Stelsels substitutie

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Substitutie

  1. \(\left\{\begin{matrix}-x-4y=19\\2x+5y=-23\end{matrix}\right.\)
  2. \(\left\{\begin{matrix}3y=3-6x\\x+2y=-4\end{matrix}\right.\)
  3. \(\left\{\begin{matrix}2y=50-6x\\6x+y=43\end{matrix}\right.\)
  4. \(\left\{\begin{matrix}-6x-y=35\\-2x+3y=15\end{matrix}\right.\)
  5. \(\left\{\begin{matrix}-x-3y=-16\\5x-2y=-5\end{matrix}\right.\)
  6. \(\left\{\begin{matrix}-5x+2y=-33\\x=3y-9\end{matrix}\right.\)
  7. \(\left\{\begin{matrix}x+6y=-10\\6x=5y+63\end{matrix}\right.\)
  8. \(\left\{\begin{matrix}2x+y=-12\\3x=-2y-20\end{matrix}\right.\)
  9. \(\left\{\begin{matrix}4y=12-4x\\-x+6y=-52\end{matrix}\right.\)
  10. \(\left\{\begin{matrix}4x-5y=22\\-3x+y=-11\end{matrix}\right.\)
  11. \(\left\{\begin{matrix}-5x+4y=16\\-x+5y=-22\end{matrix}\right.\)
  12. \(\left\{\begin{matrix}3y=-29-x\\6x-2y=-14\end{matrix}\right.\)

