Stelsels substitutie

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Substitutie

1. \(\left\{\begin{matrix}-5x-5y=45\\-x+2y=-15\end{matrix}\right.\)
2. \(\left\{\begin{matrix}-5y=-25-5x\\x+4y=-20\end{matrix}\right.\)
3. \(\left\{\begin{matrix}3y=57+6x\\-2x-y=1\end{matrix}\right.\)
4. \(\left\{\begin{matrix}3x+y=18\\2x-5y=-22\end{matrix}\right.\)
5. \(\left\{\begin{matrix}3x+2y=23\\x=5y-49\end{matrix}\right.\)
6. \(\left\{\begin{matrix}-x+2y=23\\2x-6y=-60\end{matrix}\right.\)
7. \(\left\{\begin{matrix}2x+y=3\\4x=-3y-3\end{matrix}\right.\)
8. \(\left\{\begin{matrix}3y=-30-4x\\x-6y=6\end{matrix}\right.\)
9. \(\left\{\begin{matrix}3y=11+2x\\x-6y=-37\end{matrix}\right.\)
10. \(\left\{\begin{matrix}-2y=34-4x\\6x+y=47\end{matrix}\right.\)
11. \(\left\{\begin{matrix}-6x+y=-16\\-3x=3y-36\end{matrix}\right.\)
12. \(\left\{\begin{matrix}y=-26+2x\\4x-2y=52\end{matrix}\right.\)

