Stelsels substitutie

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Substitutie

  1. \(\left\{\begin{matrix}3y=17+x\\3x-6y=-27\end{matrix}\right.\)
  2. \(\left\{\begin{matrix}3x-y=1\\-5x=2y-31\end{matrix}\right.\)
  3. \(\left\{\begin{matrix}5x+5y=5\\x+3y=21\end{matrix}\right.\)
  4. \(\left\{\begin{matrix}-y=-25+4x\\5x-6y=24\end{matrix}\right.\)
  5. \(\left\{\begin{matrix}-2x+y=-12\\-3x+6y=9\end{matrix}\right.\)
  6. \(\left\{\begin{matrix}3y=-9+6x\\6x+y=37\end{matrix}\right.\)
  7. \(\left\{\begin{matrix}4x+4y=-40\\-2x=-y-7\end{matrix}\right.\)
  8. \(\left\{\begin{matrix}y=-26+5x\\2x+5y=-22\end{matrix}\right.\)
  9. \(\left\{\begin{matrix}x+2y=7\\-3x=3y-9\end{matrix}\right.\)
  10. \(\left\{\begin{matrix}-2x+2y=-18\\-x-5y=21\end{matrix}\right.\)
  11. \(\left\{\begin{matrix}x+6y=-67\\-4x-3y=58\end{matrix}\right.\)
  12. \(\left\{\begin{matrix}-4x+4y=-12\\x=3y+13\end{matrix}\right.\)