Substitutie

Verbetersleutel

  1. \(\left\{\begin{matrix}-x-4y=19\\2x+5y=-23\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-4y-19=x\\2x+5y=-23\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}x=-4y-19\\ 2.\left(-4y-19\right)+5y=-23\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}x=-4y-19\\ -8y-38+5y=-23\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}x=-4y-19\\ -3y=-23+38=15\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=-4y-19\\ y=\frac{15}{-3}=-5\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=-4.(-5)-19=1\\ y=-5\end{matrix}\right.\\ \qquad V=\{(1,-5)\}\)
  2. \(\left\{\begin{matrix}3y=3-6x\\x+2y=-4\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}6x+3y=3\\x+2y=-4\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}6x+3y=3\\ x=-2y-4\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}6\left(-2y-4\right)+3y=3\\x=-2y-4\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-12y-24+3y=3\\x=-2y-4\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-9y=3+24=27\\x=-2y-4\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{27}{-9} = -3 \\ x=-2y-4\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = -3 \\ x=-2.(-3)-4=2\end{matrix}\right.\\ \qquad V=\{(2,-3)\}\)
  3. \(\left\{\begin{matrix}2y=50-6x\\6x+y=43\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}6x+2y=50\\6x+y=43\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}6x+2y=50\\ y=-6x+43\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}6x+2\left(-6x+43\right)=50\\y=-6x+43\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}6x-12x+86=50\\y=-6x+43\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-6x=50-86=-36\\y=-6x+43\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{-36}{-6} = 6 \\ y=-6x+43\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = 6 \\ y=-6.(6)+43=7\end{matrix}\right.\\ \qquad V=\{(6,7)\}\)
  4. \(\left\{\begin{matrix}-6x-y=35\\-2x+3y=15\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-6x-35=y\\-2x+3y=15\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}y=-6x-35\\ -2x+3\left(-6x-35\right)=15\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}y=-6x-35\\ -2x-18x-105=15\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}y=-6x-35\\ -20x=15+105=120\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=-6x-35\\ x=\frac{120}{-20}=-6\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=-6.(-6)-35=1\\ x=-6\end{matrix}\right.\\ \qquad V=\{(-6,1)\}\)
  5. \(\left\{\begin{matrix}-x-3y=-16\\5x-2y=-5\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-3y+16=x\\5x-2y=-5\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}x=-3y+16\\ 5.\left(-3y+16\right)-2y=-5\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}x=-3y+16\\ -15y+80-2y=-5\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}x=-3y+16\\ -17y=-5-80=-85\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=-3y+16\\ y=\frac{-85}{-17}=5\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=-3.(5)+16=1\\ y=5\end{matrix}\right.\\ \qquad V=\{(1,5)\}\)
  6. \(\left\{\begin{matrix}-5x+2y=-33\\x=3y-9\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-5x+2y=-33\\x-3y=-9\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}-5x+2y=-33\\ x=3y-9\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-5\left(3y-9\right)+2y=-33\\x=3y-9\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-15y+45+2y=-33\\x=3y-9\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-13y=-33-45=-78\\x=3y-9\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{-78}{-13} = 6 \\ x=3y-9\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = 6 \\ x=3.(6)-9=9\end{matrix}\right.\\ \qquad V=\{(9,6)\}\)
  7. \(\left\{\begin{matrix}x+6y=-10\\6x=5y+63\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x+6y=-10\\6x-5y=63\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}x=-6y-10\\ 6x-5y=63\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}x=-6y-10\\ 6.\left(-6y-10\right)-5y=63\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}x=-6y-10\\ -36y-60-5y=63\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}x=-6y-10\\ -41y=63+60=123\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=-6y-10\\ y=\frac{123}{-41}=-3\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=-6.(-3)-10=8\\ y=-3\end{matrix}\right.\\ \qquad V=\{(8,-3)\}\)
  8. \(\left\{\begin{matrix}2x+y=-12\\3x=-2y-20\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}2x+y=-12\\3x+2y=-20\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}y=-2x-12\\ 3x+2y=-20\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}y=-2x-12\\ 3x+2\left(-2x-12\right)=-20\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}y=-2x-12\\ 3x-4x-24=-20\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}y=-2x-12\\ -x=-20+24=4\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=-2x-12\\ x=\frac{4}{-1}=-4\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=-2.(-4)-12=-4\\ x=-4\end{matrix}\right.\\ \qquad V=\{(-4,-4)\}\)
  9. \(\left\{\begin{matrix}4y=12-4x\\-x+6y=-52\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}4x+4y=12\\-x+6y=-52\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}4x+4y=12\\ 6y+52=x\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}4\left(6y+52\right)+4y=12\\x=6y+52\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}24y+208+4y=12\\x=6y+52\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}28y=12-208=-196\\x=6y+52\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{-196}{28} = -7 \\ x=6y+52\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = -7 \\ x=6.(-7)+52=10\end{matrix}\right.\\ \qquad V=\{(10,-7)\}\)
  10. \(\left\{\begin{matrix}4x-5y=22\\-3x+y=-11\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}4x-5y=22\\ y=3x-11\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}4x-5\left(3x-11\right)=22\\y=3x-11\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}4x-15x+55=22\\y=3x-11\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-11x=22-55=-33\\y=3x-11\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{-33}{-11} = 3 \\ y=3x-11\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = 3 \\ y=3.(3)-11=-2\end{matrix}\right.\\ \qquad V=\{(3,-2)\}\)
  11. \(\left\{\begin{matrix}-5x+4y=16\\-x+5y=-22\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-5x+4y=16\\ 5y+22=x\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-5\left(5y+22\right)+4y=16\\x=5y+22\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-25y-110+4y=16\\x=5y+22\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-21y=16+110=126\\x=5y+22\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{126}{-21} = -6 \\ x=5y+22\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = -6 \\ x=5.(-6)+22=-8\end{matrix}\right.\\ \qquad V=\{(-8,-6)\}\)
  12. \(\left\{\begin{matrix}3y=-29-x\\6x-2y=-14\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x+3y=-29\\6x-2y=-14\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}x=-3y-29\\ 6x-2y=-14\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}x=-3y-29\\ 6.\left(-3y-29\right)-2y=-14\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}x=-3y-29\\ -18y-174-2y=-14\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}x=-3y-29\\ -20y=-14+174=160\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=-3y-29\\ y=\frac{160}{-20}=-8\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=-3.(-8)-29=-5\\ y=-8\end{matrix}\right.\\ \qquad V=\{(-5,-8)\}\)
Oefeningengenerator vanhoeckes.be/wiskunde 2024-05-02 01:48:28