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Substitutie

Verbetersleutel

1. \(\left\{\begin{matrix}-5x-5y=45\\-x+2y=-15\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-5x-5y=45\\ 2y+15=x\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-5\left(2y+15\right)-5y=45\\x=2y+15\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-10y-75-5y=45\\x=2y+15\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-15y=45+75=120\\x=2y+15\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{120}{-15} = -8 \\ x=2y+15\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = -8 \\ x=2.(-8)+15=-1\end{matrix}\right.\\ \qquad V=\{(-1,-8)\}\)
2. \(\left\{\begin{matrix}-5y=-25-5x\\x+4y=-20\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}5x-5y=-25\\x+4y=-20\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}5x-5y=-25\\ x=-4y-20\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}5\left(-4y-20\right)-5y=-25\\x=-4y-20\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-20y-100-5y=-25\\x=-4y-20\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-25y=-25+100=75\\x=-4y-20\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{75}{-25} = -3 \\ x=-4y-20\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = -3 \\ x=-4.(-3)-20=-8\end{matrix}\right.\\ \qquad V=\{(-8,-3)\}\)
3. \(\left\{\begin{matrix}3y=57+6x\\-2x-y=1\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-6x+3y=57\\-2x-y=1\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}-6x+3y=57\\ -2x-1=y\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-6x+3\left(-2x-1\right)=57\\y=-2x-1\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-6x-6x-3=57\\y=-2x-1\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-12x=57+3=60\\y=-2x-1\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{60}{-12} = -5 \\ y=-2x-1\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = -5 \\ y=-2.(-5)-1=9\end{matrix}\right.\\ \qquad V=\{(-5,9)\}\)
4. \(\left\{\begin{matrix}3x+y=18\\2x-5y=-22\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=-3x+18\\ 2x-5y=-22\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}y=-3x+18\\ 2x-5\left(-3x+18\right)=-22\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}y=-3x+18\\ 2x+15x-90=-22\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}y=-3x+18\\ 17x=-22+90=68\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=-3x+18\\ x=\frac{68}{17}=4\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=-3.(4)+18=6\\ x=4\end{matrix}\right.\\ \qquad V=\{(4,6)\}\)
5. \(\left\{\begin{matrix}3x+2y=23\\x=5y-49\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}3x+2y=23\\x-5y=-49\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}3x+2y=23\\ x=5y-49\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}3\left(5y-49\right)+2y=23\\x=5y-49\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}15y-147+2y=23\\x=5y-49\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}17y=23+147=170\\x=5y-49\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{170}{17} = 10 \\ x=5y-49\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = 10 \\ x=5.(10)-49=1\end{matrix}\right.\\ \qquad V=\{(1,10)\}\)
6. \(\left\{\begin{matrix}-x+2y=23\\2x-6y=-60\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}2y-23=x\\2x-6y=-60\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}x=2y-23\\ 2.\left(2y-23\right)-6y=-60\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}x=2y-23\\ 4y-46-6y=-60\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}x=2y-23\\ -2y=-60+46=-14\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=2y-23\\ y=\frac{-14}{-2}=7\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=2.(7)-23=-9\\ y=7\end{matrix}\right.\\ \qquad V=\{(-9,7)\}\)
7. \(\left\{\begin{matrix}2x+y=3\\4x=-3y-3\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}2x+y=3\\4x+3y=-3\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}y=-2x+3\\ 4x+3y=-3\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}y=-2x+3\\ 4x+3\left(-2x+3\right)=-3\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}y=-2x+3\\ 4x-6x+9=-3\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}y=-2x+3\\ -2x=-3-9=-12\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=-2x+3\\ x=\frac{-12}{-2}=6\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=-2.(6)+3=-9\\ x=6\end{matrix}\right.\\ \qquad V=\{(6,-9)\}\)
8. \(\left\{\begin{matrix}3y=-30-4x\\x-6y=6\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}4x+3y=-30\\x-6y=6\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}4x+3y=-30\\ x=6y+6\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}4\left(6y+6\right)+3y=-30\\x=6y+6\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}24y+24+3y=-30\\x=6y+6\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}27y=-30-24=-54\\x=6y+6\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{-54}{27} = -2 \\ x=6y+6\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = -2 \\ x=6.(-2)+6=-6\end{matrix}\right.\\ \qquad V=\{(-6,-2)\}\)
9. \(\left\{\begin{matrix}3y=11+2x\\x-6y=-37\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-2x+3y=11\\x-6y=-37\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}-2x+3y=11\\ x=6y-37\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-2\left(6y-37\right)+3y=11\\x=6y-37\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-12y+74+3y=11\\x=6y-37\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-9y=11-74=-63\\x=6y-37\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{-63}{-9} = 7 \\ x=6y-37\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = 7 \\ x=6.(7)-37=5\end{matrix}\right.\\ \qquad V=\{(5,7)\}\)
10. \(\left\{\begin{matrix}-2y=34-4x\\6x+y=47\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}4x-2y=34\\6x+y=47\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}4x-2y=34\\ y=-6x+47\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}4x-2\left(-6x+47\right)=34\\y=-6x+47\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}4x+12x-94=34\\y=-6x+47\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}16x=34+94=128\\y=-6x+47\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{128}{16} = 8 \\ y=-6x+47\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = 8 \\ y=-6.(8)+47=-1\end{matrix}\right.\\ \qquad V=\{(8,-1)\}\)
11. \(\left\{\begin{matrix}-6x+y=-16\\-3x=3y-36\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-6x+y=-16\\-3x-3y=-36\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}y=6x-16\\ -3x-3y=-36\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}y=6x-16\\ -3x-3\left(6x-16\right)=-36\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}y=6x-16\\ -3x-18x+48=-36\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}y=6x-16\\ -21x=-36-48=-84\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=6x-16\\ x=\frac{-84}{-21}=4\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=6.(4)-16=8\\ x=4\end{matrix}\right.\\ \qquad V=\{(4,8)\}\)
12. \(\left\{\begin{matrix}y=-26+2x\\4x-2y=52\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-2x+y=-26\\4x-2y=52\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}y=2x-26\\ 4x-2y=52\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}y=2x-26\\ 4x-2\left(2x-26\right)=52\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}y=2x-26\\ 4x-4x+52=52\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}y=2x-26\\ 0x=52-52=0\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=2x-26\\ x=\frac{0}{0}=8\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=2.(8)-26=-10\\ x=8\end{matrix}\right.\\ \qquad V=\{(8,-10)\}\)