Vgln. eerste graad (reeks 4)

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

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

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

Verbetersleutel

1. \(\text{42 is het kleinste gemene veelvoud van 7, 6 en 3} \\ \begin{align} & \frac{x}{7}+\frac{5}{6}& = & \frac{7}{3}x+2 \\\Leftrightarrow & \color{blue}{42.} (\frac{6x}{ \color{blue}{42} }+ \frac{ 35 }{ \color{blue}{42} })& = & (\frac{98}{ \color{blue}{42} }x+\frac{84}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 6x+35& = & 98x+84 \\\Leftrightarrow & 6x \color{red}{+35} \color{blue}{-35} \color{blue}{-98x} & = & \color{red}{98x} +84 \color{blue}{-98x} \color{blue}{-35} \\\Leftrightarrow & -92x& = & 49 \\\Leftrightarrow & \frac{-92x}{ \color{red}{-92} }& = & \frac{49}{-92} \\\Leftrightarrow & x = \frac{-49}{92} & & \\ & V = \left\{ \frac{-49}{92} \right\} & \\\end{align}\)
2. \(\text{130 is het kleinste gemene veelvoud van 5, 13 en 2} \\ \begin{align} & \frac{x}{5}+\frac{4}{13}& = & \frac{1}{2}x-3 \\\Leftrightarrow & \color{blue}{130.} (\frac{26x}{ \color{blue}{130} }+ \frac{ 40 }{ \color{blue}{130} })& = & (\frac{65}{ \color{blue}{130} }x-\frac{390}{ \color{blue}{130} }) \color{blue}{.130} \\\Leftrightarrow & 26x+40& = & 65x-390 \\\Leftrightarrow & 26x \color{red}{+40} \color{blue}{-40} \color{blue}{-65x} & = & \color{red}{65x} -390 \color{blue}{-65x} \color{blue}{-40} \\\Leftrightarrow & -39x& = & -430 \\\Leftrightarrow & \frac{-39x}{ \color{red}{-39} }& = & \frac{-430}{-39} \\\Leftrightarrow & x = \frac{430}{39} & & \\ & V = \left\{ \frac{430}{39} \right\} & \\\end{align}\)
3. \(\text{42 is het kleinste gemene veelvoud van 2, 7 en 3} \\ \begin{align} & \frac{x}{2}-\frac{2}{7}& = & \frac{5}{3}x+1 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }- \frac{ 12 }{ \color{blue}{42} })& = & (\frac{70}{ \color{blue}{42} }x+\frac{42}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x-12& = & 70x+42 \\\Leftrightarrow & 21x \color{red}{-12} \color{blue}{+12} \color{blue}{-70x} & = & \color{red}{70x} +42 \color{blue}{-70x} \color{blue}{+12} \\\Leftrightarrow & -49x& = & 54 \\\Leftrightarrow & \frac{-49x}{ \color{red}{-49} }& = & \frac{54}{-49} \\\Leftrightarrow & x = \frac{-54}{49} & & \\ & V = \left\{ \frac{-54}{49} \right\} & \\\end{align}\)
4. \(\text{30 is het kleinste gemene veelvoud van 6, 6 en 5} \\ \begin{align} & \frac{x}{6}+\frac{5}{6}& = & \frac{-4}{5}x-7 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }+ \frac{ 25 }{ \color{blue}{30} })& = & (\frac{-24}{ \color{blue}{30} }x-\frac{210}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x+25& = & -24x-210 \\\Leftrightarrow & 5x \color{red}{+25} \color{blue}{-25} \color{blue}{+24x} & = & \color{red}{-24x} -210 \color{blue}{+24x} \color{blue}{-25} \\\Leftrightarrow & 29x& = & -235 \\\Leftrightarrow & \frac{29x}{ \color{red}{29} }& = & \frac{-235}{29} \\\Leftrightarrow & x = \frac{-235}{29} & & \\ & V = \left\{ \frac{-235}{29} \right\} & \\\end{align}\)
