Stelsels substitutie

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Substitutie

1. \(\left\{\begin{matrix}-2x+y=-10\\-2x=-5y-26\end{matrix}\right.\)
2. \(\left\{\begin{matrix}-5x+6y=-21\\3x+y=8\end{matrix}\right.\)
3. \(\left\{\begin{matrix}-4x-4y=28\\x-3y=-15\end{matrix}\right.\)
4. \(\left\{\begin{matrix}x-y=-5\\6x=-6y-66\end{matrix}\right.\)
5. \(\left\{\begin{matrix}-6x-y=28\\5x=-4y+2\end{matrix}\right.\)
6. \(\left\{\begin{matrix}-4y=26-x\\-5x-5y=-5\end{matrix}\right.\)
7. \(\left\{\begin{matrix}x-3y=25\\-2x=5y+60\end{matrix}\right.\)
8. \(\left\{\begin{matrix}-x-6y=-69\\-2x+5y=32\end{matrix}\right.\)
9. \(\left\{\begin{matrix}-4x+3y=-58\\2x=y+26\end{matrix}\right.\)
10. \(\left\{\begin{matrix}-6x-5y=-51\\2x=y+9\end{matrix}\right.\)
11. \(\left\{\begin{matrix}-2y=46-6x\\-3x-y=-31\end{matrix}\right.\)
12. \(\left\{\begin{matrix}-6x+y=26\\5x=5y+20\end{matrix}\right.\)

