Stelsels substitutie

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Substitutie

1. \(\left\{\begin{matrix}2x+3y=-32\\6x+y=-48\end{matrix}\right.\)
2. \(\left\{\begin{matrix}3x-y=-16\\3x=2y-26\end{matrix}\right.\)
3. \(\left\{\begin{matrix}4x+3y=46\\x=-2y+14\end{matrix}\right.\)
4. \(\left\{\begin{matrix}3y=5+4x\\6x-y=-25\end{matrix}\right.\)
5. \(\left\{\begin{matrix}2x-2y=18\\-x-5y=3\end{matrix}\right.\)
6. \(\left\{\begin{matrix}5x-5y=95\\6x-y=64\end{matrix}\right.\)
7. \(\left\{\begin{matrix}4x-6y=-70\\x=5y-49\end{matrix}\right.\)
8. \(\left\{\begin{matrix}2x-5y=13\\x=-3y+1\end{matrix}\right.\)
9. \(\left\{\begin{matrix}2x-4y=-52\\-x-5y=-30\end{matrix}\right.\)
10. \(\left\{\begin{matrix}3x+4y=-43\\-x-2y=21\end{matrix}\right.\)
11. \(\left\{\begin{matrix}6x-4y=-68\\3x=y-32\end{matrix}\right.\)
12. \(\left\{\begin{matrix}-6x-5y=-11\\x-2y=-1\end{matrix}\right.\)

