Stelsels substitutie

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Substitutie

1. \(\left\{\begin{matrix}-x-5y=7\\5x=-6y+22\end{matrix}\right.\)
2. \(\left\{\begin{matrix}4x+3y=-19\\2x=-y-7\end{matrix}\right.\)
3. \(\left\{\begin{matrix}4x-5y=13\\x=-2y+13\end{matrix}\right.\)
4. \(\left\{\begin{matrix}-6x-6y=-54\\-4x+y=-31\end{matrix}\right.\)
5. \(\left\{\begin{matrix}-5y=10-6x\\3x-y=11\end{matrix}\right.\)
6. \(\left\{\begin{matrix}-2y=14+2x\\3x+y=-9\end{matrix}\right.\)
7. \(\left\{\begin{matrix}3x+6y=54\\x-3y=-12\end{matrix}\right.\)
8. \(\left\{\begin{matrix}-5x+2y=-26\\4x=-y+13\end{matrix}\right.\)
9. \(\left\{\begin{matrix}5y=4-3x\\6x+y=17\end{matrix}\right.\)
10. \(\left\{\begin{matrix}x+4y=30\\3x=6y-18\end{matrix}\right.\)
11. \(\left\{\begin{matrix}6x+3y=-36\\-4x-y=20\end{matrix}\right.\)
12. \(\left\{\begin{matrix}-5x+6y=38\\x+5y=-20\end{matrix}\right.\)

