Stelsels substitutie

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Substitutie

1. \(\left\{\begin{matrix}x+2y=-23\\-5x+3y=-2\end{matrix}\right.\)
2. \(\left\{\begin{matrix}4y=64+4x\\-2x-y=14\end{matrix}\right.\)
3. \(\left\{\begin{matrix}-x+2y=-2\\4x+3y=-14\end{matrix}\right.\)
4. \(\left\{\begin{matrix}-y=-53+5x\\5x+6y=93\end{matrix}\right.\)
5. \(\left\{\begin{matrix}3x-5y=47\\-x+6y=-46\end{matrix}\right.\)
6. \(\left\{\begin{matrix}-x+y=-20\\5x-6y=110\end{matrix}\right.\)
7. \(\left\{\begin{matrix}2y=6+5x\\6x-y=-10\end{matrix}\right.\)
8. \(\left\{\begin{matrix}-x+3y=-1\\-6x=-6y-42\end{matrix}\right.\)
9. \(\left\{\begin{matrix}-6y=-10+4x\\-3x-y=3\end{matrix}\right.\)
10. \(\left\{\begin{matrix}-2x+3y=5\\x+2y=-13\end{matrix}\right.\)
11. \(\left\{\begin{matrix}-3x-4y=-26\\x=5y-42\end{matrix}\right.\)
12. \(\left\{\begin{matrix}-6x-y=-28\\3x+2y=20\end{matrix}\right.\)

