Stelsels substitutie

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Substitutie

1. \(\left\{\begin{matrix}-3y=-40+x\\4x+5y=90\end{matrix}\right.\)
2. \(\left\{\begin{matrix}-5x+2y=-17\\x+6y=61\end{matrix}\right.\)
3. \(\left\{\begin{matrix}y=18+2x\\-6x-3y=18\end{matrix}\right.\)
4. \(\left\{\begin{matrix}-x-4y=34\\-6x+2y=22\end{matrix}\right.\)
5. \(\left\{\begin{matrix}5x+3y=-65\\x=2y+0\end{matrix}\right.\)
6. \(\left\{\begin{matrix}-6y=-36+2x\\x+2y=15\end{matrix}\right.\)
7. \(\left\{\begin{matrix}-x-5y=41\\-2x-3y=33\end{matrix}\right.\)
8. \(\left\{\begin{matrix}-x+5y=55\\-4x-2y=0\end{matrix}\right.\)
9. \(\left\{\begin{matrix}-5x-2y=4\\3x=-y-4\end{matrix}\right.\)
10. \(\left\{\begin{matrix}-4y=-11-3x\\-x+5y=0\end{matrix}\right.\)
11. \(\left\{\begin{matrix}x+6y=-23\\-2x-3y=10\end{matrix}\right.\)
12. \(\left\{\begin{matrix}-6x+6y=78\\-5x=-y+29\end{matrix}\right.\)

