Stelsels substitutie

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Substitutie

1. \(\left\{\begin{matrix}-4x+2y=-16\\3x=y+16\end{matrix}\right.\)
2. \(\left\{\begin{matrix}-4y=20-x\\3x-2y=20\end{matrix}\right.\)
3. \(\left\{\begin{matrix}6x+4y=2\\x-2y=3\end{matrix}\right.\)
4. \(\left\{\begin{matrix}-3x+y=8\\-2x-2y=-8\end{matrix}\right.\)
5. \(\left\{\begin{matrix}-4x-4y=40\\x+2y=-17\end{matrix}\right.\)
6. \(\left\{\begin{matrix}-2y=-44+5x\\3x+y=26\end{matrix}\right.\)
7. \(\left\{\begin{matrix}2x-3y=-36\\4x+y=-30\end{matrix}\right.\)
8. \(\left\{\begin{matrix}x-y=-6\\3x=-4y+31\end{matrix}\right.\)
9. \(\left\{\begin{matrix}-5x-6y=66\\-x=3y+24\end{matrix}\right.\)
10. \(\left\{\begin{matrix}-2x-4y=-34\\x-6y=-15\end{matrix}\right.\)
11. \(\left\{\begin{matrix}6x-6y=-84\\-5x+y=34\end{matrix}\right.\)
12. \(\left\{\begin{matrix}-5y=-7+x\\3x-2y=4\end{matrix}\right.\)

