Stelsels substitutie

Hoofdmenu Eentje per keer  🖨

Substitutie

1. \(\left\{\begin{matrix}4y=29+x\\-6x-6y=-6\end{matrix}\right.\)
2. \(\left\{\begin{matrix}-4x+5y=-57\\-4x=-y-21\end{matrix}\right.\)
3. \(\left\{\begin{matrix}-6y=-51-5x\\-3x+y=28\end{matrix}\right.\)
4. \(\left\{\begin{matrix}6x+5y=50\\3x+y=19\end{matrix}\right.\)
5. \(\left\{\begin{matrix}-3y=9+4x\\4x+y=-19\end{matrix}\right.\)
6. \(\left\{\begin{matrix}6x+y=-25\\-2x=-3y+25\end{matrix}\right.\)
7. \(\left\{\begin{matrix}y=-19-3x\\-6x-4y=58\end{matrix}\right.\)
8. \(\left\{\begin{matrix}6x-2y=32\\-3x-y=-8\end{matrix}\right.\)
9. \(\left\{\begin{matrix}5y=-69-3x\\-6x+y=39\end{matrix}\right.\)
10. \(\left\{\begin{matrix}-y=9+5x\\-3x-6y=0\end{matrix}\right.\)
11. \(\left\{\begin{matrix}-2x-2y=2\\-5x-y=25\end{matrix}\right.\)
12. \(\left\{\begin{matrix}-x+6y=55\\-4x-2y=12\end{matrix}\right.\)

