Stelsels substitutie

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Substitutie

1. \(\left\{\begin{matrix}-3y=1-4x\\-x+5y=-13\end{matrix}\right.\)
2. \(\left\{\begin{matrix}-4x-3y=-66\\x+2y=29\end{matrix}\right.\)
3. \(\left\{\begin{matrix}6x+4y=-12\\6x=y-42\end{matrix}\right.\)
4. \(\left\{\begin{matrix}-3x+2y=19\\4x-y=-22\end{matrix}\right.\)
5. \(\left\{\begin{matrix}-5y=48+x\\2x+4y=-48\end{matrix}\right.\)
6. \(\left\{\begin{matrix}3x+2y=15\\-5x=-y-12\end{matrix}\right.\)
7. \(\left\{\begin{matrix}3y=63+4x\\x+5y=36\end{matrix}\right.\)
8. \(\left\{\begin{matrix}2y=-2-4x\\4x+y=-11\end{matrix}\right.\)
9. \(\left\{\begin{matrix}6x-4y=-50\\-x-6y=-5\end{matrix}\right.\)
10. \(\left\{\begin{matrix}2y=-6+x\\4x-3y=19\end{matrix}\right.\)
11. \(\left\{\begin{matrix}2x+y=5\\6x=3y+57\end{matrix}\right.\)
12. \(\left\{\begin{matrix}-4y=18+6x\\-x+6y=-57\end{matrix}\right.\)