Substitutie

Verbetersleutel

  1. \(\left\{\begin{matrix}3y=17+x\\3x-6y=-27\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-x+3y=17\\3x-6y=-27\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}3y-17=x\\3x-6y=-27\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}x=3y-17\\ 3.\left(3y-17\right)-6y=-27\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}x=3y-17\\ 9y-51-6y=-27\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}x=3y-17\\ 3y=-27+51=24\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=3y-17\\ y=\frac{24}{3}=8\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=3.(8)-17=7\\ y=8\end{matrix}\right.\\ \qquad V=\{(7,8)\}\)
  2. \(\left\{\begin{matrix}3x-y=1\\-5x=2y-31\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}3x-y=1\\-5x-2y=-31\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}3x-1=y\\-5x-2y=-31\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}y=3x-1\\ -5x-2\left(3x-1\right)=-31\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}y=3x-1\\ -5x-6x+2=-31\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}y=3x-1\\ -11x=-31-2=-33\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=3x-1\\ x=\frac{-33}{-11}=3\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=3.(3)-1=8\\ x=3\end{matrix}\right.\\ \qquad V=\{(3,8)\}\)
  3. \(\left\{\begin{matrix}5x+5y=5\\x+3y=21\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}5x+5y=5\\ x=-3y+21\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}5\left(-3y+21\right)+5y=5\\x=-3y+21\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-15y+105+5y=5\\x=-3y+21\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-10y=5-105=-100\\x=-3y+21\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{-100}{-10} = 10 \\ x=-3y+21\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = 10 \\ x=-3.(10)+21=-9\end{matrix}\right.\\ \qquad V=\{(-9,10)\}\)
  4. \(\left\{\begin{matrix}-y=-25+4x\\5x-6y=24\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-4x-y=-25\\5x-6y=24\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}-4x+25=y\\5x-6y=24\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}y=-4x+25\\ 5x-6\left(-4x+25\right)=24\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}y=-4x+25\\ 5x+24x-150=24\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}y=-4x+25\\ 29x=24+150=174\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=-4x+25\\ x=\frac{174}{29}=6\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=-4.(6)+25=1\\ x=6\end{matrix}\right.\\ \qquad V=\{(6,1)\}\)
  5. \(\left\{\begin{matrix}-2x+y=-12\\-3x+6y=9\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=2x-12\\ -3x+6y=9\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}y=2x-12\\ -3x+6\left(2x-12\right)=9\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}y=2x-12\\ -3x+12x-72=9\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}y=2x-12\\ 9x=9+72=81\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=2x-12\\ x=\frac{81}{9}=9\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=2.(9)-12=6\\ x=9\end{matrix}\right.\\ \qquad V=\{(9,6)\}\)
  6. \(\left\{\begin{matrix}3y=-9+6x\\6x+y=37\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-6x+3y=-9\\6x+y=37\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}-6x+3y=-9\\ y=-6x+37\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-6x+3\left(-6x+37\right)=-9\\y=-6x+37\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-6x-18x+111=-9\\y=-6x+37\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-24x=-9-111=-120\\y=-6x+37\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{-120}{-24} = 5 \\ y=-6x+37\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = 5 \\ y=-6.(5)+37=7\end{matrix}\right.\\ \qquad V=\{(5,7)\}\)
  7. \(\left\{\begin{matrix}4x+4y=-40\\-2x=-y-7\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}4x+4y=-40\\-2x+y=-7\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}4x+4y=-40\\ y=2x-7\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}4x+4\left(2x-7\right)=-40\\y=2x-7\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}4x+8x-28=-40\\y=2x-7\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}12x=-40+28=-12\\y=2x-7\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{-12}{12} = -1 \\ y=2x-7\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = -1 \\ y=2.(-1)-7=-9\end{matrix}\right.\\ \qquad V=\{(-1,-9)\}\)
  8. \(\left\{\begin{matrix}y=-26+5x\\2x+5y=-22\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-5x+y=-26\\2x+5y=-22\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}y=5x-26\\ 2x+5y=-22\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}y=5x-26\\ 2x+5\left(5x-26\right)=-22\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}y=5x-26\\ 2x+25x-130=-22\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}y=5x-26\\ 27x=-22+130=108\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=5x-26\\ x=\frac{108}{27}=4\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=5.(4)-26=-6\\ x=4\end{matrix}\right.\\ \qquad V=\{(4,-6)\}\)
  9. \(\left\{\begin{matrix}x+2y=7\\-3x=3y-9\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x+2y=7\\-3x-3y=-9\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}x=-2y+7\\ -3x-3y=-9\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}x=-2y+7\\ -3.\left(-2y+7\right)-3y=-9\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}x=-2y+7\\ 6y-21-3y=-9\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}x=-2y+7\\ 3y=-9+21=12\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=-2y+7\\ y=\frac{12}{3}=4\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=-2.(4)+7=-1\\ y=4\end{matrix}\right.\\ \qquad V=\{(-1,4)\}\)
  10. \(\left\{\begin{matrix}-2x+2y=-18\\-x-5y=21\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-2x+2y=-18\\ -5y-21=x\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-2\left(-5y-21\right)+2y=-18\\x=-5y-21\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}10y+42+2y=-18\\x=-5y-21\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}12y=-18-42=-60\\x=-5y-21\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{-60}{12} = -5 \\ x=-5y-21\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = -5 \\ x=-5.(-5)-21=4\end{matrix}\right.\\ \qquad V=\{(4,-5)\}\)
  11. \(\left\{\begin{matrix}x+6y=-67\\-4x-3y=58\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=-6y-67\\ -4x-3y=58\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}x=-6y-67\\ -4.\left(-6y-67\right)-3y=58\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}x=-6y-67\\ 24y+268-3y=58\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}x=-6y-67\\ 21y=58-268=-210\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=-6y-67\\ y=\frac{-210}{21}=-10\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=-6.(-10)-67=-7\\ y=-10\end{matrix}\right.\\ \qquad V=\{(-7,-10)\}\)
  12. \(\left\{\begin{matrix}-4x+4y=-12\\x=3y+13\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-4x+4y=-12\\x-3y=13\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}-4x+4y=-12\\ x=3y+13\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-4\left(3y+13\right)+4y=-12\\x=3y+13\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-12y-52+4y=-12\\x=3y+13\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-8y=-12+52=40\\x=3y+13\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{40}{-8} = -5 \\ x=3y+13\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = -5 \\ x=3.(-5)+13=-2\end{matrix}\right.\\ \qquad V=\{(-2,-5)\}\)
Oefeningengenerator vanhoeckes.be/wiskunde 2025-04-03 05:29:55