Stelsels substitutie

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Substitutie

  1. \(\left\{\begin{matrix}2x-5y=-43\\-6x=y+49\end{matrix}\right.\)
  2. \(\left\{\begin{matrix}3x-5y=-40\\4x-y=-25\end{matrix}\right.\)
  3. \(\left\{\begin{matrix}2x+5y=33\\-x+2y=24\end{matrix}\right.\)
  4. \(\left\{\begin{matrix}-6x+y=-45\\-4x-5y=-81\end{matrix}\right.\)
  5. \(\left\{\begin{matrix}x-5y=-12\\-4x=4y-48\end{matrix}\right.\)
  6. \(\left\{\begin{matrix}-6y=42-3x\\-5x+y=20\end{matrix}\right.\)
  7. \(\left\{\begin{matrix}-4x+6y=-22\\-x=4y+11\end{matrix}\right.\)
  8. \(\left\{\begin{matrix}3x-4y=12\\2x=y-7\end{matrix}\right.\)
  9. \(\left\{\begin{matrix}6y=-45-x\\6x+3y=27\end{matrix}\right.\)
  10. \(\left\{\begin{matrix}5x-2y=53\\x=-4y-29\end{matrix}\right.\)
  11. \(\left\{\begin{matrix}-5y=-25+x\\6x-2y=54\end{matrix}\right.\)
  12. \(\left\{\begin{matrix}5y=54+4x\\-3x+y=24\end{matrix}\right.\)

