Stelsels combinatie

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Combinatie

1. \(\left\{\begin{matrix}6x-3y=21\\6x=5y+23\end{matrix}\right.\)
2. \(\left\{\begin{matrix}-8x-3y=73\\6y-10x=62\end{matrix}\right.\)
3. \(\left\{\begin{matrix}10y-6x=84\\5x+4y=-33\end{matrix}\right.\)
4. \(\left\{\begin{matrix}-6x+7y=82\\-7x=10y-86\end{matrix}\right.\)
5. \(\left\{\begin{matrix}-8x-5y=-38\\-8x-10y=-28\end{matrix}\right.\)
6. \(\left\{\begin{matrix}9y=-126+9x\\8x+5y=47\end{matrix}\right.\)
7. \(\left\{\begin{matrix}3x-10y=100\\3x=7y+79\end{matrix}\right.\)
8. \(\left\{\begin{matrix}-3x+5y=42\\-3x+9y=54\end{matrix}\right.\)
9. \(\left\{\begin{matrix}8y=81+3x\\8x+3y=3\end{matrix}\right.\)
10. \(\left\{\begin{matrix}4x+10y=106\\8x=-7y+121\end{matrix}\right.\)
11. \(\left\{\begin{matrix}-8x+10y=-12\\-3x=7y+17\end{matrix}\right.\)
12. \(\left\{\begin{matrix}3y=-39-7x\\-7x+9y=-33\end{matrix}\right.\)

