Stelsels substitutie

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Substitutie

1. \(\left\{\begin{matrix}5x+4y=-15\\-5x-y=0\end{matrix}\right.\)
2. \(\left\{\begin{matrix}2y=54-6x\\-6x-y=-57\end{matrix}\right.\)
3. \(\left\{\begin{matrix}-2y=-29-5x\\x-3y=15\end{matrix}\right.\)
4. \(\left\{\begin{matrix}3x+4y=26\\5x=-y+32\end{matrix}\right.\)
5. \(\left\{\begin{matrix}4x+y=21\\-2x-5y=-33\end{matrix}\right.\)
6. \(\left\{\begin{matrix}-3x-5y=21\\-4x=-y-18\end{matrix}\right.\)
7. \(\left\{\begin{matrix}-4x+6y=36\\3x+y=-5\end{matrix}\right.\)
8. \(\left\{\begin{matrix}5y=-2+3x\\-6x+y=5\end{matrix}\right.\)
9. \(\left\{\begin{matrix}4x-5y=-9\\-2x=-y-3\end{matrix}\right.\)
10. \(\left\{\begin{matrix}-x+6y=-64\\4x-3y=46\end{matrix}\right.\)
11. \(\left\{\begin{matrix}-5x+6y=104\\-2x+y=29\end{matrix}\right.\)
12. \(\left\{\begin{matrix}-6x+6y=84\\-x-5y=-10\end{matrix}\right.\)

