Stelsels substitutie

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Substitutie

1. \(\left\{\begin{matrix}-6x+2y=-4\\-x=3y-34\end{matrix}\right.\)
2. \(\left\{\begin{matrix}2x+3y=5\\6x-y=25\end{matrix}\right.\)
3. \(\left\{\begin{matrix}4x+3y=23\\-5x+y=-24\end{matrix}\right.\)
4. \(\left\{\begin{matrix}x-y=3\\5x=2y-12\end{matrix}\right.\)
5. \(\left\{\begin{matrix}y=-8-3x\\-4x-3y=-1\end{matrix}\right.\)
6. \(\left\{\begin{matrix}x-6y=12\\-2x=2y-10\end{matrix}\right.\)
7. \(\left\{\begin{matrix}-3x-3y=3\\-6x-y=51\end{matrix}\right.\)
8. \(\left\{\begin{matrix}-3x-5y=-13\\6x+y=35\end{matrix}\right.\)
9. \(\left\{\begin{matrix}y=-1-4x\\-2x-5y=-31\end{matrix}\right.\)
10. \(\left\{\begin{matrix}4x-3y=34\\4x=-y+10\end{matrix}\right.\)
11. \(\left\{\begin{matrix}5x+2y=-53\\-4x=y+40\end{matrix}\right.\)
12. \(\left\{\begin{matrix}-3y=-3+2x\\x-4y=-15\end{matrix}\right.\)

