Stelsels substitutie

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Substitutie

1. \(\left\{\begin{matrix}6x+2y=-64\\-x-4y=29\end{matrix}\right.\)
2. \(\left\{\begin{matrix}-2x-3y=23\\2x=y-19\end{matrix}\right.\)
3. \(\left\{\begin{matrix}y=-1-x\\6x-3y=-15\end{matrix}\right.\)
4. \(\left\{\begin{matrix}-2y=-10-5x\\-x+4y=-34\end{matrix}\right.\)
5. \(\left\{\begin{matrix}-5x-y=34\\4x-6y=0\end{matrix}\right.\)
6. \(\left\{\begin{matrix}6y=13+5x\\-x-5y=-16\end{matrix}\right.\)
7. \(\left\{\begin{matrix}-5y=-36-2x\\-x+6y=53\end{matrix}\right.\)
8. \(\left\{\begin{matrix}-5y=-15+5x\\-x-4y=12\end{matrix}\right.\)
9. \(\left\{\begin{matrix}x-3y=17\\3x-6y=42\end{matrix}\right.\)
10. \(\left\{\begin{matrix}-4x-4y=48\\5x=y+0\end{matrix}\right.\)
11. \(\left\{\begin{matrix}-3y=-27+6x\\-x-4y=20\end{matrix}\right.\)
12. \(\left\{\begin{matrix}-4x-6y=18\\x-2y=-15\end{matrix}\right.\)

