Stelsels substitutie

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Substitutie

  1. \(\left\{\begin{matrix}4y=58+5x\\-x-3y=-15\end{matrix}\right.\)
  2. \(\left\{\begin{matrix}-2x+3y=-4\\5x=-y-41\end{matrix}\right.\)
  3. \(\left\{\begin{matrix}-3x+2y=-30\\3x+y=3\end{matrix}\right.\)
  4. \(\left\{\begin{matrix}-6x-6y=0\\x+3y=-16\end{matrix}\right.\)
  5. \(\left\{\begin{matrix}x+6y=-38\\-6x=-3y-45\end{matrix}\right.\)
  6. \(\left\{\begin{matrix}-x+5y=-53\\6x=6y+78\end{matrix}\right.\)
  7. \(\left\{\begin{matrix}-4y=-2-x\\3x-3y=-24\end{matrix}\right.\)
  8. \(\left\{\begin{matrix}-x+y=6\\3x-5y=-16\end{matrix}\right.\)
  9. \(\left\{\begin{matrix}-y=-28+3x\\-6x+6y=-72\end{matrix}\right.\)
  10. \(\left\{\begin{matrix}-4y=66-6x\\6x+y=36\end{matrix}\right.\)
  11. \(\left\{\begin{matrix}-5x-4y=14\\5x-y=-9\end{matrix}\right.\)
  12. \(\left\{\begin{matrix}-6x+4y=-50\\-5x=y-7\end{matrix}\right.\)

