Stelsels substitutie

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Substitutie

1. \(\left\{\begin{matrix}-3x+6y=-66\\-x=-4y-42\end{matrix}\right.\)
2. \(\left\{\begin{matrix}3x-2y=-9\\-4x=y+1\end{matrix}\right.\)
3. \(\left\{\begin{matrix}-x-3y=-6\\4x=-5y-4\end{matrix}\right.\)
4. \(\left\{\begin{matrix}4x+2y=10\\6x=y+43\end{matrix}\right.\)
5. \(\left\{\begin{matrix}5y=49+x\\3x-6y=-66\end{matrix}\right.\)
6. \(\left\{\begin{matrix}5y=-40-x\\6x-4y=-36\end{matrix}\right.\)
7. \(\left\{\begin{matrix}-2y=32+5x\\3x+y=-18\end{matrix}\right.\)
8. \(\left\{\begin{matrix}-2x-3y=27\\x=2y-3\end{matrix}\right.\)
9. \(\left\{\begin{matrix}4y=18-5x\\3x+y=8\end{matrix}\right.\)
10. \(\left\{\begin{matrix}2y=2-3x\\-x+5y=-29\end{matrix}\right.\)
11. \(\left\{\begin{matrix}-y=-6+6x\\-5x+4y=-34\end{matrix}\right.\)
12. \(\left\{\begin{matrix}2x+y=9\\4x=5y+81\end{matrix}\right.\)

