Stelsels substitutie

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Substitutie

  1. \(\left\{\begin{matrix}y=34+6x\\-6x+4y=10\end{matrix}\right.\)
  2. \(\left\{\begin{matrix}-5x+6y=-90\\x+2y=-14\end{matrix}\right.\)
  3. \(\left\{\begin{matrix}6x-4y=-100\\-3x-y=20\end{matrix}\right.\)
  4. \(\left\{\begin{matrix}-y=1-4x\\-6x+6y=-42\end{matrix}\right.\)
  5. \(\left\{\begin{matrix}6x+y=13\\3x-5y=1\end{matrix}\right.\)
  6. \(\left\{\begin{matrix}5y=-7-3x\\-x+4y=25\end{matrix}\right.\)
  7. \(\left\{\begin{matrix}-2x-4y=-24\\-3x-y=-26\end{matrix}\right.\)
  8. \(\left\{\begin{matrix}-6x+2y=-54\\5x-y=43\end{matrix}\right.\)
  9. \(\left\{\begin{matrix}6x-y=35\\4x=2y+30\end{matrix}\right.\)
  10. \(\left\{\begin{matrix}4y=30+2x\\x+5y=55\end{matrix}\right.\)
  11. \(\left\{\begin{matrix}-2x+y=16\\5x+6y=11\end{matrix}\right.\)
  12. \(\left\{\begin{matrix}-x-5y=19\\2x+6y=-30\end{matrix}\right.\)

