Stelsels substitutie

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Substitutie

1. \(\left\{\begin{matrix}5x-3y=29\\-6x-y=2\end{matrix}\right.\)
2. \(\left\{\begin{matrix}-2x-3y=15\\-2x+y=11\end{matrix}\right.\)
3. \(\left\{\begin{matrix}-x-y=-3\\6x+5y=9\end{matrix}\right.\)
4. \(\left\{\begin{matrix}-3x-y=-20\\3x=-2y+22\end{matrix}\right.\)
5. \(\left\{\begin{matrix}2y=0-x\\-5x+6y=-48\end{matrix}\right.\)
6. \(\left\{\begin{matrix}4y=-48+6x\\x-3y=15\end{matrix}\right.\)
7. \(\left\{\begin{matrix}-2y=44-4x\\-6x-y=-26\end{matrix}\right.\)
8. \(\left\{\begin{matrix}x-y=1\\-6x=4y-96\end{matrix}\right.\)
9. \(\left\{\begin{matrix}-5x-6y=-4\\x-3y=-37\end{matrix}\right.\)
10. \(\left\{\begin{matrix}4y=-12-5x\\-x-2y=0\end{matrix}\right.\)
11. \(\left\{\begin{matrix}x+3y=-16\\6x=-6y-24\end{matrix}\right.\)
12. \(\left\{\begin{matrix}5x-2y=41\\-4x=y-38\end{matrix}\right.\)

