Stelsels substitutie

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Substitutie

1. \(\left\{\begin{matrix}4x+4y=36\\6x=y+47\end{matrix}\right.\)
2. \(\left\{\begin{matrix}x+4y=-14\\2x=-2y-16\end{matrix}\right.\)
3. \(\left\{\begin{matrix}y=0+4x\\-6x+5y=-28\end{matrix}\right.\)
4. \(\left\{\begin{matrix}6x+y=-50\\-4x=-5y+56\end{matrix}\right.\)
5. \(\left\{\begin{matrix}y=-14+x\\-5x+2y=-46\end{matrix}\right.\)
6. \(\left\{\begin{matrix}3x-5y=-60\\-x=-3y+28\end{matrix}\right.\)
7. \(\left\{\begin{matrix}x-5y=-33\\6x+4y=6\end{matrix}\right.\)
8. \(\left\{\begin{matrix}-x-4y=-27\\6x=-5y+48\end{matrix}\right.\)
9. \(\left\{\begin{matrix}-2x+6y=56\\2x+y=-14\end{matrix}\right.\)
10. \(\left\{\begin{matrix}6x+5y=-4\\-3x=y+17\end{matrix}\right.\)
11. \(\left\{\begin{matrix}-6x+y=-38\\-6x=5y-98\end{matrix}\right.\)
12. \(\left\{\begin{matrix}-2y=26-2x\\4x-y=40\end{matrix}\right.\)