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Combinatie

Verbetersleutel

1. \(\left\{\begin{matrix}6x-3y=21\\6x=5y+23\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}6x-3y=21\\6x-5y=23\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{array}{c|c|c}6x-3y=21& \color{red}{1.} & \color{blue}{5.} \\6x-5y=23& \color{red}{-1.} & \color{blue}{-3.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{6x-6x}-3y+5y=21-23} \\ \color{blue}{30x-18x\underline{-15y+15y}=105-69} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}2y=-2 \\12x=36\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{-2}{2}=-1 \\x=\frac{36}{12}=3\end{matrix}\right.\\ \qquad V=\{(3,-1)\}\)
2. \(\left\{\begin{matrix}-8x-3y=73\\6y-10x=62\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-8x-3y=73\\-10x+6y=62\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{array}{c|c|c}-8x-3y=73& \color{red}{5.} & \color{blue}{2.} \\-10x+6y=62& \color{red}{-4.} & \color{blue}{1.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{-40x+40x}-15y-24y=365-248} \\ \color{blue}{-16x-10x\underline{-6y+6y}=146+62} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-39y=117 \\-26x=208\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{117}{-39}=-3 \\x=\frac{208}{-26}=-8\end{matrix}\right.\\ \qquad V=\{(-8,-3)\}\)
3. \(\left\{\begin{matrix}10y-6x=84\\5x+4y=-33\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-6x+10y=84\\5x+4y=-33\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{array}{c|c|c}-6x+10y=84& \color{red}{5.} & \color{blue}{2.} \\5x+4y=-33& \color{red}{6.} & \color{blue}{-5.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{-30x+30x}+50y+24y=420-198} \\ \color{blue}{-12x-25x\underline{+20y-20y}=168+165} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}74y=222 \\-37x=333\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{222}{74}=3 \\x=\frac{333}{-37}=-9\end{matrix}\right.\\ \qquad V=\{(-9,3)\}\)
4. \(\left\{\begin{matrix}-6x+7y=82\\-7x=10y-86\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-6x+7y=82\\-7x-10y=-86\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{array}{c|c|c}-6x+7y=82& \color{red}{7.} & \color{blue}{10.} \\-7x-10y=-86& \color{red}{-6.} & \color{blue}{7.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{-42x+42x}+49y+60y=574+516} \\ \color{blue}{-60x-49x\underline{+70y-70y}=820-602} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}109y=1090 \\-109x=218\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{1090}{109}=10 \\x=\frac{218}{-109}=-2\end{matrix}\right.\\ \qquad V=\{(-2,10)\}\)
5. \(\left\{\begin{matrix}-8x-5y=-38\\-8x-10y=-28\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{array}{c|c|c}-8x-5y=-38& \color{red}{1.} & \color{blue}{2.} \\-8x-10y=-28& \color{red}{-1.} & \color{blue}{-1.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{-8x+8x}-5y+10y=-38+28} \\ \color{blue}{-16x+8x\underline{-10y+10y}=-76+28} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}5y=-10 \\-8x=-48\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{-10}{5}=-2 \\x=\frac{-48}{-8}=6\end{matrix}\right.\\ \qquad V=\{(6,-2)\}\)
6. \(\left\{\begin{matrix}9y=-126+9x\\8x+5y=47\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-9x+9y=-126\\8x+5y=47\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{array}{c|c|c}-9x+9y=-126& \color{red}{8.} & \color{blue}{5.} \\8x+5y=47& \color{red}{9.} & \color{blue}{-9.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{-72x+72x}+72y+45y=-1008+423} \\ \color{blue}{-45x-72x\underline{+45y-45y}=-630-423} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}117y=-585 \\-117x=-1053\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{-585}{117}=-5 \\x=\frac{-1053}{-117}=9\end{matrix}\right.\\ \qquad V=\{(9,-5)\}\)
7. \(\left\{\begin{matrix}3x-10y=100\\3x=7y+79\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}3x-10y=100\\3x-7y=79\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{array}{c|c|c}3x-10y=100& \color{red}{1.} & \color{blue}{7.} \\3x-7y=79& \color{red}{-1.} & \color{blue}{-10.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{3x-3x}-10y+7y=100-79} \\ \color{blue}{21x-30x\underline{-70y+70y}=700-790} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-3y=21 \\-9x=-90\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{21}{-3}=-7 \\x=\frac{-90}{-9}=10\end{matrix}\right.\\ \qquad V=\{(10,-7)\}\)
8. \(\left\{\begin{matrix}-3x+5y=42\\-3x+9y=54\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{array}{c|c|c}-3x+5y=42& \color{red}{1.} & \color{blue}{9.} \\-3x+9y=54& \color{red}{-1.} & \color{blue}{-5.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{-3x+3x}+5y-9y=42-54} \\ \color{blue}{-27x+15x\underline{+45y-45y}=378-270} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-4y=-12 \\-12x=108\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{-12}{-4}=3 \\x=\frac{108}{-12}=-9\end{matrix}\right.\\ \qquad V=\{(-9,3)\}\)
9. \(\left\{\begin{matrix}8y=81+3x\\8x+3y=3\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-3x+8y=81\\8x+3y=3\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{array}{c|c|c}-3x+8y=81& \color{red}{8.} & \color{blue}{3.} \\8x+3y=3& \color{red}{3.} & \color{blue}{-8.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{-24x+24x}+64y+9y=648+9} \\ \color{blue}{-9x-64x\underline{+24y-24y}=243-24} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}73y=657 \\-73x=219\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{657}{73}=9 \\x=\frac{219}{-73}=-3\end{matrix}\right.\\ \qquad V=\{(-3,9)\}\)
10. \(\left\{\begin{matrix}4x+10y=106\\8x=-7y+121\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}4x+10y=106\\8x+7y=121\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{array}{c|c|c}4x+10y=106& \color{red}{2.} & \color{blue}{7.} \\8x+7y=121& \color{red}{-1.} & \color{blue}{-10.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{8x-8x}+20y-7y=212-121} \\ \color{blue}{28x-80x\underline{+70y-70y}=742-1210} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}13y=91 \\-52x=-468\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{91}{13}=7 \\x=\frac{-468}{-52}=9\end{matrix}\right.\\ \qquad V=\{(9,7)\}\)
11. \(\left\{\begin{matrix}-8x+10y=-12\\-3x=7y+17\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-8x+10y=-12\\-3x-7y=17\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{array}{c|c|c}-8x+10y=-12& \color{red}{3.} & \color{blue}{7.} \\-3x-7y=17& \color{red}{-8.} & \color{blue}{10.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{-24x+24x}+30y+56y=-36-136} \\ \color{blue}{-56x-30x\underline{+70y-70y}=-84+170} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}86y=-172 \\-86x=86\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{-172}{86}=-2 \\x=\frac{86}{-86}=-1\end{matrix}\right.\\ \qquad V=\{(-1,-2)\}\)
12. \(\left\{\begin{matrix}3y=-39-7x\\-7x+9y=-33\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}7x+3y=-39\\-7x+9y=-33\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{array}{c|c|c}7x+3y=-39& \color{red}{1.} & \color{blue}{3.} \\-7x+9y=-33& \color{red}{1.} & \color{blue}{-1.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{7x-7x}+3y+9y=-39-33} \\ \color{blue}{21x+7x\underline{+9y-9y}=-117+33} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}12y=-72 \\28x=-84\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{-72}{12}=-6 \\x=\frac{-84}{28}=-3\end{matrix}\right.\\ \qquad V=\{(-3,-6)\}\)