Stelsels substitutie

Hoofdmenu Eentje per keer  🖨

Substitutie

1. \(\left\{\begin{matrix}-2x-5y=59\\x+2y=-25\end{matrix}\right.\)
2. \(\left\{\begin{matrix}-y=6+3x\\5x-3y=-38\end{matrix}\right.\)
3. \(\left\{\begin{matrix}-4y=10-6x\\x+3y=-35\end{matrix}\right.\)
4. \(\left\{\begin{matrix}5y=-38+4x\\-x+3y=-13\end{matrix}\right.\)
5. \(\left\{\begin{matrix}5y=-7-4x\\-6x+y=-49\end{matrix}\right.\)
6. \(\left\{\begin{matrix}-3x-5y=-14\\-x=3y-2\end{matrix}\right.\)
7. \(\left\{\begin{matrix}x-5y=15\\-6x=2y+70\end{matrix}\right.\)
8. \(\left\{\begin{matrix}4x-2y=-56\\-x=5y-30\end{matrix}\right.\)
9. \(\left\{\begin{matrix}-x-3y=-19\\-4x=2y-16\end{matrix}\right.\)
10. \(\left\{\begin{matrix}6y=-48+4x\\x+3y=-33\end{matrix}\right.\)
11. \(\left\{\begin{matrix}-x+3y=-12\\2x+4y=-6\end{matrix}\right.\)
12. \(\left\{\begin{matrix}-5x+2y=-55\\x+6y=-21\end{matrix}\right.\)

