Vgln. eerste graad (reeks 4)

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1. \(\frac{x}{7}+\frac{2}{15}=\frac{-2}{3}x+7\)
2. \(\frac{x}{5}-\frac{5}{11}=\frac{7}{2}x+7\)
3. \(\frac{x}{3}-\frac{4}{7}=\frac{1}{2}x+8\)
4. \(\frac{x}{6}+\frac{2}{15}=\frac{4}{5}x-3\)
5. \(\frac{x}{6}+\frac{5}{6}=\frac{-2}{5}x+3\)
6. \(\frac{x}{6}-\frac{2}{7}=\frac{6}{5}x+2\)
7. \(\frac{x}{2}-\frac{2}{7}=\frac{-4}{5}x-6\)
8. \(\frac{x}{5}-\frac{4}{13}=\frac{-5}{3}x-7\)
9. \(\frac{x}{5}-\frac{2}{15}=\frac{-3}{4}x-6\)
10. \(\frac{x}{7}-\frac{2}{7}=\frac{-7}{5}x+7\)
11. \(\frac{x}{6}-\frac{5}{16}=\frac{-7}{5}x-6\)
12. \(\frac{x}{3}+\frac{2}{15}=\frac{5}{2}x+8\)

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Reeks met breuken waarbij we ervoor kiezen om de breuken weg te werken. Neem je ZRM!

Verbetersleutel

1. \(\text{105 is het kleinste gemene veelvoud van 7, 15 en 3} \\ \begin{align} & \frac{x}{7}+\frac{2}{15}& = & \frac{-2}{3}x+7 \\\Leftrightarrow & \color{blue}{105.} (\frac{15x}{ \color{blue}{105} }+ \frac{ 14 }{ \color{blue}{105} })& = & (\frac{-70}{ \color{blue}{105} }x+\frac{735}{ \color{blue}{105} }) \color{blue}{.105} \\\Leftrightarrow & 15x+14& = & -70x+735 \\\Leftrightarrow & 15x \color{red}{+14} \color{blue}{-14} \color{blue}{+70x} & = & \color{red}{-70x} +735 \color{blue}{+70x} \color{blue}{-14} \\\Leftrightarrow & 85x& = & 721 \\\Leftrightarrow & \frac{85x}{ \color{red}{85} }& = & \frac{721}{85} \\\Leftrightarrow & x = \frac{721}{85} & & \\ & V = \left\{ \frac{721}{85} \right\} & \\\end{align}\)
2. \(\text{110 is het kleinste gemene veelvoud van 5, 11 en 2} \\ \begin{align} & \frac{x}{5}-\frac{5}{11}& = & \frac{7}{2}x+7 \\\Leftrightarrow & \color{blue}{110.} (\frac{22x}{ \color{blue}{110} }- \frac{ 50 }{ \color{blue}{110} })& = & (\frac{385}{ \color{blue}{110} }x+\frac{770}{ \color{blue}{110} }) \color{blue}{.110} \\\Leftrightarrow & 22x-50& = & 385x+770 \\\Leftrightarrow & 22x \color{red}{-50} \color{blue}{+50} \color{blue}{-385x} & = & \color{red}{385x} +770 \color{blue}{-385x} \color{blue}{+50} \\\Leftrightarrow & -363x& = & 820 \\\Leftrightarrow & \frac{-363x}{ \color{red}{-363} }& = & \frac{820}{-363} \\\Leftrightarrow & x = \frac{-820}{363} & & \\ & V = \left\{ \frac{-820}{363} \right\} & \\\end{align}\)
3. \(\text{42 is het kleinste gemene veelvoud van 3, 7 en 2} \\ \begin{align} & \frac{x}{3}-\frac{4}{7}& = & \frac{1}{2}x+8 \\\Leftrightarrow & \color{blue}{42.} (\frac{14x}{ \color{blue}{42} }- \frac{ 24 }{ \color{blue}{42} })& = & (\frac{21}{ \color{blue}{42} }x+\frac{336}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 14x-24& = & 21x+336 \\\Leftrightarrow & 14x \color{red}{-24} \color{blue}{+24} \color{blue}{-21x} & = & \color{red}{21x} +336 \color{blue}{-21x} \color{blue}{+24} \\\Leftrightarrow & -7x& = & 360 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{360}{-7} \\\Leftrightarrow & x = \frac{-360}{7} & & \\ & V = \left\{ \frac{-360}{7} \right\} & \\\end{align}\)
