Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{48 is het kleinste gemene veelvoud van 4, 16 en 3} \\ \begin{align} & \frac{x}{4}-\frac{5}{16}& = & \frac{5}{3}x+7 \\\Leftrightarrow & \color{blue}{48.} (\frac{12x}{ \color{blue}{48} }- \frac{ 15 }{ \color{blue}{48} })& = & (\frac{80}{ \color{blue}{48} }x+\frac{336}{ \color{blue}{48} }) \color{blue}{.48} \\\Leftrightarrow & 12x-15& = & 80x+336 \\\Leftrightarrow & 12x \color{red}{-15} \color{blue}{+15} \color{blue}{-80x} & = & \color{red}{80x} +336 \color{blue}{-80x} \color{blue}{+15} \\\Leftrightarrow & -68x& = & 351 \\\Leftrightarrow & \frac{-68x}{ \color{red}{-68} }& = & \frac{351}{-68} \\\Leftrightarrow & x = \frac{-351}{68} & & \\ & V = \left\{ \frac{-351}{68} \right\} & \\\end{align}\)
2. \(\text{30 is het kleinste gemene veelvoud van 5, 10 en 6} \\ \begin{align} & \frac{x}{5}-\frac{3}{10}& = & \frac{-5}{6}x+1 \\\Leftrightarrow & \color{blue}{30.} (\frac{6x}{ \color{blue}{30} }- \frac{ 9 }{ \color{blue}{30} })& = & (\frac{-25}{ \color{blue}{30} }x+\frac{30}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 6x-9& = & -25x+30 \\\Leftrightarrow & 6x \color{red}{-9} \color{blue}{+9} \color{blue}{+25x} & = & \color{red}{-25x} +30 \color{blue}{+25x} \color{blue}{+9} \\\Leftrightarrow & 31x& = & 39 \\\Leftrightarrow & \frac{31x}{ \color{red}{31} }& = & \frac{39}{31} \\\Leftrightarrow & x = \frac{39}{31} & & \\ & V = \left\{ \frac{39}{31} \right\} & \\\end{align}\)
3. \(\text{105 is het kleinste gemene veelvoud van 3, 7 en 5} \\ \begin{align} & \frac{x}{3}+\frac{4}{7}& = & \frac{3}{5}x+1 \\\Leftrightarrow & \color{blue}{105.} (\frac{35x}{ \color{blue}{105} }+ \frac{ 60 }{ \color{blue}{105} })& = & (\frac{63}{ \color{blue}{105} }x+\frac{105}{ \color{blue}{105} }) \color{blue}{.105} \\\Leftrightarrow & 35x+60& = & 63x+105 \\\Leftrightarrow & 35x \color{red}{+60} \color{blue}{-60} \color{blue}{-63x} & = & \color{red}{63x} +105 \color{blue}{-63x} \color{blue}{-60} \\\Leftrightarrow & -28x& = & 45 \\\Leftrightarrow & \frac{-28x}{ \color{red}{-28} }& = & \frac{45}{-28} \\\Leftrightarrow & x = \frac{-45}{28} & & \\ & V = \left\{ \frac{-45}{28} \right\} & \\\end{align}\)
4. \(\text{42 is het kleinste gemene veelvoud van 2, 7 en 3} \\ \begin{align} & \frac{x}{2}-\frac{2}{7}& = & \frac{-2}{3}x-2 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }- \frac{ 12 }{ \color{blue}{42} })& = & (\frac{-28}{ \color{blue}{42} }x-\frac{84}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x-12& = & -28x-84 \\\Leftrightarrow & 21x \color{red}{-12} \color{blue}{+12} \color{blue}{+28x} & = & \color{red}{-28x} -84 \color{blue}{+28x} \color{blue}{+12} \\\Leftrightarrow & 49x& = & -72 \\\Leftrightarrow & \frac{49x}{ \color{red}{49} }& = & \frac{-72}{49} \\\Leftrightarrow & x = \frac{-72}{49} & & \\ & V = \left\{ \frac{-72}{49} \right\} & \\\end{align}\)
