Stelsels substitutie

Hoofdmenu Eentje per keer  🖨

Substitutie

1. \(\left\{\begin{matrix}3y=23+2x\\x-4y=-29\end{matrix}\right.\)
2. \(\left\{\begin{matrix}5y=-40-5x\\-6x-y=53\end{matrix}\right.\)
3. \(\left\{\begin{matrix}-x+y=19\\2x-2y=-38\end{matrix}\right.\)
4. \(\left\{\begin{matrix}-5x-y=-59\\-6x=-3y-33\end{matrix}\right.\)
5. \(\left\{\begin{matrix}4x-4y=-20\\2x=-y-4\end{matrix}\right.\)
6. \(\left\{\begin{matrix}-4x-y=2\\4x-5y=34\end{matrix}\right.\)
7. \(\left\{\begin{matrix}-4x+3y=-47\\4x+y=11\end{matrix}\right.\)
8. \(\left\{\begin{matrix}-x+3y=28\\-2x+2y=16\end{matrix}\right.\)
9. \(\left\{\begin{matrix}-6x+y=-66\\-2x=-2y-32\end{matrix}\right.\)
10. \(\left\{\begin{matrix}-4x-6y=12\\4x+y=28\end{matrix}\right.\)
11. \(\left\{\begin{matrix}-6x+6y=6\\3x+y=17\end{matrix}\right.\)
12. \(\left\{\begin{matrix}2y=22-2x\\-5x-y=-47\end{matrix}\right.\)