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Substitutie

Verbetersleutel

1. \(\left\{\begin{matrix}5x+4y=-15\\-5x-y=0\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}5x+4y=-15\\ -5x+0=y\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}5x+4\left(-5x+0\right)=-15\\y=-5x+0\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}5x-20x+0=-15\\y=-5x+0\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-15x=-15+0=-15\\y=-5x+0\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{-15}{-15} = 1 \\ y=-5x+0\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = 1 \\ y=-5.(1)+0=-5\end{matrix}\right.\\ \qquad V=\{(1,-5)\}\)
2. \(\left\{\begin{matrix}2y=54-6x\\-6x-y=-57\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}6x+2y=54\\-6x-y=-57\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}6x+2y=54\\ -6x+57=y\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}6x+2\left(-6x+57\right)=54\\y=-6x+57\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}6x-12x+114=54\\y=-6x+57\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-6x=54-114=-60\\y=-6x+57\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{-60}{-6} = 10 \\ y=-6x+57\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = 10 \\ y=-6.(10)+57=-3\end{matrix}\right.\\ \qquad V=\{(10,-3)\}\)
3. \(\left\{\begin{matrix}-2y=-29-5x\\x-3y=15\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}5x-2y=-29\\x-3y=15\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}5x-2y=-29\\ x=3y+15\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}5\left(3y+15\right)-2y=-29\\x=3y+15\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}15y+75-2y=-29\\x=3y+15\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}13y=-29-75=-104\\x=3y+15\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{-104}{13} = -8 \\ x=3y+15\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = -8 \\ x=3.(-8)+15=-9\end{matrix}\right.\\ \qquad V=\{(-9,-8)\}\)
4. \(\left\{\begin{matrix}3x+4y=26\\5x=-y+32\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}3x+4y=26\\5x+y=32\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}3x+4y=26\\ y=-5x+32\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}3x+4\left(-5x+32\right)=26\\y=-5x+32\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}3x-20x+128=26\\y=-5x+32\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-17x=26-128=-102\\y=-5x+32\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{-102}{-17} = 6 \\ y=-5x+32\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = 6 \\ y=-5.(6)+32=2\end{matrix}\right.\\ \qquad V=\{(6,2)\}\)
5. \(\left\{\begin{matrix}4x+y=21\\-2x-5y=-33\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=-4x+21\\ -2x-5y=-33\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}y=-4x+21\\ -2x-5\left(-4x+21\right)=-33\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}y=-4x+21\\ -2x+20x-105=-33\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}y=-4x+21\\ 18x=-33+105=72\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=-4x+21\\ x=\frac{72}{18}=4\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=-4.(4)+21=5\\ x=4\end{matrix}\right.\\ \qquad V=\{(4,5)\}\)
6. \(\left\{\begin{matrix}-3x-5y=21\\-4x=-y-18\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-3x-5y=21\\-4x+y=-18\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}-3x-5y=21\\ y=4x-18\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-3x-5\left(4x-18\right)=21\\y=4x-18\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-3x-20x+90=21\\y=4x-18\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-23x=21-90=-69\\y=4x-18\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{-69}{-23} = 3 \\ y=4x-18\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = 3 \\ y=4.(3)-18=-6\end{matrix}\right.\\ \qquad V=\{(3,-6)\}\)
7. \(\left\{\begin{matrix}-4x+6y=36\\3x+y=-5\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-4x+6y=36\\ y=-3x-5\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-4x+6\left(-3x-5\right)=36\\y=-3x-5\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-4x-18x-30=36\\y=-3x-5\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-22x=36+30=66\\y=-3x-5\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{66}{-22} = -3 \\ y=-3x-5\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = -3 \\ y=-3.(-3)-5=4\end{matrix}\right.\\ \qquad V=\{(-3,4)\}\)
8. \(\left\{\begin{matrix}5y=-2+3x\\-6x+y=5\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-3x+5y=-2\\-6x+y=5\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}-3x+5y=-2\\ y=6x+5\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-3x+5\left(6x+5\right)=-2\\y=6x+5\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-3x+30x+25=-2\\y=6x+5\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}27x=-2-25=-27\\y=6x+5\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{-27}{27} = -1 \\ y=6x+5\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = -1 \\ y=6.(-1)+5=-1\end{matrix}\right.\\ \qquad V=\{(-1,-1)\}\)
9. \(\left\{\begin{matrix}4x-5y=-9\\-2x=-y-3\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}4x-5y=-9\\-2x+y=-3\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}4x-5y=-9\\ y=2x-3\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}4x-5\left(2x-3\right)=-9\\y=2x-3\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}4x-10x+15=-9\\y=2x-3\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-6x=-9-15=-24\\y=2x-3\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{-24}{-6} = 4 \\ y=2x-3\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = 4 \\ y=2.(4)-3=5\end{matrix}\right.\\ \qquad V=\{(4,5)\}\)
10. \(\left\{\begin{matrix}-x+6y=-64\\4x-3y=46\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}6y+64=x\\4x-3y=46\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}x=6y+64\\ 4.\left(6y+64\right)-3y=46\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}x=6y+64\\ 24y+256-3y=46\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}x=6y+64\\ 21y=46-256=-210\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=6y+64\\ y=\frac{-210}{21}=-10\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=6.(-10)+64=4\\ y=-10\end{matrix}\right.\\ \qquad V=\{(4,-10)\}\)
11. \(\left\{\begin{matrix}-5x+6y=104\\-2x+y=29\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-5x+6y=104\\ y=2x+29\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-5x+6\left(2x+29\right)=104\\y=2x+29\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-5x+12x+174=104\\y=2x+29\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}7x=104-174=-70\\y=2x+29\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{-70}{7} = -10 \\ y=2x+29\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = -10 \\ y=2.(-10)+29=9\end{matrix}\right.\\ \qquad V=\{(-10,9)\}\)
12. \(\left\{\begin{matrix}-6x+6y=84\\-x-5y=-10\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-6x+6y=84\\ -5y+10=x\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-6\left(-5y+10\right)+6y=84\\x=-5y+10\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}30y-60+6y=84\\x=-5y+10\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}36y=84+60=144\\x=-5y+10\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{144}{36} = 4 \\ x=-5y+10\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = 4 \\ x=-5.(4)+10=-10\end{matrix}\right.\\ \qquad V=\{(-10,4)\}\)