Stelsels substitutie

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Substitutie

1. \(\left\{\begin{matrix}x-5y=10\\4x-2y=-14\end{matrix}\right.\)
2. \(\left\{\begin{matrix}-4y=48-2x\\-x-5y=25\end{matrix}\right.\)
3. \(\left\{\begin{matrix}x-2y=8\\4x=5y+8\end{matrix}\right.\)
4. \(\left\{\begin{matrix}-6x-5y=-48\\-6x=y-24\end{matrix}\right.\)
5. \(\left\{\begin{matrix}x+5y=-6\\-5x-6y=-27\end{matrix}\right.\)
6. \(\left\{\begin{matrix}-y=-14+5x\\3x-5y=-42\end{matrix}\right.\)
7. \(\left\{\begin{matrix}-2x-5y=44\\-4x-y=34\end{matrix}\right.\)
8. \(\left\{\begin{matrix}-6x-y=-26\\6x+5y=58\end{matrix}\right.\)
9. \(\left\{\begin{matrix}-x+5y=13\\-2x+4y=20\end{matrix}\right.\)
10. \(\left\{\begin{matrix}3y=0+3x\\-6x-y=-70\end{matrix}\right.\)
11. \(\left\{\begin{matrix}2x+3y=8\\x+5y=11\end{matrix}\right.\)
12. \(\left\{\begin{matrix}-2x+5y=56\\2x+y=-8\end{matrix}\right.\)

