Stelsels substitutie

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Substitutie

1. \(\left\{\begin{matrix}5x-4y=-46\\-x+2y=20\end{matrix}\right.\)
2. \(\left\{\begin{matrix}4x-2y=-24\\6x=y-48\end{matrix}\right.\)
3. \(\left\{\begin{matrix}-5x+y=22\\-6x=-2y+20\end{matrix}\right.\)
4. \(\left\{\begin{matrix}-5y=-19-2x\\-x+2y=6\end{matrix}\right.\)
5. \(\left\{\begin{matrix}-4x+4y=32\\-3x+y=20\end{matrix}\right.\)
6. \(\left\{\begin{matrix}4x+3y=27\\-x-5y=6\end{matrix}\right.\)
7. \(\left\{\begin{matrix}-4y=-20+x\\-4x+2y=-8\end{matrix}\right.\)
8. \(\left\{\begin{matrix}x+2y=18\\4x=-5y+42\end{matrix}\right.\)
9. \(\left\{\begin{matrix}-3y=-6+6x\\-x+2y=19\end{matrix}\right.\)
10. \(\left\{\begin{matrix}y=-28-4x\\2x+5y=-32\end{matrix}\right.\)
11. \(\left\{\begin{matrix}5x-3y=26\\-4x-y=-31\end{matrix}\right.\)
12. \(\left\{\begin{matrix}-6x+6y=-6\\-x-3y=-33\end{matrix}\right.\)

