Stelsels substitutie

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Substitutie

1. \(\left\{\begin{matrix}2x+3y=-7\\5x=y-43\end{matrix}\right.\)
2. \(\left\{\begin{matrix}3x-4y=22\\-2x=y+0\end{matrix}\right.\)
3. \(\left\{\begin{matrix}-2x-4y=-14\\x=2y-5\end{matrix}\right.\)
4. \(\left\{\begin{matrix}5y=-5-5x\\x+2y=-6\end{matrix}\right.\)
5. \(\left\{\begin{matrix}-6x-y=34\\6x=3y-66\end{matrix}\right.\)
6. \(\left\{\begin{matrix}-y=-3-2x\\6x+2y=26\end{matrix}\right.\)
7. \(\left\{\begin{matrix}x-5y=9\\3x+3y=-9\end{matrix}\right.\)
8. \(\left\{\begin{matrix}3x-6y=18\\x-4y=8\end{matrix}\right.\)
9. \(\left\{\begin{matrix}3y=12+4x\\x+2y=-25\end{matrix}\right.\)
10. \(\left\{\begin{matrix}4x+2y=8\\-x-5y=34\end{matrix}\right.\)
11. \(\left\{\begin{matrix}-6x-2y=-30\\-x-5y=23\end{matrix}\right.\)
12. \(\left\{\begin{matrix}6x-6y=54\\5x+y=9\end{matrix}\right.\)

