Stelsels substitutie

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Substitutie

1. \(\left\{\begin{matrix}-2y=22+5x\\-x-3y=7\end{matrix}\right.\)
2. \(\left\{\begin{matrix}-5x+5y=-60\\4x+y=33\end{matrix}\right.\)
3. \(\left\{\begin{matrix}-3x+y=-4\\-3x=6y+45\end{matrix}\right.\)
4. \(\left\{\begin{matrix}x-y=14\\5x=-5y+0\end{matrix}\right.\)
5. \(\left\{\begin{matrix}-3y=-16+5x\\x-6y=-43\end{matrix}\right.\)
6. \(\left\{\begin{matrix}y=14-x\\4x+4y=56\end{matrix}\right.\)
7. \(\left\{\begin{matrix}3x+5y=-28\\-4x=-y-24\end{matrix}\right.\)
8. \(\left\{\begin{matrix}4x-3y=-3\\x=5y-39\end{matrix}\right.\)
9. \(\left\{\begin{matrix}-x-6y=42\\2x=4y+12\end{matrix}\right.\)
10. \(\left\{\begin{matrix}-6x-3y=-9\\-x=4y+23\end{matrix}\right.\)
11. \(\left\{\begin{matrix}4x-4y=-12\\x+3y=5\end{matrix}\right.\)
12. \(\left\{\begin{matrix}x-2y=30\\5x+4y=10\end{matrix}\right.\)