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Substitutie

Verbetersleutel

1. \(\left\{\begin{matrix}-x-5y=7\\5x=-6y+22\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-x-5y=7\\5x+6y=22\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}-5y-7=x\\5x+6y=22\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}x=-5y-7\\ 5.\left(-5y-7\right)+6y=22\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}x=-5y-7\\ -25y-35+6y=22\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}x=-5y-7\\ -19y=22+35=57\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=-5y-7\\ y=\frac{57}{-19}=-3\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=-5.(-3)-7=8\\ y=-3\end{matrix}\right.\\ \qquad V=\{(8,-3)\}\)
2. \(\left\{\begin{matrix}4x+3y=-19\\2x=-y-7\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}4x+3y=-19\\2x+y=-7\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}4x+3y=-19\\ y=-2x-7\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}4x+3\left(-2x-7\right)=-19\\y=-2x-7\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}4x-6x-21=-19\\y=-2x-7\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-2x=-19+21=2\\y=-2x-7\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{2}{-2} = -1 \\ y=-2x-7\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = -1 \\ y=-2.(-1)-7=-5\end{matrix}\right.\\ \qquad V=\{(-1,-5)\}\)
3. \(\left\{\begin{matrix}4x-5y=13\\x=-2y+13\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}4x-5y=13\\x+2y=13\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}4x-5y=13\\ x=-2y+13\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}4\left(-2y+13\right)-5y=13\\x=-2y+13\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-8y+52-5y=13\\x=-2y+13\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-13y=13-52=-39\\x=-2y+13\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{-39}{-13} = 3 \\ x=-2y+13\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = 3 \\ x=-2.(3)+13=7\end{matrix}\right.\\ \qquad V=\{(7,3)\}\)
4. \(\left\{\begin{matrix}-6x-6y=-54\\-4x+y=-31\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-6x-6y=-54\\ y=4x-31\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-6x-6\left(4x-31\right)=-54\\y=4x-31\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-6x-24x+186=-54\\y=4x-31\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-30x=-54-186=-240\\y=4x-31\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{-240}{-30} = 8 \\ y=4x-31\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = 8 \\ y=4.(8)-31=1\end{matrix}\right.\\ \qquad V=\{(8,1)\}\)
5. \(\left\{\begin{matrix}-5y=10-6x\\3x-y=11\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}6x-5y=10\\3x-y=11\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}6x-5y=10\\ 3x-11=y\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}6x-5\left(3x-11\right)=10\\y=3x-11\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}6x-15x+55=10\\y=3x-11\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-9x=10-55=-45\\y=3x-11\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{-45}{-9} = 5 \\ y=3x-11\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = 5 \\ y=3.(5)-11=4\end{matrix}\right.\\ \qquad V=\{(5,4)\}\)
6. \(\left\{\begin{matrix}-2y=14+2x\\3x+y=-9\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-2x-2y=14\\3x+y=-9\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}-2x-2y=14\\ y=-3x-9\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-2x-2\left(-3x-9\right)=14\\y=-3x-9\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-2x+6x+18=14\\y=-3x-9\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}4x=14-18=-4\\y=-3x-9\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{-4}{4} = -1 \\ y=-3x-9\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = -1 \\ y=-3.(-1)-9=-6\end{matrix}\right.\\ \qquad V=\{(-1,-6)\}\)
7. \(\left\{\begin{matrix}3x+6y=54\\x-3y=-12\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}3x+6y=54\\ x=3y-12\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}3\left(3y-12\right)+6y=54\\x=3y-12\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}9y-36+6y=54\\x=3y-12\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}15y=54+36=90\\x=3y-12\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{90}{15} = 6 \\ x=3y-12\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = 6 \\ x=3.(6)-12=6\end{matrix}\right.\\ \qquad V=\{(6,6)\}\)
8. \(\left\{\begin{matrix}-5x+2y=-26\\4x=-y+13\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-5x+2y=-26\\4x+y=13\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}-5x+2y=-26\\ y=-4x+13\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-5x+2\left(-4x+13\right)=-26\\y=-4x+13\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-5x-8x+26=-26\\y=-4x+13\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-13x=-26-26=-52\\y=-4x+13\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{-52}{-13} = 4 \\ y=-4x+13\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = 4 \\ y=-4.(4)+13=-3\end{matrix}\right.\\ \qquad V=\{(4,-3)\}\)
9. \(\left\{\begin{matrix}5y=4-3x\\6x+y=17\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}3x+5y=4\\6x+y=17\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}3x+5y=4\\ y=-6x+17\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}3x+5\left(-6x+17\right)=4\\y=-6x+17\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}3x-30x+85=4\\y=-6x+17\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-27x=4-85=-81\\y=-6x+17\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{-81}{-27} = 3 \\ y=-6x+17\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = 3 \\ y=-6.(3)+17=-1\end{matrix}\right.\\ \qquad V=\{(3,-1)\}\)
10. \(\left\{\begin{matrix}x+4y=30\\3x=6y-18\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x+4y=30\\3x-6y=-18\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}x=-4y+30\\ 3x-6y=-18\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}x=-4y+30\\ 3.\left(-4y+30\right)-6y=-18\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}x=-4y+30\\ -12y+90-6y=-18\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}x=-4y+30\\ -18y=-18-90=-108\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=-4y+30\\ y=\frac{-108}{-18}=6\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=-4.(6)+30=6\\ y=6\end{matrix}\right.\\ \qquad V=\{(6,6)\}\)
11. \(\left\{\begin{matrix}6x+3y=-36\\-4x-y=20\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}6x+3y=-36\\ -4x-20=y\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}6x+3\left(-4x-20\right)=-36\\y=-4x-20\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}6x-12x-60=-36\\y=-4x-20\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-6x=-36+60=24\\y=-4x-20\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{24}{-6} = -4 \\ y=-4x-20\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = -4 \\ y=-4.(-4)-20=-4\end{matrix}\right.\\ \qquad V=\{(-4,-4)\}\)
12. \(\left\{\begin{matrix}-5x+6y=38\\x+5y=-20\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-5x+6y=38\\ x=-5y-20\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-5\left(-5y-20\right)+6y=38\\x=-5y-20\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}25y+100+6y=38\\x=-5y-20\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}31y=38-100=-62\\x=-5y-20\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{-62}{31} = -2 \\ x=-5y-20\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = -2 \\ x=-5.(-2)-20=-10\end{matrix}\right.\\ \qquad V=\{(-10,-2)\}\)