Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{210 is het kleinste gemene veelvoud van 5, 7 en 6} \\ \begin{align} & \frac{x}{5}-\frac{2}{7}& = & \frac{5}{6}x-7 \\\Leftrightarrow & \color{blue}{210.} (\frac{42x}{ \color{blue}{210} }- \frac{ 60 }{ \color{blue}{210} })& = & (\frac{175}{ \color{blue}{210} }x-\frac{1470}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 42x-60& = & 175x-1470 \\\Leftrightarrow & 42x \color{red}{-60} \color{blue}{+60} \color{blue}{-175x} & = & \color{red}{175x} -1470 \color{blue}{-175x} \color{blue}{+60} \\\Leftrightarrow & -133x& = & -1410 \\\Leftrightarrow & \frac{-133x}{ \color{red}{-133} }& = & \frac{-1410}{-133} \\\Leftrightarrow & x = \frac{1410}{133} & & \\ & V = \left\{ \frac{1410}{133} \right\} & \\\end{align}\)
2. \(\text{60 is het kleinste gemene veelvoud van 4, 15 en 3} \\ \begin{align} & \frac{x}{4}+\frac{2}{15}& = & \frac{5}{3}x-1 \\\Leftrightarrow & \color{blue}{60.} (\frac{15x}{ \color{blue}{60} }+ \frac{ 8 }{ \color{blue}{60} })& = & (\frac{100}{ \color{blue}{60} }x-\frac{60}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 15x+8& = & 100x-60 \\\Leftrightarrow & 15x \color{red}{+8} \color{blue}{-8} \color{blue}{-100x} & = & \color{red}{100x} -60 \color{blue}{-100x} \color{blue}{-8} \\\Leftrightarrow & -85x& = & -68 \\\Leftrightarrow & \frac{-85x}{ \color{red}{-85} }& = & \frac{-68}{-85} \\\Leftrightarrow & x = \frac{4}{5} & & \\ & V = \left\{ \frac{4}{5} \right\} & \\\end{align}\)
3. \(\text{30 is het kleinste gemene veelvoud van 2, 15 en 3} \\ \begin{align} & \frac{x}{2}+\frac{4}{15}& = & \frac{-2}{3}x-5 \\\Leftrightarrow & \color{blue}{30.} (\frac{15x}{ \color{blue}{30} }+ \frac{ 8 }{ \color{blue}{30} })& = & (\frac{-20}{ \color{blue}{30} }x-\frac{150}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 15x+8& = & -20x-150 \\\Leftrightarrow & 15x \color{red}{+8} \color{blue}{-8} \color{blue}{+20x} & = & \color{red}{-20x} -150 \color{blue}{+20x} \color{blue}{-8} \\\Leftrightarrow & 35x& = & -158 \\\Leftrightarrow & \frac{35x}{ \color{red}{35} }& = & \frac{-158}{35} \\\Leftrightarrow & x = \frac{-158}{35} & & \\ & V = \left\{ \frac{-158}{35} \right\} & \\\end{align}\)
4. \(\text{336 is het kleinste gemene veelvoud van 7, 16 en 6} \\ \begin{align} & \frac{x}{7}+\frac{3}{16}& = & \frac{1}{6}x+6 \\\Leftrightarrow & \color{blue}{336.} (\frac{48x}{ \color{blue}{336} }+ \frac{ 63 }{ \color{blue}{336} })& = & (\frac{56}{ \color{blue}{336} }x+\frac{2016}{ \color{blue}{336} }) \color{blue}{.336} \\\Leftrightarrow & 48x+63& = & 56x+2016 \\\Leftrightarrow & 48x \color{red}{+63} \color{blue}{-63} \color{blue}{-56x} & = & \color{red}{56x} +2016 \color{blue}{-56x} \color{blue}{-63} \\\Leftrightarrow & -8x& = & 1953 \\\Leftrightarrow & \frac{-8x}{ \color{red}{-8} }& = & \frac{1953}{-8} \\\Leftrightarrow & x = \frac{-1953}{8} & & \\ & V = \left\{ \frac{-1953}{8} \right\} & \\\end{align}\)