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Substitutie

Verbetersleutel

1. \(\left\{\begin{matrix}-4x+2y=-16\\3x=y+16\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-4x+2y=-16\\3x-y=16\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}-4x+2y=-16\\ 3x-16=y\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-4x+2\left(3x-16\right)=-16\\y=3x-16\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-4x+6x-32=-16\\y=3x-16\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}2x=-16+32=16\\y=3x-16\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{16}{2} = 8 \\ y=3x-16\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = 8 \\ y=3.(8)-16=8\end{matrix}\right.\\ \qquad V=\{(8,8)\}\)
2. \(\left\{\begin{matrix}-4y=20-x\\3x-2y=20\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x-4y=20\\3x-2y=20\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}x=4y+20\\ 3x-2y=20\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}x=4y+20\\ 3.\left(4y+20\right)-2y=20\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}x=4y+20\\ 12y+60-2y=20\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}x=4y+20\\ 10y=20-60=-40\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=4y+20\\ y=\frac{-40}{10}=-4\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=4.(-4)+20=4\\ y=-4\end{matrix}\right.\\ \qquad V=\{(4,-4)\}\)
3. \(\left\{\begin{matrix}6x+4y=2\\x-2y=3\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}6x+4y=2\\ x=2y+3\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}6\left(2y+3\right)+4y=2\\x=2y+3\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}12y+18+4y=2\\x=2y+3\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}16y=2-18=-16\\x=2y+3\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{-16}{16} = -1 \\ x=2y+3\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = -1 \\ x=2.(-1)+3=1\end{matrix}\right.\\ \qquad V=\{(1,-1)\}\)
4. \(\left\{\begin{matrix}-3x+y=8\\-2x-2y=-8\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=3x+8\\ -2x-2y=-8\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}y=3x+8\\ -2x-2\left(3x+8\right)=-8\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}y=3x+8\\ -2x-6x-16=-8\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}y=3x+8\\ -8x=-8+16=8\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=3x+8\\ x=\frac{8}{-8}=-1\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=3.(-1)+8=5\\ x=-1\end{matrix}\right.\\ \qquad V=\{(-1,5)\}\)
5. \(\left\{\begin{matrix}-4x-4y=40\\x+2y=-17\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-4x-4y=40\\ x=-2y-17\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-4\left(-2y-17\right)-4y=40\\x=-2y-17\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}8y+68-4y=40\\x=-2y-17\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}4y=40-68=-28\\x=-2y-17\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{-28}{4} = -7 \\ x=-2y-17\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = -7 \\ x=-2.(-7)-17=-3\end{matrix}\right.\\ \qquad V=\{(-3,-7)\}\)
6. \(\left\{\begin{matrix}-2y=-44+5x\\3x+y=26\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-5x-2y=-44\\3x+y=26\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}-5x-2y=-44\\ y=-3x+26\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-5x-2\left(-3x+26\right)=-44\\y=-3x+26\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-5x+6x-52=-44\\y=-3x+26\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}x=-44+52=8\\y=-3x+26\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{8}{1} = 8 \\ y=-3x+26\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = 8 \\ y=-3.(8)+26=2\end{matrix}\right.\\ \qquad V=\{(8,2)\}\)
7. \(\left\{\begin{matrix}2x-3y=-36\\4x+y=-30\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}2x-3y=-36\\ y=-4x-30\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}2x-3\left(-4x-30\right)=-36\\y=-4x-30\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}2x+12x+90=-36\\y=-4x-30\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}14x=-36-90=-126\\y=-4x-30\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{-126}{14} = -9 \\ y=-4x-30\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = -9 \\ y=-4.(-9)-30=6\end{matrix}\right.\\ \qquad V=\{(-9,6)\}\)
8. \(\left\{\begin{matrix}x-y=-6\\3x=-4y+31\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x-y=-6\\3x+4y=31\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}x=y-6\\ 3x+4y=31\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}x=y-6\\ 3.\left(y-6\right)+4y=31\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}x=y-6\\ 3y-18+4y=31\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}x=y-6\\ 7y=31+18=49\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=y-6\\ y=\frac{49}{7}=7\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=.(7)-6=1\\ y=7\end{matrix}\right.\\ \qquad V=\{(1,7)\}\)
9. \(\left\{\begin{matrix}-5x-6y=66\\-x=3y+24\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-5x-6y=66\\-x-3y=24\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}-5x-6y=66\\ -3y-24=x\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-5\left(-3y-24\right)-6y=66\\x=-3y-24\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}15y+120-6y=66\\x=-3y-24\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}9y=66-120=-54\\x=-3y-24\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{-54}{9} = -6 \\ x=-3y-24\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = -6 \\ x=-3.(-6)-24=-6\end{matrix}\right.\\ \qquad V=\{(-6,-6)\}\)
10. \(\left\{\begin{matrix}-2x-4y=-34\\x-6y=-15\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-2x-4y=-34\\ x=6y-15\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-2\left(6y-15\right)-4y=-34\\x=6y-15\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-12y+30-4y=-34\\x=6y-15\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-16y=-34-30=-64\\x=6y-15\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{-64}{-16} = 4 \\ x=6y-15\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = 4 \\ x=6.(4)-15=9\end{matrix}\right.\\ \qquad V=\{(9,4)\}\)
11. \(\left\{\begin{matrix}6x-6y=-84\\-5x+y=34\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}6x-6y=-84\\ y=5x+34\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}6x-6\left(5x+34\right)=-84\\y=5x+34\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}6x-30x-204=-84\\y=5x+34\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-24x=-84+204=120\\y=5x+34\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{120}{-24} = -5 \\ y=5x+34\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = -5 \\ y=5.(-5)+34=9\end{matrix}\right.\\ \qquad V=\{(-5,9)\}\)
12. \(\left\{\begin{matrix}-5y=-7+x\\3x-2y=4\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-x-5y=-7\\3x-2y=4\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}-5y+7=x\\3x-2y=4\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}x=-5y+7\\ 3.\left(-5y+7\right)-2y=4\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}x=-5y+7\\ -15y+21-2y=4\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}x=-5y+7\\ -17y=4-21=-17\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=-5y+7\\ y=\frac{-17}{-17}=1\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=-5.(1)+7=2\\ y=1\end{matrix}\right.\\ \qquad V=\{(2,1)\}\)