Stelsels substitutie

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Substitutie

1. \(\left\{\begin{matrix}-5y=-44+3x\\-6x+y=22\end{matrix}\right.\)
2. \(\left\{\begin{matrix}y=2-x\\-3x+2y=24\end{matrix}\right.\)
3. \(\left\{\begin{matrix}2x+2y=8\\-5x=y+20\end{matrix}\right.\)
4. \(\left\{\begin{matrix}-2y=-13-3x\\5x+y=-52\end{matrix}\right.\)
5. \(\left\{\begin{matrix}-2x+4y=30\\-x+2y=15\end{matrix}\right.\)
6. \(\left\{\begin{matrix}-6x-3y=48\\x=-4y-43\end{matrix}\right.\)
7. \(\left\{\begin{matrix}3x-6y=21\\-5x=y+31\end{matrix}\right.\)
8. \(\left\{\begin{matrix}-x-4y=20\\4x=-2y-24\end{matrix}\right.\)
9. \(\left\{\begin{matrix}-5y=42+2x\\-6x+y=-34\end{matrix}\right.\)
10. \(\left\{\begin{matrix}-x+6y=22\\3x-3y=-6\end{matrix}\right.\)
11. \(\left\{\begin{matrix}-5x+3y=56\\5x+y=-48\end{matrix}\right.\)
12. \(\left\{\begin{matrix}2y=37-5x\\-6x+y=-24\end{matrix}\right.\)

