Stelsels combinatie

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Combinatie

1. \(\left\{\begin{matrix}-8x-10y=34\\5x-3y=-12\end{matrix}\right.\)
2. \(\left\{\begin{matrix}10y=40-10x\\-6x-7y=-29\end{matrix}\right.\)
3. \(\left\{\begin{matrix}-8x+9y=31\\-3y+10x=19\end{matrix}\right.\)
4. \(\left\{\begin{matrix}-2y+8x=24\\8x-10y=56\end{matrix}\right.\)
5. \(\left\{\begin{matrix}7x+4y=-66\\10x=8y-12\end{matrix}\right.\)
6. \(\left\{\begin{matrix}9x-5y=112\\2x=7y+72\end{matrix}\right.\)
7. \(\left\{\begin{matrix}-4x+4y=68\\4x=6y-86\end{matrix}\right.\)
8. \(\left\{\begin{matrix}-10x-8y=10\\7x=-5y-1\end{matrix}\right.\)
9. \(\left\{\begin{matrix}-4x-7y=-18\\-9y+8x=82\end{matrix}\right.\)
10. \(\left\{\begin{matrix}-2x-10y=102\\2x=-6y-62\end{matrix}\right.\)
11. \(\left\{\begin{matrix}-3x-7y=36\\-7x-2y=-45\end{matrix}\right.\)
12. \(\left\{\begin{matrix}-3y=-16-4x\\-6x+9y=42\end{matrix}\right.\)

