Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{30 is het kleinste gemene veelvoud van 6, 10 en 5} \\ \begin{align} & \frac{x}{6}+\frac{3}{10}& = & \frac{1}{5}x-7 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }+ \frac{ 9 }{ \color{blue}{30} })& = & (\frac{6}{ \color{blue}{30} }x-\frac{210}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x+9& = & 6x-210 \\\Leftrightarrow & 5x \color{red}{+9} \color{blue}{-9} \color{blue}{-6x} & = & \color{red}{6x} -210 \color{blue}{-6x} \color{blue}{-9} \\\Leftrightarrow & -x& = & -219 \\\Leftrightarrow & \frac{-x}{ \color{red}{-1} }& = & \frac{-219}{-1} \\\Leftrightarrow & x = 219 & & \\ & V = \left\{ 219 \right\} & \\\end{align}\)
2. \(\text{220 is het kleinste gemene veelvoud van 4, 11 en 5} \\ \begin{align} & \frac{x}{4}+\frac{2}{11}& = & \frac{1}{5}x+3 \\\Leftrightarrow & \color{blue}{220.} (\frac{55x}{ \color{blue}{220} }+ \frac{ 40 }{ \color{blue}{220} })& = & (\frac{44}{ \color{blue}{220} }x+\frac{660}{ \color{blue}{220} }) \color{blue}{.220} \\\Leftrightarrow & 55x+40& = & 44x+660 \\\Leftrightarrow & 55x \color{red}{+40} \color{blue}{-40} \color{blue}{-44x} & = & \color{red}{44x} +660 \color{blue}{-44x} \color{blue}{-40} \\\Leftrightarrow & 11x& = & 620 \\\Leftrightarrow & \frac{11x}{ \color{red}{11} }& = & \frac{620}{11} \\\Leftrightarrow & x = \frac{620}{11} & & \\ & V = \left\{ \frac{620}{11} \right\} & \\\end{align}\)
3. \(\text{78 is het kleinste gemene veelvoud van 2, 13 en 3} \\ \begin{align} & \frac{x}{2}-\frac{4}{13}& = & \frac{4}{3}x+8 \\\Leftrightarrow & \color{blue}{78.} (\frac{39x}{ \color{blue}{78} }- \frac{ 24 }{ \color{blue}{78} })& = & (\frac{104}{ \color{blue}{78} }x+\frac{624}{ \color{blue}{78} }) \color{blue}{.78} \\\Leftrightarrow & 39x-24& = & 104x+624 \\\Leftrightarrow & 39x \color{red}{-24} \color{blue}{+24} \color{blue}{-104x} & = & \color{red}{104x} +624 \color{blue}{-104x} \color{blue}{+24} \\\Leftrightarrow & -65x& = & 648 \\\Leftrightarrow & \frac{-65x}{ \color{red}{-65} }& = & \frac{648}{-65} \\\Leftrightarrow & x = \frac{-648}{65} & & \\ & V = \left\{ \frac{-648}{65} \right\} & \\\end{align}\)
4. \(\text{210 is het kleinste gemene veelvoud van 5, 7 en 6} \\ \begin{align} & \frac{x}{5}-\frac{2}{7}& = & \frac{5}{6}x-5 \\\Leftrightarrow & \color{blue}{210.} (\frac{42x}{ \color{blue}{210} }- \frac{ 60 }{ \color{blue}{210} })& = & (\frac{175}{ \color{blue}{210} }x-\frac{1050}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 42x-60& = & 175x-1050 \\\Leftrightarrow & 42x \color{red}{-60} \color{blue}{+60} \color{blue}{-175x} & = & \color{red}{175x} -1050 \color{blue}{-175x} \color{blue}{+60} \\\Leftrightarrow & -133x& = & -990 \\\Leftrightarrow & \frac{-133x}{ \color{red}{-133} }& = & \frac{-990}{-133} \\\Leftrightarrow & x = \frac{990}{133} & & \\ & V = \left\{ \frac{990}{133} \right\} & \\\end{align}\)