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Substitutie

Verbetersleutel

1. \(\left\{\begin{matrix}x-5y=10\\4x-2y=-14\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=5y+10\\ 4x-2y=-14\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}x=5y+10\\ 4.\left(5y+10\right)-2y=-14\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}x=5y+10\\ 20y+40-2y=-14\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}x=5y+10\\ 18y=-14-40=-54\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=5y+10\\ y=\frac{-54}{18}=-3\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=5.(-3)+10=-5\\ y=-3\end{matrix}\right.\\ \qquad V=\{(-5,-3)\}\)
2. \(\left\{\begin{matrix}-4y=48-2x\\-x-5y=25\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}2x-4y=48\\-x-5y=25\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}2x-4y=48\\ -5y-25=x\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}2\left(-5y-25\right)-4y=48\\x=-5y-25\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-10y-50-4y=48\\x=-5y-25\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-14y=48+50=98\\x=-5y-25\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{98}{-14} = -7 \\ x=-5y-25\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = -7 \\ x=-5.(-7)-25=10\end{matrix}\right.\\ \qquad V=\{(10,-7)\}\)
3. \(\left\{\begin{matrix}x-2y=8\\4x=5y+8\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x-2y=8\\4x-5y=8\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}x=2y+8\\ 4x-5y=8\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}x=2y+8\\ 4.\left(2y+8\right)-5y=8\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}x=2y+8\\ 8y+32-5y=8\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}x=2y+8\\ 3y=8-32=-24\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=2y+8\\ y=\frac{-24}{3}=-8\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=2.(-8)+8=-8\\ y=-8\end{matrix}\right.\\ \qquad V=\{(-8,-8)\}\)
4. \(\left\{\begin{matrix}-6x-5y=-48\\-6x=y-24\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-6x-5y=-48\\-6x-y=-24\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}-6x-5y=-48\\ -6x+24=y\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-6x-5\left(-6x+24\right)=-48\\y=-6x+24\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-6x+30x-120=-48\\y=-6x+24\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}24x=-48+120=72\\y=-6x+24\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{72}{24} = 3 \\ y=-6x+24\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = 3 \\ y=-6.(3)+24=6\end{matrix}\right.\\ \qquad V=\{(3,6)\}\)
5. \(\left\{\begin{matrix}x+5y=-6\\-5x-6y=-27\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=-5y-6\\ -5x-6y=-27\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}x=-5y-6\\ -5.\left(-5y-6\right)-6y=-27\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}x=-5y-6\\ 25y+30-6y=-27\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}x=-5y-6\\ 19y=-27-30=-57\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=-5y-6\\ y=\frac{-57}{19}=-3\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=-5.(-3)-6=9\\ y=-3\end{matrix}\right.\\ \qquad V=\{(9,-3)\}\)
6. \(\left\{\begin{matrix}-y=-14+5x\\3x-5y=-42\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-5x-y=-14\\3x-5y=-42\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}-5x+14=y\\3x-5y=-42\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}y=-5x+14\\ 3x-5\left(-5x+14\right)=-42\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}y=-5x+14\\ 3x+25x-70=-42\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}y=-5x+14\\ 28x=-42+70=28\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=-5x+14\\ x=\frac{28}{28}=1\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=-5.(1)+14=9\\ x=1\end{matrix}\right.\\ \qquad V=\{(1,9)\}\)
7. \(\left\{\begin{matrix}-2x-5y=44\\-4x-y=34\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-2x-5y=44\\ -4x-34=y\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-2x-5\left(-4x-34\right)=44\\y=-4x-34\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-2x+20x+170=44\\y=-4x-34\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}18x=44-170=-126\\y=-4x-34\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{-126}{18} = -7 \\ y=-4x-34\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = -7 \\ y=-4.(-7)-34=-6\end{matrix}\right.\\ \qquad V=\{(-7,-6)\}\)
8. \(\left\{\begin{matrix}-6x-y=-26\\6x+5y=58\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-6x+26=y\\6x+5y=58\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}y=-6x+26\\ 6x+5\left(-6x+26\right)=58\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}y=-6x+26\\ 6x-30x+130=58\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}y=-6x+26\\ -24x=58-130=-72\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=-6x+26\\ x=\frac{-72}{-24}=3\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=-6.(3)+26=8\\ x=3\end{matrix}\right.\\ \qquad V=\{(3,8)\}\)
9. \(\left\{\begin{matrix}-x+5y=13\\-2x+4y=20\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}5y-13=x\\-2x+4y=20\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}x=5y-13\\ -2.\left(5y-13\right)+4y=20\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}x=5y-13\\ -10y+26+4y=20\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}x=5y-13\\ -6y=20-26=-6\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=5y-13\\ y=\frac{-6}{-6}=1\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=5.(1)-13=-8\\ y=1\end{matrix}\right.\\ \qquad V=\{(-8,1)\}\)
10. \(\left\{\begin{matrix}3y=0+3x\\-6x-y=-70\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-3x+3y=0\\-6x-y=-70\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}-3x+3y=0\\ -6x+70=y\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-3x+3\left(-6x+70\right)=0\\y=-6x+70\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-3x-18x+210=0\\y=-6x+70\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-21x=0-210=-210\\y=-6x+70\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{-210}{-21} = 10 \\ y=-6x+70\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = 10 \\ y=-6.(10)+70=10\end{matrix}\right.\\ \qquad V=\{(10,10)\}\)
11. \(\left\{\begin{matrix}2x+3y=8\\x+5y=11\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}2x+3y=8\\ x=-5y+11\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}2\left(-5y+11\right)+3y=8\\x=-5y+11\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-10y+22+3y=8\\x=-5y+11\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-7y=8-22=-14\\x=-5y+11\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{-14}{-7} = 2 \\ x=-5y+11\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = 2 \\ x=-5.(2)+11=1\end{matrix}\right.\\ \qquad V=\{(1,2)\}\)
12. \(\left\{\begin{matrix}-2x+5y=56\\2x+y=-8\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-2x+5y=56\\ y=-2x-8\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-2x+5\left(-2x-8\right)=56\\y=-2x-8\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-2x-10x-40=56\\y=-2x-8\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-12x=56+40=96\\y=-2x-8\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{96}{-12} = -8 \\ y=-2x-8\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = -8 \\ y=-2.(-8)-8=8\end{matrix}\right.\\ \qquad V=\{(-8,8)\}\)