Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{220 is het kleinste gemene veelvoud van 4, 11 en 5} \\ \begin{align} & \frac{x}{4}-\frac{5}{11}& = & \frac{3}{5}x+8 \\\Leftrightarrow & \color{blue}{220.} (\frac{55x}{ \color{blue}{220} }- \frac{ 100 }{ \color{blue}{220} })& = & (\frac{132}{ \color{blue}{220} }x+\frac{1760}{ \color{blue}{220} }) \color{blue}{.220} \\\Leftrightarrow & 55x-100& = & 132x+1760 \\\Leftrightarrow & 55x \color{red}{-100} \color{blue}{+100} \color{blue}{-132x} & = & \color{red}{132x} +1760 \color{blue}{-132x} \color{blue}{+100} \\\Leftrightarrow & -77x& = & 1860 \\\Leftrightarrow & \frac{-77x}{ \color{red}{-77} }& = & \frac{1860}{-77} \\\Leftrightarrow & x = \frac{-1860}{77} & & \\ & V = \left\{ \frac{-1860}{77} \right\} & \\\end{align}\)
2. \(\text{330 is het kleinste gemene veelvoud van 6, 11 en 5} \\ \begin{align} & \frac{x}{6}+\frac{2}{11}& = & \frac{6}{5}x-1 \\\Leftrightarrow & \color{blue}{330.} (\frac{55x}{ \color{blue}{330} }+ \frac{ 60 }{ \color{blue}{330} })& = & (\frac{396}{ \color{blue}{330} }x-\frac{330}{ \color{blue}{330} }) \color{blue}{.330} \\\Leftrightarrow & 55x+60& = & 396x-330 \\\Leftrightarrow & 55x \color{red}{+60} \color{blue}{-60} \color{blue}{-396x} & = & \color{red}{396x} -330 \color{blue}{-396x} \color{blue}{-60} \\\Leftrightarrow & -341x& = & -390 \\\Leftrightarrow & \frac{-341x}{ \color{red}{-341} }& = & \frac{-390}{-341} \\\Leftrightarrow & x = \frac{390}{341} & & \\ & V = \left\{ \frac{390}{341} \right\} & \\\end{align}\)
3. \(\text{260 is het kleinste gemene veelvoud van 4, 13 en 5} \\ \begin{align} & \frac{x}{4}-\frac{4}{13}& = & \frac{-2}{5}x-7 \\\Leftrightarrow & \color{blue}{260.} (\frac{65x}{ \color{blue}{260} }- \frac{ 80 }{ \color{blue}{260} })& = & (\frac{-104}{ \color{blue}{260} }x-\frac{1820}{ \color{blue}{260} }) \color{blue}{.260} \\\Leftrightarrow & 65x-80& = & -104x-1820 \\\Leftrightarrow & 65x \color{red}{-80} \color{blue}{+80} \color{blue}{+104x} & = & \color{red}{-104x} -1820 \color{blue}{+104x} \color{blue}{+80} \\\Leftrightarrow & 169x& = & -1740 \\\Leftrightarrow & \frac{169x}{ \color{red}{169} }& = & \frac{-1740}{169} \\\Leftrightarrow & x = \frac{-1740}{169} & & \\ & V = \left\{ \frac{-1740}{169} \right\} & \\\end{align}\)
4. \(\text{330 is het kleinste gemene veelvoud van 6, 11 en 5} \\ \begin{align} & \frac{x}{6}+\frac{5}{11}& = & \frac{-2}{5}x+6 \\\Leftrightarrow & \color{blue}{330.} (\frac{55x}{ \color{blue}{330} }+ \frac{ 150 }{ \color{blue}{330} })& = & (\frac{-132}{ \color{blue}{330} }x+\frac{1980}{ \color{blue}{330} }) \color{blue}{.330} \\\Leftrightarrow & 55x+150& = & -132x+1980 \\\Leftrightarrow & 55x \color{red}{+150} \color{blue}{-150} \color{blue}{+132x} & = & \color{red}{-132x} +1980 \color{blue}{+132x} \color{blue}{-150} \\\Leftrightarrow & 187x& = & 1830 \\\Leftrightarrow & \frac{187x}{ \color{red}{187} }& = & \frac{1830}{187} \\\Leftrightarrow & x = \frac{1830}{187} & & \\ & V = \left\{ \frac{1830}{187} \right\} & \\\end{align}\)