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Substitutie

Verbetersleutel

1. \(\left\{\begin{matrix}5x-4y=-46\\-x+2y=20\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}5x-4y=-46\\ 2y-20=x\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}5\left(2y-20\right)-4y=-46\\x=2y-20\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}10y-100-4y=-46\\x=2y-20\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}6y=-46+100=54\\x=2y-20\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{54}{6} = 9 \\ x=2y-20\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = 9 \\ x=2.(9)-20=-2\end{matrix}\right.\\ \qquad V=\{(-2,9)\}\)
2. \(\left\{\begin{matrix}4x-2y=-24\\6x=y-48\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}4x-2y=-24\\6x-y=-48\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}4x-2y=-24\\ 6x+48=y\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}4x-2\left(6x+48\right)=-24\\y=6x+48\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}4x-12x-96=-24\\y=6x+48\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-8x=-24+96=72\\y=6x+48\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{72}{-8} = -9 \\ y=6x+48\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = -9 \\ y=6.(-9)+48=-6\end{matrix}\right.\\ \qquad V=\{(-9,-6)\}\)
3. \(\left\{\begin{matrix}-5x+y=22\\-6x=-2y+20\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-5x+y=22\\-6x+2y=20\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}y=5x+22\\ -6x+2y=20\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}y=5x+22\\ -6x+2\left(5x+22\right)=20\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}y=5x+22\\ -6x+10x+44=20\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}y=5x+22\\ 4x=20-44=-24\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=5x+22\\ x=\frac{-24}{4}=-6\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=5.(-6)+22=-8\\ x=-6\end{matrix}\right.\\ \qquad V=\{(-6,-8)\}\)
4. \(\left\{\begin{matrix}-5y=-19-2x\\-x+2y=6\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}2x-5y=-19\\-x+2y=6\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}2x-5y=-19\\ 2y-6=x\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}2\left(2y-6\right)-5y=-19\\x=2y-6\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}4y-12-5y=-19\\x=2y-6\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-y=-19+12=-7\\x=2y-6\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{-7}{-1} = 7 \\ x=2y-6\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = 7 \\ x=2.(7)-6=8\end{matrix}\right.\\ \qquad V=\{(8,7)\}\)
5. \(\left\{\begin{matrix}-4x+4y=32\\-3x+y=20\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-4x+4y=32\\ y=3x+20\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-4x+4\left(3x+20\right)=32\\y=3x+20\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-4x+12x+80=32\\y=3x+20\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}8x=32-80=-48\\y=3x+20\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{-48}{8} = -6 \\ y=3x+20\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = -6 \\ y=3.(-6)+20=2\end{matrix}\right.\\ \qquad V=\{(-6,2)\}\)
6. \(\left\{\begin{matrix}4x+3y=27\\-x-5y=6\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}4x+3y=27\\ -5y-6=x\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}4\left(-5y-6\right)+3y=27\\x=-5y-6\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-20y-24+3y=27\\x=-5y-6\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-17y=27+24=51\\x=-5y-6\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{51}{-17} = -3 \\ x=-5y-6\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = -3 \\ x=-5.(-3)-6=9\end{matrix}\right.\\ \qquad V=\{(9,-3)\}\)
7. \(\left\{\begin{matrix}-4y=-20+x\\-4x+2y=-8\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-x-4y=-20\\-4x+2y=-8\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}-4y+20=x\\-4x+2y=-8\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}x=-4y+20\\ -4.\left(-4y+20\right)+2y=-8\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}x=-4y+20\\ 16y-80+2y=-8\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}x=-4y+20\\ 18y=-8+80=72\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=-4y+20\\ y=\frac{72}{18}=4\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=-4.(4)+20=4\\ y=4\end{matrix}\right.\\ \qquad V=\{(4,4)\}\)
8. \(\left\{\begin{matrix}x+2y=18\\4x=-5y+42\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x+2y=18\\4x+5y=42\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}x=-2y+18\\ 4x+5y=42\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}x=-2y+18\\ 4.\left(-2y+18\right)+5y=42\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}x=-2y+18\\ -8y+72+5y=42\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}x=-2y+18\\ -3y=42-72=-30\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=-2y+18\\ y=\frac{-30}{-3}=10\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=-2.(10)+18=-2\\ y=10\end{matrix}\right.\\ \qquad V=\{(-2,10)\}\)
9. \(\left\{\begin{matrix}-3y=-6+6x\\-x+2y=19\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-6x-3y=-6\\-x+2y=19\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}-6x-3y=-6\\ 2y-19=x\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-6\left(2y-19\right)-3y=-6\\x=2y-19\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-12y+114-3y=-6\\x=2y-19\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-15y=-6-114=-120\\x=2y-19\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{-120}{-15} = 8 \\ x=2y-19\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = 8 \\ x=2.(8)-19=-3\end{matrix}\right.\\ \qquad V=\{(-3,8)\}\)
10. \(\left\{\begin{matrix}y=-28-4x\\2x+5y=-32\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}4x+y=-28\\2x+5y=-32\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}y=-4x-28\\ 2x+5y=-32\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}y=-4x-28\\ 2x+5\left(-4x-28\right)=-32\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}y=-4x-28\\ 2x-20x-140=-32\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}y=-4x-28\\ -18x=-32+140=108\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=-4x-28\\ x=\frac{108}{-18}=-6\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=-4.(-6)-28=-4\\ x=-6\end{matrix}\right.\\ \qquad V=\{(-6,-4)\}\)
11. \(\left\{\begin{matrix}5x-3y=26\\-4x-y=-31\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}5x-3y=26\\ -4x+31=y\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}5x-3\left(-4x+31\right)=26\\y=-4x+31\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}5x+12x-93=26\\y=-4x+31\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}17x=26+93=119\\y=-4x+31\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{119}{17} = 7 \\ y=-4x+31\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = 7 \\ y=-4.(7)+31=3\end{matrix}\right.\\ \qquad V=\{(7,3)\}\)
12. \(\left\{\begin{matrix}-6x+6y=-6\\-x-3y=-33\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-6x+6y=-6\\ -3y+33=x\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-6\left(-3y+33\right)+6y=-6\\x=-3y+33\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}18y-198+6y=-6\\x=-3y+33\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}24y=-6+198=192\\x=-3y+33\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{192}{24} = 8 \\ x=-3y+33\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = 8 \\ x=-3.(8)+33=9\end{matrix}\right.\\ \qquad V=\{(9,8)\}\)