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Substitutie

Verbetersleutel

1. \(\left\{\begin{matrix}2x+3y=-32\\6x+y=-48\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}2x+3y=-32\\ y=-6x-48\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}2x+3\left(-6x-48\right)=-32\\y=-6x-48\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}2x-18x-144=-32\\y=-6x-48\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-16x=-32+144=112\\y=-6x-48\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{112}{-16} = -7 \\ y=-6x-48\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = -7 \\ y=-6.(-7)-48=-6\end{matrix}\right.\\ \qquad V=\{(-7,-6)\}\)
2. \(\left\{\begin{matrix}3x-y=-16\\3x=2y-26\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}3x-y=-16\\3x-2y=-26\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}3x+16=y\\3x-2y=-26\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}y=3x+16\\ 3x-2\left(3x+16\right)=-26\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}y=3x+16\\ 3x-6x-32=-26\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}y=3x+16\\ -3x=-26+32=6\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=3x+16\\ x=\frac{6}{-3}=-2\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=3.(-2)+16=10\\ x=-2\end{matrix}\right.\\ \qquad V=\{(-2,10)\}\)
3. \(\left\{\begin{matrix}4x+3y=46\\x=-2y+14\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}4x+3y=46\\x+2y=14\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}4x+3y=46\\ x=-2y+14\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}4\left(-2y+14\right)+3y=46\\x=-2y+14\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-8y+56+3y=46\\x=-2y+14\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-5y=46-56=-10\\x=-2y+14\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{-10}{-5} = 2 \\ x=-2y+14\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = 2 \\ x=-2.(2)+14=10\end{matrix}\right.\\ \qquad V=\{(10,2)\}\)
4. \(\left\{\begin{matrix}3y=5+4x\\6x-y=-25\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-4x+3y=5\\6x-y=-25\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}-4x+3y=5\\ 6x+25=y\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-4x+3\left(6x+25\right)=5\\y=6x+25\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-4x+18x+75=5\\y=6x+25\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}14x=5-75=-70\\y=6x+25\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{-70}{14} = -5 \\ y=6x+25\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = -5 \\ y=6.(-5)+25=-5\end{matrix}\right.\\ \qquad V=\{(-5,-5)\}\)
5. \(\left\{\begin{matrix}2x-2y=18\\-x-5y=3\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}2x-2y=18\\ -5y-3=x\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}2\left(-5y-3\right)-2y=18\\x=-5y-3\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-10y-6-2y=18\\x=-5y-3\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-12y=18+6=24\\x=-5y-3\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{24}{-12} = -2 \\ x=-5y-3\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = -2 \\ x=-5.(-2)-3=7\end{matrix}\right.\\ \qquad V=\{(7,-2)\}\)
6. \(\left\{\begin{matrix}5x-5y=95\\6x-y=64\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}5x-5y=95\\ 6x-64=y\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}5x-5\left(6x-64\right)=95\\y=6x-64\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}5x-30x+320=95\\y=6x-64\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-25x=95-320=-225\\y=6x-64\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{-225}{-25} = 9 \\ y=6x-64\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = 9 \\ y=6.(9)-64=-10\end{matrix}\right.\\ \qquad V=\{(9,-10)\}\)
7. \(\left\{\begin{matrix}4x-6y=-70\\x=5y-49\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}4x-6y=-70\\x-5y=-49\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}4x-6y=-70\\ x=5y-49\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}4\left(5y-49\right)-6y=-70\\x=5y-49\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}20y-196-6y=-70\\x=5y-49\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}14y=-70+196=126\\x=5y-49\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{126}{14} = 9 \\ x=5y-49\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = 9 \\ x=5.(9)-49=-4\end{matrix}\right.\\ \qquad V=\{(-4,9)\}\)
8. \(\left\{\begin{matrix}2x-5y=13\\x=-3y+1\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}2x-5y=13\\x+3y=1\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}2x-5y=13\\ x=-3y+1\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}2\left(-3y+1\right)-5y=13\\x=-3y+1\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-6y+2-5y=13\\x=-3y+1\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-11y=13-2=11\\x=-3y+1\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{11}{-11} = -1 \\ x=-3y+1\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = -1 \\ x=-3.(-1)+1=4\end{matrix}\right.\\ \qquad V=\{(4,-1)\}\)
9. \(\left\{\begin{matrix}2x-4y=-52\\-x-5y=-30\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}2x-4y=-52\\ -5y+30=x\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}2\left(-5y+30\right)-4y=-52\\x=-5y+30\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-10y+60-4y=-52\\x=-5y+30\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-14y=-52-60=-112\\x=-5y+30\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{-112}{-14} = 8 \\ x=-5y+30\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = 8 \\ x=-5.(8)+30=-10\end{matrix}\right.\\ \qquad V=\{(-10,8)\}\)
10. \(\left\{\begin{matrix}3x+4y=-43\\-x-2y=21\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}3x+4y=-43\\ -2y-21=x\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}3\left(-2y-21\right)+4y=-43\\x=-2y-21\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-6y-63+4y=-43\\x=-2y-21\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-2y=-43+63=20\\x=-2y-21\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{20}{-2} = -10 \\ x=-2y-21\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = -10 \\ x=-2.(-10)-21=-1\end{matrix}\right.\\ \qquad V=\{(-1,-10)\}\)
11. \(\left\{\begin{matrix}6x-4y=-68\\3x=y-32\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}6x-4y=-68\\3x-y=-32\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}6x-4y=-68\\ 3x+32=y\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}6x-4\left(3x+32\right)=-68\\y=3x+32\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}6x-12x-128=-68\\y=3x+32\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-6x=-68+128=60\\y=3x+32\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{60}{-6} = -10 \\ y=3x+32\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = -10 \\ y=3.(-10)+32=2\end{matrix}\right.\\ \qquad V=\{(-10,2)\}\)
12. \(\left\{\begin{matrix}-6x-5y=-11\\x-2y=-1\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-6x-5y=-11\\ x=2y-1\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-6\left(2y-1\right)-5y=-11\\x=2y-1\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-12y+6-5y=-11\\x=2y-1\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-17y=-11-6=-17\\x=2y-1\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{-17}{-17} = 1 \\ x=2y-1\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = 1 \\ x=2.(1)-1=1\end{matrix}\right.\\ \qquad V=\{(1,1)\}\)