Substitutie

Verbetersleutel

  1. \(\left\{\begin{matrix}y=34+6x\\-6x+4y=10\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-6x+y=34\\-6x+4y=10\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}y=6x+34\\ -6x+4y=10\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}y=6x+34\\ -6x+4\left(6x+34\right)=10\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}y=6x+34\\ -6x+24x+136=10\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}y=6x+34\\ 18x=10-136=-126\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=6x+34\\ x=\frac{-126}{18}=-7\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=6.(-7)+34=-8\\ x=-7\end{matrix}\right.\\ \qquad V=\{(-7,-8)\}\)
  2. \(\left\{\begin{matrix}-5x+6y=-90\\x+2y=-14\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-5x+6y=-90\\ x=-2y-14\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-5\left(-2y-14\right)+6y=-90\\x=-2y-14\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}10y+70+6y=-90\\x=-2y-14\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}16y=-90-70=-160\\x=-2y-14\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{-160}{16} = -10 \\ x=-2y-14\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = -10 \\ x=-2.(-10)-14=6\end{matrix}\right.\\ \qquad V=\{(6,-10)\}\)
  3. \(\left\{\begin{matrix}6x-4y=-100\\-3x-y=20\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}6x-4y=-100\\ -3x-20=y\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}6x-4\left(-3x-20\right)=-100\\y=-3x-20\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}6x+12x+80=-100\\y=-3x-20\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}18x=-100-80=-180\\y=-3x-20\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{-180}{18} = -10 \\ y=-3x-20\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = -10 \\ y=-3.(-10)-20=10\end{matrix}\right.\\ \qquad V=\{(-10,10)\}\)
  4. \(\left\{\begin{matrix}-y=1-4x\\-6x+6y=-42\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}4x-y=1\\-6x+6y=-42\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}4x-1=y\\-6x+6y=-42\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}y=4x-1\\ -6x+6\left(4x-1\right)=-42\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}y=4x-1\\ -6x+24x-6=-42\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}y=4x-1\\ 18x=-42+6=-36\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=4x-1\\ x=\frac{-36}{18}=-2\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=4.(-2)-1=-9\\ x=-2\end{matrix}\right.\\ \qquad V=\{(-2,-9)\}\)
  5. \(\left\{\begin{matrix}6x+y=13\\3x-5y=1\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=-6x+13\\ 3x-5y=1\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}y=-6x+13\\ 3x-5\left(-6x+13\right)=1\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}y=-6x+13\\ 3x+30x-65=1\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}y=-6x+13\\ 33x=1+65=66\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=-6x+13\\ x=\frac{66}{33}=2\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=-6.(2)+13=1\\ x=2\end{matrix}\right.\\ \qquad V=\{(2,1)\}\)
  6. \(\left\{\begin{matrix}5y=-7-3x\\-x+4y=25\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}3x+5y=-7\\-x+4y=25\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}3x+5y=-7\\ 4y-25=x\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}3\left(4y-25\right)+5y=-7\\x=4y-25\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}12y-75+5y=-7\\x=4y-25\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}17y=-7+75=68\\x=4y-25\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{68}{17} = 4 \\ x=4y-25\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = 4 \\ x=4.(4)-25=-9\end{matrix}\right.\\ \qquad V=\{(-9,4)\}\)
  7. \(\left\{\begin{matrix}-2x-4y=-24\\-3x-y=-26\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-2x-4y=-24\\ -3x+26=y\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-2x-4\left(-3x+26\right)=-24\\y=-3x+26\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-2x+12x-104=-24\\y=-3x+26\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}10x=-24+104=80\\y=-3x+26\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{80}{10} = 8 \\ y=-3x+26\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = 8 \\ y=-3.(8)+26=2\end{matrix}\right.\\ \qquad V=\{(8,2)\}\)
  8. \(\left\{\begin{matrix}-6x+2y=-54\\5x-y=43\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-6x+2y=-54\\ 5x-43=y\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-6x+2\left(5x-43\right)=-54\\y=5x-43\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-6x+10x-86=-54\\y=5x-43\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}4x=-54+86=32\\y=5x-43\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{32}{4} = 8 \\ y=5x-43\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = 8 \\ y=5.(8)-43=-3\end{matrix}\right.\\ \qquad V=\{(8,-3)\}\)
  9. \(\left\{\begin{matrix}6x-y=35\\4x=2y+30\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}6x-y=35\\4x-2y=30\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}6x-35=y\\4x-2y=30\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}y=6x-35\\ 4x-2\left(6x-35\right)=30\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}y=6x-35\\ 4x-12x+70=30\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}y=6x-35\\ -8x=30-70=-40\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=6x-35\\ x=\frac{-40}{-8}=5\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=6.(5)-35=-5\\ x=5\end{matrix}\right.\\ \qquad V=\{(5,-5)\}\)
  10. \(\left\{\begin{matrix}4y=30+2x\\x+5y=55\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-2x+4y=30\\x+5y=55\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}-2x+4y=30\\ x=-5y+55\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-2\left(-5y+55\right)+4y=30\\x=-5y+55\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}10y-110+4y=30\\x=-5y+55\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}14y=30+110=140\\x=-5y+55\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{140}{14} = 10 \\ x=-5y+55\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = 10 \\ x=-5.(10)+55=5\end{matrix}\right.\\ \qquad V=\{(5,10)\}\)
  11. \(\left\{\begin{matrix}-2x+y=16\\5x+6y=11\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=2x+16\\ 5x+6y=11\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}y=2x+16\\ 5x+6\left(2x+16\right)=11\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}y=2x+16\\ 5x+12x+96=11\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}y=2x+16\\ 17x=11-96=-85\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=2x+16\\ x=\frac{-85}{17}=-5\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=2.(-5)+16=6\\ x=-5\end{matrix}\right.\\ \qquad V=\{(-5,6)\}\)
  12. \(\left\{\begin{matrix}-x-5y=19\\2x+6y=-30\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-5y-19=x\\2x+6y=-30\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}x=-5y-19\\ 2.\left(-5y-19\right)+6y=-30\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}x=-5y-19\\ -10y-38+6y=-30\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}x=-5y-19\\ -4y=-30+38=8\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=-5y-19\\ y=\frac{8}{-4}=-2\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=-5.(-2)-19=-9\\ y=-2\end{matrix}\right.\\ \qquad V=\{(-9,-2)\}\)
Oefeningengenerator vanhoeckes.be/wiskunde 2025-05-19 10:25:00