Stelsels substitutie

Hoofdmenu Eentje per keer  🖨

Substitutie

1. \(\left\{\begin{matrix}2x+2y=-16\\-5x+y=34\end{matrix}\right.\)
2. \(\left\{\begin{matrix}-3y=-9+3x\\x-2y=15\end{matrix}\right.\)
3. \(\left\{\begin{matrix}6x+5y=-61\\6x=-y-41\end{matrix}\right.\)
4. \(\left\{\begin{matrix}-6y=84+3x\\-2x+y=6\end{matrix}\right.\)
5. \(\left\{\begin{matrix}-4x-2y=8\\-6x+y=-12\end{matrix}\right.\)
6. \(\left\{\begin{matrix}5x-2y=-26\\-x+6y=-34\end{matrix}\right.\)
7. \(\left\{\begin{matrix}6x-y=43\\-4x-2y=-26\end{matrix}\right.\)
8. \(\left\{\begin{matrix}-4x-5y=52\\x+3y=-20\end{matrix}\right.\)
9. \(\left\{\begin{matrix}-4x+y=-6\\-4x=-5y+34\end{matrix}\right.\)
10. \(\left\{\begin{matrix}-4y=-57-5x\\2x-y=-21\end{matrix}\right.\)
11. \(\left\{\begin{matrix}-4x+2y=-26\\-x=3y-3\end{matrix}\right.\)
12. \(\left\{\begin{matrix}-6y=-5+x\\-5x+3y=41\end{matrix}\right.\)

