Stelsels substitutie

Hoofdmenu Eentje per keer  🖨

Substitutie

1. \(\left\{\begin{matrix}-3x-4y=9\\-x-6y=31\end{matrix}\right.\)
2. \(\left\{\begin{matrix}4x-5y=-22\\-4x=y-14\end{matrix}\right.\)
3. \(\left\{\begin{matrix}-3y=-66+5x\\-x+6y=33\end{matrix}\right.\)
4. \(\left\{\begin{matrix}x-5y=50\\-3x-3y=12\end{matrix}\right.\)
5. \(\left\{\begin{matrix}4x-6y=-58\\-2x+y=23\end{matrix}\right.\)
6. \(\left\{\begin{matrix}x-3y=-28\\-5x-2y=-30\end{matrix}\right.\)
7. \(\left\{\begin{matrix}2x+y=-27\\3x=5y+18\end{matrix}\right.\)
8. \(\left\{\begin{matrix}2y=-32-2x\\3x-y=-12\end{matrix}\right.\)
9. \(\left\{\begin{matrix}-6x+4y=100\\x+5y=40\end{matrix}\right.\)
10. \(\left\{\begin{matrix}-6x-3y=-39\\5x-y=50\end{matrix}\right.\)
11. \(\left\{\begin{matrix}-3x+4y=61\\-2x+y=24\end{matrix}\right.\)
12. \(\left\{\begin{matrix}2x+4y=4\\x-6y=26\end{matrix}\right.\)