Substitutie

Verbetersleutel

  1. \(\left\{\begin{matrix}4y=58+5x\\-x-3y=-15\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-5x+4y=58\\-x-3y=-15\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}-5x+4y=58\\ -3y+15=x\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-5\left(-3y+15\right)+4y=58\\x=-3y+15\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}15y-75+4y=58\\x=-3y+15\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}19y=58+75=133\\x=-3y+15\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{133}{19} = 7 \\ x=-3y+15\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = 7 \\ x=-3.(7)+15=-6\end{matrix}\right.\\ \qquad V=\{(-6,7)\}\)
  2. \(\left\{\begin{matrix}-2x+3y=-4\\5x=-y-41\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-2x+3y=-4\\5x+y=-41\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}-2x+3y=-4\\ y=-5x-41\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-2x+3\left(-5x-41\right)=-4\\y=-5x-41\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-2x-15x-123=-4\\y=-5x-41\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-17x=-4+123=119\\y=-5x-41\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{119}{-17} = -7 \\ y=-5x-41\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = -7 \\ y=-5.(-7)-41=-6\end{matrix}\right.\\ \qquad V=\{(-7,-6)\}\)
  3. \(\left\{\begin{matrix}-3x+2y=-30\\3x+y=3\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-3x+2y=-30\\ y=-3x+3\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-3x+2\left(-3x+3\right)=-30\\y=-3x+3\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-3x-6x+6=-30\\y=-3x+3\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-9x=-30-6=-36\\y=-3x+3\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{-36}{-9} = 4 \\ y=-3x+3\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = 4 \\ y=-3.(4)+3=-9\end{matrix}\right.\\ \qquad V=\{(4,-9)\}\)
  4. \(\left\{\begin{matrix}-6x-6y=0\\x+3y=-16\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-6x-6y=0\\ x=-3y-16\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-6\left(-3y-16\right)-6y=0\\x=-3y-16\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}18y+96-6y=0\\x=-3y-16\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}12y=0-96=-96\\x=-3y-16\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{-96}{12} = -8 \\ x=-3y-16\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = -8 \\ x=-3.(-8)-16=8\end{matrix}\right.\\ \qquad V=\{(8,-8)\}\)
  5. \(\left\{\begin{matrix}x+6y=-38\\-6x=-3y-45\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x+6y=-38\\-6x+3y=-45\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}x=-6y-38\\ -6x+3y=-45\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}x=-6y-38\\ -6.\left(-6y-38\right)+3y=-45\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}x=-6y-38\\ 36y+228+3y=-45\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}x=-6y-38\\ 39y=-45-228=-273\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=-6y-38\\ y=\frac{-273}{39}=-7\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=-6.(-7)-38=4\\ y=-7\end{matrix}\right.\\ \qquad V=\{(4,-7)\}\)
  6. \(\left\{\begin{matrix}-x+5y=-53\\6x=6y+78\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-x+5y=-53\\6x-6y=78\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}5y+53=x\\6x-6y=78\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}x=5y+53\\ 6.\left(5y+53\right)-6y=78\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}x=5y+53\\ 30y+318-6y=78\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}x=5y+53\\ 24y=78-318=-240\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=5y+53\\ y=\frac{-240}{24}=-10\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=5.(-10)+53=3\\ y=-10\end{matrix}\right.\\ \qquad V=\{(3,-10)\}\)
  7. \(\left\{\begin{matrix}-4y=-2-x\\3x-3y=-24\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x-4y=-2\\3x-3y=-24\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}x=4y-2\\ 3x-3y=-24\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}x=4y-2\\ 3.\left(4y-2\right)-3y=-24\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}x=4y-2\\ 12y-6-3y=-24\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}x=4y-2\\ 9y=-24+6=-18\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=4y-2\\ y=\frac{-18}{9}=-2\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=4.(-2)-2=-10\\ y=-2\end{matrix}\right.\\ \qquad V=\{(-10,-2)\}\)
  8. \(\left\{\begin{matrix}-x+y=6\\3x-5y=-16\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y-6=x\\3x-5y=-16\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}x=y-6\\ 3.\left(y-6\right)-5y=-16\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}x=y-6\\ 3y-18-5y=-16\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}x=y-6\\ -2y=-16+18=2\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=y-6\\ y=\frac{2}{-2}=-1\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=.(-1)-6=-7\\ y=-1\end{matrix}\right.\\ \qquad V=\{(-7,-1)\}\)
  9. \(\left\{\begin{matrix}-y=-28+3x\\-6x+6y=-72\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-3x-y=-28\\-6x+6y=-72\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}-3x+28=y\\-6x+6y=-72\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}y=-3x+28\\ -6x+6\left(-3x+28\right)=-72\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}y=-3x+28\\ -6x-18x+168=-72\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}y=-3x+28\\ -24x=-72-168=-240\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=-3x+28\\ x=\frac{-240}{-24}=10\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=-3.(10)+28=-2\\ x=10\end{matrix}\right.\\ \qquad V=\{(10,-2)\}\)
  10. \(\left\{\begin{matrix}-4y=66-6x\\6x+y=36\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}6x-4y=66\\6x+y=36\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}6x-4y=66\\ y=-6x+36\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}6x-4\left(-6x+36\right)=66\\y=-6x+36\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}6x+24x-144=66\\y=-6x+36\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}30x=66+144=210\\y=-6x+36\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{210}{30} = 7 \\ y=-6x+36\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = 7 \\ y=-6.(7)+36=-6\end{matrix}\right.\\ \qquad V=\{(7,-6)\}\)
  11. \(\left\{\begin{matrix}-5x-4y=14\\5x-y=-9\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-5x-4y=14\\ 5x+9=y\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-5x-4\left(5x+9\right)=14\\y=5x+9\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-5x-20x-36=14\\y=5x+9\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-25x=14+36=50\\y=5x+9\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{50}{-25} = -2 \\ y=5x+9\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = -2 \\ y=5.(-2)+9=-1\end{matrix}\right.\\ \qquad V=\{(-2,-1)\}\)
  12. \(\left\{\begin{matrix}-6x+4y=-50\\-5x=y-7\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-6x+4y=-50\\-5x-y=-7\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}-6x+4y=-50\\ -5x+7=y\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-6x+4\left(-5x+7\right)=-50\\y=-5x+7\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-6x-20x+28=-50\\y=-5x+7\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-26x=-50-28=-78\\y=-5x+7\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{-78}{-26} = 3 \\ y=-5x+7\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = 3 \\ y=-5.(3)+7=-8\end{matrix}\right.\\ \qquad V=\{(3,-8)\}\)
Oefeningengenerator vanhoeckes.be/wiskunde 2024-11-21 09:36:53