Stelsels substitutie

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Substitutie

1. \(\left\{\begin{matrix}4x+y=-28\\-4x+3y=12\end{matrix}\right.\)
2. \(\left\{\begin{matrix}5x+5y=-10\\-x-3y=18\end{matrix}\right.\)
3. \(\left\{\begin{matrix}-2x-6y=-6\\x-5y=11\end{matrix}\right.\)
4. \(\left\{\begin{matrix}-x-2y=9\\4x+4y=-16\end{matrix}\right.\)
5. \(\left\{\begin{matrix}5x+6y=-57\\-4x=-y+5\end{matrix}\right.\)
6. \(\left\{\begin{matrix}5x+y=23\\-6x+5y=-40\end{matrix}\right.\)
7. \(\left\{\begin{matrix}y=-25+5x\\-3x-5y=13\end{matrix}\right.\)
8. \(\left\{\begin{matrix}-5x-2y=-31\\x=-6y-5\end{matrix}\right.\)
9. \(\left\{\begin{matrix}-6y=-25+x\\3x+2y=11\end{matrix}\right.\)
10. \(\left\{\begin{matrix}4y=-24+4x\\x-3y=14\end{matrix}\right.\)
11. \(\left\{\begin{matrix}-3x-5y=39\\x=6y+33\end{matrix}\right.\)
12. \(\left\{\begin{matrix}-6x+y=64\\-5x=5y-5\end{matrix}\right.\)