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Substitutie

Verbetersleutel

1. \(\left\{\begin{matrix}5x-3y=29\\-6x-y=2\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}5x-3y=29\\ -6x-2=y\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}5x-3\left(-6x-2\right)=29\\y=-6x-2\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}5x+18x+6=29\\y=-6x-2\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}23x=29-6=23\\y=-6x-2\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{23}{23} = 1 \\ y=-6x-2\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = 1 \\ y=-6.(1)-2=-8\end{matrix}\right.\\ \qquad V=\{(1,-8)\}\)
2. \(\left\{\begin{matrix}-2x-3y=15\\-2x+y=11\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-2x-3y=15\\ y=2x+11\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-2x-3\left(2x+11\right)=15\\y=2x+11\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-2x-6x-33=15\\y=2x+11\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-8x=15+33=48\\y=2x+11\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{48}{-8} = -6 \\ y=2x+11\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = -6 \\ y=2.(-6)+11=-1\end{matrix}\right.\\ \qquad V=\{(-6,-1)\}\)
3. \(\left\{\begin{matrix}-x-y=-3\\6x+5y=9\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-y+3=x\\6x+5y=9\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}x=-y+3\\ 6.\left(-y+3\right)+5y=9\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}x=-y+3\\ -6y+18+5y=9\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}x=-y+3\\ -y=9-18=-9\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=-y+3\\ y=\frac{-9}{-1}=9\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=-.(9)+3=-6\\ y=9\end{matrix}\right.\\ \qquad V=\{(-6,9)\}\)
4. \(\left\{\begin{matrix}-3x-y=-20\\3x=-2y+22\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-3x-y=-20\\3x+2y=22\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}-3x+20=y\\3x+2y=22\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}y=-3x+20\\ 3x+2\left(-3x+20\right)=22\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}y=-3x+20\\ 3x-6x+40=22\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}y=-3x+20\\ -3x=22-40=-18\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=-3x+20\\ x=\frac{-18}{-3}=6\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=-3.(6)+20=2\\ x=6\end{matrix}\right.\\ \qquad V=\{(6,2)\}\)
5. \(\left\{\begin{matrix}2y=0-x\\-5x+6y=-48\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x+2y=0\\-5x+6y=-48\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}x=-2y+0\\ -5x+6y=-48\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}x=-2y+0\\ -5.\left(-2y+0\right)+6y=-48\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}x=-2y+0\\ 10y+0+6y=-48\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}x=-2y+0\\ 16y=-48+0=-48\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=-2y+0\\ y=\frac{-48}{16}=-3\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=-2.(-3)+0=6\\ y=-3\end{matrix}\right.\\ \qquad V=\{(6,-3)\}\)
6. \(\left\{\begin{matrix}4y=-48+6x\\x-3y=15\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-6x+4y=-48\\x-3y=15\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}-6x+4y=-48\\ x=3y+15\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-6\left(3y+15\right)+4y=-48\\x=3y+15\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-18y-90+4y=-48\\x=3y+15\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-14y=-48+90=42\\x=3y+15\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{42}{-14} = -3 \\ x=3y+15\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = -3 \\ x=3.(-3)+15=6\end{matrix}\right.\\ \qquad V=\{(6,-3)\}\)
7. \(\left\{\begin{matrix}-2y=44-4x\\-6x-y=-26\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}4x-2y=44\\-6x-y=-26\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}4x-2y=44\\ -6x+26=y\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}4x-2\left(-6x+26\right)=44\\y=-6x+26\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}4x+12x-52=44\\y=-6x+26\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}16x=44+52=96\\y=-6x+26\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{96}{16} = 6 \\ y=-6x+26\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = 6 \\ y=-6.(6)+26=-10\end{matrix}\right.\\ \qquad V=\{(6,-10)\}\)
8. \(\left\{\begin{matrix}x-y=1\\-6x=4y-96\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x-y=1\\-6x-4y=-96\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}x=y+1\\ -6x-4y=-96\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}x=y+1\\ -6.\left(y+1\right)-4y=-96\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}x=y+1\\ -6y-6-4y=-96\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}x=y+1\\ -10y=-96+6=-90\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=y+1\\ y=\frac{-90}{-10}=9\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=.(9)+1=10\\ y=9\end{matrix}\right.\\ \qquad V=\{(10,9)\}\)
9. \(\left\{\begin{matrix}-5x-6y=-4\\x-3y=-37\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-5x-6y=-4\\ x=3y-37\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-5\left(3y-37\right)-6y=-4\\x=3y-37\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-15y+185-6y=-4\\x=3y-37\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-21y=-4-185=-189\\x=3y-37\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{-189}{-21} = 9 \\ x=3y-37\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = 9 \\ x=3.(9)-37=-10\end{matrix}\right.\\ \qquad V=\{(-10,9)\}\)
10. \(\left\{\begin{matrix}4y=-12-5x\\-x-2y=0\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}5x+4y=-12\\-x-2y=0\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}5x+4y=-12\\ -2y+0=x\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}5\left(-2y+0\right)+4y=-12\\x=-2y+0\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-10y+0+4y=-12\\x=-2y+0\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-6y=-12+0=-12\\x=-2y+0\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{-12}{-6} = 2 \\ x=-2y+0\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = 2 \\ x=-2.(2)+0=-4\end{matrix}\right.\\ \qquad V=\{(-4,2)\}\)
11. \(\left\{\begin{matrix}x+3y=-16\\6x=-6y-24\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x+3y=-16\\6x+6y=-24\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}x=-3y-16\\ 6x+6y=-24\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}x=-3y-16\\ 6.\left(-3y-16\right)+6y=-24\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}x=-3y-16\\ -18y-96+6y=-24\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}x=-3y-16\\ -12y=-24+96=72\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=-3y-16\\ y=\frac{72}{-12}=-6\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=-3.(-6)-16=2\\ y=-6\end{matrix}\right.\\ \qquad V=\{(2,-6)\}\)
12. \(\left\{\begin{matrix}5x-2y=41\\-4x=y-38\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}5x-2y=41\\-4x-y=-38\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}5x-2y=41\\ -4x+38=y\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}5x-2\left(-4x+38\right)=41\\y=-4x+38\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}5x+8x-76=41\\y=-4x+38\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}13x=41+76=117\\y=-4x+38\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{117}{13} = 9 \\ y=-4x+38\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = 9 \\ y=-4.(9)+38=2\end{matrix}\right.\\ \qquad V=\{(9,2)\}\)