Stelsels substitutie

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Substitutie

1. \(\left\{\begin{matrix}y=-23+3x\\5x-6y=73\end{matrix}\right.\)
2. \(\left\{\begin{matrix}-4y=10-6x\\x+5y=-21\end{matrix}\right.\)
3. \(\left\{\begin{matrix}-y=-40+5x\\6x+5y=29\end{matrix}\right.\)
4. \(\left\{\begin{matrix}2x+y=-10\\3x+5y=-22\end{matrix}\right.\)
5. \(\left\{\begin{matrix}-4x-2y=28\\-4x=-y+10\end{matrix}\right.\)
6. \(\left\{\begin{matrix}2x+2y=-22\\-x=-3y-25\end{matrix}\right.\)
7. \(\left\{\begin{matrix}y=-2+x\\3x-3y=6\end{matrix}\right.\)
8. \(\left\{\begin{matrix}-3y=11+4x\\2x+y=-7\end{matrix}\right.\)
9. \(\left\{\begin{matrix}x+6y=58\\-3x+3y=36\end{matrix}\right.\)
10. \(\left\{\begin{matrix}6x+2y=-50\\x+4y=-12\end{matrix}\right.\)
11. \(\left\{\begin{matrix}6x-2y=-38\\x=6y-46\end{matrix}\right.\)
12. \(\left\{\begin{matrix}-x-2y=-6\\4x=4y-24\end{matrix}\right.\)