---

Substitutie

Verbetersleutel

1. \(\left\{\begin{matrix}4y=29+x\\-6x-6y=-6\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-x+4y=29\\-6x-6y=-6\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}4y-29=x\\-6x-6y=-6\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}x=4y-29\\ -6.\left(4y-29\right)-6y=-6\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}x=4y-29\\ -24y+174-6y=-6\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}x=4y-29\\ -30y=-6-174=-180\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=4y-29\\ y=\frac{-180}{-30}=6\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=4.(6)-29=-5\\ y=6\end{matrix}\right.\\ \qquad V=\{(-5,6)\}\)
2. \(\left\{\begin{matrix}-4x+5y=-57\\-4x=-y-21\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-4x+5y=-57\\-4x+y=-21\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}-4x+5y=-57\\ y=4x-21\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-4x+5\left(4x-21\right)=-57\\y=4x-21\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-4x+20x-105=-57\\y=4x-21\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}16x=-57+105=48\\y=4x-21\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{48}{16} = 3 \\ y=4x-21\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = 3 \\ y=4.(3)-21=-9\end{matrix}\right.\\ \qquad V=\{(3,-9)\}\)
3. \(\left\{\begin{matrix}-6y=-51-5x\\-3x+y=28\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}5x-6y=-51\\-3x+y=28\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}5x-6y=-51\\ y=3x+28\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}5x-6\left(3x+28\right)=-51\\y=3x+28\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}5x-18x-168=-51\\y=3x+28\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-13x=-51+168=117\\y=3x+28\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{117}{-13} = -9 \\ y=3x+28\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = -9 \\ y=3.(-9)+28=1\end{matrix}\right.\\ \qquad V=\{(-9,1)\}\)
4. \(\left\{\begin{matrix}6x+5y=50\\3x+y=19\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}6x+5y=50\\ y=-3x+19\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}6x+5\left(-3x+19\right)=50\\y=-3x+19\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}6x-15x+95=50\\y=-3x+19\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-9x=50-95=-45\\y=-3x+19\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{-45}{-9} = 5 \\ y=-3x+19\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = 5 \\ y=-3.(5)+19=4\end{matrix}\right.\\ \qquad V=\{(5,4)\}\)
5. \(\left\{\begin{matrix}-3y=9+4x\\4x+y=-19\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-4x-3y=9\\4x+y=-19\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}-4x-3y=9\\ y=-4x-19\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-4x-3\left(-4x-19\right)=9\\y=-4x-19\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-4x+12x+57=9\\y=-4x-19\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}8x=9-57=-48\\y=-4x-19\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{-48}{8} = -6 \\ y=-4x-19\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = -6 \\ y=-4.(-6)-19=5\end{matrix}\right.\\ \qquad V=\{(-6,5)\}\)
6. \(\left\{\begin{matrix}6x+y=-25\\-2x=-3y+25\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}6x+y=-25\\-2x+3y=25\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}y=-6x-25\\ -2x+3y=25\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}y=-6x-25\\ -2x+3\left(-6x-25\right)=25\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}y=-6x-25\\ -2x-18x-75=25\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}y=-6x-25\\ -20x=25+75=100\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=-6x-25\\ x=\frac{100}{-20}=-5\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=-6.(-5)-25=5\\ x=-5\end{matrix}\right.\\ \qquad V=\{(-5,5)\}\)
7. \(\left\{\begin{matrix}y=-19-3x\\-6x-4y=58\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}3x+y=-19\\-6x-4y=58\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}y=-3x-19\\ -6x-4y=58\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}y=-3x-19\\ -6x-4\left(-3x-19\right)=58\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}y=-3x-19\\ -6x+12x+76=58\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}y=-3x-19\\ 6x=58-76=-18\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=-3x-19\\ x=\frac{-18}{6}=-3\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=-3.(-3)-19=-10\\ x=-3\end{matrix}\right.\\ \qquad V=\{(-3,-10)\}\)
8. \(\left\{\begin{matrix}6x-2y=32\\-3x-y=-8\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}6x-2y=32\\ -3x+8=y\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}6x-2\left(-3x+8\right)=32\\y=-3x+8\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}6x+6x-16=32\\y=-3x+8\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}12x=32+16=48\\y=-3x+8\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{48}{12} = 4 \\ y=-3x+8\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = 4 \\ y=-3.(4)+8=-4\end{matrix}\right.\\ \qquad V=\{(4,-4)\}\)
9. \(\left\{\begin{matrix}5y=-69-3x\\-6x+y=39\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}3x+5y=-69\\-6x+y=39\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}3x+5y=-69\\ y=6x+39\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}3x+5\left(6x+39\right)=-69\\y=6x+39\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}3x+30x+195=-69\\y=6x+39\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}33x=-69-195=-264\\y=6x+39\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{-264}{33} = -8 \\ y=6x+39\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = -8 \\ y=6.(-8)+39=-9\end{matrix}\right.\\ \qquad V=\{(-8,-9)\}\)
10. \(\left\{\begin{matrix}-y=9+5x\\-3x-6y=0\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-5x-y=9\\-3x-6y=0\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}-5x-9=y\\-3x-6y=0\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}y=-5x-9\\ -3x-6\left(-5x-9\right)=0\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}y=-5x-9\\ -3x+30x+54=0\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}y=-5x-9\\ 27x=0-54=-54\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=-5x-9\\ x=\frac{-54}{27}=-2\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=-5.(-2)-9=1\\ x=-2\end{matrix}\right.\\ \qquad V=\{(-2,1)\}\)
11. \(\left\{\begin{matrix}-2x-2y=2\\-5x-y=25\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-2x-2y=2\\ -5x-25=y\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-2x-2\left(-5x-25\right)=2\\y=-5x-25\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-2x+10x+50=2\\y=-5x-25\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}8x=2-50=-48\\y=-5x-25\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{-48}{8} = -6 \\ y=-5x-25\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = -6 \\ y=-5.(-6)-25=5\end{matrix}\right.\\ \qquad V=\{(-6,5)\}\)
12. \(\left\{\begin{matrix}-x+6y=55\\-4x-2y=12\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}6y-55=x\\-4x-2y=12\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}x=6y-55\\ -4.\left(6y-55\right)-2y=12\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}x=6y-55\\ -24y+220-2y=12\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}x=6y-55\\ -26y=12-220=-208\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=6y-55\\ y=\frac{-208}{-26}=8\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=6.(8)-55=-7\\ y=8\end{matrix}\right.\\ \qquad V=\{(-7,8)\}\)