Stelsels substitutie

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Substitutie

  1. \(\left\{\begin{matrix}3x-3y=-3\\6x+y=8\end{matrix}\right.\)
  2. \(\left\{\begin{matrix}-y=-64-6x\\4x+3y=-28\end{matrix}\right.\)
  3. \(\left\{\begin{matrix}4x+5y=3\\-2x=y+9\end{matrix}\right.\)
  4. \(\left\{\begin{matrix}-4x+2y=-30\\-x=-4y-39\end{matrix}\right.\)
  5. \(\left\{\begin{matrix}-x-3y=23\\6x-6y=102\end{matrix}\right.\)
  6. \(\left\{\begin{matrix}5x+3y=-12\\-2x=y+6\end{matrix}\right.\)
  7. \(\left\{\begin{matrix}-4x+5y=16\\-x=-3y-3\end{matrix}\right.\)
  8. \(\left\{\begin{matrix}-4y=-11-x\\-6x-5y=-79\end{matrix}\right.\)
  9. \(\left\{\begin{matrix}-6x+6y=24\\-x-4y=-26\end{matrix}\right.\)
  10. \(\left\{\begin{matrix}5x+y=-29\\-5x=5y+25\end{matrix}\right.\)
  11. \(\left\{\begin{matrix}y=27-6x\\2x-3y=-1\end{matrix}\right.\)
  12. \(\left\{\begin{matrix}4y=-63-5x\\x+2y=-21\end{matrix}\right.\)

