Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{240 is het kleinste gemene veelvoud van 5, 16 en 6} \\ \begin{align} & \frac{x}{5}-\frac{3}{16}& = & \frac{5}{6}x+1 \\\Leftrightarrow & \color{blue}{240.} (\frac{48x}{ \color{blue}{240} }- \frac{ 45 }{ \color{blue}{240} })& = & (\frac{200}{ \color{blue}{240} }x+\frac{240}{ \color{blue}{240} }) \color{blue}{.240} \\\Leftrightarrow & 48x-45& = & 200x+240 \\\Leftrightarrow & 48x \color{red}{-45} \color{blue}{+45} \color{blue}{-200x} & = & \color{red}{200x} +240 \color{blue}{-200x} \color{blue}{+45} \\\Leftrightarrow & -152x& = & 285 \\\Leftrightarrow & \frac{-152x}{ \color{red}{-152} }& = & \frac{285}{-152} \\\Leftrightarrow & x = \frac{-15}{8} & & \\ & V = \left\{ \frac{-15}{8} \right\} & \\\end{align}\)
2. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}+\frac{2}{7}& = & \frac{7}{5}x+6 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }+ \frac{ 60 }{ \color{blue}{210} })& = & (\frac{294}{ \color{blue}{210} }x+\frac{1260}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x+60& = & 294x+1260 \\\Leftrightarrow & 35x \color{red}{+60} \color{blue}{-60} \color{blue}{-294x} & = & \color{red}{294x} +1260 \color{blue}{-294x} \color{blue}{-60} \\\Leftrightarrow & -259x& = & 1200 \\\Leftrightarrow & \frac{-259x}{ \color{red}{-259} }& = & \frac{1200}{-259} \\\Leftrightarrow & x = \frac{-1200}{259} & & \\ & V = \left\{ \frac{-1200}{259} \right\} & \\\end{align}\)
3. \(\text{21 is het kleinste gemene veelvoud van 7, 7 en 3} \\ \begin{align} & \frac{x}{7}+\frac{3}{7}& = & \frac{2}{3}x-4 \\\Leftrightarrow & \color{blue}{21.} (\frac{3x}{ \color{blue}{21} }+ \frac{ 9 }{ \color{blue}{21} })& = & (\frac{14}{ \color{blue}{21} }x-\frac{84}{ \color{blue}{21} }) \color{blue}{.21} \\\Leftrightarrow & 3x+9& = & 14x-84 \\\Leftrightarrow & 3x \color{red}{+9} \color{blue}{-9} \color{blue}{-14x} & = & \color{red}{14x} -84 \color{blue}{-14x} \color{blue}{-9} \\\Leftrightarrow & -11x& = & -93 \\\Leftrightarrow & \frac{-11x}{ \color{red}{-11} }& = & \frac{-93}{-11} \\\Leftrightarrow & x = \frac{93}{11} & & \\ & V = \left\{ \frac{93}{11} \right\} & \\\end{align}\)
4. \(\text{48 is het kleinste gemene veelvoud van 3, 16 en 2} \\ \begin{align} & \frac{x}{3}+\frac{5}{16}& = & \frac{1}{2}x-5 \\\Leftrightarrow & \color{blue}{48.} (\frac{16x}{ \color{blue}{48} }+ \frac{ 15 }{ \color{blue}{48} })& = & (\frac{24}{ \color{blue}{48} }x-\frac{240}{ \color{blue}{48} }) \color{blue}{.48} \\\Leftrightarrow & 16x+15& = & 24x-240 \\\Leftrightarrow & 16x \color{red}{+15} \color{blue}{-15} \color{blue}{-24x} & = & \color{red}{24x} -240 \color{blue}{-24x} \color{blue}{-15} \\\Leftrightarrow & -8x& = & -255 \\\Leftrightarrow & \frac{-8x}{ \color{red}{-8} }& = & \frac{-255}{-8} \\\Leftrightarrow & x = \frac{255}{8} & & \\ & V = \left\{ \frac{255}{8} \right\} & \\\end{align}\)
