. \(\left\{\begin{matrix}y=-3-5x\\5x+6y=32\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}5x+y=-3\\5x+6y=32\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}y=-5x-3\\ 5x+6y=32\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}y=-5x-3\\ 5x+6\left(-5x-3\right)=32\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}y=-5x-3\\ 5x-30x-18=32\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}y=-5x-3\\ -25x=32+18=50\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=-5x-3\\ x=\frac{50}{-25}=-2\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=-5.(-2)-3=7\\ x=-2\end{matrix}\right.\\ \qquad V=\{(-2,7)\}\)
  2. \(\left\{\begin{matrix}y=-14+6x\\-2x+6y=18\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-6x+y=-14\\-2x+6y=18\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}y=6x-14\\ -2x+6y=18\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}y=6x-14\\ -2x+6\left(6x-14\right)=18\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}y=6x-14\\ -2x+36x-84=18\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}y=6x-14\\ 34x=18+84=102\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=6x-14\\ x=\frac{102}{34}=3\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=6.(3)-14=4\\ x=3\end{matrix}\right.\\ \qquad V=\{(3,4)\}\)
  3. \(\left\{\begin{matrix}x+6y=-14\\3x+3y=-12\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=-6y-14\\ 3x+3y=-12\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}x=-6y-14\\ 3.\left(-6y-14\right)+3y=-12\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}x=-6y-14\\ -18y-42+3y=-12\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}x=-6y-14\\ -15y=-12+42=30\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=-6y-14\\ y=\frac{30}{-15}=-2\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=-6.(-2)-14=-2\\ y=-2\end{matrix}\right.\\ \qquad V=\{(-2,-2)\}\)
  4. \(\left\{\begin{matrix}-4y=-17+3x\\-x-2y=-5\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-3x-4y=-17\\-x-2y=-5\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}-3x-4y=-17\\ -2y+5=x\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-3\left(-2y+5\right)-4y=-17\\x=-2y+5\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}6y-15-4y=-17\\x=-2y+5\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}2y=-17+15=-2\\x=-2y+5\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{-2}{2} = -1 \\ x=-2y+5\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = -1 \\ x=-2.(-1)+5=7\end{matrix}\right.\\ \qquad V=\{(7,-1)\}\)
  5. \(\left\{\begin{matrix}3y=3-6x\\-x+2y=-8\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}6x+3y=3\\-x+2y=-8\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}6x+3y=3\\ 2y+8=x\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}6\left(2y+8\right)+3y=3\\x=2y+8\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}12y+48+3y=3\\x=2y+8\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}15y=3-48=-45\\x=2y+8\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{-45}{15} = -3 \\ x=2y+8\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = -3 \\ x=2.(-3)+8=2\end{matrix}\right.\\ \qquad V=\{(2,-3)\}\)
  6. \(\left\{\begin{matrix}x-5y=35\\2x=-3y-34\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x-5y=35\\2x+3y=-34\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}x=5y+35\\ 2x+3y=-34\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}x=5y+35\\ 2.\left(5y+35\right)+3y=-34\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}x=5y+35\\ 10y+70+3y=-34\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}x=5y+35\\ 13y=-34-70=-104\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=5y+35\\ y=\frac{-104}{13}=-8\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=5.(-8)+35=-5\\ y=-8\end{matrix}\right.\\ \qquad V=\{(-5,-8)\}\)
  7. \(\left\{\begin{matrix}3y=-13+x\\6x-6y=18\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-x+3y=-13\\6x-6y=18\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}3y+13=x\\6x-6y=18\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}x=3y+13\\ 6.\left(3y+13\right)-6y=18\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}x=3y+13\\ 18y+78-6y=18\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}x=3y+13\\ 12y=18-78=-60\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=3y+13\\ y=\frac{-60}{12}=-5\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=3.(-5)+13=-2\\ y=-5\end{matrix}\right.\\ \qquad V=\{(-2,-5)\}\)
  8. \(\left\{\begin{matrix}-4x-5y=12\\x-6y=26\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-4x-5y=12\\ x=6y+26\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-4\left(6y+26\right)-5y=12\\x=6y+26\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-24y-104-5y=12\\x=6y+26\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-29y=12+104=116\\x=6y+26\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{116}{-29} = -4 \\ x=6y+26\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = -4 \\ x=6.(-4)+26=2\end{matrix}\right.\\ \qquad V=\{(2,-4)\}\)
  9. \(\left\{\begin{matrix}-3x+4y=54\\2x=-y-3\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-3x+4y=54\\2x+y=-3\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}-3x+4y=54\\ y=-2x-3\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-3x+4\left(-2x-3\right)=54\\y=-2x-3\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-3x-8x-12=54\\y=-2x-3\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-11x=54+12=66\\y=-2x-3\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{66}{-11} = -6 \\ y=-2x-3\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = -6 \\ y=-2.(-6)-3=9\end{matrix}\right.\\ \qquad V=\{(-6,9)\}\)
  10. \(\left\{\begin{matrix}6x+5y=-85\\-x=3y+25\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}6x+5y=-85\\-x-3y=25\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}6x+5y=-85\\ -3y-25=x\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}6\left(-3y-25\right)+5y=-85\\x=-3y-25\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-18y-150+5y=-85\\x=-3y-25\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-13y=-85+150=65\\x=-3y-25\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{65}{-13} = -5 \\ x=-3y-25\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = -5 \\ x=-3.(-5)-25=-10\end{matrix}\right.\\ \qquad V=\{(-10,-5)\}\)
  11. \(\left\{\begin{matrix}-6y=36-6x\\5x+y=-18\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}6x-6y=36\\5x+y=-18\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}6x-6y=36\\ y=-5x-18\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}6x-6\left(-5x-18\right)=36\\y=-5x-18\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}6x+30x+108=36\\y=-5x-18\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}36x=36-108=-72\\y=-5x-18\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{-72}{36} = -2 \\ y=-5x-18\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = -2 \\ y=-5.(-2)-18=-8\end{matrix}\right.\\ \qquad V=\{(-2,-8)\}\)
  12. \(\left\{\begin{matrix}4y=34+x\\2x-3y=-18\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-x+4y=34\\2x-3y=-18\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}4y-34=x\\2x-3y=-18\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}x=4y-34\\ 2.\left(4y-34\right)-3y=-18\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}x=4y-34\\ 8y-68-3y=-18\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}x=4y-34\\ 5y=-18+68=50\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=4y-34\\ y=\frac{50}{5}=10\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=4.(10)-34=6\\ y=10\end{matrix}\right.\\ \qquad V=\{(6,10)\}\)
Oefeningengenerator vanhoeckes.be/wiskunde 2024-04-19 05:21:18