4. \(\text{30 is het kleinste gemene veelvoud van 6, 15 en 5} \\ \begin{align} & \frac{x}{6}+\frac{2}{15}& = & \frac{4}{5}x-3 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }+ \frac{ 4 }{ \color{blue}{30} })& = & (\frac{24}{ \color{blue}{30} }x-\frac{90}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x+4& = & 24x-90 \\\Leftrightarrow & 5x \color{red}{+4} \color{blue}{-4} \color{blue}{-24x} & = & \color{red}{24x} -90 \color{blue}{-24x} \color{blue}{-4} \\\Leftrightarrow & -19x& = & -94 \\\Leftrightarrow & \frac{-19x}{ \color{red}{-19} }& = & \frac{-94}{-19} \\\Leftrightarrow & x = \frac{94}{19} & & \\ & V = \left\{ \frac{94}{19} \right\} & \\\end{align}\)
5. \(\text{30 is het kleinste gemene veelvoud van 6, 6 en 5} \\ \begin{align} & \frac{x}{6}+\frac{5}{6}& = & \frac{-2}{5}x+3 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }+ \frac{ 25 }{ \color{blue}{30} })& = & (\frac{-12}{ \color{blue}{30} }x+\frac{90}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x+25& = & -12x+90 \\\Leftrightarrow & 5x \color{red}{+25} \color{blue}{-25} \color{blue}{+12x} & = & \color{red}{-12x} +90 \color{blue}{+12x} \color{blue}{-25} \\\Leftrightarrow & 17x& = & 65 \\\Leftrightarrow & \frac{17x}{ \color{red}{17} }& = & \frac{65}{17} \\\Leftrightarrow & x = \frac{65}{17} & & \\ & V = \left\{ \frac{65}{17} \right\} & \\\end{align}\)
6. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}-\frac{2}{7}& = & \frac{6}{5}x+2 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }- \frac{ 60 }{ \color{blue}{210} })& = & (\frac{252}{ \color{blue}{210} }x+\frac{420}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x-60& = & 252x+420 \\\Leftrightarrow & 35x \color{red}{-60} \color{blue}{+60} \color{blue}{-252x} & = & \color{red}{252x} +420 \color{blue}{-252x} \color{blue}{+60} \\\Leftrightarrow & -217x& = & 480 \\\Leftrightarrow & \frac{-217x}{ \color{red}{-217} }& = & \frac{480}{-217} \\\Leftrightarrow & x = \frac{-480}{217} & & \\ & V = \left\{ \frac{-480}{217} \right\} & \\\end{align}\)
7. \(\text{70 is het kleinste gemene veelvoud van 2, 7 en 5} \\ \begin{align} & \frac{x}{2}-\frac{2}{7}& = & \frac{-4}{5}x-6 \\\Leftrightarrow & \color{blue}{70.} (\frac{35x}{ \color{blue}{70} }- \frac{ 20 }{ \color{blue}{70} })& = & (\frac{-56}{ \color{blue}{70} }x-\frac{420}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 35x-20& = & -56x-420 \\\Leftrightarrow & 35x \color{red}{-20} \color{blue}{+20} \color{blue}{+56x} & = & \color{red}{-56x} -420 \color{blue}{+56x} \color{blue}{+20} \\\Leftrightarrow & 91x& = & -400 \\\Leftrightarrow & \frac{91x}{ \color{red}{91} }& = & \frac{-400}{91} \\\Leftrightarrow & x = \frac{-400}{91} & & \\ & V = \left\{ \frac{-400}{91} \right\} & \\\end{align}\)
8. \(\text{195 is het kleinste gemene veelvoud van 5, 13 en 3} \\ \begin{align} & \frac{x}{5}-\frac{4}{13}& = & \frac{-5}{3}x-7 \\\Leftrightarrow & \color{blue}{195.} (\frac{39x}{ \color{blue}{195} }- \frac{ 60 }{ \color{blue}{195} })& = & (\frac{-325}{ \color{blue}{195} }x-\frac{1365}{ \color{blue}{195} }) \color{blue}{.195} \\\Leftrightarrow & 39x-60& = & -325x-1365 \\\Leftrightarrow & 39x \color{red}{-60} \color{blue}{+60} \color{blue}{+325x} & = & \color{red}{-325x} -1365 \color{blue}{+325x} \color{blue}{+60} \\\Leftrightarrow & 364x& = & -1305 \\\Leftrightarrow & \frac{364x}{ \color{red}{364} }& = & \frac{-1305}{364} \\\Leftrightarrow & x = \frac{-1305}{364} & & \\ & V = \left\{ \frac{-1305}{364} \right\} & \\\end{align}\)