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Substitutie

Verbetersleutel

1. \(\left\{\begin{matrix}-3y=-40+x\\4x+5y=90\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-x-3y=-40\\4x+5y=90\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}-3y+40=x\\4x+5y=90\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}x=-3y+40\\ 4.\left(-3y+40\right)+5y=90\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}x=-3y+40\\ -12y+160+5y=90\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}x=-3y+40\\ -7y=90-160=-70\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=-3y+40\\ y=\frac{-70}{-7}=10\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=-3.(10)+40=10\\ y=10\end{matrix}\right.\\ \qquad V=\{(10,10)\}\)
2. \(\left\{\begin{matrix}-5x+2y=-17\\x+6y=61\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-5x+2y=-17\\ x=-6y+61\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-5\left(-6y+61\right)+2y=-17\\x=-6y+61\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}30y-305+2y=-17\\x=-6y+61\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}32y=-17+305=288\\x=-6y+61\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{288}{32} = 9 \\ x=-6y+61\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = 9 \\ x=-6.(9)+61=7\end{matrix}\right.\\ \qquad V=\{(7,9)\}\)
3. \(\left\{\begin{matrix}y=18+2x\\-6x-3y=18\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-2x+y=18\\-6x-3y=18\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}y=2x+18\\ -6x-3y=18\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}y=2x+18\\ -6x-3\left(2x+18\right)=18\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}y=2x+18\\ -6x-6x-54=18\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}y=2x+18\\ -12x=18+54=72\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=2x+18\\ x=\frac{72}{-12}=-6\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=2.(-6)+18=6\\ x=-6\end{matrix}\right.\\ \qquad V=\{(-6,6)\}\)
4. \(\left\{\begin{matrix}-x-4y=34\\-6x+2y=22\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-4y-34=x\\-6x+2y=22\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}x=-4y-34\\ -6.\left(-4y-34\right)+2y=22\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}x=-4y-34\\ 24y+204+2y=22\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}x=-4y-34\\ 26y=22-204=-182\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=-4y-34\\ y=\frac{-182}{26}=-7\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=-4.(-7)-34=-6\\ y=-7\end{matrix}\right.\\ \qquad V=\{(-6,-7)\}\)
5. \(\left\{\begin{matrix}5x+3y=-65\\x=2y+0\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}5x+3y=-65\\x-2y=0\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}5x+3y=-65\\ x=2y+0\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}5\left(2y+0\right)+3y=-65\\x=2y+0\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}10y+0+3y=-65\\x=2y+0\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}13y=-65+0=-65\\x=2y+0\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{-65}{13} = -5 \\ x=2y+0\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = -5 \\ x=2.(-5)+0=-10\end{matrix}\right.\\ \qquad V=\{(-10,-5)\}\)
6. \(\left\{\begin{matrix}-6y=-36+2x\\x+2y=15\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-2x-6y=-36\\x+2y=15\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}-2x-6y=-36\\ x=-2y+15\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-2\left(-2y+15\right)-6y=-36\\x=-2y+15\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}4y-30-6y=-36\\x=-2y+15\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-2y=-36+30=-6\\x=-2y+15\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{-6}{-2} = 3 \\ x=-2y+15\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = 3 \\ x=-2.(3)+15=9\end{matrix}\right.\\ \qquad V=\{(9,3)\}\)
7. \(\left\{\begin{matrix}-x-5y=41\\-2x-3y=33\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-5y-41=x\\-2x-3y=33\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}x=-5y-41\\ -2.\left(-5y-41\right)-3y=33\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}x=-5y-41\\ 10y+82-3y=33\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}x=-5y-41\\ 7y=33-82=-49\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=-5y-41\\ y=\frac{-49}{7}=-7\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=-5.(-7)-41=-6\\ y=-7\end{matrix}\right.\\ \qquad V=\{(-6,-7)\}\)
8. \(\left\{\begin{matrix}-x+5y=55\\-4x-2y=0\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}5y-55=x\\-4x-2y=0\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}x=5y-55\\ -4.\left(5y-55\right)-2y=0\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}x=5y-55\\ -20y+220-2y=0\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}x=5y-55\\ -22y=0-220=-220\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=5y-55\\ y=\frac{-220}{-22}=10\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=5.(10)-55=-5\\ y=10\end{matrix}\right.\\ \qquad V=\{(-5,10)\}\)
9. \(\left\{\begin{matrix}-5x-2y=4\\3x=-y-4\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-5x-2y=4\\3x+y=-4\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}-5x-2y=4\\ y=-3x-4\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-5x-2\left(-3x-4\right)=4\\y=-3x-4\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-5x+6x+8=4\\y=-3x-4\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}x=4-8=-4\\y=-3x-4\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{-4}{1} = -4 \\ y=-3x-4\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = -4 \\ y=-3.(-4)-4=8\end{matrix}\right.\\ \qquad V=\{(-4,8)\}\)
10. \(\left\{\begin{matrix}-4y=-11-3x\\-x+5y=0\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}3x-4y=-11\\-x+5y=0\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}3x-4y=-11\\ 5y+0=x\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}3\left(5y+0\right)-4y=-11\\x=5y+0\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}15y+0-4y=-11\\x=5y+0\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}11y=-11+0=-11\\x=5y+0\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{-11}{11} = -1 \\ x=5y+0\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = -1 \\ x=5.(-1)+0=-5\end{matrix}\right.\\ \qquad V=\{(-5,-1)\}\)
11. \(\left\{\begin{matrix}x+6y=-23\\-2x-3y=10\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=-6y-23\\ -2x-3y=10\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}x=-6y-23\\ -2.\left(-6y-23\right)-3y=10\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}x=-6y-23\\ 12y+46-3y=10\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}x=-6y-23\\ 9y=10-46=-36\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=-6y-23\\ y=\frac{-36}{9}=-4\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=-6.(-4)-23=1\\ y=-4\end{matrix}\right.\\ \qquad V=\{(1,-4)\}\)
12. \(\left\{\begin{matrix}-6x+6y=78\\-5x=-y+29\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-6x+6y=78\\-5x+y=29\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}-6x+6y=78\\ y=5x+29\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-6x+6\left(5x+29\right)=78\\y=5x+29\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-6x+30x+174=78\\y=5x+29\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}24x=78-174=-96\\y=5x+29\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{-96}{24} = -4 \\ y=5x+29\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = -4 \\ y=5.(-4)+29=9\end{matrix}\right.\\ \qquad V=\{(-4,9)\}\)