5. \(\text{60 is het kleinste gemene veelvoud van 5, 12 en 2} \\ \begin{align} & \frac{x}{5}-\frac{5}{12}& = & \frac{1}{2}x+8 \\\Leftrightarrow & \color{blue}{60.} (\frac{12x}{ \color{blue}{60} }- \frac{ 25 }{ \color{blue}{60} })& = & (\frac{30}{ \color{blue}{60} }x+\frac{480}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 12x-25& = & 30x+480 \\\Leftrightarrow & 12x \color{red}{-25} \color{blue}{+25} \color{blue}{-30x} & = & \color{red}{30x} +480 \color{blue}{-30x} \color{blue}{+25} \\\Leftrightarrow & -18x& = & 505 \\\Leftrightarrow & \frac{-18x}{ \color{red}{-18} }& = & \frac{505}{-18} \\\Leftrightarrow & x = \frac{-505}{18} & & \\ & V = \left\{ \frac{-505}{18} \right\} & \\\end{align}\)
6. \(\text{30 is het kleinste gemene veelvoud van 5, 10 en 6} \\ \begin{align} & \frac{x}{5}-\frac{3}{10}& = & \frac{1}{6}x-8 \\\Leftrightarrow & \color{blue}{30.} (\frac{6x}{ \color{blue}{30} }- \frac{ 9 }{ \color{blue}{30} })& = & (\frac{5}{ \color{blue}{30} }x-\frac{240}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 6x-9& = & 5x-240 \\\Leftrightarrow & 6x \color{red}{-9} \color{blue}{+9} \color{blue}{-5x} & = & \color{red}{5x} -240 \color{blue}{-5x} \color{blue}{+9} \\\Leftrightarrow & x& = & -231 \\ & V = \left\{ -231 \right\} & \\\end{align}\)
7. \(\text{48 is het kleinste gemene veelvoud van 2, 16 en 3} \\ \begin{align} & \frac{x}{2}-\frac{5}{16}& = & \frac{5}{3}x+2 \\\Leftrightarrow & \color{blue}{48.} (\frac{24x}{ \color{blue}{48} }- \frac{ 15 }{ \color{blue}{48} })& = & (\frac{80}{ \color{blue}{48} }x+\frac{96}{ \color{blue}{48} }) \color{blue}{.48} \\\Leftrightarrow & 24x-15& = & 80x+96 \\\Leftrightarrow & 24x \color{red}{-15} \color{blue}{+15} \color{blue}{-80x} & = & \color{red}{80x} +96 \color{blue}{-80x} \color{blue}{+15} \\\Leftrightarrow & -56x& = & 111 \\\Leftrightarrow & \frac{-56x}{ \color{red}{-56} }& = & \frac{111}{-56} \\\Leftrightarrow & x = \frac{-111}{56} & & \\ & V = \left\{ \frac{-111}{56} \right\} & \\\end{align}\)
8. \(\text{12 is het kleinste gemene veelvoud van 4, 6 en 3} \\ \begin{align} & \frac{x}{4}-\frac{5}{6}& = & \frac{2}{3}x+7 \\\Leftrightarrow & \color{blue}{12.} (\frac{3x}{ \color{blue}{12} }- \frac{ 10 }{ \color{blue}{12} })& = & (\frac{8}{ \color{blue}{12} }x+\frac{84}{ \color{blue}{12} }) \color{blue}{.12} \\\Leftrightarrow & 3x-10& = & 8x+84 \\\Leftrightarrow & 3x \color{red}{-10} \color{blue}{+10} \color{blue}{-8x} & = & \color{red}{8x} +84 \color{blue}{-8x} \color{blue}{+10} \\\Leftrightarrow & -5x& = & 94 \\\Leftrightarrow & \frac{-5x}{ \color{red}{-5} }& = & \frac{94}{-5} \\\Leftrightarrow & x = \frac{-94}{5} & & \\ & V = \left\{ \frac{-94}{5} \right\} & \\\end{align}\)