5. \(\text{24 is het kleinste gemene veelvoud van 2, 8 en 3} \\ \begin{align} & \frac{x}{2}-\frac{3}{8}& = & \frac{4}{3}x-6 \\\Leftrightarrow & \color{blue}{24.} (\frac{12x}{ \color{blue}{24} }- \frac{ 9 }{ \color{blue}{24} })& = & (\frac{32}{ \color{blue}{24} }x-\frac{144}{ \color{blue}{24} }) \color{blue}{.24} \\\Leftrightarrow & 12x-9& = & 32x-144 \\\Leftrightarrow & 12x \color{red}{-9} \color{blue}{+9} \color{blue}{-32x} & = & \color{red}{32x} -144 \color{blue}{-32x} \color{blue}{+9} \\\Leftrightarrow & -20x& = & -135 \\\Leftrightarrow & \frac{-20x}{ \color{red}{-20} }& = & \frac{-135}{-20} \\\Leftrightarrow & x = \frac{27}{4} & & \\ & V = \left\{ \frac{27}{4} \right\} & \\\end{align}\)
6. \(\text{80 is het kleinste gemene veelvoud van 5, 16 en 2} \\ \begin{align} & \frac{x}{5}-\frac{5}{16}& = & \frac{1}{2}x-3 \\\Leftrightarrow & \color{blue}{80.} (\frac{16x}{ \color{blue}{80} }- \frac{ 25 }{ \color{blue}{80} })& = & (\frac{40}{ \color{blue}{80} }x-\frac{240}{ \color{blue}{80} }) \color{blue}{.80} \\\Leftrightarrow & 16x-25& = & 40x-240 \\\Leftrightarrow & 16x \color{red}{-25} \color{blue}{+25} \color{blue}{-40x} & = & \color{red}{40x} -240 \color{blue}{-40x} \color{blue}{+25} \\\Leftrightarrow & -24x& = & -215 \\\Leftrightarrow & \frac{-24x}{ \color{red}{-24} }& = & \frac{-215}{-24} \\\Leftrightarrow & x = \frac{215}{24} & & \\ & V = \left\{ \frac{215}{24} \right\} & \\\end{align}\)
7. \(\text{30 is het kleinste gemene veelvoud van 2, 10 en 3} \\ \begin{align} & \frac{x}{2}+\frac{3}{10}& = & \frac{-5}{3}x+3 \\\Leftrightarrow & \color{blue}{30.} (\frac{15x}{ \color{blue}{30} }+ \frac{ 9 }{ \color{blue}{30} })& = & (\frac{-50}{ \color{blue}{30} }x+\frac{90}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 15x+9& = & -50x+90 \\\Leftrightarrow & 15x \color{red}{+9} \color{blue}{-9} \color{blue}{+50x} & = & \color{red}{-50x} +90 \color{blue}{+50x} \color{blue}{-9} \\\Leftrightarrow & 65x& = & 81 \\\Leftrightarrow & \frac{65x}{ \color{red}{65} }& = & \frac{81}{65} \\\Leftrightarrow & x = \frac{81}{65} & & \\ & V = \left\{ \frac{81}{65} \right\} & \\\end{align}\)
8. \(\text{195 is het kleinste gemene veelvoud van 5, 13 en 3} \\ \begin{align} & \frac{x}{5}-\frac{3}{13}& = & \frac{-8}{3}x-6 \\\Leftrightarrow & \color{blue}{195.} (\frac{39x}{ \color{blue}{195} }- \frac{ 45 }{ \color{blue}{195} })& = & (\frac{-520}{ \color{blue}{195} }x-\frac{1170}{ \color{blue}{195} }) \color{blue}{.195} \\\Leftrightarrow & 39x-45& = & -520x-1170 \\\Leftrightarrow & 39x \color{red}{-45} \color{blue}{+45} \color{blue}{+520x} & = & \color{red}{-520x} -1170 \color{blue}{+520x} \color{blue}{+45} \\\Leftrightarrow & 559x& = & -1125 \\\Leftrightarrow & \frac{559x}{ \color{red}{559} }& = & \frac{-1125}{559} \\\Leftrightarrow & x = \frac{-1125}{559} & & \\ & V = \left\{ \frac{-1125}{559} \right\} & \\\end{align}\)