9. \(\text{60 is het kleinste gemene veelvoud van 5, 15 en 4} \\ \begin{align} & \frac{x}{5}-\frac{2}{15}& = & \frac{-3}{4}x-6 \\\Leftrightarrow & \color{blue}{60.} (\frac{12x}{ \color{blue}{60} }- \frac{ 8 }{ \color{blue}{60} })& = & (\frac{-45}{ \color{blue}{60} }x-\frac{360}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 12x-8& = & -45x-360 \\\Leftrightarrow & 12x \color{red}{-8} \color{blue}{+8} \color{blue}{+45x} & = & \color{red}{-45x} -360 \color{blue}{+45x} \color{blue}{+8} \\\Leftrightarrow & 57x& = & -352 \\\Leftrightarrow & \frac{57x}{ \color{red}{57} }& = & \frac{-352}{57} \\\Leftrightarrow & x = \frac{-352}{57} & & \\ & V = \left\{ \frac{-352}{57} \right\} & \\\end{align}\)
10. \(\text{35 is het kleinste gemene veelvoud van 7, 7 en 5} \\ \begin{align} & \frac{x}{7}-\frac{2}{7}& = & \frac{-7}{5}x+7 \\\Leftrightarrow & \color{blue}{35.} (\frac{5x}{ \color{blue}{35} }- \frac{ 10 }{ \color{blue}{35} })& = & (\frac{-49}{ \color{blue}{35} }x+\frac{245}{ \color{blue}{35} }) \color{blue}{.35} \\\Leftrightarrow & 5x-10& = & -49x+245 \\\Leftrightarrow & 5x \color{red}{-10} \color{blue}{+10} \color{blue}{+49x} & = & \color{red}{-49x} +245 \color{blue}{+49x} \color{blue}{+10} \\\Leftrightarrow & 54x& = & 255 \\\Leftrightarrow & \frac{54x}{ \color{red}{54} }& = & \frac{255}{54} \\\Leftrightarrow & x = \frac{85}{18} & & \\ & V = \left\{ \frac{85}{18} \right\} & \\\end{align}\)
11. \(\text{240 is het kleinste gemene veelvoud van 6, 16 en 5} \\ \begin{align} & \frac{x}{6}-\frac{5}{16}& = & \frac{-7}{5}x-6 \\\Leftrightarrow & \color{blue}{240.} (\frac{40x}{ \color{blue}{240} }- \frac{ 75 }{ \color{blue}{240} })& = & (\frac{-336}{ \color{blue}{240} }x-\frac{1440}{ \color{blue}{240} }) \color{blue}{.240} \\\Leftrightarrow & 40x-75& = & -336x-1440 \\\Leftrightarrow & 40x \color{red}{-75} \color{blue}{+75} \color{blue}{+336x} & = & \color{red}{-336x} -1440 \color{blue}{+336x} \color{blue}{+75} \\\Leftrightarrow & 376x& = & -1365 \\\Leftrightarrow & \frac{376x}{ \color{red}{376} }& = & \frac{-1365}{376} \\\Leftrightarrow & x = \frac{-1365}{376} & & \\ & V = \left\{ \frac{-1365}{376} \right\} & \\\end{align}\)
12. \(\text{30 is het kleinste gemene veelvoud van 3, 15 en 2} \\ \begin{align} & \frac{x}{3}+\frac{2}{15}& = & \frac{5}{2}x+8 \\\Leftrightarrow & \color{blue}{30.} (\frac{10x}{ \color{blue}{30} }+ \frac{ 4 }{ \color{blue}{30} })& = & (\frac{75}{ \color{blue}{30} }x+\frac{240}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 10x+4& = & 75x+240 \\\Leftrightarrow & 10x \color{red}{+4} \color{blue}{-4} \color{blue}{-75x} & = & \color{red}{75x} +240 \color{blue}{-75x} \color{blue}{-4} \\\Leftrightarrow & -65x& = & 236 \\\Leftrightarrow & \frac{-65x}{ \color{red}{-65} }& = & \frac{236}{-65} \\\Leftrightarrow & x = \frac{-236}{65} & & \\ & V = \left\{ \frac{-236}{65} \right\} & \\\end{align}\)