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Substitutie

Verbetersleutel

1. \(\left\{\begin{matrix}6x+2y=-64\\-x-4y=29\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}6x+2y=-64\\ -4y-29=x\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}6\left(-4y-29\right)+2y=-64\\x=-4y-29\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-24y-174+2y=-64\\x=-4y-29\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-22y=-64+174=110\\x=-4y-29\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{110}{-22} = -5 \\ x=-4y-29\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = -5 \\ x=-4.(-5)-29=-9\end{matrix}\right.\\ \qquad V=\{(-9,-5)\}\)
2. \(\left\{\begin{matrix}-2x-3y=23\\2x=y-19\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-2x-3y=23\\2x-y=-19\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}-2x-3y=23\\ 2x+19=y\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-2x-3\left(2x+19\right)=23\\y=2x+19\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-2x-6x-57=23\\y=2x+19\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-8x=23+57=80\\y=2x+19\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{80}{-8} = -10 \\ y=2x+19\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = -10 \\ y=2.(-10)+19=-1\end{matrix}\right.\\ \qquad V=\{(-10,-1)\}\)
3. \(\left\{\begin{matrix}y=-1-x\\6x-3y=-15\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x+y=-1\\6x-3y=-15\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}x=-y-1\\ 6x-3y=-15\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}x=-y-1\\ 6.\left(-y-1\right)-3y=-15\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}x=-y-1\\ -6y-6-3y=-15\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}x=-y-1\\ -9y=-15+6=-9\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=-y-1\\ y=\frac{-9}{-9}=1\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=-.(1)-1=-2\\ y=1\end{matrix}\right.\\ \qquad V=\{(-2,1)\}\)
4. \(\left\{\begin{matrix}-2y=-10-5x\\-x+4y=-34\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}5x-2y=-10\\-x+4y=-34\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}5x-2y=-10\\ 4y+34=x\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}5\left(4y+34\right)-2y=-10\\x=4y+34\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}20y+170-2y=-10\\x=4y+34\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}18y=-10-170=-180\\x=4y+34\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{-180}{18} = -10 \\ x=4y+34\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = -10 \\ x=4.(-10)+34=-6\end{matrix}\right.\\ \qquad V=\{(-6,-10)\}\)
5. \(\left\{\begin{matrix}-5x-y=34\\4x-6y=0\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-5x-34=y\\4x-6y=0\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}y=-5x-34\\ 4x-6\left(-5x-34\right)=0\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}y=-5x-34\\ 4x+30x+204=0\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}y=-5x-34\\ 34x=0-204=-204\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=-5x-34\\ x=\frac{-204}{34}=-6\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=-5.(-6)-34=-4\\ x=-6\end{matrix}\right.\\ \qquad V=\{(-6,-4)\}\)
6. \(\left\{\begin{matrix}6y=13+5x\\-x-5y=-16\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-5x+6y=13\\-x-5y=-16\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}-5x+6y=13\\ -5y+16=x\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-5\left(-5y+16\right)+6y=13\\x=-5y+16\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}25y-80+6y=13\\x=-5y+16\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}31y=13+80=93\\x=-5y+16\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{93}{31} = 3 \\ x=-5y+16\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = 3 \\ x=-5.(3)+16=1\end{matrix}\right.\\ \qquad V=\{(1,3)\}\)
7. \(\left\{\begin{matrix}-5y=-36-2x\\-x+6y=53\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}2x-5y=-36\\-x+6y=53\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}2x-5y=-36\\ 6y-53=x\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}2\left(6y-53\right)-5y=-36\\x=6y-53\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}12y-106-5y=-36\\x=6y-53\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}7y=-36+106=70\\x=6y-53\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{70}{7} = 10 \\ x=6y-53\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = 10 \\ x=6.(10)-53=7\end{matrix}\right.\\ \qquad V=\{(7,10)\}\)
8. \(\left\{\begin{matrix}-5y=-15+5x\\-x-4y=12\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-5x-5y=-15\\-x-4y=12\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}-5x-5y=-15\\ -4y-12=x\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-5\left(-4y-12\right)-5y=-15\\x=-4y-12\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}20y+60-5y=-15\\x=-4y-12\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}15y=-15-60=-75\\x=-4y-12\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{-75}{15} = -5 \\ x=-4y-12\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = -5 \\ x=-4.(-5)-12=8\end{matrix}\right.\\ \qquad V=\{(8,-5)\}\)
9. \(\left\{\begin{matrix}x-3y=17\\3x-6y=42\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=3y+17\\ 3x-6y=42\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}x=3y+17\\ 3.\left(3y+17\right)-6y=42\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}x=3y+17\\ 9y+51-6y=42\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}x=3y+17\\ 3y=42-51=-9\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=3y+17\\ y=\frac{-9}{3}=-3\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=3.(-3)+17=8\\ y=-3\end{matrix}\right.\\ \qquad V=\{(8,-3)\}\)
10. \(\left\{\begin{matrix}-4x-4y=48\\5x=y+0\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-4x-4y=48\\5x-y=0\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}-4x-4y=48\\ 5x+0=y\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-4x-4\left(5x+0\right)=48\\y=5x+0\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-4x-20x+0=48\\y=5x+0\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-24x=48+0=48\\y=5x+0\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{48}{-24} = -2 \\ y=5x+0\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = -2 \\ y=5.(-2)+0=-10\end{matrix}\right.\\ \qquad V=\{(-2,-10)\}\)
11. \(\left\{\begin{matrix}-3y=-27+6x\\-x-4y=20\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-6x-3y=-27\\-x-4y=20\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}-6x-3y=-27\\ -4y-20=x\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-6\left(-4y-20\right)-3y=-27\\x=-4y-20\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}24y+120-3y=-27\\x=-4y-20\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}21y=-27-120=-147\\x=-4y-20\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{-147}{21} = -7 \\ x=-4y-20\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = -7 \\ x=-4.(-7)-20=8\end{matrix}\right.\\ \qquad V=\{(8,-7)\}\)
12. \(\left\{\begin{matrix}-4x-6y=18\\x-2y=-15\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-4x-6y=18\\ x=2y-15\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-4\left(2y-15\right)-6y=18\\x=2y-15\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-8y+60-6y=18\\x=2y-15\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-14y=18-60=-42\\x=2y-15\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{-42}{-14} = 3 \\ x=2y-15\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = 3 \\ x=2.(3)-15=-9\end{matrix}\right.\\ \qquad V=\{(-9,3)\}\)