5. \(\text{156 is het kleinste gemene veelvoud van 4, 13 en 3} \\ \begin{align} & \frac{x}{4}-\frac{4}{13}& = & \frac{2}{3}x-3 \\\Leftrightarrow & \color{blue}{156.} (\frac{39x}{ \color{blue}{156} }- \frac{ 48 }{ \color{blue}{156} })& = & (\frac{104}{ \color{blue}{156} }x-\frac{468}{ \color{blue}{156} }) \color{blue}{.156} \\\Leftrightarrow & 39x-48& = & 104x-468 \\\Leftrightarrow & 39x \color{red}{-48} \color{blue}{+48} \color{blue}{-104x} & = & \color{red}{104x} -468 \color{blue}{-104x} \color{blue}{+48} \\\Leftrightarrow & -65x& = & -420 \\\Leftrightarrow & \frac{-65x}{ \color{red}{-65} }& = & \frac{-420}{-65} \\\Leftrightarrow & x = \frac{84}{13} & & \\ & V = \left\{ \frac{84}{13} \right\} & \\\end{align}\)
6. \(\text{24 is het kleinste gemene veelvoud van 3, 8 en 2} \\ \begin{align} & \frac{x}{3}-\frac{3}{8}& = & \frac{7}{2}x-2 \\\Leftrightarrow & \color{blue}{24.} (\frac{8x}{ \color{blue}{24} }- \frac{ 9 }{ \color{blue}{24} })& = & (\frac{84}{ \color{blue}{24} }x-\frac{48}{ \color{blue}{24} }) \color{blue}{.24} \\\Leftrightarrow & 8x-9& = & 84x-48 \\\Leftrightarrow & 8x \color{red}{-9} \color{blue}{+9} \color{blue}{-84x} & = & \color{red}{84x} -48 \color{blue}{-84x} \color{blue}{+9} \\\Leftrightarrow & -76x& = & -39 \\\Leftrightarrow & \frac{-76x}{ \color{red}{-76} }& = & \frac{-39}{-76} \\\Leftrightarrow & x = \frac{39}{76} & & \\ & V = \left\{ \frac{39}{76} \right\} & \\\end{align}\)
7. \(\text{120 is het kleinste gemene veelvoud van 6, 8 en 5} \\ \begin{align} & \frac{x}{6}-\frac{5}{8}& = & \frac{2}{5}x-3 \\\Leftrightarrow & \color{blue}{120.} (\frac{20x}{ \color{blue}{120} }- \frac{ 75 }{ \color{blue}{120} })& = & (\frac{48}{ \color{blue}{120} }x-\frac{360}{ \color{blue}{120} }) \color{blue}{.120} \\\Leftrightarrow & 20x-75& = & 48x-360 \\\Leftrightarrow & 20x \color{red}{-75} \color{blue}{+75} \color{blue}{-48x} & = & \color{red}{48x} -360 \color{blue}{-48x} \color{blue}{+75} \\\Leftrightarrow & -28x& = & -285 \\\Leftrightarrow & \frac{-28x}{ \color{red}{-28} }& = & \frac{-285}{-28} \\\Leftrightarrow & x = \frac{285}{28} & & \\ & V = \left\{ \frac{285}{28} \right\} & \\\end{align}\)
8. \(\text{165 is het kleinste gemene veelvoud van 3, 11 en 5} \\ \begin{align} & \frac{x}{3}+\frac{5}{11}& = & \frac{7}{5}x+8 \\\Leftrightarrow & \color{blue}{165.} (\frac{55x}{ \color{blue}{165} }+ \frac{ 75 }{ \color{blue}{165} })& = & (\frac{231}{ \color{blue}{165} }x+\frac{1320}{ \color{blue}{165} }) \color{blue}{.165} \\\Leftrightarrow & 55x+75& = & 231x+1320 \\\Leftrightarrow & 55x \color{red}{+75} \color{blue}{-75} \color{blue}{-231x} & = & \color{red}{231x} +1320 \color{blue}{-231x} \color{blue}{-75} \\\Leftrightarrow & -176x& = & 1245 \\\Leftrightarrow & \frac{-176x}{ \color{red}{-176} }& = & \frac{1245}{-176} \\\Leftrightarrow & x = \frac{-1245}{176} & & \\ & V = \left\{ \frac{-1245}{176} \right\} & \\\end{align}\)