5. \(\text{390 is het kleinste gemene veelvoud van 5, 13 en 6} \\ \begin{align} & \frac{x}{5}-\frac{3}{13}& = & \frac{1}{6}x-7 \\\Leftrightarrow & \color{blue}{390.} (\frac{78x}{ \color{blue}{390} }- \frac{ 90 }{ \color{blue}{390} })& = & (\frac{65}{ \color{blue}{390} }x-\frac{2730}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 78x-90& = & 65x-2730 \\\Leftrightarrow & 78x \color{red}{-90} \color{blue}{+90} \color{blue}{-65x} & = & \color{red}{65x} -2730 \color{blue}{-65x} \color{blue}{+90} \\\Leftrightarrow & 13x& = & -2640 \\\Leftrightarrow & \frac{13x}{ \color{red}{13} }& = & \frac{-2640}{13} \\\Leftrightarrow & x = \frac{-2640}{13} & & \\ & V = \left\{ \frac{-2640}{13} \right\} & \\\end{align}\)
6. \(\text{455 is het kleinste gemene veelvoud van 7, 13 en 5} \\ \begin{align} & \frac{x}{7}+\frac{4}{13}& = & \frac{-4}{5}x-2 \\\Leftrightarrow & \color{blue}{455.} (\frac{65x}{ \color{blue}{455} }+ \frac{ 140 }{ \color{blue}{455} })& = & (\frac{-364}{ \color{blue}{455} }x-\frac{910}{ \color{blue}{455} }) \color{blue}{.455} \\\Leftrightarrow & 65x+140& = & -364x-910 \\\Leftrightarrow & 65x \color{red}{+140} \color{blue}{-140} \color{blue}{+364x} & = & \color{red}{-364x} -910 \color{blue}{+364x} \color{blue}{-140} \\\Leftrightarrow & 429x& = & -1050 \\\Leftrightarrow & \frac{429x}{ \color{red}{429} }& = & \frac{-1050}{429} \\\Leftrightarrow & x = \frac{-350}{143} & & \\ & V = \left\{ \frac{-350}{143} \right\} & \\\end{align}\)
7. \(\text{6 is het kleinste gemene veelvoud van 3, 6 en 2} \\ \begin{align} & \frac{x}{3}-\frac{5}{6}& = & \frac{1}{2}x-2 \\\Leftrightarrow & \color{blue}{6.} (\frac{2x}{ \color{blue}{6} }- \frac{ 5 }{ \color{blue}{6} })& = & (\frac{3}{ \color{blue}{6} }x-\frac{12}{ \color{blue}{6} }) \color{blue}{.6} \\\Leftrightarrow & 2x-5& = & 3x-12 \\\Leftrightarrow & 2x \color{red}{-5} \color{blue}{+5} \color{blue}{-3x} & = & \color{red}{3x} -12 \color{blue}{-3x} \color{blue}{+5} \\\Leftrightarrow & -x& = & -7 \\\Leftrightarrow & \frac{-x}{ \color{red}{-1} }& = & \frac{-7}{-1} \\\Leftrightarrow & x = 7 & & \\ & V = \left\{ 7 \right\} & \\\end{align}\)
8. \(\text{240 is het kleinste gemene veelvoud van 6, 16 en 5} \\ \begin{align} & \frac{x}{6}+\frac{3}{16}& = & \frac{-4}{5}x-8 \\\Leftrightarrow & \color{blue}{240.} (\frac{40x}{ \color{blue}{240} }+ \frac{ 45 }{ \color{blue}{240} })& = & (\frac{-192}{ \color{blue}{240} }x-\frac{1920}{ \color{blue}{240} }) \color{blue}{.240} \\\Leftrightarrow & 40x+45& = & -192x-1920 \\\Leftrightarrow & 40x \color{red}{+45} \color{blue}{-45} \color{blue}{+192x} & = & \color{red}{-192x} -1920 \color{blue}{+192x} \color{blue}{-45} \\\Leftrightarrow & 232x& = & -1965 \\\Leftrightarrow & \frac{232x}{ \color{red}{232} }& = & \frac{-1965}{232} \\\Leftrightarrow & x = \frac{-1965}{232} & & \\ & V = \left\{ \frac{-1965}{232} \right\} & \\\end{align}\)
