Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{30 is het kleinste gemene veelvoud van 6, 10 en 5} \\ \begin{align} & \frac{x}{6}+\frac{3}{10}& = & \frac{3}{5}x+7 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }+ \frac{ 9 }{ \color{blue}{30} })& = & (\frac{18}{ \color{blue}{30} }x+\frac{210}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x+9& = & 18x+210 \\\Leftrightarrow & 5x \color{red}{+9} \color{blue}{-9} \color{blue}{-18x} & = & \color{red}{18x} +210 \color{blue}{-18x} \color{blue}{-9} \\\Leftrightarrow & -13x& = & 201 \\\Leftrightarrow & \frac{-13x}{ \color{red}{-13} }& = & \frac{201}{-13} \\\Leftrightarrow & x = \frac{-201}{13} & & \\ & V = \left\{ \frac{-201}{13} \right\} & \\\end{align}\)
2. \(\text{48 is het kleinste gemene veelvoud van 2, 16 en 3} \\ \begin{align} & \frac{x}{2}-\frac{3}{16}& = & \frac{-2}{3}x+4 \\\Leftrightarrow & \color{blue}{48.} (\frac{24x}{ \color{blue}{48} }- \frac{ 9 }{ \color{blue}{48} })& = & (\frac{-32}{ \color{blue}{48} }x+\frac{192}{ \color{blue}{48} }) \color{blue}{.48} \\\Leftrightarrow & 24x-9& = & -32x+192 \\\Leftrightarrow & 24x \color{red}{-9} \color{blue}{+9} \color{blue}{+32x} & = & \color{red}{-32x} +192 \color{blue}{+32x} \color{blue}{+9} \\\Leftrightarrow & 56x& = & 201 \\\Leftrightarrow & \frac{56x}{ \color{red}{56} }& = & \frac{201}{56} \\\Leftrightarrow & x = \frac{201}{56} & & \\ & V = \left\{ \frac{201}{56} \right\} & \\\end{align}\)
3. \(\text{48 is het kleinste gemene veelvoud van 3, 16 en 2} \\ \begin{align} & \frac{x}{3}+\frac{3}{16}& = & \frac{1}{2}x-3 \\\Leftrightarrow & \color{blue}{48.} (\frac{16x}{ \color{blue}{48} }+ \frac{ 9 }{ \color{blue}{48} })& = & (\frac{24}{ \color{blue}{48} }x-\frac{144}{ \color{blue}{48} }) \color{blue}{.48} \\\Leftrightarrow & 16x+9& = & 24x-144 \\\Leftrightarrow & 16x \color{red}{+9} \color{blue}{-9} \color{blue}{-24x} & = & \color{red}{24x} -144 \color{blue}{-24x} \color{blue}{-9} \\\Leftrightarrow & -8x& = & -153 \\\Leftrightarrow & \frac{-8x}{ \color{red}{-8} }& = & \frac{-153}{-8} \\\Leftrightarrow & x = \frac{153}{8} & & \\ & V = \left\{ \frac{153}{8} \right\} & \\\end{align}\)
4. \(\text{336 is het kleinste gemene veelvoud van 7, 16 en 6} \\ \begin{align} & \frac{x}{7}+\frac{3}{16}& = & \frac{-5}{6}x+5 \\\Leftrightarrow & \color{blue}{336.} (\frac{48x}{ \color{blue}{336} }+ \frac{ 63 }{ \color{blue}{336} })& = & (\frac{-280}{ \color{blue}{336} }x+\frac{1680}{ \color{blue}{336} }) \color{blue}{.336} \\\Leftrightarrow & 48x+63& = & -280x+1680 \\\Leftrightarrow & 48x \color{red}{+63} \color{blue}{-63} \color{blue}{+280x} & = & \color{red}{-280x} +1680 \color{blue}{+280x} \color{blue}{-63} \\\Leftrightarrow & 328x& = & 1617 \\\Leftrightarrow & \frac{328x}{ \color{red}{328} }& = & \frac{1617}{328} \\\Leftrightarrow & x = \frac{1617}{328} & & \\ & V = \left\{ \frac{1617}{328} \right\} & \\\end{align}\)