9. \(\text{21 is het kleinste gemene veelvoud van 7, 7 en 3} \\ \begin{align} & \frac{x}{7}+\frac{2}{7}& = & \frac{-2}{3}x-1 \\\Leftrightarrow & \color{blue}{21.} (\frac{3x}{ \color{blue}{21} }+ \frac{ 6 }{ \color{blue}{21} })& = & (\frac{-14}{ \color{blue}{21} }x-\frac{21}{ \color{blue}{21} }) \color{blue}{.21} \\\Leftrightarrow & 3x+6& = & -14x-21 \\\Leftrightarrow & 3x \color{red}{+6} \color{blue}{-6} \color{blue}{+14x} & = & \color{red}{-14x} -21 \color{blue}{+14x} \color{blue}{-6} \\\Leftrightarrow & 17x& = & -27 \\\Leftrightarrow & \frac{17x}{ \color{red}{17} }& = & \frac{-27}{17} \\\Leftrightarrow & x = \frac{-27}{17} & & \\ & V = \left\{ \frac{-27}{17} \right\} & \\\end{align}\)
10. \(\text{36 is het kleinste gemene veelvoud van 4, 9 en 3} \\ \begin{align} & \frac{x}{4}-\frac{4}{9}& = & \frac{-8}{3}x+6 \\\Leftrightarrow & \color{blue}{36.} (\frac{9x}{ \color{blue}{36} }- \frac{ 16 }{ \color{blue}{36} })& = & (\frac{-96}{ \color{blue}{36} }x+\frac{216}{ \color{blue}{36} }) \color{blue}{.36} \\\Leftrightarrow & 9x-16& = & -96x+216 \\\Leftrightarrow & 9x \color{red}{-16} \color{blue}{+16} \color{blue}{+96x} & = & \color{red}{-96x} +216 \color{blue}{+96x} \color{blue}{+16} \\\Leftrightarrow & 105x& = & 232 \\\Leftrightarrow & \frac{105x}{ \color{red}{105} }& = & \frac{232}{105} \\\Leftrightarrow & x = \frac{232}{105} & & \\ & V = \left\{ \frac{232}{105} \right\} & \\\end{align}\)
11. \(\text{385 is het kleinste gemene veelvoud van 7, 11 en 5} \\ \begin{align} & \frac{x}{7}+\frac{5}{11}& = & \frac{7}{5}x+5 \\\Leftrightarrow & \color{blue}{385.} (\frac{55x}{ \color{blue}{385} }+ \frac{ 175 }{ \color{blue}{385} })& = & (\frac{539}{ \color{blue}{385} }x+\frac{1925}{ \color{blue}{385} }) \color{blue}{.385} \\\Leftrightarrow & 55x+175& = & 539x+1925 \\\Leftrightarrow & 55x \color{red}{+175} \color{blue}{-175} \color{blue}{-539x} & = & \color{red}{539x} +1925 \color{blue}{-539x} \color{blue}{-175} \\\Leftrightarrow & -484x& = & 1750 \\\Leftrightarrow & \frac{-484x}{ \color{red}{-484} }& = & \frac{1750}{-484} \\\Leftrightarrow & x = \frac{-875}{242} & & \\ & V = \left\{ \frac{-875}{242} \right\} & \\\end{align}\)
12. \(\text{70 is het kleinste gemene veelvoud van 5, 7 en 2} \\ \begin{align} & \frac{x}{5}+\frac{2}{7}& = & \frac{1}{2}x+4 \\\Leftrightarrow & \color{blue}{70.} (\frac{14x}{ \color{blue}{70} }+ \frac{ 20 }{ \color{blue}{70} })& = & (\frac{35}{ \color{blue}{70} }x+\frac{280}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 14x+20& = & 35x+280 \\\Leftrightarrow & 14x \color{red}{+20} \color{blue}{-20} \color{blue}{-35x} & = & \color{red}{35x} +280 \color{blue}{-35x} \color{blue}{-20} \\\Leftrightarrow & -21x& = & 260 \\\Leftrightarrow & \frac{-21x}{ \color{red}{-21} }& = & \frac{260}{-21} \\\Leftrightarrow & x = \frac{-260}{21} & & \\ & V = \left\{ \frac{-260}{21} \right\} & \\\end{align}\)