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Substitutie

Verbetersleutel

1. \(\left\{\begin{matrix}-3y=1-4x\\-x+5y=-13\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}4x-3y=1\\-x+5y=-13\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}4x-3y=1\\ 5y+13=x\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}4\left(5y+13\right)-3y=1\\x=5y+13\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}20y+52-3y=1\\x=5y+13\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}17y=1-52=-51\\x=5y+13\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{-51}{17} = -3 \\ x=5y+13\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = -3 \\ x=5.(-3)+13=-2\end{matrix}\right.\\ \qquad V=\{(-2,-3)\}\)
2. \(\left\{\begin{matrix}-4x-3y=-66\\x+2y=29\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-4x-3y=-66\\ x=-2y+29\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-4\left(-2y+29\right)-3y=-66\\x=-2y+29\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}8y-116-3y=-66\\x=-2y+29\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}5y=-66+116=50\\x=-2y+29\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{50}{5} = 10 \\ x=-2y+29\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = 10 \\ x=-2.(10)+29=9\end{matrix}\right.\\ \qquad V=\{(9,10)\}\)
3. \(\left\{\begin{matrix}6x+4y=-12\\6x=y-42\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}6x+4y=-12\\6x-y=-42\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}6x+4y=-12\\ 6x+42=y\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}6x+4\left(6x+42\right)=-12\\y=6x+42\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}6x+24x+168=-12\\y=6x+42\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}30x=-12-168=-180\\y=6x+42\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{-180}{30} = -6 \\ y=6x+42\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = -6 \\ y=6.(-6)+42=6\end{matrix}\right.\\ \qquad V=\{(-6,6)\}\)
4. \(\left\{\begin{matrix}-3x+2y=19\\4x-y=-22\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-3x+2y=19\\ 4x+22=y\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-3x+2\left(4x+22\right)=19\\y=4x+22\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-3x+8x+44=19\\y=4x+22\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}5x=19-44=-25\\y=4x+22\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{-25}{5} = -5 \\ y=4x+22\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = -5 \\ y=4.(-5)+22=2\end{matrix}\right.\\ \qquad V=\{(-5,2)\}\)
5. \(\left\{\begin{matrix}-5y=48+x\\2x+4y=-48\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-x-5y=48\\2x+4y=-48\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}-5y-48=x\\2x+4y=-48\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}x=-5y-48\\ 2.\left(-5y-48\right)+4y=-48\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}x=-5y-48\\ -10y-96+4y=-48\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}x=-5y-48\\ -6y=-48+96=48\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=-5y-48\\ y=\frac{48}{-6}=-8\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=-5.(-8)-48=-8\\ y=-8\end{matrix}\right.\\ \qquad V=\{(-8,-8)\}\)
6. \(\left\{\begin{matrix}3x+2y=15\\-5x=-y-12\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}3x+2y=15\\-5x+y=-12\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}3x+2y=15\\ y=5x-12\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}3x+2\left(5x-12\right)=15\\y=5x-12\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}3x+10x-24=15\\y=5x-12\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}13x=15+24=39\\y=5x-12\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{39}{13} = 3 \\ y=5x-12\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = 3 \\ y=5.(3)-12=3\end{matrix}\right.\\ \qquad V=\{(3,3)\}\)
7. \(\left\{\begin{matrix}3y=63+4x\\x+5y=36\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-4x+3y=63\\x+5y=36\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}-4x+3y=63\\ x=-5y+36\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-4\left(-5y+36\right)+3y=63\\x=-5y+36\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}20y-144+3y=63\\x=-5y+36\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}23y=63+144=207\\x=-5y+36\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{207}{23} = 9 \\ x=-5y+36\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = 9 \\ x=-5.(9)+36=-9\end{matrix}\right.\\ \qquad V=\{(-9,9)\}\)
8. \(\left\{\begin{matrix}2y=-2-4x\\4x+y=-11\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}4x+2y=-2\\4x+y=-11\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}4x+2y=-2\\ y=-4x-11\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}4x+2\left(-4x-11\right)=-2\\y=-4x-11\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}4x-8x-22=-2\\y=-4x-11\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-4x=-2+22=20\\y=-4x-11\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{20}{-4} = -5 \\ y=-4x-11\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = -5 \\ y=-4.(-5)-11=9\end{matrix}\right.\\ \qquad V=\{(-5,9)\}\)
9. \(\left\{\begin{matrix}6x-4y=-50\\-x-6y=-5\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}6x-4y=-50\\ -6y+5=x\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}6\left(-6y+5\right)-4y=-50\\x=-6y+5\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-36y+30-4y=-50\\x=-6y+5\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-40y=-50-30=-80\\x=-6y+5\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{-80}{-40} = 2 \\ x=-6y+5\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = 2 \\ x=-6.(2)+5=-7\end{matrix}\right.\\ \qquad V=\{(-7,2)\}\)
10. \(\left\{\begin{matrix}2y=-6+x\\4x-3y=19\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-x+2y=-6\\4x-3y=19\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}2y+6=x\\4x-3y=19\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}x=2y+6\\ 4.\left(2y+6\right)-3y=19\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}x=2y+6\\ 8y+24-3y=19\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}x=2y+6\\ 5y=19-24=-5\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=2y+6\\ y=\frac{-5}{5}=-1\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=2.(-1)+6=4\\ y=-1\end{matrix}\right.\\ \qquad V=\{(4,-1)\}\)
11. \(\left\{\begin{matrix}2x+y=5\\6x=3y+57\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}2x+y=5\\6x-3y=57\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}y=-2x+5\\ 6x-3y=57\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}y=-2x+5\\ 6x-3\left(-2x+5\right)=57\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}y=-2x+5\\ 6x+6x-15=57\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}y=-2x+5\\ 12x=57+15=72\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=-2x+5\\ x=\frac{72}{12}=6\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=-2.(6)+5=-7\\ x=6\end{matrix}\right.\\ \qquad V=\{(6,-7)\}\)
12. \(\left\{\begin{matrix}-4y=18+6x\\-x+6y=-57\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-6x-4y=18\\-x+6y=-57\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}-6x-4y=18\\ 6y+57=x\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-6\left(6y+57\right)-4y=18\\x=6y+57\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-36y-342-4y=18\\x=6y+57\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-40y=18+342=360\\x=6y+57\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{360}{-40} = -9 \\ x=6y+57\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = -9 \\ x=6.(-9)+57=3\end{matrix}\right.\\ \qquad V=\{(3,-9)\}\)