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Substitutie

Verbetersleutel

1. \(\left\{\begin{matrix}-3x+6y=-66\\-x=-4y-42\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-3x+6y=-66\\-x+4y=-42\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}-3x+6y=-66\\ 4y+42=x\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-3\left(4y+42\right)+6y=-66\\x=4y+42\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-12y-126+6y=-66\\x=4y+42\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-6y=-66+126=60\\x=4y+42\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{60}{-6} = -10 \\ x=4y+42\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = -10 \\ x=4.(-10)+42=2\end{matrix}\right.\\ \qquad V=\{(2,-10)\}\)
2. \(\left\{\begin{matrix}3x-2y=-9\\-4x=y+1\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}3x-2y=-9\\-4x-y=1\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}3x-2y=-9\\ -4x-1=y\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}3x-2\left(-4x-1\right)=-9\\y=-4x-1\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}3x+8x+2=-9\\y=-4x-1\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}11x=-9-2=-11\\y=-4x-1\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{-11}{11} = -1 \\ y=-4x-1\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = -1 \\ y=-4.(-1)-1=3\end{matrix}\right.\\ \qquad V=\{(-1,3)\}\)
3. \(\left\{\begin{matrix}-x-3y=-6\\4x=-5y-4\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-x-3y=-6\\4x+5y=-4\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}-3y+6=x\\4x+5y=-4\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}x=-3y+6\\ 4.\left(-3y+6\right)+5y=-4\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}x=-3y+6\\ -12y+24+5y=-4\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}x=-3y+6\\ -7y=-4-24=-28\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=-3y+6\\ y=\frac{-28}{-7}=4\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=-3.(4)+6=-6\\ y=4\end{matrix}\right.\\ \qquad V=\{(-6,4)\}\)
4. \(\left\{\begin{matrix}4x+2y=10\\6x=y+43\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}4x+2y=10\\6x-y=43\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}4x+2y=10\\ 6x-43=y\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}4x+2\left(6x-43\right)=10\\y=6x-43\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}4x+12x-86=10\\y=6x-43\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}16x=10+86=96\\y=6x-43\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{96}{16} = 6 \\ y=6x-43\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = 6 \\ y=6.(6)-43=-7\end{matrix}\right.\\ \qquad V=\{(6,-7)\}\)
5. \(\left\{\begin{matrix}5y=49+x\\3x-6y=-66\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-x+5y=49\\3x-6y=-66\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}5y-49=x\\3x-6y=-66\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}x=5y-49\\ 3.\left(5y-49\right)-6y=-66\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}x=5y-49\\ 15y-147-6y=-66\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}x=5y-49\\ 9y=-66+147=81\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=5y-49\\ y=\frac{81}{9}=9\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=5.(9)-49=-4\\ y=9\end{matrix}\right.\\ \qquad V=\{(-4,9)\}\)
6. \(\left\{\begin{matrix}5y=-40-x\\6x-4y=-36\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x+5y=-40\\6x-4y=-36\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}x=-5y-40\\ 6x-4y=-36\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}x=-5y-40\\ 6.\left(-5y-40\right)-4y=-36\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}x=-5y-40\\ -30y-240-4y=-36\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}x=-5y-40\\ -34y=-36+240=204\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=-5y-40\\ y=\frac{204}{-34}=-6\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=-5.(-6)-40=-10\\ y=-6\end{matrix}\right.\\ \qquad V=\{(-10,-6)\}\)
7. \(\left\{\begin{matrix}-2y=32+5x\\3x+y=-18\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-5x-2y=32\\3x+y=-18\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}-5x-2y=32\\ y=-3x-18\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-5x-2\left(-3x-18\right)=32\\y=-3x-18\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-5x+6x+36=32\\y=-3x-18\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}x=32-36=-4\\y=-3x-18\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{-4}{1} = -4 \\ y=-3x-18\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = -4 \\ y=-3.(-4)-18=-6\end{matrix}\right.\\ \qquad V=\{(-4,-6)\}\)
8. \(\left\{\begin{matrix}-2x-3y=27\\x=2y-3\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-2x-3y=27\\x-2y=-3\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}-2x-3y=27\\ x=2y-3\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-2\left(2y-3\right)-3y=27\\x=2y-3\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-4y+6-3y=27\\x=2y-3\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-7y=27-6=21\\x=2y-3\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{21}{-7} = -3 \\ x=2y-3\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = -3 \\ x=2.(-3)-3=-9\end{matrix}\right.\\ \qquad V=\{(-9,-3)\}\)
9. \(\left\{\begin{matrix}4y=18-5x\\3x+y=8\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}5x+4y=18\\3x+y=8\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}5x+4y=18\\ y=-3x+8\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}5x+4\left(-3x+8\right)=18\\y=-3x+8\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}5x-12x+32=18\\y=-3x+8\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-7x=18-32=-14\\y=-3x+8\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{-14}{-7} = 2 \\ y=-3x+8\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = 2 \\ y=-3.(2)+8=2\end{matrix}\right.\\ \qquad V=\{(2,2)\}\)
10. \(\left\{\begin{matrix}2y=2-3x\\-x+5y=-29\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}3x+2y=2\\-x+5y=-29\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}3x+2y=2\\ 5y+29=x\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}3\left(5y+29\right)+2y=2\\x=5y+29\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}15y+87+2y=2\\x=5y+29\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}17y=2-87=-85\\x=5y+29\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{-85}{17} = -5 \\ x=5y+29\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = -5 \\ x=5.(-5)+29=4\end{matrix}\right.\\ \qquad V=\{(4,-5)\}\)
11. \(\left\{\begin{matrix}-y=-6+6x\\-5x+4y=-34\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-6x-y=-6\\-5x+4y=-34\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}-6x+6=y\\-5x+4y=-34\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}y=-6x+6\\ -5x+4\left(-6x+6\right)=-34\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}y=-6x+6\\ -5x-24x+24=-34\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}y=-6x+6\\ -29x=-34-24=-58\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=-6x+6\\ x=\frac{-58}{-29}=2\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=-6.(2)+6=-6\\ x=2\end{matrix}\right.\\ \qquad V=\{(2,-6)\}\)
12. \(\left\{\begin{matrix}2x+y=9\\4x=5y+81\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}2x+y=9\\4x-5y=81\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}y=-2x+9\\ 4x-5y=81\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}y=-2x+9\\ 4x-5\left(-2x+9\right)=81\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}y=-2x+9\\ 4x+10x-45=81\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}y=-2x+9\\ 14x=81+45=126\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=-2x+9\\ x=\frac{126}{14}=9\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=-2.(9)+9=-9\\ x=9\end{matrix}\right.\\ \qquad V=\{(9,-9)\}\)