9. \(\text{126 is het kleinste gemene veelvoud van 7, 9 en 2} \\ \begin{align} & \frac{x}{7}-\frac{4}{9}& = & \frac{1}{2}x-4 \\\Leftrightarrow & \color{blue}{126.} (\frac{18x}{ \color{blue}{126} }- \frac{ 56 }{ \color{blue}{126} })& = & (\frac{63}{ \color{blue}{126} }x-\frac{504}{ \color{blue}{126} }) \color{blue}{.126} \\\Leftrightarrow & 18x-56& = & 63x-504 \\\Leftrightarrow & 18x \color{red}{-56} \color{blue}{+56} \color{blue}{-63x} & = & \color{red}{63x} -504 \color{blue}{-63x} \color{blue}{+56} \\\Leftrightarrow & -45x& = & -448 \\\Leftrightarrow & \frac{-45x}{ \color{red}{-45} }& = & \frac{-448}{-45} \\\Leftrightarrow & x = \frac{448}{45} & & \\ & V = \left\{ \frac{448}{45} \right\} & \\\end{align}\)
10. \(\text{78 is het kleinste gemene veelvoud van 3, 13 en 2} \\ \begin{align} & \frac{x}{3}+\frac{3}{13}& = & \frac{1}{2}x-5 \\\Leftrightarrow & \color{blue}{78.} (\frac{26x}{ \color{blue}{78} }+ \frac{ 18 }{ \color{blue}{78} })& = & (\frac{39}{ \color{blue}{78} }x-\frac{390}{ \color{blue}{78} }) \color{blue}{.78} \\\Leftrightarrow & 26x+18& = & 39x-390 \\\Leftrightarrow & 26x \color{red}{+18} \color{blue}{-18} \color{blue}{-39x} & = & \color{red}{39x} -390 \color{blue}{-39x} \color{blue}{-18} \\\Leftrightarrow & -13x& = & -408 \\\Leftrightarrow & \frac{-13x}{ \color{red}{-13} }& = & \frac{-408}{-13} \\\Leftrightarrow & x = \frac{408}{13} & & \\ & V = \left\{ \frac{408}{13} \right\} & \\\end{align}\)
11. \(\text{78 is het kleinste gemene veelvoud van 2, 13 en 3} \\ \begin{align} & \frac{x}{2}+\frac{2}{13}& = & \frac{-5}{3}x-1 \\\Leftrightarrow & \color{blue}{78.} (\frac{39x}{ \color{blue}{78} }+ \frac{ 12 }{ \color{blue}{78} })& = & (\frac{-130}{ \color{blue}{78} }x-\frac{78}{ \color{blue}{78} }) \color{blue}{.78} \\\Leftrightarrow & 39x+12& = & -130x-78 \\\Leftrightarrow & 39x \color{red}{+12} \color{blue}{-12} \color{blue}{+130x} & = & \color{red}{-130x} -78 \color{blue}{+130x} \color{blue}{-12} \\\Leftrightarrow & 169x& = & -90 \\\Leftrightarrow & \frac{169x}{ \color{red}{169} }& = & \frac{-90}{169} \\\Leftrightarrow & x = \frac{-90}{169} & & \\ & V = \left\{ \frac{-90}{169} \right\} & \\\end{align}\)
12. \(\text{78 is het kleinste gemene veelvoud van 3, 13 en 2} \\ \begin{align} & \frac{x}{3}+\frac{2}{13}& = & \frac{1}{2}x+2 \\\Leftrightarrow & \color{blue}{78.} (\frac{26x}{ \color{blue}{78} }+ \frac{ 12 }{ \color{blue}{78} })& = & (\frac{39}{ \color{blue}{78} }x+\frac{156}{ \color{blue}{78} }) \color{blue}{.78} \\\Leftrightarrow & 26x+12& = & 39x+156 \\\Leftrightarrow & 26x \color{red}{+12} \color{blue}{-12} \color{blue}{-39x} & = & \color{red}{39x} +156 \color{blue}{-39x} \color{blue}{-12} \\\Leftrightarrow & -13x& = & 144 \\\Leftrightarrow & \frac{-13x}{ \color{red}{-13} }& = & \frac{144}{-13} \\\Leftrightarrow & x = \frac{-144}{13} & & \\ & V = \left\{ \frac{-144}{13} \right\} & \\\end{align}\)