9. \(\text{24 is het kleinste gemene veelvoud van 3, 8 en 2} \\ \begin{align} & \frac{x}{3}+\frac{3}{8}& = & \frac{1}{2}x-2 \\\Leftrightarrow & \color{blue}{24.} (\frac{8x}{ \color{blue}{24} }+ \frac{ 9 }{ \color{blue}{24} })& = & (\frac{12}{ \color{blue}{24} }x-\frac{48}{ \color{blue}{24} }) \color{blue}{.24} \\\Leftrightarrow & 8x+9& = & 12x-48 \\\Leftrightarrow & 8x \color{red}{+9} \color{blue}{-9} \color{blue}{-12x} & = & \color{red}{12x} -48 \color{blue}{-12x} \color{blue}{-9} \\\Leftrightarrow & -4x& = & -57 \\\Leftrightarrow & \frac{-4x}{ \color{red}{-4} }& = & \frac{-57}{-4} \\\Leftrightarrow & x = \frac{57}{4} & & \\ & V = \left\{ \frac{57}{4} \right\} & \\\end{align}\)
10. \(\text{385 is het kleinste gemene veelvoud van 7, 11 en 5} \\ \begin{align} & \frac{x}{7}+\frac{4}{11}& = & \frac{7}{5}x+7 \\\Leftrightarrow & \color{blue}{385.} (\frac{55x}{ \color{blue}{385} }+ \frac{ 140 }{ \color{blue}{385} })& = & (\frac{539}{ \color{blue}{385} }x+\frac{2695}{ \color{blue}{385} }) \color{blue}{.385} \\\Leftrightarrow & 55x+140& = & 539x+2695 \\\Leftrightarrow & 55x \color{red}{+140} \color{blue}{-140} \color{blue}{-539x} & = & \color{red}{539x} +2695 \color{blue}{-539x} \color{blue}{-140} \\\Leftrightarrow & -484x& = & 2555 \\\Leftrightarrow & \frac{-484x}{ \color{red}{-484} }& = & \frac{2555}{-484} \\\Leftrightarrow & x = \frac{-2555}{484} & & \\ & V = \left\{ \frac{-2555}{484} \right\} & \\\end{align}\)
11. \(\text{84 is het kleinste gemene veelvoud van 3, 7 en 4} \\ \begin{align} & \frac{x}{3}-\frac{4}{7}& = & \frac{7}{4}x+2 \\\Leftrightarrow & \color{blue}{84.} (\frac{28x}{ \color{blue}{84} }- \frac{ 48 }{ \color{blue}{84} })& = & (\frac{147}{ \color{blue}{84} }x+\frac{168}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 28x-48& = & 147x+168 \\\Leftrightarrow & 28x \color{red}{-48} \color{blue}{+48} \color{blue}{-147x} & = & \color{red}{147x} +168 \color{blue}{-147x} \color{blue}{+48} \\\Leftrightarrow & -119x& = & 216 \\\Leftrightarrow & \frac{-119x}{ \color{red}{-119} }& = & \frac{216}{-119} \\\Leftrightarrow & x = \frac{-216}{119} & & \\ & V = \left\{ \frac{-216}{119} \right\} & \\\end{align}\)
12. \(\text{30 is het kleinste gemene veelvoud van 5, 6 en 2} \\ \begin{align} & \frac{x}{5}-\frac{5}{6}& = & \frac{1}{2}x+5 \\\Leftrightarrow & \color{blue}{30.} (\frac{6x}{ \color{blue}{30} }- \frac{ 25 }{ \color{blue}{30} })& = & (\frac{15}{ \color{blue}{30} }x+\frac{150}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 6x-25& = & 15x+150 \\\Leftrightarrow & 6x \color{red}{-25} \color{blue}{+25} \color{blue}{-15x} & = & \color{red}{15x} +150 \color{blue}{-15x} \color{blue}{+25} \\\Leftrightarrow & -9x& = & 175 \\\Leftrightarrow & \frac{-9x}{ \color{red}{-9} }& = & \frac{175}{-9} \\\Leftrightarrow & x = \frac{-175}{9} & & \\ & V = \left\{ \frac{-175}{9} \right\} & \\\end{align}\)