---

Substitutie

Verbetersleutel

1. \(\left\{\begin{matrix}2x+2y=-16\\-5x+y=34\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}2x+2y=-16\\ y=5x+34\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}2x+2\left(5x+34\right)=-16\\y=5x+34\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}2x+10x+68=-16\\y=5x+34\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}12x=-16-68=-84\\y=5x+34\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{-84}{12} = -7 \\ y=5x+34\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = -7 \\ y=5.(-7)+34=-1\end{matrix}\right.\\ \qquad V=\{(-7,-1)\}\)
2. \(\left\{\begin{matrix}-3y=-9+3x\\x-2y=15\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-3x-3y=-9\\x-2y=15\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}-3x-3y=-9\\ x=2y+15\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-3\left(2y+15\right)-3y=-9\\x=2y+15\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-6y-45-3y=-9\\x=2y+15\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-9y=-9+45=36\\x=2y+15\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{36}{-9} = -4 \\ x=2y+15\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = -4 \\ x=2.(-4)+15=7\end{matrix}\right.\\ \qquad V=\{(7,-4)\}\)
3. \(\left\{\begin{matrix}6x+5y=-61\\6x=-y-41\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}6x+5y=-61\\6x+y=-41\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}6x+5y=-61\\ y=-6x-41\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}6x+5\left(-6x-41\right)=-61\\y=-6x-41\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}6x-30x-205=-61\\y=-6x-41\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-24x=-61+205=144\\y=-6x-41\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{144}{-24} = -6 \\ y=-6x-41\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = -6 \\ y=-6.(-6)-41=-5\end{matrix}\right.\\ \qquad V=\{(-6,-5)\}\)
4. \(\left\{\begin{matrix}-6y=84+3x\\-2x+y=6\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-3x-6y=84\\-2x+y=6\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}-3x-6y=84\\ y=2x+6\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-3x-6\left(2x+6\right)=84\\y=2x+6\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-3x-12x-36=84\\y=2x+6\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-15x=84+36=120\\y=2x+6\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{120}{-15} = -8 \\ y=2x+6\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = -8 \\ y=2.(-8)+6=-10\end{matrix}\right.\\ \qquad V=\{(-8,-10)\}\)
5. \(\left\{\begin{matrix}-4x-2y=8\\-6x+y=-12\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-4x-2y=8\\ y=6x-12\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-4x-2\left(6x-12\right)=8\\y=6x-12\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-4x-12x+24=8\\y=6x-12\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-16x=8-24=-16\\y=6x-12\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{-16}{-16} = 1 \\ y=6x-12\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = 1 \\ y=6.(1)-12=-6\end{matrix}\right.\\ \qquad V=\{(1,-6)\}\)
6. \(\left\{\begin{matrix}5x-2y=-26\\-x+6y=-34\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}5x-2y=-26\\ 6y+34=x\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}5\left(6y+34\right)-2y=-26\\x=6y+34\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}30y+170-2y=-26\\x=6y+34\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}28y=-26-170=-196\\x=6y+34\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{-196}{28} = -7 \\ x=6y+34\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = -7 \\ x=6.(-7)+34=-8\end{matrix}\right.\\ \qquad V=\{(-8,-7)\}\)
7. \(\left\{\begin{matrix}6x-y=43\\-4x-2y=-26\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}6x-43=y\\-4x-2y=-26\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}y=6x-43\\ -4x-2\left(6x-43\right)=-26\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}y=6x-43\\ -4x-12x+86=-26\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}y=6x-43\\ -16x=-26-86=-112\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=6x-43\\ x=\frac{-112}{-16}=7\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=6.(7)-43=-1\\ x=7\end{matrix}\right.\\ \qquad V=\{(7,-1)\}\)
8. \(\left\{\begin{matrix}-4x-5y=52\\x+3y=-20\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-4x-5y=52\\ x=-3y-20\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-4\left(-3y-20\right)-5y=52\\x=-3y-20\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}12y+80-5y=52\\x=-3y-20\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}7y=52-80=-28\\x=-3y-20\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{-28}{7} = -4 \\ x=-3y-20\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = -4 \\ x=-3.(-4)-20=-8\end{matrix}\right.\\ \qquad V=\{(-8,-4)\}\)
9. \(\left\{\begin{matrix}-4x+y=-6\\-4x=-5y+34\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-4x+y=-6\\-4x+5y=34\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}y=4x-6\\ -4x+5y=34\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}y=4x-6\\ -4x+5\left(4x-6\right)=34\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}y=4x-6\\ -4x+20x-30=34\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}y=4x-6\\ 16x=34+30=64\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=4x-6\\ x=\frac{64}{16}=4\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=4.(4)-6=10\\ x=4\end{matrix}\right.\\ \qquad V=\{(4,10)\}\)
10. \(\left\{\begin{matrix}-4y=-57-5x\\2x-y=-21\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}5x-4y=-57\\2x-y=-21\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}5x-4y=-57\\ 2x+21=y\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}5x-4\left(2x+21\right)=-57\\y=2x+21\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}5x-8x-84=-57\\y=2x+21\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-3x=-57+84=27\\y=2x+21\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{27}{-3} = -9 \\ y=2x+21\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = -9 \\ y=2.(-9)+21=3\end{matrix}\right.\\ \qquad V=\{(-9,3)\}\)
11. \(\left\{\begin{matrix}-4x+2y=-26\\-x=3y-3\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-4x+2y=-26\\-x-3y=-3\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}-4x+2y=-26\\ -3y+3=x\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-4\left(-3y+3\right)+2y=-26\\x=-3y+3\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}12y-12+2y=-26\\x=-3y+3\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}14y=-26+12=-14\\x=-3y+3\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{-14}{14} = -1 \\ x=-3y+3\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = -1 \\ x=-3.(-1)+3=6\end{matrix}\right.\\ \qquad V=\{(6,-1)\}\)
12. \(\left\{\begin{matrix}-6y=-5+x\\-5x+3y=41\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-x-6y=-5\\-5x+3y=41\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}-6y+5=x\\-5x+3y=41\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}x=-6y+5\\ -5.\left(-6y+5\right)+3y=41\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}x=-6y+5\\ 30y-25+3y=41\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}x=-6y+5\\ 33y=41+25=66\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=-6y+5\\ y=\frac{66}{33}=2\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=-6.(2)+5=-7\\ y=2\end{matrix}\right.\\ \qquad V=\{(-7,2)\}\)