5. \(\text{210 is het kleinste gemene veelvoud van 5, 7 en 6} \\ \begin{align} & \frac{x}{5}+\frac{2}{7}& = & \frac{1}{6}x-4 \\\Leftrightarrow & \color{blue}{210.} (\frac{42x}{ \color{blue}{210} }+ \frac{ 60 }{ \color{blue}{210} })& = & (\frac{35}{ \color{blue}{210} }x-\frac{840}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 42x+60& = & 35x-840 \\\Leftrightarrow & 42x \color{red}{+60} \color{blue}{-60} \color{blue}{-35x} & = & \color{red}{35x} -840 \color{blue}{-35x} \color{blue}{-60} \\\Leftrightarrow & 7x& = & -900 \\\Leftrightarrow & \frac{7x}{ \color{red}{7} }& = & \frac{-900}{7} \\\Leftrightarrow & x = \frac{-900}{7} & & \\ & V = \left\{ \frac{-900}{7} \right\} & \\\end{align}\)
6. \(\text{30 is het kleinste gemene veelvoud van 3, 10 en 2} \\ \begin{align} & \frac{x}{3}+\frac{3}{10}& = & \frac{5}{2}x+3 \\\Leftrightarrow & \color{blue}{30.} (\frac{10x}{ \color{blue}{30} }+ \frac{ 9 }{ \color{blue}{30} })& = & (\frac{75}{ \color{blue}{30} }x+\frac{90}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 10x+9& = & 75x+90 \\\Leftrightarrow & 10x \color{red}{+9} \color{blue}{-9} \color{blue}{-75x} & = & \color{red}{75x} +90 \color{blue}{-75x} \color{blue}{-9} \\\Leftrightarrow & -65x& = & 81 \\\Leftrightarrow & \frac{-65x}{ \color{red}{-65} }& = & \frac{81}{-65} \\\Leftrightarrow & x = \frac{-81}{65} & & \\ & V = \left\{ \frac{-81}{65} \right\} & \\\end{align}\)
7. \(\text{12 is het kleinste gemene veelvoud van 4, 6 en 3} \\ \begin{align} & \frac{x}{4}+\frac{5}{6}& = & \frac{-2}{3}x-6 \\\Leftrightarrow & \color{blue}{12.} (\frac{3x}{ \color{blue}{12} }+ \frac{ 10 }{ \color{blue}{12} })& = & (\frac{-8}{ \color{blue}{12} }x-\frac{72}{ \color{blue}{12} }) \color{blue}{.12} \\\Leftrightarrow & 3x+10& = & -8x-72 \\\Leftrightarrow & 3x \color{red}{+10} \color{blue}{-10} \color{blue}{+8x} & = & \color{red}{-8x} -72 \color{blue}{+8x} \color{blue}{-10} \\\Leftrightarrow & 11x& = & -82 \\\Leftrightarrow & \frac{11x}{ \color{red}{11} }& = & \frac{-82}{11} \\\Leftrightarrow & x = \frac{-82}{11} & & \\ & V = \left\{ \frac{-82}{11} \right\} & \\\end{align}\)
8. \(\text{110 is het kleinste gemene veelvoud van 5, 11 en 2} \\ \begin{align} & \frac{x}{5}+\frac{4}{11}& = & \frac{1}{2}x-1 \\\Leftrightarrow & \color{blue}{110.} (\frac{22x}{ \color{blue}{110} }+ \frac{ 40 }{ \color{blue}{110} })& = & (\frac{55}{ \color{blue}{110} }x-\frac{110}{ \color{blue}{110} }) \color{blue}{.110} \\\Leftrightarrow & 22x+40& = & 55x-110 \\\Leftrightarrow & 22x \color{red}{+40} \color{blue}{-40} \color{blue}{-55x} & = & \color{red}{55x} -110 \color{blue}{-55x} \color{blue}{-40} \\\Leftrightarrow & -33x& = & -150 \\\Leftrightarrow & \frac{-33x}{ \color{red}{-33} }& = & \frac{-150}{-33} \\\Leftrightarrow & x = \frac{50}{11} & & \\ & V = \left\{ \frac{50}{11} \right\} & \\\end{align}\)