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Substitutie

Verbetersleutel

1. \(\left\{\begin{matrix}4x+4y=36\\6x=y+47\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}4x+4y=36\\6x-y=47\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}4x+4y=36\\ 6x-47=y\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}4x+4\left(6x-47\right)=36\\y=6x-47\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}4x+24x-188=36\\y=6x-47\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}28x=36+188=224\\y=6x-47\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{224}{28} = 8 \\ y=6x-47\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = 8 \\ y=6.(8)-47=1\end{matrix}\right.\\ \qquad V=\{(8,1)\}\)
2. \(\left\{\begin{matrix}x+4y=-14\\2x=-2y-16\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x+4y=-14\\2x+2y=-16\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}x=-4y-14\\ 2x+2y=-16\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}x=-4y-14\\ 2.\left(-4y-14\right)+2y=-16\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}x=-4y-14\\ -8y-28+2y=-16\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}x=-4y-14\\ -6y=-16+28=12\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=-4y-14\\ y=\frac{12}{-6}=-2\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=-4.(-2)-14=-6\\ y=-2\end{matrix}\right.\\ \qquad V=\{(-6,-2)\}\)
3. \(\left\{\begin{matrix}y=0+4x\\-6x+5y=-28\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-4x+y=0\\-6x+5y=-28\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}y=4x+0\\ -6x+5y=-28\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}y=4x+0\\ -6x+5\left(4x+0\right)=-28\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}y=4x+0\\ -6x+20x+0=-28\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}y=4x+0\\ 14x=-28+0=-28\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=4x+0\\ x=\frac{-28}{14}=-2\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=4.(-2)+0=-8\\ x=-2\end{matrix}\right.\\ \qquad V=\{(-2,-8)\}\)
4. \(\left\{\begin{matrix}6x+y=-50\\-4x=-5y+56\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}6x+y=-50\\-4x+5y=56\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}y=-6x-50\\ -4x+5y=56\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}y=-6x-50\\ -4x+5\left(-6x-50\right)=56\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}y=-6x-50\\ -4x-30x-250=56\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}y=-6x-50\\ -34x=56+250=306\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=-6x-50\\ x=\frac{306}{-34}=-9\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=-6.(-9)-50=4\\ x=-9\end{matrix}\right.\\ \qquad V=\{(-9,4)\}\)
5. \(\left\{\begin{matrix}y=-14+x\\-5x+2y=-46\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-x+y=-14\\-5x+2y=-46\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}y+14=x\\-5x+2y=-46\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}x=y+14\\ -5.\left(y+14\right)+2y=-46\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}x=y+14\\ -5y-70+2y=-46\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}x=y+14\\ -3y=-46+70=24\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=y+14\\ y=\frac{24}{-3}=-8\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=.(-8)+14=6\\ y=-8\end{matrix}\right.\\ \qquad V=\{(6,-8)\}\)
6. \(\left\{\begin{matrix}3x-5y=-60\\-x=-3y+28\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}3x-5y=-60\\-x+3y=28\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}3x-5y=-60\\ 3y-28=x\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}3\left(3y-28\right)-5y=-60\\x=3y-28\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}9y-84-5y=-60\\x=3y-28\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}4y=-60+84=24\\x=3y-28\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{24}{4} = 6 \\ x=3y-28\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = 6 \\ x=3.(6)-28=-10\end{matrix}\right.\\ \qquad V=\{(-10,6)\}\)
7. \(\left\{\begin{matrix}x-5y=-33\\6x+4y=6\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=5y-33\\ 6x+4y=6\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}x=5y-33\\ 6.\left(5y-33\right)+4y=6\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}x=5y-33\\ 30y-198+4y=6\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}x=5y-33\\ 34y=6+198=204\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=5y-33\\ y=\frac{204}{34}=6\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=5.(6)-33=-3\\ y=6\end{matrix}\right.\\ \qquad V=\{(-3,6)\}\)
8. \(\left\{\begin{matrix}-x-4y=-27\\6x=-5y+48\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-x-4y=-27\\6x+5y=48\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}-4y+27=x\\6x+5y=48\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}x=-4y+27\\ 6.\left(-4y+27\right)+5y=48\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}x=-4y+27\\ -24y+162+5y=48\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}x=-4y+27\\ -19y=48-162=-114\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=-4y+27\\ y=\frac{-114}{-19}=6\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=-4.(6)+27=3\\ y=6\end{matrix}\right.\\ \qquad V=\{(3,6)\}\)
9. \(\left\{\begin{matrix}-2x+6y=56\\2x+y=-14\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-2x+6y=56\\ y=-2x-14\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-2x+6\left(-2x-14\right)=56\\y=-2x-14\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-2x-12x-84=56\\y=-2x-14\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-14x=56+84=140\\y=-2x-14\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{140}{-14} = -10 \\ y=-2x-14\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = -10 \\ y=-2.(-10)-14=6\end{matrix}\right.\\ \qquad V=\{(-10,6)\}\)
10. \(\left\{\begin{matrix}6x+5y=-4\\-3x=y+17\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}6x+5y=-4\\-3x-y=17\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}6x+5y=-4\\ -3x-17=y\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}6x+5\left(-3x-17\right)=-4\\y=-3x-17\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}6x-15x-85=-4\\y=-3x-17\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-9x=-4+85=81\\y=-3x-17\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{81}{-9} = -9 \\ y=-3x-17\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = -9 \\ y=-3.(-9)-17=10\end{matrix}\right.\\ \qquad V=\{(-9,10)\}\)
11. \(\left\{\begin{matrix}-6x+y=-38\\-6x=5y-98\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-6x+y=-38\\-6x-5y=-98\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}y=6x-38\\ -6x-5y=-98\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}y=6x-38\\ -6x-5\left(6x-38\right)=-98\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}y=6x-38\\ -6x-30x+190=-98\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}y=6x-38\\ -36x=-98-190=-288\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=6x-38\\ x=\frac{-288}{-36}=8\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=6.(8)-38=10\\ x=8\end{matrix}\right.\\ \qquad V=\{(8,10)\}\)
12. \(\left\{\begin{matrix}-2y=26-2x\\4x-y=40\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}2x-2y=26\\4x-y=40\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}2x-2y=26\\ 4x-40=y\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}2x-2\left(4x-40\right)=26\\y=4x-40\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}2x-8x+80=26\\y=4x-40\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-6x=26-80=-54\\y=4x-40\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{-54}{-6} = 9 \\ y=4x-40\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = 9 \\ y=4.(9)-40=-4\end{matrix}\right.\\ \qquad V=\{(9,-4)\}\)