---

Substitutie

Verbetersleutel

1. \(\left\{\begin{matrix}-3x-4y=9\\-x-6y=31\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-3x-4y=9\\ -6y-31=x\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-3\left(-6y-31\right)-4y=9\\x=-6y-31\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}18y+93-4y=9\\x=-6y-31\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}14y=9-93=-84\\x=-6y-31\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{-84}{14} = -6 \\ x=-6y-31\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = -6 \\ x=-6.(-6)-31=5\end{matrix}\right.\\ \qquad V=\{(5,-6)\}\)
2. \(\left\{\begin{matrix}4x-5y=-22\\-4x=y-14\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}4x-5y=-22\\-4x-y=-14\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}4x-5y=-22\\ -4x+14=y\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}4x-5\left(-4x+14\right)=-22\\y=-4x+14\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}4x+20x-70=-22\\y=-4x+14\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}24x=-22+70=48\\y=-4x+14\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{48}{24} = 2 \\ y=-4x+14\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = 2 \\ y=-4.(2)+14=6\end{matrix}\right.\\ \qquad V=\{(2,6)\}\)
3. \(\left\{\begin{matrix}-3y=-66+5x\\-x+6y=33\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-5x-3y=-66\\-x+6y=33\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}-5x-3y=-66\\ 6y-33=x\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-5\left(6y-33\right)-3y=-66\\x=6y-33\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-30y+165-3y=-66\\x=6y-33\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-33y=-66-165=-231\\x=6y-33\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{-231}{-33} = 7 \\ x=6y-33\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = 7 \\ x=6.(7)-33=9\end{matrix}\right.\\ \qquad V=\{(9,7)\}\)
4. \(\left\{\begin{matrix}x-5y=50\\-3x-3y=12\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=5y+50\\ -3x-3y=12\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}x=5y+50\\ -3.\left(5y+50\right)-3y=12\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}x=5y+50\\ -15y-150-3y=12\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}x=5y+50\\ -18y=12+150=162\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=5y+50\\ y=\frac{162}{-18}=-9\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=5.(-9)+50=5\\ y=-9\end{matrix}\right.\\ \qquad V=\{(5,-9)\}\)
5. \(\left\{\begin{matrix}4x-6y=-58\\-2x+y=23\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}4x-6y=-58\\ y=2x+23\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}4x-6\left(2x+23\right)=-58\\y=2x+23\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}4x-12x-138=-58\\y=2x+23\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-8x=-58+138=80\\y=2x+23\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{80}{-8} = -10 \\ y=2x+23\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = -10 \\ y=2.(-10)+23=3\end{matrix}\right.\\ \qquad V=\{(-10,3)\}\)
6. \(\left\{\begin{matrix}x-3y=-28\\-5x-2y=-30\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=3y-28\\ -5x-2y=-30\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}x=3y-28\\ -5.\left(3y-28\right)-2y=-30\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}x=3y-28\\ -15y+140-2y=-30\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}x=3y-28\\ -17y=-30-140=-170\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=3y-28\\ y=\frac{-170}{-17}=10\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=3.(10)-28=2\\ y=10\end{matrix}\right.\\ \qquad V=\{(2,10)\}\)
7. \(\left\{\begin{matrix}2x+y=-27\\3x=5y+18\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}2x+y=-27\\3x-5y=18\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}y=-2x-27\\ 3x-5y=18\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}y=-2x-27\\ 3x-5\left(-2x-27\right)=18\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}y=-2x-27\\ 3x+10x+135=18\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}y=-2x-27\\ 13x=18-135=-117\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=-2x-27\\ x=\frac{-117}{13}=-9\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=-2.(-9)-27=-9\\ x=-9\end{matrix}\right.\\ \qquad V=\{(-9,-9)\}\)
8. \(\left\{\begin{matrix}2y=-32-2x\\3x-y=-12\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}2x+2y=-32\\3x-y=-12\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}2x+2y=-32\\ 3x+12=y\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}2x+2\left(3x+12\right)=-32\\y=3x+12\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}2x+6x+24=-32\\y=3x+12\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}8x=-32-24=-56\\y=3x+12\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{-56}{8} = -7 \\ y=3x+12\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = -7 \\ y=3.(-7)+12=-9\end{matrix}\right.\\ \qquad V=\{(-7,-9)\}\)
9. \(\left\{\begin{matrix}-6x+4y=100\\x+5y=40\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-6x+4y=100\\ x=-5y+40\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-6\left(-5y+40\right)+4y=100\\x=-5y+40\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}30y-240+4y=100\\x=-5y+40\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}34y=100+240=340\\x=-5y+40\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{340}{34} = 10 \\ x=-5y+40\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = 10 \\ x=-5.(10)+40=-10\end{matrix}\right.\\ \qquad V=\{(-10,10)\}\)
10. \(\left\{\begin{matrix}-6x-3y=-39\\5x-y=50\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-6x-3y=-39\\ 5x-50=y\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-6x-3\left(5x-50\right)=-39\\y=5x-50\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-6x-15x+150=-39\\y=5x-50\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-21x=-39-150=-189\\y=5x-50\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{-189}{-21} = 9 \\ y=5x-50\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = 9 \\ y=5.(9)-50=-5\end{matrix}\right.\\ \qquad V=\{(9,-5)\}\)
11. \(\left\{\begin{matrix}-3x+4y=61\\-2x+y=24\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-3x+4y=61\\ y=2x+24\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-3x+4\left(2x+24\right)=61\\y=2x+24\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-3x+8x+96=61\\y=2x+24\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}5x=61-96=-35\\y=2x+24\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{-35}{5} = -7 \\ y=2x+24\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = -7 \\ y=2.(-7)+24=10\end{matrix}\right.\\ \qquad V=\{(-7,10)\}\)
12. \(\left\{\begin{matrix}2x+4y=4\\x-6y=26\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}2x+4y=4\\ x=6y+26\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}2\left(6y+26\right)+4y=4\\x=6y+26\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}12y+52+4y=4\\x=6y+26\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}16y=4-52=-48\\x=6y+26\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{-48}{16} = -3 \\ x=6y+26\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = -3 \\ x=6.(-3)+26=8\end{matrix}\right.\\ \qquad V=\{(8,-3)\}\)