Stelsels substitutie

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Substitutie

1. \(\left\{\begin{matrix}-6x+y=-7\\2x+6y=-4\end{matrix}\right.\)
2. \(\left\{\begin{matrix}-3x+4y=52\\-2x+y=18\end{matrix}\right.\)
3. \(\left\{\begin{matrix}-2x+y=-9\\5x=6y+40\end{matrix}\right.\)
4. \(\left\{\begin{matrix}-2y=-4+3x\\-x+6y=52\end{matrix}\right.\)
5. \(\left\{\begin{matrix}-5y=7+4x\\-x-4y=-12\end{matrix}\right.\)
6. \(\left\{\begin{matrix}-5y=-56+6x\\-x-2y=-14\end{matrix}\right.\)
7. \(\left\{\begin{matrix}-5x-5y=-5\\-6x+y=-34\end{matrix}\right.\)
8. \(\left\{\begin{matrix}y=-6+3x\\6x+6y=60\end{matrix}\right.\)
9. \(\left\{\begin{matrix}6x-6y=-24\\3x+y=-36\end{matrix}\right.\)
10. \(\left\{\begin{matrix}-4x-4y=-8\\-x=3y-20\end{matrix}\right.\)
11. \(\left\{\begin{matrix}-6y=-12-5x\\x-4y=6\end{matrix}\right.\)
12. \(\left\{\begin{matrix}6y=32+5x\\-5x-y=53\end{matrix}\right.\)