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Substitutie

Verbetersleutel

1. \(\left\{\begin{matrix}y=-23+3x\\5x-6y=73\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-3x+y=-23\\5x-6y=73\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}y=3x-23\\ 5x-6y=73\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}y=3x-23\\ 5x-6\left(3x-23\right)=73\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}y=3x-23\\ 5x-18x+138=73\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}y=3x-23\\ -13x=73-138=-65\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=3x-23\\ x=\frac{-65}{-13}=5\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=3.(5)-23=-8\\ x=5\end{matrix}\right.\\ \qquad V=\{(5,-8)\}\)
2. \(\left\{\begin{matrix}-4y=10-6x\\x+5y=-21\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}6x-4y=10\\x+5y=-21\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}6x-4y=10\\ x=-5y-21\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}6\left(-5y-21\right)-4y=10\\x=-5y-21\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-30y-126-4y=10\\x=-5y-21\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-34y=10+126=136\\x=-5y-21\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{136}{-34} = -4 \\ x=-5y-21\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = -4 \\ x=-5.(-4)-21=-1\end{matrix}\right.\\ \qquad V=\{(-1,-4)\}\)
3. \(\left\{\begin{matrix}-y=-40+5x\\6x+5y=29\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-5x-y=-40\\6x+5y=29\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}-5x+40=y\\6x+5y=29\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}y=-5x+40\\ 6x+5\left(-5x+40\right)=29\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}y=-5x+40\\ 6x-25x+200=29\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}y=-5x+40\\ -19x=29-200=-171\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=-5x+40\\ x=\frac{-171}{-19}=9\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=-5.(9)+40=-5\\ x=9\end{matrix}\right.\\ \qquad V=\{(9,-5)\}\)
4. \(\left\{\begin{matrix}2x+y=-10\\3x+5y=-22\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=-2x-10\\ 3x+5y=-22\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}y=-2x-10\\ 3x+5\left(-2x-10\right)=-22\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}y=-2x-10\\ 3x-10x-50=-22\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}y=-2x-10\\ -7x=-22+50=28\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=-2x-10\\ x=\frac{28}{-7}=-4\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=-2.(-4)-10=-2\\ x=-4\end{matrix}\right.\\ \qquad V=\{(-4,-2)\}\)
5. \(\left\{\begin{matrix}-4x-2y=28\\-4x=-y+10\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-4x-2y=28\\-4x+y=10\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}-4x-2y=28\\ y=4x+10\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-4x-2\left(4x+10\right)=28\\y=4x+10\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-4x-8x-20=28\\y=4x+10\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-12x=28+20=48\\y=4x+10\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{48}{-12} = -4 \\ y=4x+10\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = -4 \\ y=4.(-4)+10=-6\end{matrix}\right.\\ \qquad V=\{(-4,-6)\}\)
6. \(\left\{\begin{matrix}2x+2y=-22\\-x=-3y-25\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}2x+2y=-22\\-x+3y=-25\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}2x+2y=-22\\ 3y+25=x\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}2\left(3y+25\right)+2y=-22\\x=3y+25\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}6y+50+2y=-22\\x=3y+25\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}8y=-22-50=-72\\x=3y+25\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{-72}{8} = -9 \\ x=3y+25\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = -9 \\ x=3.(-9)+25=-2\end{matrix}\right.\\ \qquad V=\{(-2,-9)\}\)
7. \(\left\{\begin{matrix}y=-2+x\\3x-3y=6\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-x+y=-2\\3x-3y=6\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}y+2=x\\3x-3y=6\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}x=y+2\\ 3.\left(y+2\right)-3y=6\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}x=y+2\\ 3y+6-3y=6\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}x=y+2\\ 0y=6-6=0\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=y+2\\ y=\frac{0}{0}=8\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=.(8)+2=10\\ y=8\end{matrix}\right.\\ \qquad V=\{(10,8)\}\)
8. \(\left\{\begin{matrix}-3y=11+4x\\2x+y=-7\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-4x-3y=11\\2x+y=-7\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}-4x-3y=11\\ y=-2x-7\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-4x-3\left(-2x-7\right)=11\\y=-2x-7\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-4x+6x+21=11\\y=-2x-7\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}2x=11-21=-10\\y=-2x-7\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{-10}{2} = -5 \\ y=-2x-7\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = -5 \\ y=-2.(-5)-7=3\end{matrix}\right.\\ \qquad V=\{(-5,3)\}\)
9. \(\left\{\begin{matrix}x+6y=58\\-3x+3y=36\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=-6y+58\\ -3x+3y=36\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}x=-6y+58\\ -3.\left(-6y+58\right)+3y=36\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}x=-6y+58\\ 18y-174+3y=36\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}x=-6y+58\\ 21y=36+174=210\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=-6y+58\\ y=\frac{210}{21}=10\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=-6.(10)+58=-2\\ y=10\end{matrix}\right.\\ \qquad V=\{(-2,10)\}\)
10. \(\left\{\begin{matrix}6x+2y=-50\\x+4y=-12\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}6x+2y=-50\\ x=-4y-12\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}6\left(-4y-12\right)+2y=-50\\x=-4y-12\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-24y-72+2y=-50\\x=-4y-12\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-22y=-50+72=22\\x=-4y-12\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{22}{-22} = -1 \\ x=-4y-12\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = -1 \\ x=-4.(-1)-12=-8\end{matrix}\right.\\ \qquad V=\{(-8,-1)\}\)
11. \(\left\{\begin{matrix}6x-2y=-38\\x=6y-46\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}6x-2y=-38\\x-6y=-46\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}6x-2y=-38\\ x=6y-46\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}6\left(6y-46\right)-2y=-38\\x=6y-46\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}36y-276-2y=-38\\x=6y-46\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}34y=-38+276=238\\x=6y-46\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{238}{34} = 7 \\ x=6y-46\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = 7 \\ x=6.(7)-46=-4\end{matrix}\right.\\ \qquad V=\{(-4,7)\}\)
12. \(\left\{\begin{matrix}-x-2y=-6\\4x=4y-24\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-x-2y=-6\\4x-4y=-24\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}-2y+6=x\\4x-4y=-24\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}x=-2y+6\\ 4.\left(-2y+6\right)-4y=-24\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}x=-2y+6\\ -8y+24-4y=-24\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}x=-2y+6\\ -12y=-24-24=-48\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=-2y+6\\ y=\frac{-48}{-12}=4\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=-2.(4)+6=-2\\ y=4\end{matrix}\right.\\ \qquad V=\{(-2,4)\}\)