Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{260 is het kleinste gemene veelvoud van 5, 13 en 4} \\ \begin{align} & \frac{x}{5}-\frac{2}{13}& = & \frac{7}{4}x-6 \\\Leftrightarrow & \color{blue}{260.} (\frac{52x}{ \color{blue}{260} }- \frac{ 40 }{ \color{blue}{260} })& = & (\frac{455}{ \color{blue}{260} }x-\frac{1560}{ \color{blue}{260} }) \color{blue}{.260} \\\Leftrightarrow & 52x-40& = & 455x-1560 \\\Leftrightarrow & 52x \color{red}{-40} \color{blue}{+40} \color{blue}{-455x} & = & \color{red}{455x} -1560 \color{blue}{-455x} \color{blue}{+40} \\\Leftrightarrow & -403x& = & -1520 \\\Leftrightarrow & \frac{-403x}{ \color{red}{-403} }& = & \frac{-1520}{-403} \\\Leftrightarrow & x = \frac{1520}{403} & & \\ & V = \left\{ \frac{1520}{403} \right\} & \\\end{align}\)
2. \(\text{10 is het kleinste gemene veelvoud van 2, 10 en 5} \\ \begin{align} & \frac{x}{2}-\frac{3}{10}& = & \frac{7}{5}x+4 \\\Leftrightarrow & \color{blue}{10.} (\frac{5x}{ \color{blue}{10} }- \frac{ 3 }{ \color{blue}{10} })& = & (\frac{14}{ \color{blue}{10} }x+\frac{40}{ \color{blue}{10} }) \color{blue}{.10} \\\Leftrightarrow & 5x-3& = & 14x+40 \\\Leftrightarrow & 5x \color{red}{-3} \color{blue}{+3} \color{blue}{-14x} & = & \color{red}{14x} +40 \color{blue}{-14x} \color{blue}{+3} \\\Leftrightarrow & -9x& = & 43 \\\Leftrightarrow & \frac{-9x}{ \color{red}{-9} }& = & \frac{43}{-9} \\\Leftrightarrow & x = \frac{-43}{9} & & \\ & V = \left\{ \frac{-43}{9} \right\} & \\\end{align}\)
3. \(\text{60 is het kleinste gemene veelvoud van 5, 12 en 6} \\ \begin{align} & \frac{x}{5}+\frac{5}{12}& = & \frac{-5}{6}x+7 \\\Leftrightarrow & \color{blue}{60.} (\frac{12x}{ \color{blue}{60} }+ \frac{ 25 }{ \color{blue}{60} })& = & (\frac{-50}{ \color{blue}{60} }x+\frac{420}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 12x+25& = & -50x+420 \\\Leftrightarrow & 12x \color{red}{+25} \color{blue}{-25} \color{blue}{+50x} & = & \color{red}{-50x} +420 \color{blue}{+50x} \color{blue}{-25} \\\Leftrightarrow & 62x& = & 395 \\\Leftrightarrow & \frac{62x}{ \color{red}{62} }& = & \frac{395}{62} \\\Leftrightarrow & x = \frac{395}{62} & & \\ & V = \left\{ \frac{395}{62} \right\} & \\\end{align}\)
4. \(\text{260 is het kleinste gemene veelvoud van 5, 13 en 4} \\ \begin{align} & \frac{x}{5}-\frac{4}{13}& = & \frac{5}{4}x+5 \\\Leftrightarrow & \color{blue}{260.} (\frac{52x}{ \color{blue}{260} }- \frac{ 80 }{ \color{blue}{260} })& = & (\frac{325}{ \color{blue}{260} }x+\frac{1300}{ \color{blue}{260} }) \color{blue}{.260} \\\Leftrightarrow & 52x-80& = & 325x+1300 \\\Leftrightarrow & 52x \color{red}{-80} \color{blue}{+80} \color{blue}{-325x} & = & \color{red}{325x} +1300 \color{blue}{-325x} \color{blue}{+80} \\\Leftrightarrow & -273x& = & 1380 \\\Leftrightarrow & \frac{-273x}{ \color{red}{-273} }& = & \frac{1380}{-273} \\\Leftrightarrow & x = \frac{-460}{91} & & \\ & V = \left\{ \frac{-460}{91} \right\} & \\\end{align}\)