Substitutie

Verbetersleutel

  1. \(\left\{\begin{matrix}2x-5y=-43\\-6x=y+49\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}2x-5y=-43\\-6x-y=49\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}2x-5y=-43\\ -6x-49=y\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}2x-5\left(-6x-49\right)=-43\\y=-6x-49\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}2x+30x+245=-43\\y=-6x-49\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}32x=-43-245=-288\\y=-6x-49\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{-288}{32} = -9 \\ y=-6x-49\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = -9 \\ y=-6.(-9)-49=5\end{matrix}\right.\\ \qquad V=\{(-9,5)\}\)
  2. \(\left\{\begin{matrix}3x-5y=-40\\4x-y=-25\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}3x-5y=-40\\ 4x+25=y\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}3x-5\left(4x+25\right)=-40\\y=4x+25\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}3x-20x-125=-40\\y=4x+25\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-17x=-40+125=85\\y=4x+25\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{85}{-17} = -5 \\ y=4x+25\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = -5 \\ y=4.(-5)+25=5\end{matrix}\right.\\ \qquad V=\{(-5,5)\}\)
  3. \(\left\{\begin{matrix}2x+5y=33\\-x+2y=24\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}2x+5y=33\\ 2y-24=x\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}2\left(2y-24\right)+5y=33\\x=2y-24\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}4y-48+5y=33\\x=2y-24\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}9y=33+48=81\\x=2y-24\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{81}{9} = 9 \\ x=2y-24\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = 9 \\ x=2.(9)-24=-6\end{matrix}\right.\\ \qquad V=\{(-6,9)\}\)
  4. \(\left\{\begin{matrix}-6x+y=-45\\-4x-5y=-81\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=6x-45\\ -4x-5y=-81\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}y=6x-45\\ -4x-5\left(6x-45\right)=-81\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}y=6x-45\\ -4x-30x+225=-81\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}y=6x-45\\ -34x=-81-225=-306\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=6x-45\\ x=\frac{-306}{-34}=9\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=6.(9)-45=9\\ x=9\end{matrix}\right.\\ \qquad V=\{(9,9)\}\)
  5. \(\left\{\begin{matrix}x-5y=-12\\-4x=4y-48\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x-5y=-12\\-4x-4y=-48\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}x=5y-12\\ -4x-4y=-48\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}x=5y-12\\ -4.\left(5y-12\right)-4y=-48\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}x=5y-12\\ -20y+48-4y=-48\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}x=5y-12\\ -24y=-48-48=-96\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=5y-12\\ y=\frac{-96}{-24}=4\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=5.(4)-12=8\\ y=4\end{matrix}\right.\\ \qquad V=\{(8,4)\}\)
  6. \(\left\{\begin{matrix}-6y=42-3x\\-5x+y=20\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}3x-6y=42\\-5x+y=20\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}3x-6y=42\\ y=5x+20\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}3x-6\left(5x+20\right)=42\\y=5x+20\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}3x-30x-120=42\\y=5x+20\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-27x=42+120=162\\y=5x+20\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{162}{-27} = -6 \\ y=5x+20\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = -6 \\ y=5.(-6)+20=-10\end{matrix}\right.\\ \qquad V=\{(-6,-10)\}\)
  7. \(\left\{\begin{matrix}-4x+6y=-22\\-x=4y+11\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-4x+6y=-22\\-x-4y=11\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}-4x+6y=-22\\ -4y-11=x\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-4\left(-4y-11\right)+6y=-22\\x=-4y-11\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}16y+44+6y=-22\\x=-4y-11\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}22y=-22-44=-66\\x=-4y-11\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{-66}{22} = -3 \\ x=-4y-11\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = -3 \\ x=-4.(-3)-11=1\end{matrix}\right.\\ \qquad V=\{(1,-3)\}\)
  8. \(\left\{\begin{matrix}3x-4y=12\\2x=y-7\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}3x-4y=12\\2x-y=-7\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}3x-4y=12\\ 2x+7=y\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}3x-4\left(2x+7\right)=12\\y=2x+7\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}3x-8x-28=12\\y=2x+7\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-5x=12+28=40\\y=2x+7\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{40}{-5} = -8 \\ y=2x+7\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = -8 \\ y=2.(-8)+7=-9\end{matrix}\right.\\ \qquad V=\{(-8,-9)\}\)
  9. \(\left\{\begin{matrix}6y=-45-x\\6x+3y=27\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x+6y=-45\\6x+3y=27\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}x=-6y-45\\ 6x+3y=27\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}x=-6y-45\\ 6.\left(-6y-45\right)+3y=27\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}x=-6y-45\\ -36y-270+3y=27\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}x=-6y-45\\ -33y=27+270=297\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=-6y-45\\ y=\frac{297}{-33}=-9\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=-6.(-9)-45=9\\ y=-9\end{matrix}\right.\\ \qquad V=\{(9,-9)\}\)
  10. \(\left\{\begin{matrix}5x-2y=53\\x=-4y-29\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}5x-2y=53\\x+4y=-29\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}5x-2y=53\\ x=-4y-29\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}5\left(-4y-29\right)-2y=53\\x=-4y-29\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-20y-145-2y=53\\x=-4y-29\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-22y=53+145=198\\x=-4y-29\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{198}{-22} = -9 \\ x=-4y-29\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = -9 \\ x=-4.(-9)-29=7\end{matrix}\right.\\ \qquad V=\{(7,-9)\}\)
  11. \(\left\{\begin{matrix}-5y=-25+x\\6x-2y=54\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-x-5y=-25\\6x-2y=54\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}-5y+25=x\\6x-2y=54\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}x=-5y+25\\ 6.\left(-5y+25\right)-2y=54\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}x=-5y+25\\ -30y+150-2y=54\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}x=-5y+25\\ -32y=54-150=-96\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=-5y+25\\ y=\frac{-96}{-32}=3\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=-5.(3)+25=10\\ y=3\end{matrix}\right.\\ \qquad V=\{(10,3)\}\)
  12. \(\left\{\begin{matrix}5y=54+4x\\-3x+y=24\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-4x+5y=54\\-3x+y=24\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}-4x+5y=54\\ y=3x+24\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-4x+5\left(3x+24\right)=54\\y=3x+24\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-4x+15x+120=54\\y=3x+24\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}11x=54-120=-66\\y=3x+24\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{-66}{11} = -6 \\ y=3x+24\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = -6 \\ y=3.(-6)+24=6\end{matrix}\right.\\ \qquad V=\{(-6,6)\}\)
Oefeningengenerator vanhoeckes.be/wiskunde 2024-05-03 14:08:37