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Substitutie

Verbetersleutel

1. \(\left\{\begin{matrix}-2x+y=-10\\-2x=-5y-26\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-2x+y=-10\\-2x+5y=-26\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}y=2x-10\\ -2x+5y=-26\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}y=2x-10\\ -2x+5\left(2x-10\right)=-26\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}y=2x-10\\ -2x+10x-50=-26\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}y=2x-10\\ 8x=-26+50=24\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=2x-10\\ x=\frac{24}{8}=3\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=2.(3)-10=-4\\ x=3\end{matrix}\right.\\ \qquad V=\{(3,-4)\}\)
2. \(\left\{\begin{matrix}-5x+6y=-21\\3x+y=8\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-5x+6y=-21\\ y=-3x+8\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-5x+6\left(-3x+8\right)=-21\\y=-3x+8\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-5x-18x+48=-21\\y=-3x+8\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-23x=-21-48=-69\\y=-3x+8\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{-69}{-23} = 3 \\ y=-3x+8\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = 3 \\ y=-3.(3)+8=-1\end{matrix}\right.\\ \qquad V=\{(3,-1)\}\)
3. \(\left\{\begin{matrix}-4x-4y=28\\x-3y=-15\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-4x-4y=28\\ x=3y-15\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-4\left(3y-15\right)-4y=28\\x=3y-15\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-12y+60-4y=28\\x=3y-15\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-16y=28-60=-32\\x=3y-15\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{-32}{-16} = 2 \\ x=3y-15\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = 2 \\ x=3.(2)-15=-9\end{matrix}\right.\\ \qquad V=\{(-9,2)\}\)
4. \(\left\{\begin{matrix}x-y=-5\\6x=-6y-66\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x-y=-5\\6x+6y=-66\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}x=y-5\\ 6x+6y=-66\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}x=y-5\\ 6.\left(y-5\right)+6y=-66\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}x=y-5\\ 6y-30+6y=-66\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}x=y-5\\ 12y=-66+30=-36\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=y-5\\ y=\frac{-36}{12}=-3\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=.(-3)-5=-8\\ y=-3\end{matrix}\right.\\ \qquad V=\{(-8,-3)\}\)
5. \(\left\{\begin{matrix}-6x-y=28\\5x=-4y+2\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-6x-y=28\\5x+4y=2\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}-6x-28=y\\5x+4y=2\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}y=-6x-28\\ 5x+4\left(-6x-28\right)=2\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}y=-6x-28\\ 5x-24x-112=2\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}y=-6x-28\\ -19x=2+112=114\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=-6x-28\\ x=\frac{114}{-19}=-6\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=-6.(-6)-28=8\\ x=-6\end{matrix}\right.\\ \qquad V=\{(-6,8)\}\)
6. \(\left\{\begin{matrix}-4y=26-x\\-5x-5y=-5\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x-4y=26\\-5x-5y=-5\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}x=4y+26\\ -5x-5y=-5\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}x=4y+26\\ -5.\left(4y+26\right)-5y=-5\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}x=4y+26\\ -20y-130-5y=-5\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}x=4y+26\\ -25y=-5+130=125\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=4y+26\\ y=\frac{125}{-25}=-5\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=4.(-5)+26=6\\ y=-5\end{matrix}\right.\\ \qquad V=\{(6,-5)\}\)
7. \(\left\{\begin{matrix}x-3y=25\\-2x=5y+60\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x-3y=25\\-2x-5y=60\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}x=3y+25\\ -2x-5y=60\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}x=3y+25\\ -2.\left(3y+25\right)-5y=60\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}x=3y+25\\ -6y-50-5y=60\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}x=3y+25\\ -11y=60+50=110\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=3y+25\\ y=\frac{110}{-11}=-10\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=3.(-10)+25=-5\\ y=-10\end{matrix}\right.\\ \qquad V=\{(-5,-10)\}\)
8. \(\left\{\begin{matrix}-x-6y=-69\\-2x+5y=32\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-6y+69=x\\-2x+5y=32\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}x=-6y+69\\ -2.\left(-6y+69\right)+5y=32\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}x=-6y+69\\ 12y-138+5y=32\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}x=-6y+69\\ 17y=32+138=170\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=-6y+69\\ y=\frac{170}{17}=10\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=-6.(10)+69=9\\ y=10\end{matrix}\right.\\ \qquad V=\{(9,10)\}\)
9. \(\left\{\begin{matrix}-4x+3y=-58\\2x=y+26\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-4x+3y=-58\\2x-y=26\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}-4x+3y=-58\\ 2x-26=y\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-4x+3\left(2x-26\right)=-58\\y=2x-26\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-4x+6x-78=-58\\y=2x-26\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}2x=-58+78=20\\y=2x-26\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{20}{2} = 10 \\ y=2x-26\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = 10 \\ y=2.(10)-26=-6\end{matrix}\right.\\ \qquad V=\{(10,-6)\}\)
10. \(\left\{\begin{matrix}-6x-5y=-51\\2x=y+9\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-6x-5y=-51\\2x-y=9\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}-6x-5y=-51\\ 2x-9=y\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-6x-5\left(2x-9\right)=-51\\y=2x-9\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-6x-10x+45=-51\\y=2x-9\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-16x=-51-45=-96\\y=2x-9\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{-96}{-16} = 6 \\ y=2x-9\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = 6 \\ y=2.(6)-9=3\end{matrix}\right.\\ \qquad V=\{(6,3)\}\)
11. \(\left\{\begin{matrix}-2y=46-6x\\-3x-y=-31\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}6x-2y=46\\-3x-y=-31\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}6x-2y=46\\ -3x+31=y\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}6x-2\left(-3x+31\right)=46\\y=-3x+31\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}6x+6x-62=46\\y=-3x+31\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}12x=46+62=108\\y=-3x+31\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{108}{12} = 9 \\ y=-3x+31\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = 9 \\ y=-3.(9)+31=4\end{matrix}\right.\\ \qquad V=\{(9,4)\}\)
12. \(\left\{\begin{matrix}-6x+y=26\\5x=5y+20\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-6x+y=26\\5x-5y=20\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}y=6x+26\\ 5x-5y=20\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}y=6x+26\\ 5x-5\left(6x+26\right)=20\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}y=6x+26\\ 5x-30x-130=20\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}y=6x+26\\ -25x=20+130=150\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=6x+26\\ x=\frac{150}{-25}=-6\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=6.(-6)+26=-10\\ x=-6\end{matrix}\right.\\ \qquad V=\{(-6,-10)\}\)