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Substitutie

Verbetersleutel

1. \(\left\{\begin{matrix}2x+3y=-7\\5x=y-43\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}2x+3y=-7\\5x-y=-43\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}2x+3y=-7\\ 5x+43=y\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}2x+3\left(5x+43\right)=-7\\y=5x+43\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}2x+15x+129=-7\\y=5x+43\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}17x=-7-129=-136\\y=5x+43\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{-136}{17} = -8 \\ y=5x+43\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = -8 \\ y=5.(-8)+43=3\end{matrix}\right.\\ \qquad V=\{(-8,3)\}\)
2. \(\left\{\begin{matrix}3x-4y=22\\-2x=y+0\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}3x-4y=22\\-2x-y=0\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}3x-4y=22\\ -2x+0=y\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}3x-4\left(-2x+0\right)=22\\y=-2x+0\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}3x+8x+0=22\\y=-2x+0\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}11x=22+0=22\\y=-2x+0\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{22}{11} = 2 \\ y=-2x+0\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = 2 \\ y=-2.(2)+0=-4\end{matrix}\right.\\ \qquad V=\{(2,-4)\}\)
3. \(\left\{\begin{matrix}-2x-4y=-14\\x=2y-5\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-2x-4y=-14\\x-2y=-5\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}-2x-4y=-14\\ x=2y-5\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-2\left(2y-5\right)-4y=-14\\x=2y-5\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-4y+10-4y=-14\\x=2y-5\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-8y=-14-10=-24\\x=2y-5\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{-24}{-8} = 3 \\ x=2y-5\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = 3 \\ x=2.(3)-5=1\end{matrix}\right.\\ \qquad V=\{(1,3)\}\)
4. \(\left\{\begin{matrix}5y=-5-5x\\x+2y=-6\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}5x+5y=-5\\x+2y=-6\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}5x+5y=-5\\ x=-2y-6\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}5\left(-2y-6\right)+5y=-5\\x=-2y-6\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-10y-30+5y=-5\\x=-2y-6\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-5y=-5+30=25\\x=-2y-6\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{25}{-5} = -5 \\ x=-2y-6\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = -5 \\ x=-2.(-5)-6=4\end{matrix}\right.\\ \qquad V=\{(4,-5)\}\)
5. \(\left\{\begin{matrix}-6x-y=34\\6x=3y-66\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-6x-y=34\\6x-3y=-66\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}-6x-34=y\\6x-3y=-66\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}y=-6x-34\\ 6x-3\left(-6x-34\right)=-66\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}y=-6x-34\\ 6x+18x+102=-66\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}y=-6x-34\\ 24x=-66-102=-168\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=-6x-34\\ x=\frac{-168}{24}=-7\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=-6.(-7)-34=8\\ x=-7\end{matrix}\right.\\ \qquad V=\{(-7,8)\}\)
6. \(\left\{\begin{matrix}-y=-3-2x\\6x+2y=26\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}2x-y=-3\\6x+2y=26\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}2x+3=y\\6x+2y=26\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}y=2x+3\\ 6x+2\left(2x+3\right)=26\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}y=2x+3\\ 6x+4x+6=26\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}y=2x+3\\ 10x=26-6=20\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=2x+3\\ x=\frac{20}{10}=2\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=2.(2)+3=7\\ x=2\end{matrix}\right.\\ \qquad V=\{(2,7)\}\)
7. \(\left\{\begin{matrix}x-5y=9\\3x+3y=-9\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=5y+9\\ 3x+3y=-9\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}x=5y+9\\ 3.\left(5y+9\right)+3y=-9\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}x=5y+9\\ 15y+27+3y=-9\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}x=5y+9\\ 18y=-9-27=-36\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=5y+9\\ y=\frac{-36}{18}=-2\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=5.(-2)+9=-1\\ y=-2\end{matrix}\right.\\ \qquad V=\{(-1,-2)\}\)
8. \(\left\{\begin{matrix}3x-6y=18\\x-4y=8\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}3x-6y=18\\ x=4y+8\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}3\left(4y+8\right)-6y=18\\x=4y+8\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}12y+24-6y=18\\x=4y+8\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}6y=18-24=-6\\x=4y+8\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{-6}{6} = -1 \\ x=4y+8\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = -1 \\ x=4.(-1)+8=4\end{matrix}\right.\\ \qquad V=\{(4,-1)\}\)
9. \(\left\{\begin{matrix}3y=12+4x\\x+2y=-25\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-4x+3y=12\\x+2y=-25\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}-4x+3y=12\\ x=-2y-25\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-4\left(-2y-25\right)+3y=12\\x=-2y-25\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}8y+100+3y=12\\x=-2y-25\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}11y=12-100=-88\\x=-2y-25\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{-88}{11} = -8 \\ x=-2y-25\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = -8 \\ x=-2.(-8)-25=-9\end{matrix}\right.\\ \qquad V=\{(-9,-8)\}\)
10. \(\left\{\begin{matrix}4x+2y=8\\-x-5y=34\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}4x+2y=8\\ -5y-34=x\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}4\left(-5y-34\right)+2y=8\\x=-5y-34\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-20y-136+2y=8\\x=-5y-34\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-18y=8+136=144\\x=-5y-34\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{144}{-18} = -8 \\ x=-5y-34\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = -8 \\ x=-5.(-8)-34=6\end{matrix}\right.\\ \qquad V=\{(6,-8)\}\)
11. \(\left\{\begin{matrix}-6x-2y=-30\\-x-5y=23\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-6x-2y=-30\\ -5y-23=x\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-6\left(-5y-23\right)-2y=-30\\x=-5y-23\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}30y+138-2y=-30\\x=-5y-23\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}28y=-30-138=-168\\x=-5y-23\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{-168}{28} = -6 \\ x=-5y-23\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = -6 \\ x=-5.(-6)-23=7\end{matrix}\right.\\ \qquad V=\{(7,-6)\}\)
12. \(\left\{\begin{matrix}6x-6y=54\\5x+y=9\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}6x-6y=54\\ y=-5x+9\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}6x-6\left(-5x+9\right)=54\\y=-5x+9\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}6x+30x-54=54\\y=-5x+9\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}36x=54+54=108\\y=-5x+9\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{108}{36} = 3 \\ y=-5x+9\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = 3 \\ y=-5.(3)+9=-6\end{matrix}\right.\\ \qquad V=\{(3,-6)\}\)