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Combinatie

Verbetersleutel

1. \(\left\{\begin{matrix}-8x-10y=34\\5x-3y=-12\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{array}{c|c|c}-8x-10y=34& \color{red}{5.} & \color{blue}{3.} \\5x-3y=-12& \color{red}{8.} & \color{blue}{-10.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{-40x+40x}-50y-24y=170-96} \\ \color{blue}{-24x-50x\underline{-30y+30y}=102+120} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-74y=74 \\-74x=222\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{74}{-74}=-1 \\x=\frac{222}{-74}=-3\end{matrix}\right.\\ \qquad V=\{(-3,-1)\}\)
2. \(\left\{\begin{matrix}10y=40-10x\\-6x-7y=-29\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}10x+10y=40\\-6x-7y=-29\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{array}{c|c|c}10x+10y=40& \color{red}{3.} & \color{blue}{7.} \\-6x-7y=-29& \color{red}{5.} & \color{blue}{10.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{30x-30x}+30y-35y=120-145} \\ \color{blue}{70x-60x\underline{+70y-70y}=280-290} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-5y=-25 \\10x=-10\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{-25}{-5}=5 \\x=\frac{-10}{10}=-1\end{matrix}\right.\\ \qquad V=\{(-1,5)\}\)
3. \(\left\{\begin{matrix}-8x+9y=31\\-3y+10x=19\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-8x+9y=31\\10x-3y=19\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{array}{c|c|c}-8x+9y=31& \color{red}{5.} & \color{blue}{1.} \\10x-3y=19& \color{red}{4.} & \color{blue}{3.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{-40x+40x}+45y-12y=155+76} \\ \color{blue}{-8x+30x\underline{+9y-9y}=31+57} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}33y=231 \\22x=88\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{231}{33}=7 \\x=\frac{88}{22}=4\end{matrix}\right.\\ \qquad V=\{(4,7)\}\)
4. \(\left\{\begin{matrix}-2y+8x=24\\8x-10y=56\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}8x-2y=24\\8x-10y=56\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{array}{c|c|c}8x-2y=24& \color{red}{1.} & \color{blue}{5.} \\8x-10y=56& \color{red}{-1.} & \color{blue}{-1.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{8x-8x}-2y+10y=24-56} \\ \color{blue}{40x-8x\underline{-10y+10y}=120-56} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}8y=-32 \\32x=64\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{-32}{8}=-4 \\x=\frac{64}{32}=2\end{matrix}\right.\\ \qquad V=\{(2,-4)\}\)
5. \(\left\{\begin{matrix}7x+4y=-66\\10x=8y-12\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}7x+4y=-66\\10x-8y=-12\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{array}{c|c|c}7x+4y=-66& \color{red}{10.} & \color{blue}{2.} \\10x-8y=-12& \color{red}{-7.} & \color{blue}{1.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{70x-70x}+40y+56y=-660+84} \\ \color{blue}{14x+10x\underline{+8y-8y}=-132-12} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}96y=-576 \\24x=-144\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{-576}{96}=-6 \\x=\frac{-144}{24}=-6\end{matrix}\right.\\ \qquad V=\{(-6,-6)\}\)
6. \(\left\{\begin{matrix}9x-5y=112\\2x=7y+72\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}9x-5y=112\\2x-7y=72\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{array}{c|c|c}9x-5y=112& \color{red}{2.} & \color{blue}{7.} \\2x-7y=72& \color{red}{-9.} & \color{blue}{-5.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{18x-18x}-10y+63y=224-648} \\ \color{blue}{63x-10x\underline{-35y+35y}=784-360} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}53y=-424 \\53x=424\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{-424}{53}=-8 \\x=\frac{424}{53}=8\end{matrix}\right.\\ \qquad V=\{(8,-8)\}\)
7. \(\left\{\begin{matrix}-4x+4y=68\\4x=6y-86\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-4x+4y=68\\4x-6y=-86\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{array}{c|c|c}-4x+4y=68& \color{red}{1.} & \color{blue}{3.} \\4x-6y=-86& \color{red}{1.} & \color{blue}{2.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{-4x+4x}+4y-6y=68-86} \\ \color{blue}{-12x+8x\underline{+12y-12y}=204-172} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-2y=-18 \\-4x=32\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{-18}{-2}=9 \\x=\frac{32}{-4}=-8\end{matrix}\right.\\ \qquad V=\{(-8,9)\}\)
8. \(\left\{\begin{matrix}-10x-8y=10\\7x=-5y-1\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-10x-8y=10\\7x+5y=-1\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{array}{c|c|c}-10x-8y=10& \color{red}{7.} & \color{blue}{5.} \\7x+5y=-1& \color{red}{10.} & \color{blue}{8.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{-70x+70x}-56y+50y=70-10} \\ \color{blue}{-50x+56x\underline{-40y+40y}=50-8} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-6y=60 \\6x=42\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{60}{-6}=-10 \\x=\frac{42}{6}=7\end{matrix}\right.\\ \qquad V=\{(7,-10)\}\)
9. \(\left\{\begin{matrix}-4x-7y=-18\\-9y+8x=82\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-4x-7y=-18\\8x-9y=82\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{array}{c|c|c}-4x-7y=-18& \color{red}{2.} & \color{blue}{9.} \\8x-9y=82& \color{red}{1.} & \color{blue}{-7.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{-8x+8x}-14y-9y=-36+82} \\ \color{blue}{-36x-56x\underline{-63y+63y}=-162-574} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-23y=46 \\-92x=-736\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{46}{-23}=-2 \\x=\frac{-736}{-92}=8\end{matrix}\right.\\ \qquad V=\{(8,-2)\}\)
10. \(\left\{\begin{matrix}-2x-10y=102\\2x=-6y-62\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-2x-10y=102\\2x+6y=-62\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{array}{c|c|c}-2x-10y=102& \color{red}{1.} & \color{blue}{3.} \\2x+6y=-62& \color{red}{1.} & \color{blue}{5.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{-2x+2x}-10y+6y=102-62} \\ \color{blue}{-6x+10x\underline{-30y+30y}=306-310} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-4y=40 \\4x=-4\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{40}{-4}=-10 \\x=\frac{-4}{4}=-1\end{matrix}\right.\\ \qquad V=\{(-1,-10)\}\)
11. \(\left\{\begin{matrix}-3x-7y=36\\-7x-2y=-45\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{array}{c|c|c}-3x-7y=36& \color{red}{7.} & \color{blue}{2.} \\-7x-2y=-45& \color{red}{-3.} & \color{blue}{-7.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{-21x+21x}-49y+6y=252+135} \\ \color{blue}{-6x+49x\underline{-14y+14y}=72+315} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}-43y=387 \\43x=387\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{387}{-43}=-9 \\x=\frac{387}{43}=9\end{matrix}\right.\\ \qquad V=\{(9,-9)\}\)
12. \(\left\{\begin{matrix}-3y=-16-4x\\-6x+9y=42\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}4x-3y=-16\\-6x+9y=42\end{matrix}\right.\text{(Niet verplichte stap, proper schrijven)}\\ \Leftrightarrow \left\{\begin{array}{c|c|c}4x-3y=-16& \color{red}{3.} & \color{blue}{3.} \\-6x+9y=42& \color{red}{2.} & \color{blue}{1.} \end{array}\right.\\ \Leftrightarrow \left\{\begin{matrix} \color{red}{\underline{12x-12x}-9y+18y=-48+84} \\ \color{blue}{12x-6x\underline{-9y+9y}=-48+42} \end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}9y=36 \\6x=-6\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{matrix}y=\frac{36}{9}=4 \\x=\frac{-6}{6}=-1\end{matrix}\right.\\ \qquad V=\{(-1,4)\}\)