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Substitutie

Verbetersleutel

1. \(\left\{\begin{matrix}-6x+y=-7\\2x+6y=-4\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=6x-7\\ 2x+6y=-4\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}y=6x-7\\ 2x+6\left(6x-7\right)=-4\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}y=6x-7\\ 2x+36x-42=-4\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}y=6x-7\\ 38x=-4+42=38\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=6x-7\\ x=\frac{38}{38}=1\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=6.(1)-7=-1\\ x=1\end{matrix}\right.\\ \qquad V=\{(1,-1)\}\)
2. \(\left\{\begin{matrix}-3x+4y=52\\-2x+y=18\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-3x+4y=52\\ y=2x+18\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-3x+4\left(2x+18\right)=52\\y=2x+18\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-3x+8x+72=52\\y=2x+18\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}5x=52-72=-20\\y=2x+18\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{-20}{5} = -4 \\ y=2x+18\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = -4 \\ y=2.(-4)+18=10\end{matrix}\right.\\ \qquad V=\{(-4,10)\}\)
3. \(\left\{\begin{matrix}-2x+y=-9\\5x=6y+40\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-2x+y=-9\\5x-6y=40\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}y=2x-9\\ 5x-6y=40\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}y=2x-9\\ 5x-6\left(2x-9\right)=40\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}y=2x-9\\ 5x-12x+54=40\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}y=2x-9\\ -7x=40-54=-14\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=2x-9\\ x=\frac{-14}{-7}=2\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=2.(2)-9=-5\\ x=2\end{matrix}\right.\\ \qquad V=\{(2,-5)\}\)
4. \(\left\{\begin{matrix}-2y=-4+3x\\-x+6y=52\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-3x-2y=-4\\-x+6y=52\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}-3x-2y=-4\\ 6y-52=x\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-3\left(6y-52\right)-2y=-4\\x=6y-52\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-18y+156-2y=-4\\x=6y-52\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-20y=-4-156=-160\\x=6y-52\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{-160}{-20} = 8 \\ x=6y-52\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = 8 \\ x=6.(8)-52=-4\end{matrix}\right.\\ \qquad V=\{(-4,8)\}\)
5. \(\left\{\begin{matrix}-5y=7+4x\\-x-4y=-12\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-4x-5y=7\\-x-4y=-12\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}-4x-5y=7\\ -4y+12=x\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-4\left(-4y+12\right)-5y=7\\x=-4y+12\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}16y-48-5y=7\\x=-4y+12\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}11y=7+48=55\\x=-4y+12\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{55}{11} = 5 \\ x=-4y+12\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = 5 \\ x=-4.(5)+12=-8\end{matrix}\right.\\ \qquad V=\{(-8,5)\}\)
6. \(\left\{\begin{matrix}-5y=-56+6x\\-x-2y=-14\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-6x-5y=-56\\-x-2y=-14\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}-6x-5y=-56\\ -2y+14=x\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-6\left(-2y+14\right)-5y=-56\\x=-2y+14\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}12y-84-5y=-56\\x=-2y+14\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}7y=-56+84=28\\x=-2y+14\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{28}{7} = 4 \\ x=-2y+14\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = 4 \\ x=-2.(4)+14=6\end{matrix}\right.\\ \qquad V=\{(6,4)\}\)
7. \(\left\{\begin{matrix}-5x-5y=-5\\-6x+y=-34\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-5x-5y=-5\\ y=6x-34\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-5x-5\left(6x-34\right)=-5\\y=6x-34\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-5x-30x+170=-5\\y=6x-34\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-35x=-5-170=-175\\y=6x-34\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{-175}{-35} = 5 \\ y=6x-34\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = 5 \\ y=6.(5)-34=-4\end{matrix}\right.\\ \qquad V=\{(5,-4)\}\)
8. \(\left\{\begin{matrix}y=-6+3x\\6x+6y=60\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-3x+y=-6\\6x+6y=60\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}y=3x-6\\ 6x+6y=60\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}y=3x-6\\ 6x+6\left(3x-6\right)=60\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}y=3x-6\\ 6x+18x-36=60\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}y=3x-6\\ 24x=60+36=96\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=3x-6\\ x=\frac{96}{24}=4\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=3.(4)-6=6\\ x=4\end{matrix}\right.\\ \qquad V=\{(4,6)\}\)
9. \(\left\{\begin{matrix}6x-6y=-24\\3x+y=-36\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}6x-6y=-24\\ y=-3x-36\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}6x-6\left(-3x-36\right)=-24\\y=-3x-36\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}6x+18x+216=-24\\y=-3x-36\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}24x=-24-216=-240\\y=-3x-36\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{-240}{24} = -10 \\ y=-3x-36\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = -10 \\ y=-3.(-10)-36=-6\end{matrix}\right.\\ \qquad V=\{(-10,-6)\}\)
10. \(\left\{\begin{matrix}-4x-4y=-8\\-x=3y-20\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-4x-4y=-8\\-x-3y=-20\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}-4x-4y=-8\\ -3y+20=x\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-4\left(-3y+20\right)-4y=-8\\x=-3y+20\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}12y-80-4y=-8\\x=-3y+20\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}8y=-8+80=72\\x=-3y+20\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{72}{8} = 9 \\ x=-3y+20\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = 9 \\ x=-3.(9)+20=-7\end{matrix}\right.\\ \qquad V=\{(-7,9)\}\)
11. \(\left\{\begin{matrix}-6y=-12-5x\\x-4y=6\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}5x-6y=-12\\x-4y=6\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}5x-6y=-12\\ x=4y+6\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}5\left(4y+6\right)-6y=-12\\x=4y+6\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}20y+30-6y=-12\\x=4y+6\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}14y=-12-30=-42\\x=4y+6\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{-42}{14} = -3 \\ x=4y+6\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = -3 \\ x=4.(-3)+6=-6\end{matrix}\right.\\ \qquad V=\{(-6,-3)\}\)
12. \(\left\{\begin{matrix}6y=32+5x\\-5x-y=53\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-5x+6y=32\\-5x-y=53\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}-5x+6y=32\\ -5x-53=y\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-5x+6\left(-5x-53\right)=32\\y=-5x-53\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-5x-30x-318=32\\y=-5x-53\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-35x=32+318=350\\y=-5x-53\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{350}{-35} = -10 \\ y=-5x-53\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = -10 \\ y=-5.(-10)-53=-3\end{matrix}\right.\\ \qquad V=\{(-10,-3)\}\)