Stelsels substitutie

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Substitutie

1. \(\left\{\begin{matrix}-y=-38+6x\\-4x-5y=-34\end{matrix}\right.\)
2. \(\left\{\begin{matrix}-2y=-27-x\\5x+5y=0\end{matrix}\right.\)
3. \(\left\{\begin{matrix}-x-4y=-32\\-4x+3y=62\end{matrix}\right.\)
4. \(\left\{\begin{matrix}-x+3y=27\\4x=-2y+32\end{matrix}\right.\)
5. \(\left\{\begin{matrix}-5x+3y=-48\\-x=2y+6\end{matrix}\right.\)
6. \(\left\{\begin{matrix}-2y=-32+2x\\6x+y=51\end{matrix}\right.\)
7. \(\left\{\begin{matrix}6x+4y=-14\\x+3y=21\end{matrix}\right.\)
8. \(\left\{\begin{matrix}-2x+y=-6\\-5x=-2y-10\end{matrix}\right.\)
9. \(\left\{\begin{matrix}-2x-4y=38\\x-5y=37\end{matrix}\right.\)
10. \(\left\{\begin{matrix}3x+y=-25\\-4x=6y+52\end{matrix}\right.\)
11. \(\left\{\begin{matrix}6y=20-4x\\-x+4y=-16\end{matrix}\right.\)
12. \(\left\{\begin{matrix}-2x-3y=-7\\2x=-y+1\end{matrix}\right.\)