5. \(\text{132 is het kleinste gemene veelvoud van 3, 11 en 4} \\ \begin{align} & \frac{x}{3}-\frac{4}{11}& = & \frac{-3}{4}x-8 \\\Leftrightarrow & \color{blue}{132.} (\frac{44x}{ \color{blue}{132} }- \frac{ 48 }{ \color{blue}{132} })& = & (\frac{-99}{ \color{blue}{132} }x-\frac{1056}{ \color{blue}{132} }) \color{blue}{.132} \\\Leftrightarrow & 44x-48& = & -99x-1056 \\\Leftrightarrow & 44x \color{red}{-48} \color{blue}{+48} \color{blue}{+99x} & = & \color{red}{-99x} -1056 \color{blue}{+99x} \color{blue}{+48} \\\Leftrightarrow & 143x& = & -1008 \\\Leftrightarrow & \frac{143x}{ \color{red}{143} }& = & \frac{-1008}{143} \\\Leftrightarrow & x = \frac{-1008}{143} & & \\ & V = \left\{ \frac{-1008}{143} \right\} & \\\end{align}\)
6. \(\text{60 is het kleinste gemene veelvoud van 5, 12 en 3} \\ \begin{align} & \frac{x}{5}-\frac{5}{12}& = & \frac{1}{3}x-1 \\\Leftrightarrow & \color{blue}{60.} (\frac{12x}{ \color{blue}{60} }- \frac{ 25 }{ \color{blue}{60} })& = & (\frac{20}{ \color{blue}{60} }x-\frac{60}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 12x-25& = & 20x-60 \\\Leftrightarrow & 12x \color{red}{-25} \color{blue}{+25} \color{blue}{-20x} & = & \color{red}{20x} -60 \color{blue}{-20x} \color{blue}{+25} \\\Leftrightarrow & -8x& = & -35 \\\Leftrightarrow & \frac{-8x}{ \color{red}{-8} }& = & \frac{-35}{-8} \\\Leftrightarrow & x = \frac{35}{8} & & \\ & V = \left\{ \frac{35}{8} \right\} & \\\end{align}\)
7. \(\text{70 is het kleinste gemene veelvoud van 5, 7 en 2} \\ \begin{align} & \frac{x}{5}+\frac{3}{7}& = & \frac{1}{2}x+7 \\\Leftrightarrow & \color{blue}{70.} (\frac{14x}{ \color{blue}{70} }+ \frac{ 30 }{ \color{blue}{70} })& = & (\frac{35}{ \color{blue}{70} }x+\frac{490}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 14x+30& = & 35x+490 \\\Leftrightarrow & 14x \color{red}{+30} \color{blue}{-30} \color{blue}{-35x} & = & \color{red}{35x} +490 \color{blue}{-35x} \color{blue}{-30} \\\Leftrightarrow & -21x& = & 460 \\\Leftrightarrow & \frac{-21x}{ \color{red}{-21} }& = & \frac{460}{-21} \\\Leftrightarrow & x = \frac{-460}{21} & & \\ & V = \left\{ \frac{-460}{21} \right\} & \\\end{align}\)
8. \(\text{18 is het kleinste gemene veelvoud van 2, 9 en 3} \\ \begin{align} & \frac{x}{2}+\frac{2}{9}& = & \frac{-2}{3}x+1 \\\Leftrightarrow & \color{blue}{18.} (\frac{9x}{ \color{blue}{18} }+ \frac{ 4 }{ \color{blue}{18} })& = & (\frac{-12}{ \color{blue}{18} }x+\frac{18}{ \color{blue}{18} }) \color{blue}{.18} \\\Leftrightarrow & 9x+4& = & -12x+18 \\\Leftrightarrow & 9x \color{red}{+4} \color{blue}{-4} \color{blue}{+12x} & = & \color{red}{-12x} +18 \color{blue}{+12x} \color{blue}{-4} \\\Leftrightarrow & 21x& = & 14 \\\Leftrightarrow & \frac{21x}{ \color{red}{21} }& = & \frac{14}{21} \\\Leftrightarrow & x = \frac{2}{3} & & \\ & V = \left\{ \frac{2}{3} \right\} & \\\end{align}\)