5. \(\text{210 is het kleinste gemene veelvoud van 5, 7 en 6} \\ \begin{align} & \frac{x}{5}-\frac{2}{7}& = & \frac{1}{6}x+2 \\\Leftrightarrow & \color{blue}{210.} (\frac{42x}{ \color{blue}{210} }- \frac{ 60 }{ \color{blue}{210} })& = & (\frac{35}{ \color{blue}{210} }x+\frac{420}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 42x-60& = & 35x+420 \\\Leftrightarrow & 42x \color{red}{-60} \color{blue}{+60} \color{blue}{-35x} & = & \color{red}{35x} +420 \color{blue}{-35x} \color{blue}{+60} \\\Leftrightarrow & 7x& = & 480 \\\Leftrightarrow & \frac{7x}{ \color{red}{7} }& = & \frac{480}{7} \\\Leftrightarrow & x = \frac{480}{7} & & \\ & V = \left\{ \frac{480}{7} \right\} & \\\end{align}\)
6. \(\text{12 is het kleinste gemene veelvoud van 4, 6 en 3} \\ \begin{align} & \frac{x}{4}-\frac{5}{6}& = & \frac{-5}{3}x+2 \\\Leftrightarrow & \color{blue}{12.} (\frac{3x}{ \color{blue}{12} }- \frac{ 10 }{ \color{blue}{12} })& = & (\frac{-20}{ \color{blue}{12} }x+\frac{24}{ \color{blue}{12} }) \color{blue}{.12} \\\Leftrightarrow & 3x-10& = & -20x+24 \\\Leftrightarrow & 3x \color{red}{-10} \color{blue}{+10} \color{blue}{+20x} & = & \color{red}{-20x} +24 \color{blue}{+20x} \color{blue}{+10} \\\Leftrightarrow & 23x& = & 34 \\\Leftrightarrow & \frac{23x}{ \color{red}{23} }& = & \frac{34}{23} \\\Leftrightarrow & x = \frac{34}{23} & & \\ & V = \left\{ \frac{34}{23} \right\} & \\\end{align}\)
7. \(\text{182 is het kleinste gemene veelvoud van 7, 13 en 2} \\ \begin{align} & \frac{x}{7}-\frac{3}{13}& = & \frac{1}{2}x+6 \\\Leftrightarrow & \color{blue}{182.} (\frac{26x}{ \color{blue}{182} }- \frac{ 42 }{ \color{blue}{182} })& = & (\frac{91}{ \color{blue}{182} }x+\frac{1092}{ \color{blue}{182} }) \color{blue}{.182} \\\Leftrightarrow & 26x-42& = & 91x+1092 \\\Leftrightarrow & 26x \color{red}{-42} \color{blue}{+42} \color{blue}{-91x} & = & \color{red}{91x} +1092 \color{blue}{-91x} \color{blue}{+42} \\\Leftrightarrow & -65x& = & 1134 \\\Leftrightarrow & \frac{-65x}{ \color{red}{-65} }& = & \frac{1134}{-65} \\\Leftrightarrow & x = \frac{-1134}{65} & & \\ & V = \left\{ \frac{-1134}{65} \right\} & \\\end{align}\)
8. \(\text{10 is het kleinste gemene veelvoud van 5, 10 en 2} \\ \begin{align} & \frac{x}{5}-\frac{3}{10}& = & \frac{1}{2}x-1 \\\Leftrightarrow & \color{blue}{10.} (\frac{2x}{ \color{blue}{10} }- \frac{ 3 }{ \color{blue}{10} })& = & (\frac{5}{ \color{blue}{10} }x-\frac{10}{ \color{blue}{10} }) \color{blue}{.10} \\\Leftrightarrow & 2x-3& = & 5x-10 \\\Leftrightarrow & 2x \color{red}{-3} \color{blue}{+3} \color{blue}{-5x} & = & \color{red}{5x} -10 \color{blue}{-5x} \color{blue}{+3} \\\Leftrightarrow & -3x& = & -7 \\\Leftrightarrow & \frac{-3x}{ \color{red}{-3} }& = & \frac{-7}{-3} \\\Leftrightarrow & x = \frac{7}{3} & & \\ & V = \left\{ \frac{7}{3} \right\} & \\\end{align}\)