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Substitutie

Verbetersleutel

1. \(\left\{\begin{matrix}-y=-38+6x\\-4x-5y=-34\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-6x-y=-38\\-4x-5y=-34\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}-6x+38=y\\-4x-5y=-34\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}y=-6x+38\\ -4x-5\left(-6x+38\right)=-34\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}y=-6x+38\\ -4x+30x-190=-34\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}y=-6x+38\\ 26x=-34+190=156\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=-6x+38\\ x=\frac{156}{26}=6\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=-6.(6)+38=2\\ x=6\end{matrix}\right.\\ \qquad V=\{(6,2)\}\)
2. \(\left\{\begin{matrix}-2y=-27-x\\5x+5y=0\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x-2y=-27\\5x+5y=0\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}x=2y-27\\ 5x+5y=0\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}x=2y-27\\ 5.\left(2y-27\right)+5y=0\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}x=2y-27\\ 10y-135+5y=0\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}x=2y-27\\ 15y=0+135=135\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=2y-27\\ y=\frac{135}{15}=9\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=2.(9)-27=-9\\ y=9\end{matrix}\right.\\ \qquad V=\{(-9,9)\}\)
3. \(\left\{\begin{matrix}-x-4y=-32\\-4x+3y=62\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-4y+32=x\\-4x+3y=62\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}x=-4y+32\\ -4.\left(-4y+32\right)+3y=62\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}x=-4y+32\\ 16y-128+3y=62\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}x=-4y+32\\ 19y=62+128=190\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=-4y+32\\ y=\frac{190}{19}=10\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=-4.(10)+32=-8\\ y=10\end{matrix}\right.\\ \qquad V=\{(-8,10)\}\)
4. \(\left\{\begin{matrix}-x+3y=27\\4x=-2y+32\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-x+3y=27\\4x+2y=32\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}3y-27=x\\4x+2y=32\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}x=3y-27\\ 4.\left(3y-27\right)+2y=32\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}x=3y-27\\ 12y-108+2y=32\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}x=3y-27\\ 14y=32+108=140\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=3y-27\\ y=\frac{140}{14}=10\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=3.(10)-27=3\\ y=10\end{matrix}\right.\\ \qquad V=\{(3,10)\}\)
5. \(\left\{\begin{matrix}-5x+3y=-48\\-x=2y+6\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-5x+3y=-48\\-x-2y=6\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}-5x+3y=-48\\ -2y-6=x\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-5\left(-2y-6\right)+3y=-48\\x=-2y-6\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}10y+30+3y=-48\\x=-2y-6\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}13y=-48-30=-78\\x=-2y-6\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{-78}{13} = -6 \\ x=-2y-6\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = -6 \\ x=-2.(-6)-6=6\end{matrix}\right.\\ \qquad V=\{(6,-6)\}\)
6. \(\left\{\begin{matrix}-2y=-32+2x\\6x+y=51\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-2x-2y=-32\\6x+y=51\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}-2x-2y=-32\\ y=-6x+51\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-2x-2\left(-6x+51\right)=-32\\y=-6x+51\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-2x+12x-102=-32\\y=-6x+51\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}10x=-32+102=70\\y=-6x+51\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{70}{10} = 7 \\ y=-6x+51\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = 7 \\ y=-6.(7)+51=9\end{matrix}\right.\\ \qquad V=\{(7,9)\}\)
7. \(\left\{\begin{matrix}6x+4y=-14\\x+3y=21\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}6x+4y=-14\\ x=-3y+21\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}6\left(-3y+21\right)+4y=-14\\x=-3y+21\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-18y+126+4y=-14\\x=-3y+21\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-14y=-14-126=-140\\x=-3y+21\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{-140}{-14} = 10 \\ x=-3y+21\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = 10 \\ x=-3.(10)+21=-9\end{matrix}\right.\\ \qquad V=\{(-9,10)\}\)
8. \(\left\{\begin{matrix}-2x+y=-6\\-5x=-2y-10\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-2x+y=-6\\-5x+2y=-10\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}y=2x-6\\ -5x+2y=-10\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}y=2x-6\\ -5x+2\left(2x-6\right)=-10\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}y=2x-6\\ -5x+4x-12=-10\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}y=2x-6\\ -x=-10+12=2\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=2x-6\\ x=\frac{2}{-1}=-2\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=2.(-2)-6=-10\\ x=-2\end{matrix}\right.\\ \qquad V=\{(-2,-10)\}\)
9. \(\left\{\begin{matrix}-2x-4y=38\\x-5y=37\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-2x-4y=38\\ x=5y+37\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-2\left(5y+37\right)-4y=38\\x=5y+37\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-10y-74-4y=38\\x=5y+37\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-14y=38+74=112\\x=5y+37\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{112}{-14} = -8 \\ x=5y+37\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = -8 \\ x=5.(-8)+37=-3\end{matrix}\right.\\ \qquad V=\{(-3,-8)\}\)
10. \(\left\{\begin{matrix}3x+y=-25\\-4x=6y+52\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}3x+y=-25\\-4x-6y=52\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}y=-3x-25\\ -4x-6y=52\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}y=-3x-25\\ -4x-6\left(-3x-25\right)=52\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}y=-3x-25\\ -4x+18x+150=52\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}y=-3x-25\\ 14x=52-150=-98\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=-3x-25\\ x=\frac{-98}{14}=-7\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=-3.(-7)-25=-4\\ x=-7\end{matrix}\right.\\ \qquad V=\{(-7,-4)\}\)
11. \(\left\{\begin{matrix}6y=20-4x\\-x+4y=-16\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}4x+6y=20\\-x+4y=-16\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}4x+6y=20\\ 4y+16=x\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}4\left(4y+16\right)+6y=20\\x=4y+16\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}16y+64+6y=20\\x=4y+16\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}22y=20-64=-44\\x=4y+16\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{-44}{22} = -2 \\ x=4y+16\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = -2 \\ x=4.(-2)+16=8\end{matrix}\right.\\ \qquad V=\{(8,-2)\}\)
12. \(\left\{\begin{matrix}-2x-3y=-7\\2x=-y+1\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-2x-3y=-7\\2x+y=1\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}-2x-3y=-7\\ y=-2x+1\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-2x-3\left(-2x+1\right)=-7\\y=-2x+1\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-2x+6x-3=-7\\y=-2x+1\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}4x=-7+3=-4\\y=-2x+1\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{-4}{4} = -1 \\ y=-2x+1\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = -1 \\ y=-2.(-1)+1=3\end{matrix}\right.\\ \qquad V=\{(-1,3)\}\)