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Substitutie

Verbetersleutel

1. \(\left\{\begin{matrix}-5y=-44+3x\\-6x+y=22\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-3x-5y=-44\\-6x+y=22\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}-3x-5y=-44\\ y=6x+22\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-3x-5\left(6x+22\right)=-44\\y=6x+22\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-3x-30x-110=-44\\y=6x+22\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-33x=-44+110=66\\y=6x+22\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{66}{-33} = -2 \\ y=6x+22\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = -2 \\ y=6.(-2)+22=10\end{matrix}\right.\\ \qquad V=\{(-2,10)\}\)
2. \(\left\{\begin{matrix}y=2-x\\-3x+2y=24\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x+y=2\\-3x+2y=24\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}x=-y+2\\ -3x+2y=24\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}x=-y+2\\ -3.\left(-y+2\right)+2y=24\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}x=-y+2\\ 3y-6+2y=24\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}x=-y+2\\ 5y=24+6=30\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=-y+2\\ y=\frac{30}{5}=6\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=-.(6)+2=-4\\ y=6\end{matrix}\right.\\ \qquad V=\{(-4,6)\}\)
3. \(\left\{\begin{matrix}2x+2y=8\\-5x=y+20\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}2x+2y=8\\-5x-y=20\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}2x+2y=8\\ -5x-20=y\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}2x+2\left(-5x-20\right)=8\\y=-5x-20\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}2x-10x-40=8\\y=-5x-20\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-8x=8+40=48\\y=-5x-20\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{48}{-8} = -6 \\ y=-5x-20\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = -6 \\ y=-5.(-6)-20=10\end{matrix}\right.\\ \qquad V=\{(-6,10)\}\)
4. \(\left\{\begin{matrix}-2y=-13-3x\\5x+y=-52\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}3x-2y=-13\\5x+y=-52\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}3x-2y=-13\\ y=-5x-52\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}3x-2\left(-5x-52\right)=-13\\y=-5x-52\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}3x+10x+104=-13\\y=-5x-52\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}13x=-13-104=-117\\y=-5x-52\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{-117}{13} = -9 \\ y=-5x-52\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = -9 \\ y=-5.(-9)-52=-7\end{matrix}\right.\\ \qquad V=\{(-9,-7)\}\)
5. \(\left\{\begin{matrix}-2x+4y=30\\-x+2y=15\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-2x+4y=30\\ 2y-15=x\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-2\left(2y-15\right)+4y=30\\x=2y-15\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-4y+30+4y=30\\x=2y-15\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}0y=30-30=0\\x=2y-15\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{0}{0} = 4 \\ x=2y-15\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = 4 \\ x=2.(4)-15=-7\end{matrix}\right.\\ \qquad V=\{(-7,4)\}\)
6. \(\left\{\begin{matrix}-6x-3y=48\\x=-4y-43\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-6x-3y=48\\x+4y=-43\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}-6x-3y=48\\ x=-4y-43\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-6\left(-4y-43\right)-3y=48\\x=-4y-43\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}24y+258-3y=48\\x=-4y-43\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}21y=48-258=-210\\x=-4y-43\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{-210}{21} = -10 \\ x=-4y-43\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = -10 \\ x=-4.(-10)-43=-3\end{matrix}\right.\\ \qquad V=\{(-3,-10)\}\)
7. \(\left\{\begin{matrix}3x-6y=21\\-5x=y+31\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}3x-6y=21\\-5x-y=31\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}3x-6y=21\\ -5x-31=y\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}3x-6\left(-5x-31\right)=21\\y=-5x-31\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}3x+30x+186=21\\y=-5x-31\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}33x=21-186=-165\\y=-5x-31\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{-165}{33} = -5 \\ y=-5x-31\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = -5 \\ y=-5.(-5)-31=-6\end{matrix}\right.\\ \qquad V=\{(-5,-6)\}\)
8. \(\left\{\begin{matrix}-x-4y=20\\4x=-2y-24\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-x-4y=20\\4x+2y=-24\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}-4y-20=x\\4x+2y=-24\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}x=-4y-20\\ 4.\left(-4y-20\right)+2y=-24\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}x=-4y-20\\ -16y-80+2y=-24\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}x=-4y-20\\ -14y=-24+80=56\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=-4y-20\\ y=\frac{56}{-14}=-4\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=-4.(-4)-20=-4\\ y=-4\end{matrix}\right.\\ \qquad V=\{(-4,-4)\}\)
9. \(\left\{\begin{matrix}-5y=42+2x\\-6x+y=-34\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-2x-5y=42\\-6x+y=-34\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}-2x-5y=42\\ y=6x-34\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-2x-5\left(6x-34\right)=42\\y=6x-34\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-2x-30x+170=42\\y=6x-34\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-32x=42-170=-128\\y=6x-34\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{-128}{-32} = 4 \\ y=6x-34\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = 4 \\ y=6.(4)-34=-10\end{matrix}\right.\\ \qquad V=\{(4,-10)\}\)
10. \(\left\{\begin{matrix}-x+6y=22\\3x-3y=-6\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}6y-22=x\\3x-3y=-6\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}x=6y-22\\ 3.\left(6y-22\right)-3y=-6\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}x=6y-22\\ 18y-66-3y=-6\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}x=6y-22\\ 15y=-6+66=60\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=6y-22\\ y=\frac{60}{15}=4\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=6.(4)-22=2\\ y=4\end{matrix}\right.\\ \qquad V=\{(2,4)\}\)
11. \(\left\{\begin{matrix}-5x+3y=56\\5x+y=-48\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-5x+3y=56\\ y=-5x-48\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-5x+3\left(-5x-48\right)=56\\y=-5x-48\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-5x-15x-144=56\\y=-5x-48\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-20x=56+144=200\\y=-5x-48\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{200}{-20} = -10 \\ y=-5x-48\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = -10 \\ y=-5.(-10)-48=2\end{matrix}\right.\\ \qquad V=\{(-10,2)\}\)
12. \(\left\{\begin{matrix}2y=37-5x\\-6x+y=-24\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}5x+2y=37\\-6x+y=-24\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}5x+2y=37\\ y=6x-24\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}5x+2\left(6x-24\right)=37\\y=6x-24\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}5x+12x-48=37\\y=6x-24\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}17x=37+48=85\\y=6x-24\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{85}{17} = 5 \\ y=6x-24\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = 5 \\ y=6.(5)-24=6\end{matrix}\right.\\ \qquad V=\{(5,6)\}\)