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Substitutie

Verbetersleutel

1. \(\left\{\begin{matrix}-6x+2y=-4\\-x=3y-34\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-6x+2y=-4\\-x-3y=-34\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}-6x+2y=-4\\ -3y+34=x\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-6\left(-3y+34\right)+2y=-4\\x=-3y+34\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}18y-204+2y=-4\\x=-3y+34\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}20y=-4+204=200\\x=-3y+34\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{200}{20} = 10 \\ x=-3y+34\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = 10 \\ x=-3.(10)+34=4\end{matrix}\right.\\ \qquad V=\{(4,10)\}\)
2. \(\left\{\begin{matrix}2x+3y=5\\6x-y=25\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}2x+3y=5\\ 6x-25=y\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}2x+3\left(6x-25\right)=5\\y=6x-25\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}2x+18x-75=5\\y=6x-25\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}20x=5+75=80\\y=6x-25\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{80}{20} = 4 \\ y=6x-25\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = 4 \\ y=6.(4)-25=-1\end{matrix}\right.\\ \qquad V=\{(4,-1)\}\)
3. \(\left\{\begin{matrix}4x+3y=23\\-5x+y=-24\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}4x+3y=23\\ y=5x-24\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}4x+3\left(5x-24\right)=23\\y=5x-24\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}4x+15x-72=23\\y=5x-24\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}19x=23+72=95\\y=5x-24\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{95}{19} = 5 \\ y=5x-24\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = 5 \\ y=5.(5)-24=1\end{matrix}\right.\\ \qquad V=\{(5,1)\}\)
4. \(\left\{\begin{matrix}x-y=3\\5x=2y-12\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x-y=3\\5x-2y=-12\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}x=y+3\\ 5x-2y=-12\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}x=y+3\\ 5.\left(y+3\right)-2y=-12\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}x=y+3\\ 5y+15-2y=-12\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}x=y+3\\ 3y=-12-15=-27\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=y+3\\ y=\frac{-27}{3}=-9\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=.(-9)+3=-6\\ y=-9\end{matrix}\right.\\ \qquad V=\{(-6,-9)\}\)
5. \(\left\{\begin{matrix}y=-8-3x\\-4x-3y=-1\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}3x+y=-8\\-4x-3y=-1\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}y=-3x-8\\ -4x-3y=-1\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}y=-3x-8\\ -4x-3\left(-3x-8\right)=-1\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}y=-3x-8\\ -4x+9x+24=-1\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}y=-3x-8\\ 5x=-1-24=-25\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=-3x-8\\ x=\frac{-25}{5}=-5\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=-3.(-5)-8=7\\ x=-5\end{matrix}\right.\\ \qquad V=\{(-5,7)\}\)
6. \(\left\{\begin{matrix}x-6y=12\\-2x=2y-10\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x-6y=12\\-2x-2y=-10\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}x=6y+12\\ -2x-2y=-10\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}x=6y+12\\ -2.\left(6y+12\right)-2y=-10\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}x=6y+12\\ -12y-24-2y=-10\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}x=6y+12\\ -14y=-10+24=14\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=6y+12\\ y=\frac{14}{-14}=-1\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=6.(-1)+12=6\\ y=-1\end{matrix}\right.\\ \qquad V=\{(6,-1)\}\)
7. \(\left\{\begin{matrix}-3x-3y=3\\-6x-y=51\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-3x-3y=3\\ -6x-51=y\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-3x-3\left(-6x-51\right)=3\\y=-6x-51\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-3x+18x+153=3\\y=-6x-51\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}15x=3-153=-150\\y=-6x-51\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{-150}{15} = -10 \\ y=-6x-51\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = -10 \\ y=-6.(-10)-51=9\end{matrix}\right.\\ \qquad V=\{(-10,9)\}\)
8. \(\left\{\begin{matrix}-3x-5y=-13\\6x+y=35\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-3x-5y=-13\\ y=-6x+35\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-3x-5\left(-6x+35\right)=-13\\y=-6x+35\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-3x+30x-175=-13\\y=-6x+35\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}27x=-13+175=162\\y=-6x+35\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{162}{27} = 6 \\ y=-6x+35\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = 6 \\ y=-6.(6)+35=-1\end{matrix}\right.\\ \qquad V=\{(6,-1)\}\)
9. \(\left\{\begin{matrix}y=-1-4x\\-2x-5y=-31\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}4x+y=-1\\-2x-5y=-31\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}y=-4x-1\\ -2x-5y=-31\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}y=-4x-1\\ -2x-5\left(-4x-1\right)=-31\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}y=-4x-1\\ -2x+20x+5=-31\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}y=-4x-1\\ 18x=-31-5=-36\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=-4x-1\\ x=\frac{-36}{18}=-2\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=-4.(-2)-1=7\\ x=-2\end{matrix}\right.\\ \qquad V=\{(-2,7)\}\)
10. \(\left\{\begin{matrix}4x-3y=34\\4x=-y+10\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}4x-3y=34\\4x+y=10\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}4x-3y=34\\ y=-4x+10\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}4x-3\left(-4x+10\right)=34\\y=-4x+10\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}4x+12x-30=34\\y=-4x+10\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}16x=34+30=64\\y=-4x+10\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{64}{16} = 4 \\ y=-4x+10\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = 4 \\ y=-4.(4)+10=-6\end{matrix}\right.\\ \qquad V=\{(4,-6)\}\)
11. \(\left\{\begin{matrix}5x+2y=-53\\-4x=y+40\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}5x+2y=-53\\-4x-y=40\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}5x+2y=-53\\ -4x-40=y\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}5x+2\left(-4x-40\right)=-53\\y=-4x-40\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}5x-8x-80=-53\\y=-4x-40\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-3x=-53+80=27\\y=-4x-40\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{27}{-3} = -9 \\ y=-4x-40\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = -9 \\ y=-4.(-9)-40=-4\end{matrix}\right.\\ \qquad V=\{(-9,-4)\}\)
12. \(\left\{\begin{matrix}-3y=-3+2x\\x-4y=-15\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-2x-3y=-3\\x-4y=-15\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}-2x-3y=-3\\ x=4y-15\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-2\left(4y-15\right)-3y=-3\\x=4y-15\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-8y+30-3y=-3\\x=4y-15\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-11y=-3-30=-33\\x=4y-15\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{-33}{-11} = 3 \\ x=4y-15\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = 3 \\ x=4.(3)-15=-3\end{matrix}\right.\\ \qquad V=\{(-3,3)\}\)