Substitutie

Verbetersleutel

  1. \(\left\{\begin{matrix}3x-3y=-3\\6x+y=8\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}3x-3y=-3\\ y=-6x+8\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}3x-3\left(-6x+8\right)=-3\\y=-6x+8\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}3x+18x-24=-3\\y=-6x+8\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}21x=-3+24=21\\y=-6x+8\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{21}{21} = 1 \\ y=-6x+8\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = 1 \\ y=-6.(1)+8=2\end{matrix}\right.\\ \qquad V=\{(1,2)\}\)
  2. \(\left\{\begin{matrix}-y=-64-6x\\4x+3y=-28\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}6x-y=-64\\4x+3y=-28\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}6x+64=y\\4x+3y=-28\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}y=6x+64\\ 4x+3\left(6x+64\right)=-28\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}y=6x+64\\ 4x+18x+192=-28\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}y=6x+64\\ 22x=-28-192=-220\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=6x+64\\ x=\frac{-220}{22}=-10\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=6.(-10)+64=4\\ x=-10\end{matrix}\right.\\ \qquad V=\{(-10,4)\}\)
  3. \(\left\{\begin{matrix}4x+5y=3\\-2x=y+9\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}4x+5y=3\\-2x-y=9\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}4x+5y=3\\ -2x-9=y\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}4x+5\left(-2x-9\right)=3\\y=-2x-9\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}4x-10x-45=3\\y=-2x-9\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-6x=3+45=48\\y=-2x-9\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{48}{-6} = -8 \\ y=-2x-9\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = -8 \\ y=-2.(-8)-9=7\end{matrix}\right.\\ \qquad V=\{(-8,7)\}\)
  4. \(\left\{\begin{matrix}-4x+2y=-30\\-x=-4y-39\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-4x+2y=-30\\-x+4y=-39\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}-4x+2y=-30\\ 4y+39=x\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-4\left(4y+39\right)+2y=-30\\x=4y+39\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-16y-156+2y=-30\\x=4y+39\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-14y=-30+156=126\\x=4y+39\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{126}{-14} = -9 \\ x=4y+39\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = -9 \\ x=4.(-9)+39=3\end{matrix}\right.\\ \qquad V=\{(3,-9)\}\)
  5. \(\left\{\begin{matrix}-x-3y=23\\6x-6y=102\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-3y-23=x\\6x-6y=102\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}x=-3y-23\\ 6.\left(-3y-23\right)-6y=102\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}x=-3y-23\\ -18y-138-6y=102\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}x=-3y-23\\ -24y=102+138=240\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=-3y-23\\ y=\frac{240}{-24}=-10\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=-3.(-10)-23=7\\ y=-10\end{matrix}\right.\\ \qquad V=\{(7,-10)\}\)
  6. \(\left\{\begin{matrix}5x+3y=-12\\-2x=y+6\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}5x+3y=-12\\-2x-y=6\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}5x+3y=-12\\ -2x-6=y\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}5x+3\left(-2x-6\right)=-12\\y=-2x-6\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}5x-6x-18=-12\\y=-2x-6\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-x=-12+18=6\\y=-2x-6\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{6}{-1} = -6 \\ y=-2x-6\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = -6 \\ y=-2.(-6)-6=6\end{matrix}\right.\\ \qquad V=\{(-6,6)\}\)
  7. \(\left\{\begin{matrix}-4x+5y=16\\-x=-3y-3\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-4x+5y=16\\-x+3y=-3\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}-4x+5y=16\\ 3y+3=x\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-4\left(3y+3\right)+5y=16\\x=3y+3\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-12y-12+5y=16\\x=3y+3\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-7y=16+12=28\\x=3y+3\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{28}{-7} = -4 \\ x=3y+3\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = -4 \\ x=3.(-4)+3=-9\end{matrix}\right.\\ \qquad V=\{(-9,-4)\}\)
  8. \(\left\{\begin{matrix}-4y=-11-x\\-6x-5y=-79\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x-4y=-11\\-6x-5y=-79\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}x=4y-11\\ -6x-5y=-79\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}x=4y-11\\ -6.\left(4y-11\right)-5y=-79\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}x=4y-11\\ -24y+66-5y=-79\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}x=4y-11\\ -29y=-79-66=-145\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=4y-11\\ y=\frac{-145}{-29}=5\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=4.(5)-11=9\\ y=5\end{matrix}\right.\\ \qquad V=\{(9,5)\}\)
  9. \(\left\{\begin{matrix}-6x+6y=24\\-x-4y=-26\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-6x+6y=24\\ -4y+26=x\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-6\left(-4y+26\right)+6y=24\\x=-4y+26\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}24y-156+6y=24\\x=-4y+26\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}30y=24+156=180\\x=-4y+26\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{180}{30} = 6 \\ x=-4y+26\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = 6 \\ x=-4.(6)+26=2\end{matrix}\right.\\ \qquad V=\{(2,6)\}\)
  10. \(\left\{\begin{matrix}5x+y=-29\\-5x=5y+25\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}5x+y=-29\\-5x-5y=25\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}y=-5x-29\\ -5x-5y=25\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}y=-5x-29\\ -5x-5\left(-5x-29\right)=25\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}y=-5x-29\\ -5x+25x+145=25\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}y=-5x-29\\ 20x=25-145=-120\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=-5x-29\\ x=\frac{-120}{20}=-6\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=-5.(-6)-29=1\\ x=-6\end{matrix}\right.\\ \qquad V=\{(-6,1)\}\)
  11. \(\left\{\begin{matrix}y=27-6x\\2x-3y=-1\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}6x+y=27\\2x-3y=-1\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}y=-6x+27\\ 2x-3y=-1\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}y=-6x+27\\ 2x-3\left(-6x+27\right)=-1\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}y=-6x+27\\ 2x+18x-81=-1\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}y=-6x+27\\ 20x=-1+81=80\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=-6x+27\\ x=\frac{80}{20}=4\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=-6.(4)+27=3\\ x=4\end{matrix}\right.\\ \qquad V=\{(4,3)\}\)
  12. \(\left\{\begin{matrix}4y=-63-5x\\x+2y=-21\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}5x+4y=-63\\x+2y=-21\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}5x+4y=-63\\ x=-2y-21\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}5\left(-2y-21\right)+4y=-63\\x=-2y-21\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-10y-105+4y=-63\\x=-2y-21\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-6y=-63+105=42\\x=-2y-21\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{42}{-6} = -7 \\ x=-2y-21\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = -7 \\ x=-2.(-7)-21=-7\end{matrix}\right.\\ \qquad V=\{(-7,-7)\}\)
Oefeningengenerator vanhoeckes.be/wiskunde 2025-05-30 05:01:45