---

Substitutie

Verbetersleutel

1. \(\left\{\begin{matrix}3y=23+2x\\x-4y=-29\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-2x+3y=23\\x-4y=-29\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}-2x+3y=23\\ x=4y-29\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-2\left(4y-29\right)+3y=23\\x=4y-29\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-8y+58+3y=23\\x=4y-29\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-5y=23-58=-35\\x=4y-29\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{-35}{-5} = 7 \\ x=4y-29\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = 7 \\ x=4.(7)-29=-1\end{matrix}\right.\\ \qquad V=\{(-1,7)\}\)
2. \(\left\{\begin{matrix}5y=-40-5x\\-6x-y=53\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}5x+5y=-40\\-6x-y=53\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}5x+5y=-40\\ -6x-53=y\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}5x+5\left(-6x-53\right)=-40\\y=-6x-53\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}5x-30x-265=-40\\y=-6x-53\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-25x=-40+265=225\\y=-6x-53\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{225}{-25} = -9 \\ y=-6x-53\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = -9 \\ y=-6.(-9)-53=1\end{matrix}\right.\\ \qquad V=\{(-9,1)\}\)
3. \(\left\{\begin{matrix}-x+y=19\\2x-2y=-38\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y-19=x\\2x-2y=-38\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}x=y-19\\ 2.\left(y-19\right)-2y=-38\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}x=y-19\\ 2y-38-2y=-38\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}x=y-19\\ 0y=-38+38=0\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=y-19\\ y=\frac{0}{0}=9\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=.(9)-19=-10\\ y=9\end{matrix}\right.\\ \qquad V=\{(-10,9)\}\)
4. \(\left\{\begin{matrix}-5x-y=-59\\-6x=-3y-33\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-5x-y=-59\\-6x+3y=-33\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}-5x+59=y\\-6x+3y=-33\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}y=-5x+59\\ -6x+3\left(-5x+59\right)=-33\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}y=-5x+59\\ -6x-15x+177=-33\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}y=-5x+59\\ -21x=-33-177=-210\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=-5x+59\\ x=\frac{-210}{-21}=10\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=-5.(10)+59=9\\ x=10\end{matrix}\right.\\ \qquad V=\{(10,9)\}\)
5. \(\left\{\begin{matrix}4x-4y=-20\\2x=-y-4\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}4x-4y=-20\\2x+y=-4\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}4x-4y=-20\\ y=-2x-4\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}4x-4\left(-2x-4\right)=-20\\y=-2x-4\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}4x+8x+16=-20\\y=-2x-4\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}12x=-20-16=-36\\y=-2x-4\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{-36}{12} = -3 \\ y=-2x-4\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = -3 \\ y=-2.(-3)-4=2\end{matrix}\right.\\ \qquad V=\{(-3,2)\}\)
6. \(\left\{\begin{matrix}-4x-y=2\\4x-5y=34\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-4x-2=y\\4x-5y=34\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}y=-4x-2\\ 4x-5\left(-4x-2\right)=34\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}y=-4x-2\\ 4x+20x+10=34\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}y=-4x-2\\ 24x=34-10=24\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=-4x-2\\ x=\frac{24}{24}=1\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=-4.(1)-2=-6\\ x=1\end{matrix}\right.\\ \qquad V=\{(1,-6)\}\)
7. \(\left\{\begin{matrix}-4x+3y=-47\\4x+y=11\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-4x+3y=-47\\ y=-4x+11\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-4x+3\left(-4x+11\right)=-47\\y=-4x+11\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-4x-12x+33=-47\\y=-4x+11\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-16x=-47-33=-80\\y=-4x+11\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{-80}{-16} = 5 \\ y=-4x+11\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = 5 \\ y=-4.(5)+11=-9\end{matrix}\right.\\ \qquad V=\{(5,-9)\}\)
8. \(\left\{\begin{matrix}-x+3y=28\\-2x+2y=16\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}3y-28=x\\-2x+2y=16\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}x=3y-28\\ -2.\left(3y-28\right)+2y=16\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}x=3y-28\\ -6y+56+2y=16\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}x=3y-28\\ -4y=16-56=-40\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=3y-28\\ y=\frac{-40}{-4}=10\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=3.(10)-28=2\\ y=10\end{matrix}\right.\\ \qquad V=\{(2,10)\}\)
9. \(\left\{\begin{matrix}-6x+y=-66\\-2x=-2y-32\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-6x+y=-66\\-2x+2y=-32\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}y=6x-66\\ -2x+2y=-32\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}y=6x-66\\ -2x+2\left(6x-66\right)=-32\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}y=6x-66\\ -2x+12x-132=-32\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}y=6x-66\\ 10x=-32+132=100\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=6x-66\\ x=\frac{100}{10}=10\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=6.(10)-66=-6\\ x=10\end{matrix}\right.\\ \qquad V=\{(10,-6)\}\)
10. \(\left\{\begin{matrix}-4x-6y=12\\4x+y=28\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-4x-6y=12\\ y=-4x+28\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-4x-6\left(-4x+28\right)=12\\y=-4x+28\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-4x+24x-168=12\\y=-4x+28\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}20x=12+168=180\\y=-4x+28\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{180}{20} = 9 \\ y=-4x+28\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = 9 \\ y=-4.(9)+28=-8\end{matrix}\right.\\ \qquad V=\{(9,-8)\}\)
11. \(\left\{\begin{matrix}-6x+6y=6\\3x+y=17\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-6x+6y=6\\ y=-3x+17\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-6x+6\left(-3x+17\right)=6\\y=-3x+17\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-6x-18x+102=6\\y=-3x+17\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-24x=6-102=-96\\y=-3x+17\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{-96}{-24} = 4 \\ y=-3x+17\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = 4 \\ y=-3.(4)+17=5\end{matrix}\right.\\ \qquad V=\{(4,5)\}\)
12. \(\left\{\begin{matrix}2y=22-2x\\-5x-y=-47\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}2x+2y=22\\-5x-y=-47\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}2x+2y=22\\ -5x+47=y\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}2x+2\left(-5x+47\right)=22\\y=-5x+47\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}2x-10x+94=22\\y=-5x+47\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-8x=22-94=-72\\y=-5x+47\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{-72}{-8} = 9 \\ y=-5x+47\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = 9 \\ y=-5.(9)+47=2\end{matrix}\right.\\ \qquad V=\{(9,2)\}\)