9. \(\text{30 is het kleinste gemene veelvoud van 2, 15 en 3} \\ \begin{align} & \frac{x}{2}-\frac{2}{15}& = & \frac{7}{3}x+4 \\\Leftrightarrow & \color{blue}{30.} (\frac{15x}{ \color{blue}{30} }- \frac{ 4 }{ \color{blue}{30} })& = & (\frac{70}{ \color{blue}{30} }x+\frac{120}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 15x-4& = & 70x+120 \\\Leftrightarrow & 15x \color{red}{-4} \color{blue}{+4} \color{blue}{-70x} & = & \color{red}{70x} +120 \color{blue}{-70x} \color{blue}{+4} \\\Leftrightarrow & -55x& = & 124 \\\Leftrightarrow & \frac{-55x}{ \color{red}{-55} }& = & \frac{124}{-55} \\\Leftrightarrow & x = \frac{-124}{55} & & \\ & V = \left\{ \frac{-124}{55} \right\} & \\\end{align}\)
10. \(\text{80 is het kleinste gemene veelvoud van 4, 16 en 5} \\ \begin{align} & \frac{x}{4}-\frac{3}{16}& = & \frac{6}{5}x-3 \\\Leftrightarrow & \color{blue}{80.} (\frac{20x}{ \color{blue}{80} }- \frac{ 15 }{ \color{blue}{80} })& = & (\frac{96}{ \color{blue}{80} }x-\frac{240}{ \color{blue}{80} }) \color{blue}{.80} \\\Leftrightarrow & 20x-15& = & 96x-240 \\\Leftrightarrow & 20x \color{red}{-15} \color{blue}{+15} \color{blue}{-96x} & = & \color{red}{96x} -240 \color{blue}{-96x} \color{blue}{+15} \\\Leftrightarrow & -76x& = & -225 \\\Leftrightarrow & \frac{-76x}{ \color{red}{-76} }& = & \frac{-225}{-76} \\\Leftrightarrow & x = \frac{225}{76} & & \\ & V = \left\{ \frac{225}{76} \right\} & \\\end{align}\)
11. \(\text{560 is het kleinste gemene veelvoud van 7, 16 en 5} \\ \begin{align} & \frac{x}{7}-\frac{5}{16}& = & \frac{-2}{5}x+4 \\\Leftrightarrow & \color{blue}{560.} (\frac{80x}{ \color{blue}{560} }- \frac{ 175 }{ \color{blue}{560} })& = & (\frac{-224}{ \color{blue}{560} }x+\frac{2240}{ \color{blue}{560} }) \color{blue}{.560} \\\Leftrightarrow & 80x-175& = & -224x+2240 \\\Leftrightarrow & 80x \color{red}{-175} \color{blue}{+175} \color{blue}{+224x} & = & \color{red}{-224x} +2240 \color{blue}{+224x} \color{blue}{+175} \\\Leftrightarrow & 304x& = & 2415 \\\Leftrightarrow & \frac{304x}{ \color{red}{304} }& = & \frac{2415}{304} \\\Leftrightarrow & x = \frac{2415}{304} & & \\ & V = \left\{ \frac{2415}{304} \right\} & \\\end{align}\)
12. \(\text{60 is het kleinste gemene veelvoud van 4, 15 en 3} \\ \begin{align} & \frac{x}{4}+\frac{4}{15}& = & \frac{4}{3}x+7 \\\Leftrightarrow & \color{blue}{60.} (\frac{15x}{ \color{blue}{60} }+ \frac{ 16 }{ \color{blue}{60} })& = & (\frac{80}{ \color{blue}{60} }x+\frac{420}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 15x+16& = & 80x+420 \\\Leftrightarrow & 15x \color{red}{+16} \color{blue}{-16} \color{blue}{-80x} & = & \color{red}{80x} +420 \color{blue}{-80x} \color{blue}{-16} \\\Leftrightarrow & -65x& = & 404 \\\Leftrightarrow & \frac{-65x}{ \color{red}{-65} }& = & \frac{404}{-65} \\\Leftrightarrow & x = \frac{-404}{65} & & \\ & V = \left\{ \frac{-404}{65} \right\} & \\\end{align}\)