---

Substitutie

Verbetersleutel

1. \(\left\{\begin{matrix}-2x-5y=59\\x+2y=-25\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-2x-5y=59\\ x=-2y-25\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-2\left(-2y-25\right)-5y=59\\x=-2y-25\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}4y+50-5y=59\\x=-2y-25\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-y=59-50=9\\x=-2y-25\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{9}{-1} = -9 \\ x=-2y-25\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = -9 \\ x=-2.(-9)-25=-7\end{matrix}\right.\\ \qquad V=\{(-7,-9)\}\)
2. \(\left\{\begin{matrix}-y=6+3x\\5x-3y=-38\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-3x-y=6\\5x-3y=-38\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}-3x-6=y\\5x-3y=-38\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}y=-3x-6\\ 5x-3\left(-3x-6\right)=-38\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}y=-3x-6\\ 5x+9x+18=-38\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}y=-3x-6\\ 14x=-38-18=-56\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=-3x-6\\ x=\frac{-56}{14}=-4\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=-3.(-4)-6=6\\ x=-4\end{matrix}\right.\\ \qquad V=\{(-4,6)\}\)
3. \(\left\{\begin{matrix}-4y=10-6x\\x+3y=-35\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}6x-4y=10\\x+3y=-35\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}6x-4y=10\\ x=-3y-35\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}6\left(-3y-35\right)-4y=10\\x=-3y-35\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-18y-210-4y=10\\x=-3y-35\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-22y=10+210=220\\x=-3y-35\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{220}{-22} = -10 \\ x=-3y-35\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = -10 \\ x=-3.(-10)-35=-5\end{matrix}\right.\\ \qquad V=\{(-5,-10)\}\)
4. \(\left\{\begin{matrix}5y=-38+4x\\-x+3y=-13\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-4x+5y=-38\\-x+3y=-13\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}-4x+5y=-38\\ 3y+13=x\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-4\left(3y+13\right)+5y=-38\\x=3y+13\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-12y-52+5y=-38\\x=3y+13\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-7y=-38+52=14\\x=3y+13\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{14}{-7} = -2 \\ x=3y+13\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = -2 \\ x=3.(-2)+13=7\end{matrix}\right.\\ \qquad V=\{(7,-2)\}\)
5. \(\left\{\begin{matrix}5y=-7-4x\\-6x+y=-49\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}4x+5y=-7\\-6x+y=-49\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}4x+5y=-7\\ y=6x-49\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}4x+5\left(6x-49\right)=-7\\y=6x-49\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}4x+30x-245=-7\\y=6x-49\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}34x=-7+245=238\\y=6x-49\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{238}{34} = 7 \\ y=6x-49\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = 7 \\ y=6.(7)-49=-7\end{matrix}\right.\\ \qquad V=\{(7,-7)\}\)
6. \(\left\{\begin{matrix}-3x-5y=-14\\-x=3y-2\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-3x-5y=-14\\-x-3y=-2\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}-3x-5y=-14\\ -3y+2=x\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-3\left(-3y+2\right)-5y=-14\\x=-3y+2\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}9y-6-5y=-14\\x=-3y+2\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}4y=-14+6=-8\\x=-3y+2\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{-8}{4} = -2 \\ x=-3y+2\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = -2 \\ x=-3.(-2)+2=8\end{matrix}\right.\\ \qquad V=\{(8,-2)\}\)
7. \(\left\{\begin{matrix}x-5y=15\\-6x=2y+70\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x-5y=15\\-6x-2y=70\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}x=5y+15\\ -6x-2y=70\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}x=5y+15\\ -6.\left(5y+15\right)-2y=70\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}x=5y+15\\ -30y-90-2y=70\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}x=5y+15\\ -32y=70+90=160\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=5y+15\\ y=\frac{160}{-32}=-5\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=5.(-5)+15=-10\\ y=-5\end{matrix}\right.\\ \qquad V=\{(-10,-5)\}\)
8. \(\left\{\begin{matrix}4x-2y=-56\\-x=5y-30\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}4x-2y=-56\\-x-5y=-30\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}4x-2y=-56\\ -5y+30=x\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}4\left(-5y+30\right)-2y=-56\\x=-5y+30\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-20y+120-2y=-56\\x=-5y+30\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-22y=-56-120=-176\\x=-5y+30\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{-176}{-22} = 8 \\ x=-5y+30\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = 8 \\ x=-5.(8)+30=-10\end{matrix}\right.\\ \qquad V=\{(-10,8)\}\)
9. \(\left\{\begin{matrix}-x-3y=-19\\-4x=2y-16\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-x-3y=-19\\-4x-2y=-16\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}-3y+19=x\\-4x-2y=-16\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}x=-3y+19\\ -4.\left(-3y+19\right)-2y=-16\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}x=-3y+19\\ 12y-76-2y=-16\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}x=-3y+19\\ 10y=-16+76=60\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=-3y+19\\ y=\frac{60}{10}=6\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=-3.(6)+19=1\\ y=6\end{matrix}\right.\\ \qquad V=\{(1,6)\}\)
10. \(\left\{\begin{matrix}6y=-48+4x\\x+3y=-33\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-4x+6y=-48\\x+3y=-33\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}-4x+6y=-48\\ x=-3y-33\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-4\left(-3y-33\right)+6y=-48\\x=-3y-33\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}12y+132+6y=-48\\x=-3y-33\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}18y=-48-132=-180\\x=-3y-33\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{-180}{18} = -10 \\ x=-3y-33\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = -10 \\ x=-3.(-10)-33=-3\end{matrix}\right.\\ \qquad V=\{(-3,-10)\}\)
11. \(\left\{\begin{matrix}-x+3y=-12\\2x+4y=-6\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}3y+12=x\\2x+4y=-6\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}x=3y+12\\ 2.\left(3y+12\right)+4y=-6\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}x=3y+12\\ 6y+24+4y=-6\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}x=3y+12\\ 10y=-6-24=-30\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=3y+12\\ y=\frac{-30}{10}=-3\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=3.(-3)+12=3\\ y=-3\end{matrix}\right.\\ \qquad V=\{(3,-3)\}\)
12. \(\left\{\begin{matrix}-5x+2y=-55\\x+6y=-21\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-5x+2y=-55\\ x=-6y-21\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-5\left(-6y-21\right)+2y=-55\\x=-6y-21\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}30y+105+2y=-55\\x=-6y-21\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}32y=-55-105=-160\\x=-6y-21\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{-160}{32} = -5 \\ x=-6y-21\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = -5 \\ x=-6.(-5)-21=9\end{matrix}\right.\\ \qquad V=\{(9,-5)\}\)