9. \(\text{84 is het kleinste gemene veelvoud van 3, 7 en 4} \\ \begin{align} & \frac{x}{3}-\frac{2}{7}& = & \frac{1}{4}x-6 \\\Leftrightarrow & \color{blue}{84.} (\frac{28x}{ \color{blue}{84} }- \frac{ 24 }{ \color{blue}{84} })& = & (\frac{21}{ \color{blue}{84} }x-\frac{504}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 28x-24& = & 21x-504 \\\Leftrightarrow & 28x \color{red}{-24} \color{blue}{+24} \color{blue}{-21x} & = & \color{red}{21x} -504 \color{blue}{-21x} \color{blue}{+24} \\\Leftrightarrow & 7x& = & -480 \\\Leftrightarrow & \frac{7x}{ \color{red}{7} }& = & \frac{-480}{7} \\\Leftrightarrow & x = \frac{-480}{7} & & \\ & V = \left\{ \frac{-480}{7} \right\} & \\\end{align}\)
10. \(\text{6 is het kleinste gemene veelvoud van 3, 6 en 2} \\ \begin{align} & \frac{x}{3}-\frac{5}{6}& = & \frac{7}{2}x+7 \\\Leftrightarrow & \color{blue}{6.} (\frac{2x}{ \color{blue}{6} }- \frac{ 5 }{ \color{blue}{6} })& = & (\frac{21}{ \color{blue}{6} }x+\frac{42}{ \color{blue}{6} }) \color{blue}{.6} \\\Leftrightarrow & 2x-5& = & 21x+42 \\\Leftrightarrow & 2x \color{red}{-5} \color{blue}{+5} \color{blue}{-21x} & = & \color{red}{21x} +42 \color{blue}{-21x} \color{blue}{+5} \\\Leftrightarrow & -19x& = & 47 \\\Leftrightarrow & \frac{-19x}{ \color{red}{-19} }& = & \frac{47}{-19} \\\Leftrightarrow & x = \frac{-47}{19} & & \\ & V = \left\{ \frac{-47}{19} \right\} & \\\end{align}\)
11. \(\text{260 is het kleinste gemene veelvoud van 5, 13 en 4} \\ \begin{align} & \frac{x}{5}+\frac{5}{13}& = & \frac{-3}{4}x+6 \\\Leftrightarrow & \color{blue}{260.} (\frac{52x}{ \color{blue}{260} }+ \frac{ 100 }{ \color{blue}{260} })& = & (\frac{-195}{ \color{blue}{260} }x+\frac{1560}{ \color{blue}{260} }) \color{blue}{.260} \\\Leftrightarrow & 52x+100& = & -195x+1560 \\\Leftrightarrow & 52x \color{red}{+100} \color{blue}{-100} \color{blue}{+195x} & = & \color{red}{-195x} +1560 \color{blue}{+195x} \color{blue}{-100} \\\Leftrightarrow & 247x& = & 1460 \\\Leftrightarrow & \frac{247x}{ \color{red}{247} }& = & \frac{1460}{247} \\\Leftrightarrow & x = \frac{1460}{247} & & \\ & V = \left\{ \frac{1460}{247} \right\} & \\\end{align}\)
12. \(\text{42 is het kleinste gemene veelvoud van 2, 7 en 3} \\ \begin{align} & \frac{x}{2}+\frac{4}{7}& = & \frac{1}{3}x-2 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }+ \frac{ 24 }{ \color{blue}{42} })& = & (\frac{14}{ \color{blue}{42} }x-\frac{84}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x+24& = & 14x-84 \\\Leftrightarrow & 21x \color{red}{+24} \color{blue}{-24} \color{blue}{-14x} & = & \color{red}{14x} -84 \color{blue}{-14x} \color{blue}{-24} \\\Leftrightarrow & 7x& = & -108 \\\Leftrightarrow & \frac{7x}{ \color{red}{7} }& = & \frac{-108}{7} \\\Leftrightarrow & x = \frac{-108}{7} & & \\ & V = \left\{ \frac{-108}{7} \right\} & \\\end{align}\)