9. \(\text{90 is het kleinste gemene veelvoud van 6, 9 en 5} \\ \begin{align} & \frac{x}{6}+\frac{4}{9}& = & \frac{1}{5}x+1 \\\Leftrightarrow & \color{blue}{90.} (\frac{15x}{ \color{blue}{90} }+ \frac{ 40 }{ \color{blue}{90} })& = & (\frac{18}{ \color{blue}{90} }x+\frac{90}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 15x+40& = & 18x+90 \\\Leftrightarrow & 15x \color{red}{+40} \color{blue}{-40} \color{blue}{-18x} & = & \color{red}{18x} +90 \color{blue}{-18x} \color{blue}{-40} \\\Leftrightarrow & -3x& = & 50 \\\Leftrightarrow & \frac{-3x}{ \color{red}{-3} }& = & \frac{50}{-3} \\\Leftrightarrow & x = \frac{-50}{3} & & \\ & V = \left\{ \frac{-50}{3} \right\} & \\\end{align}\)
10. \(\text{330 is het kleinste gemene veelvoud van 6, 11 en 5} \\ \begin{align} & \frac{x}{6}-\frac{5}{11}& = & \frac{6}{5}x+5 \\\Leftrightarrow & \color{blue}{330.} (\frac{55x}{ \color{blue}{330} }- \frac{ 150 }{ \color{blue}{330} })& = & (\frac{396}{ \color{blue}{330} }x+\frac{1650}{ \color{blue}{330} }) \color{blue}{.330} \\\Leftrightarrow & 55x-150& = & 396x+1650 \\\Leftrightarrow & 55x \color{red}{-150} \color{blue}{+150} \color{blue}{-396x} & = & \color{red}{396x} +1650 \color{blue}{-396x} \color{blue}{+150} \\\Leftrightarrow & -341x& = & 1800 \\\Leftrightarrow & \frac{-341x}{ \color{red}{-341} }& = & \frac{1800}{-341} \\\Leftrightarrow & x = \frac{-1800}{341} & & \\ & V = \left\{ \frac{-1800}{341} \right\} & \\\end{align}\)
11. \(\text{18 is het kleinste gemene veelvoud van 2, 9 en 3} \\ \begin{align} & \frac{x}{2}-\frac{2}{9}& = & \frac{1}{3}x+6 \\\Leftrightarrow & \color{blue}{18.} (\frac{9x}{ \color{blue}{18} }- \frac{ 4 }{ \color{blue}{18} })& = & (\frac{6}{ \color{blue}{18} }x+\frac{108}{ \color{blue}{18} }) \color{blue}{.18} \\\Leftrightarrow & 9x-4& = & 6x+108 \\\Leftrightarrow & 9x \color{red}{-4} \color{blue}{+4} \color{blue}{-6x} & = & \color{red}{6x} +108 \color{blue}{-6x} \color{blue}{+4} \\\Leftrightarrow & 3x& = & 112 \\\Leftrightarrow & \frac{3x}{ \color{red}{3} }& = & \frac{112}{3} \\\Leftrightarrow & x = \frac{112}{3} & & \\ & V = \left\{ \frac{112}{3} \right\} & \\\end{align}\)
12. \(\text{210 is het kleinste gemene veelvoud van 5, 7 en 6} \\ \begin{align} & \frac{x}{5}+\frac{3}{7}& = & \frac{1}{6}x-1 \\\Leftrightarrow & \color{blue}{210.} (\frac{42x}{ \color{blue}{210} }+ \frac{ 90 }{ \color{blue}{210} })& = & (\frac{35}{ \color{blue}{210} }x-\frac{210}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 42x+90& = & 35x-210 \\\Leftrightarrow & 42x \color{red}{+90} \color{blue}{-90} \color{blue}{-35x} & = & \color{red}{35x} -210 \color{blue}{-35x} \color{blue}{-90} \\\Leftrightarrow & 7x& = & -300 \\\Leftrightarrow & \frac{7x}{ \color{red}{7} }& = & \frac{-300}{7} \\\Leftrightarrow & x = \frac{-300}{7} & & \\ & V = \left\{ \frac{-300}{7} \right\} & \\\end{align}\)