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Substitutie

Verbetersleutel

1. \(\left\{\begin{matrix}-2y=22+5x\\-x-3y=7\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-5x-2y=22\\-x-3y=7\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}-5x-2y=22\\ -3y-7=x\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-5\left(-3y-7\right)-2y=22\\x=-3y-7\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}15y+35-2y=22\\x=-3y-7\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}13y=22-35=-13\\x=-3y-7\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{-13}{13} = -1 \\ x=-3y-7\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = -1 \\ x=-3.(-1)-7=-4\end{matrix}\right.\\ \qquad V=\{(-4,-1)\}\)
2. \(\left\{\begin{matrix}-5x+5y=-60\\4x+y=33\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-5x+5y=-60\\ y=-4x+33\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-5x+5\left(-4x+33\right)=-60\\y=-4x+33\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-5x-20x+165=-60\\y=-4x+33\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-25x=-60-165=-225\\y=-4x+33\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{-225}{-25} = 9 \\ y=-4x+33\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = 9 \\ y=-4.(9)+33=-3\end{matrix}\right.\\ \qquad V=\{(9,-3)\}\)
3. \(\left\{\begin{matrix}-3x+y=-4\\-3x=6y+45\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-3x+y=-4\\-3x-6y=45\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}y=3x-4\\ -3x-6y=45\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}y=3x-4\\ -3x-6\left(3x-4\right)=45\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}y=3x-4\\ -3x-18x+24=45\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}y=3x-4\\ -21x=45-24=21\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=3x-4\\ x=\frac{21}{-21}=-1\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=3.(-1)-4=-7\\ x=-1\end{matrix}\right.\\ \qquad V=\{(-1,-7)\}\)
4. \(\left\{\begin{matrix}x-y=14\\5x=-5y+0\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x-y=14\\5x+5y=0\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}x=y+14\\ 5x+5y=0\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}x=y+14\\ 5.\left(y+14\right)+5y=0\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}x=y+14\\ 5y+70+5y=0\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}x=y+14\\ 10y=0-70=-70\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=y+14\\ y=\frac{-70}{10}=-7\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=.(-7)+14=7\\ y=-7\end{matrix}\right.\\ \qquad V=\{(7,-7)\}\)
5. \(\left\{\begin{matrix}-3y=-16+5x\\x-6y=-43\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-5x-3y=-16\\x-6y=-43\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}-5x-3y=-16\\ x=6y-43\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-5\left(6y-43\right)-3y=-16\\x=6y-43\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-30y+215-3y=-16\\x=6y-43\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-33y=-16-215=-231\\x=6y-43\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{-231}{-33} = 7 \\ x=6y-43\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = 7 \\ x=6.(7)-43=-1\end{matrix}\right.\\ \qquad V=\{(-1,7)\}\)
6. \(\left\{\begin{matrix}y=14-x\\4x+4y=56\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x+y=14\\4x+4y=56\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}x=-y+14\\ 4x+4y=56\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}x=-y+14\\ 4.\left(-y+14\right)+4y=56\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}x=-y+14\\ -4y+56+4y=56\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}x=-y+14\\ 0y=56-56=0\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=-y+14\\ y=\frac{0}{0}=9\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=-.(9)+14=5\\ y=9\end{matrix}\right.\\ \qquad V=\{(5,9)\}\)
7. \(\left\{\begin{matrix}3x+5y=-28\\-4x=-y-24\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}3x+5y=-28\\-4x+y=-24\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}3x+5y=-28\\ y=4x-24\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}3x+5\left(4x-24\right)=-28\\y=4x-24\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}3x+20x-120=-28\\y=4x-24\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}23x=-28+120=92\\y=4x-24\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{92}{23} = 4 \\ y=4x-24\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = 4 \\ y=4.(4)-24=-8\end{matrix}\right.\\ \qquad V=\{(4,-8)\}\)
8. \(\left\{\begin{matrix}4x-3y=-3\\x=5y-39\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}4x-3y=-3\\x-5y=-39\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}4x-3y=-3\\ x=5y-39\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}4\left(5y-39\right)-3y=-3\\x=5y-39\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}20y-156-3y=-3\\x=5y-39\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}17y=-3+156=153\\x=5y-39\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{153}{17} = 9 \\ x=5y-39\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = 9 \\ x=5.(9)-39=6\end{matrix}\right.\\ \qquad V=\{(6,9)\}\)
9. \(\left\{\begin{matrix}-x-6y=42\\2x=4y+12\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-x-6y=42\\2x-4y=12\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}-6y-42=x\\2x-4y=12\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}x=-6y-42\\ 2.\left(-6y-42\right)-4y=12\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}x=-6y-42\\ -12y-84-4y=12\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}x=-6y-42\\ -16y=12+84=96\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=-6y-42\\ y=\frac{96}{-16}=-6\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=-6.(-6)-42=-6\\ y=-6\end{matrix}\right.\\ \qquad V=\{(-6,-6)\}\)
10. \(\left\{\begin{matrix}-6x-3y=-9\\-x=4y+23\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-6x-3y=-9\\-x-4y=23\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}-6x-3y=-9\\ -4y-23=x\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-6\left(-4y-23\right)-3y=-9\\x=-4y-23\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}24y+138-3y=-9\\x=-4y-23\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}21y=-9-138=-147\\x=-4y-23\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{-147}{21} = -7 \\ x=-4y-23\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = -7 \\ x=-4.(-7)-23=5\end{matrix}\right.\\ \qquad V=\{(5,-7)\}\)
11. \(\left\{\begin{matrix}4x-4y=-12\\x+3y=5\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}4x-4y=-12\\ x=-3y+5\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}4\left(-3y+5\right)-4y=-12\\x=-3y+5\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-12y+20-4y=-12\\x=-3y+5\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-16y=-12-20=-32\\x=-3y+5\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{-32}{-16} = 2 \\ x=-3y+5\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = 2 \\ x=-3.(2)+5=-1\end{matrix}\right.\\ \qquad V=\{(-1,2)\}\)
12. \(\left\{\begin{matrix}x-2y=30\\5x+4y=10\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=2y+30\\ 5x+4y=10\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}x=2y+30\\ 5.\left(2y+30\right)+4y=10\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}x=2y+30\\ 10y+150+4y=10\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}x=2y+30\\ 14y=10-150=-140\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=2y+30\\ y=\frac{-140}{14}=-10\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=2.(-10)+30=10\\ y=-10\end{matrix}\right.\\ \qquad V=\{(10,-10)\}\)