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Substitutie

Verbetersleutel

1. \(\left\{\begin{matrix}4x+y=-28\\-4x+3y=12\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=-4x-28\\ -4x+3y=12\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}y=-4x-28\\ -4x+3\left(-4x-28\right)=12\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}y=-4x-28\\ -4x-12x-84=12\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}y=-4x-28\\ -16x=12+84=96\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=-4x-28\\ x=\frac{96}{-16}=-6\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=-4.(-6)-28=-4\\ x=-6\end{matrix}\right.\\ \qquad V=\{(-6,-4)\}\)
2. \(\left\{\begin{matrix}5x+5y=-10\\-x-3y=18\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}5x+5y=-10\\ -3y-18=x\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}5\left(-3y-18\right)+5y=-10\\x=-3y-18\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-15y-90+5y=-10\\x=-3y-18\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-10y=-10+90=80\\x=-3y-18\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{80}{-10} = -8 \\ x=-3y-18\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = -8 \\ x=-3.(-8)-18=6\end{matrix}\right.\\ \qquad V=\{(6,-8)\}\)
3. \(\left\{\begin{matrix}-2x-6y=-6\\x-5y=11\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-2x-6y=-6\\ x=5y+11\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-2\left(5y+11\right)-6y=-6\\x=5y+11\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-10y-22-6y=-6\\x=5y+11\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-16y=-6+22=16\\x=5y+11\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{16}{-16} = -1 \\ x=5y+11\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = -1 \\ x=5.(-1)+11=6\end{matrix}\right.\\ \qquad V=\{(6,-1)\}\)
4. \(\left\{\begin{matrix}-x-2y=9\\4x+4y=-16\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-2y-9=x\\4x+4y=-16\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}x=-2y-9\\ 4.\left(-2y-9\right)+4y=-16\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}x=-2y-9\\ -8y-36+4y=-16\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}x=-2y-9\\ -4y=-16+36=20\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=-2y-9\\ y=\frac{20}{-4}=-5\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=-2.(-5)-9=1\\ y=-5\end{matrix}\right.\\ \qquad V=\{(1,-5)\}\)
5. \(\left\{\begin{matrix}5x+6y=-57\\-4x=-y+5\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}5x+6y=-57\\-4x+y=5\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}5x+6y=-57\\ y=4x+5\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}5x+6\left(4x+5\right)=-57\\y=4x+5\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}5x+24x+30=-57\\y=4x+5\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}29x=-57-30=-87\\y=4x+5\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = \frac{-87}{29} = -3 \\ y=4x+5\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x = -3 \\ y=4.(-3)+5=-7\end{matrix}\right.\\ \qquad V=\{(-3,-7)\}\)
6. \(\left\{\begin{matrix}5x+y=23\\-6x+5y=-40\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=-5x+23\\ -6x+5y=-40\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}y=-5x+23\\ -6x+5\left(-5x+23\right)=-40\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}y=-5x+23\\ -6x-25x+115=-40\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}y=-5x+23\\ -31x=-40-115=-155\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=-5x+23\\ x=\frac{-155}{-31}=5\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=-5.(5)+23=-2\\ x=5\end{matrix}\right.\\ \qquad V=\{(5,-2)\}\)
7. \(\left\{\begin{matrix}y=-25+5x\\-3x-5y=13\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-5x+y=-25\\-3x-5y=13\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}y=5x-25\\ -3x-5y=13\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}y=5x-25\\ -3x-5\left(5x-25\right)=13\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}y=5x-25\\ -3x-25x+125=13\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}y=5x-25\\ -28x=13-125=-112\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=5x-25\\ x=\frac{-112}{-28}=4\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=5.(4)-25=-5\\ x=4\end{matrix}\right.\\ \qquad V=\{(4,-5)\}\)
8. \(\left\{\begin{matrix}-5x-2y=-31\\x=-6y-5\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-5x-2y=-31\\x+6y=-5\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}-5x-2y=-31\\ x=-6y-5\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-5\left(-6y-5\right)-2y=-31\\x=-6y-5\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}30y+25-2y=-31\\x=-6y-5\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}28y=-31-25=-56\\x=-6y-5\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{-56}{28} = -2 \\ x=-6y-5\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = -2 \\ x=-6.(-2)-5=7\end{matrix}\right.\\ \qquad V=\{(7,-2)\}\)
9. \(\left\{\begin{matrix}-6y=-25+x\\3x+2y=11\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-x-6y=-25\\3x+2y=11\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}-6y+25=x\\3x+2y=11\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}x=-6y+25\\ 3.\left(-6y+25\right)+2y=11\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}x=-6y+25\\ -18y+75+2y=11\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}x=-6y+25\\ -16y=11-75=-64\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=-6y+25\\ y=\frac{-64}{-16}=4\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}x=-6.(4)+25=1\\ y=4\end{matrix}\right.\\ \qquad V=\{(1,4)\}\)
10. \(\left\{\begin{matrix}4y=-24+4x\\x-3y=14\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-4x+4y=-24\\x-3y=14\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}-4x+4y=-24\\ x=3y+14\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-4\left(3y+14\right)+4y=-24\\x=3y+14\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-12y-56+4y=-24\\x=3y+14\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-8y=-24+56=32\\x=3y+14\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{32}{-8} = -4 \\ x=3y+14\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = -4 \\ x=3.(-4)+14=2\end{matrix}\right.\\ \qquad V=\{(2,-4)\}\)
11. \(\left\{\begin{matrix}-3x-5y=39\\x=6y+33\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-3x-5y=39\\x-6y=33\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}-3x-5y=39\\ x=6y+33\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}-3\left(6y+33\right)-5y=39\\x=6y+33\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}-18y-99-5y=39\\x=6y+33\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}-23y=39+99=138\\x=6y+33\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = \frac{138}{-23} = -6 \\ x=6y+33\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y = -6 \\ x=6.(-6)+33=-3\end{matrix}\right.\\ \qquad V=\{(-3,-6)\}\)
12. \(\left\{\begin{matrix}-6x+y=64\\-5x=5y-5\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-6x+y=64\\-5x-5y=-5\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{matrix}y=6x+64\\ -5x-5y=-5\end{matrix}\right.\text{(Afzonderen onbekende met coëff. 1)}\\ \Leftrightarrow \left\{\begin{matrix}y=6x+64\\ -5x-5\left(6x+64\right)=-5\end{matrix}\right.\text{(Substitutie!)}\\ \Leftrightarrow \left\{\begin{matrix}y=6x+64\\ -5x-30x-320=-5\end{matrix}\right.\text{(Distributie)}\\ \Leftrightarrow \left\{\begin{matrix}y=6x+64\\ -35x=-5+320=315\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=6x+64\\ x=\frac{315}{-35}=-9\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=6.(-9)+64=10\\ x=-9\end{matrix}\right.\\ \qquad V=\{(-9,10)\}\)