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Substitutie

Verbetersleutel

1. \(\left\{\begin{matrix}x+2y=-23\\-5x+3y=-2\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=-2y-23\\ -5x+3y=-2\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}x=-2y-23\\ -5.\left(-2y-23\right)+3y=-2\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}x=-2y-23\\ 10y+115+3y=-2\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}x=-2y-23\\ 13y=-2-115=-117\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=-2y-23\\ y=\frac{-117}{13}=-9\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=-2.(-9)-23=-5\\ y=-9\end{matrix}\right.\\ \qquad V=\{(-5,-9)\}\)
2. \(\left\{\begin{matrix}4y=64+4x\\-2x-y=14\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-4x+4y=64\\-2x-y=14\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}-4x+4y=64\\ -2x-14=y\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-4x+4\left(-2x-14\right)=64\\y=-2x-14\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-4x-8x-56=64\\y=-2x-14\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-12x=64+56=120\\y=-2x-14\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{120}{-12} = -10 \\ y=-2x-14\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = -10 \\ y=-2.(-10)-14=6\end{matrix}\right.\\ \qquad V=\{(-10,6)\}\)
3. \(\left\{\begin{matrix}-x+2y=-2\\4x+3y=-14\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}2y+2=x\\4x+3y=-14\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}x=2y+2\\ 4.\left(2y+2\right)+3y=-14\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}x=2y+2\\ 8y+8+3y=-14\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}x=2y+2\\ 11y=-14-8=-22\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=2y+2\\ y=\frac{-22}{11}=-2\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=2.(-2)+2=-2\\ y=-2\end{matrix}\right.\\ \qquad V=\{(-2,-2)\}\)
4. \(\left\{\begin{matrix}-y=-53+5x\\5x+6y=93\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-5x-y=-53\\5x+6y=93\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}-5x+53=y\\5x+6y=93\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}y=-5x+53\\ 5x+6\left(-5x+53\right)=93\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}y=-5x+53\\ 5x-30x+318=93\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}y=-5x+53\\ -25x=93-318=-225\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=-5x+53\\ x=\frac{-225}{-25}=9\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=-5.(9)+53=8\\ x=9\end{matrix}\right.\\ \qquad V=\{(9,8)\}\)
5. \(\left\{\begin{matrix}3x-5y=47\\-x+6y=-46\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}3x-5y=47\\ 6y+46=x\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}3\left(6y+46\right)-5y=47\\x=6y+46\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}18y+138-5y=47\\x=6y+46\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}13y=47-138=-91\\x=6y+46\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{-91}{13} = -7 \\ x=6y+46\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = -7 \\ x=6.(-7)+46=4\end{matrix}\right.\\ \qquad V=\{(4,-7)\}\)
6. \(\left\{\begin{matrix}-x+y=-20\\5x-6y=110\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y+20=x\\5x-6y=110\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}x=y+20\\ 5.\left(y+20\right)-6y=110\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}x=y+20\\ 5y+100-6y=110\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}x=y+20\\ -y=110-100=10\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=y+20\\ y=\frac{10}{-1}=-10\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=.(-10)+20=10\\ y=-10\end{matrix}\right.\\ \qquad V=\{(10,-10)\}\)
7. \(\left\{\begin{matrix}2y=6+5x\\6x-y=-10\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-5x+2y=6\\6x-y=-10\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}-5x+2y=6\\ 6x+10=y\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-5x+2\left(6x+10\right)=6\\y=6x+10\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-5x+12x+20=6\\y=6x+10\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}7x=6-20=-14\\y=6x+10\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{-14}{7} = -2 \\ y=6x+10\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = -2 \\ y=6.(-2)+10=-2\end{matrix}\right.\\ \qquad V=\{(-2,-2)\}\)
8. \(\left\{\begin{matrix}-x+3y=-1\\-6x=-6y-42\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-x+3y=-1\\-6x+6y=-42\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}3y+1=x\\-6x+6y=-42\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}x=3y+1\\ -6.\left(3y+1\right)+6y=-42\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}x=3y+1\\ -18y-6+6y=-42\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}x=3y+1\\ -12y=-42+6=-36\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=3y+1\\ y=\frac{-36}{-12}=3\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=3.(3)+1=10\\ y=3\end{matrix}\right.\\ \qquad V=\{(10,3)\}\)
9. \(\left\{\begin{matrix}-6y=-10+4x\\-3x-y=3\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-4x-6y=-10\\-3x-y=3\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}-4x-6y=-10\\ -3x-3=y\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-4x-6\left(-3x-3\right)=-10\\y=-3x-3\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-4x+18x+18=-10\\y=-3x-3\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}14x=-10-18=-28\\y=-3x-3\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{-28}{14} = -2 \\ y=-3x-3\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = -2 \\ y=-3.(-2)-3=3\end{matrix}\right.\\ \qquad V=\{(-2,3)\}\)
10. \(\left\{\begin{matrix}-2x+3y=5\\x+2y=-13\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-2x+3y=5\\ x=-2y-13\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-2\left(-2y-13\right)+3y=5\\x=-2y-13\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}4y+26+3y=5\\x=-2y-13\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}7y=5-26=-21\\x=-2y-13\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{-21}{7} = -3 \\ x=-2y-13\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = -3 \\ x=-2.(-3)-13=-7\end{matrix}\right.\\ \qquad V=\{(-7,-3)\}\)
11. \(\left\{\begin{matrix}-3x-4y=-26\\x=5y-42\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-3x-4y=-26\\x-5y=-42\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}-3x-4y=-26\\ x=5y-42\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-3\left(5y-42\right)-4y=-26\\x=5y-42\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-15y+126-4y=-26\\x=5y-42\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-19y=-26-126=-152\\x=5y-42\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{-152}{-19} = 8 \\ x=5y-42\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = 8 \\ x=5.(8)-42=-2\end{matrix}\right.\\ \qquad V=\{(-2,8)\}\)
12. \(\left\{\begin{matrix}-6x-y=-28\\3x+2y=20\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-6x+28=y\\3x+2y=20\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}y=-6x+28\\ 3x+2\left(-6x+28\right)=20\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}y=-6x+28\\ 3x-12x+56=20\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}y=-6x+28\\ -9x=20-56=-36\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=-6x+28\\ x=\frac{-36}{-9}=4\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=-6.(4)+28=4\\ x=4\end{matrix}\right.\\ \qquad V=\{(4,4)\}\)