9. \(\text{70 is het kleinste gemene veelvoud van 5, 7 en 2} \\ \begin{align} & \frac{x}{5}+\frac{4}{7}& = & \frac{1}{2}x-2 \\\Leftrightarrow & \color{blue}{70.} (\frac{14x}{ \color{blue}{70} }+ \frac{ 40 }{ \color{blue}{70} })& = & (\frac{35}{ \color{blue}{70} }x-\frac{140}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 14x+40& = & 35x-140 \\\Leftrightarrow & 14x \color{red}{+40} \color{blue}{-40} \color{blue}{-35x} & = & \color{red}{35x} -140 \color{blue}{-35x} \color{blue}{-40} \\\Leftrightarrow & -21x& = & -180 \\\Leftrightarrow & \frac{-21x}{ \color{red}{-21} }& = & \frac{-180}{-21} \\\Leftrightarrow & x = \frac{60}{7} & & \\ & V = \left\{ \frac{60}{7} \right\} & \\\end{align}\)
10. \(\text{24 is het kleinste gemene veelvoud van 3, 8 en 2} \\ \begin{align} & \frac{x}{3}+\frac{3}{8}& = & \frac{1}{2}x-2 \\\Leftrightarrow & \color{blue}{24.} (\frac{8x}{ \color{blue}{24} }+ \frac{ 9 }{ \color{blue}{24} })& = & (\frac{12}{ \color{blue}{24} }x-\frac{48}{ \color{blue}{24} }) \color{blue}{.24} \\\Leftrightarrow & 8x+9& = & 12x-48 \\\Leftrightarrow & 8x \color{red}{+9} \color{blue}{-9} \color{blue}{-12x} & = & \color{red}{12x} -48 \color{blue}{-12x} \color{blue}{-9} \\\Leftrightarrow & -4x& = & -57 \\\Leftrightarrow & \frac{-4x}{ \color{red}{-4} }& = & \frac{-57}{-4} \\\Leftrightarrow & x = \frac{57}{4} & & \\ & V = \left\{ \frac{57}{4} \right\} & \\\end{align}\)
11. \(\text{30 is het kleinste gemene veelvoud van 2, 15 en 3} \\ \begin{align} & \frac{x}{2}+\frac{4}{15}& = & \frac{5}{3}x+3 \\\Leftrightarrow & \color{blue}{30.} (\frac{15x}{ \color{blue}{30} }+ \frac{ 8 }{ \color{blue}{30} })& = & (\frac{50}{ \color{blue}{30} }x+\frac{90}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 15x+8& = & 50x+90 \\\Leftrightarrow & 15x \color{red}{+8} \color{blue}{-8} \color{blue}{-50x} & = & \color{red}{50x} +90 \color{blue}{-50x} \color{blue}{-8} \\\Leftrightarrow & -35x& = & 82 \\\Leftrightarrow & \frac{-35x}{ \color{red}{-35} }& = & \frac{82}{-35} \\\Leftrightarrow & x = \frac{-82}{35} & & \\ & V = \left\{ \frac{-82}{35} \right\} & \\\end{align}\)
12. \(\text{260 is het kleinste gemene veelvoud van 4, 13 en 5} \\ \begin{align} & \frac{x}{4}-\frac{2}{13}& = & \frac{-4}{5}x-2 \\\Leftrightarrow & \color{blue}{260.} (\frac{65x}{ \color{blue}{260} }- \frac{ 40 }{ \color{blue}{260} })& = & (\frac{-208}{ \color{blue}{260} }x-\frac{520}{ \color{blue}{260} }) \color{blue}{.260} \\\Leftrightarrow & 65x-40& = & -208x-520 \\\Leftrightarrow & 65x \color{red}{-40} \color{blue}{+40} \color{blue}{+208x} & = & \color{red}{-208x} -520 \color{blue}{+208x} \color{blue}{+40} \\\Leftrightarrow & 273x& = & -480 \\\Leftrightarrow & \frac{273x}{ \color{red}{273} }& = & \frac{-480}{273} \\\Leftrightarrow & x = \frac{-160}{91} & & \\ & V = \left\{ \frac{-160}{91} \right\} & \\\end{align}\)