5. \(\text{48 is het kleinste gemene veelvoud van 2, 16 en 3} \\ \begin{align} & \frac{x}{2}-\frac{3}{16}& = & \frac{-5}{3}x+7 \\\Leftrightarrow & \color{blue}{48.} (\frac{24x}{ \color{blue}{48} }- \frac{ 9 }{ \color{blue}{48} })& = & (\frac{-80}{ \color{blue}{48} }x+\frac{336}{ \color{blue}{48} }) \color{blue}{.48} \\\Leftrightarrow & 24x-9& = & -80x+336 \\\Leftrightarrow & 24x \color{red}{-9} \color{blue}{+9} \color{blue}{+80x} & = & \color{red}{-80x} +336 \color{blue}{+80x} \color{blue}{+9} \\\Leftrightarrow & 104x& = & 345 \\\Leftrightarrow & \frac{104x}{ \color{red}{104} }& = & \frac{345}{104} \\\Leftrightarrow & x = \frac{345}{104} & & \\ & V = \left\{ \frac{345}{104} \right\} & \\\end{align}\)
6. \(\text{156 is het kleinste gemene veelvoud van 3, 13 en 4} \\ \begin{align} & \frac{x}{3}-\frac{3}{13}& = & \frac{-7}{4}x+4 \\\Leftrightarrow & \color{blue}{156.} (\frac{52x}{ \color{blue}{156} }- \frac{ 36 }{ \color{blue}{156} })& = & (\frac{-273}{ \color{blue}{156} }x+\frac{624}{ \color{blue}{156} }) \color{blue}{.156} \\\Leftrightarrow & 52x-36& = & -273x+624 \\\Leftrightarrow & 52x \color{red}{-36} \color{blue}{+36} \color{blue}{+273x} & = & \color{red}{-273x} +624 \color{blue}{+273x} \color{blue}{+36} \\\Leftrightarrow & 325x& = & 660 \\\Leftrightarrow & \frac{325x}{ \color{red}{325} }& = & \frac{660}{325} \\\Leftrightarrow & x = \frac{132}{65} & & \\ & V = \left\{ \frac{132}{65} \right\} & \\\end{align}\)
7. \(\text{18 is het kleinste gemene veelvoud van 3, 9 en 2} \\ \begin{align} & \frac{x}{3}-\frac{2}{9}& = & \frac{1}{2}x+2 \\\Leftrightarrow & \color{blue}{18.} (\frac{6x}{ \color{blue}{18} }- \frac{ 4 }{ \color{blue}{18} })& = & (\frac{9}{ \color{blue}{18} }x+\frac{36}{ \color{blue}{18} }) \color{blue}{.18} \\\Leftrightarrow & 6x-4& = & 9x+36 \\\Leftrightarrow & 6x \color{red}{-4} \color{blue}{+4} \color{blue}{-9x} & = & \color{red}{9x} +36 \color{blue}{-9x} \color{blue}{+4} \\\Leftrightarrow & -3x& = & 40 \\\Leftrightarrow & \frac{-3x}{ \color{red}{-3} }& = & \frac{40}{-3} \\\Leftrightarrow & x = \frac{-40}{3} & & \\ & V = \left\{ \frac{-40}{3} \right\} & \\\end{align}\)
8. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}-\frac{3}{7}& = & \frac{4}{3}x+5 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }- \frac{ 36 }{ \color{blue}{84} })& = & (\frac{112}{ \color{blue}{84} }x+\frac{420}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x-36& = & 112x+420 \\\Leftrightarrow & 21x \color{red}{-36} \color{blue}{+36} \color{blue}{-112x} & = & \color{red}{112x} +420 \color{blue}{-112x} \color{blue}{+36} \\\Leftrightarrow & -91x& = & 456 \\\Leftrightarrow & \frac{-91x}{ \color{red}{-91} }& = & \frac{456}{-91} \\\Leftrightarrow & x = \frac{-456}{91} & & \\ & V = \left\{ \frac{-456}{91} \right\} & \\\end{align}\)
9. \(\text{30 is het kleinste gemene veelvoud van 6, 15 en 5} \\ \begin{align} & \frac{x}{6}-\frac{4}{15}& = & \frac{-4}{5}x+5 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }- \frac{ 8 }{ \color{blue}{30} })& = & (\frac{-24}{ \color{blue}{30} }x+\frac{150}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x-8& = & -24x+150 \\\Leftrightarrow & 5x \color{red}{-8} \color{blue}{+8} \color{blue}{+24x} & = & \color{red}{-24x} +150 \color{blue}{+24x} \color{blue}{+8} \\\Leftrightarrow & 29x& = & 158 \\\Leftrightarrow & \frac{29x}{ \color{red}{29} }& = & \frac{158}{29} \\\Leftrightarrow & x = \frac{158}{29} & & \\ & V = \left\{ \frac{158}{29} \right\} & \\\end{align}\)
10. \(\text{195 is het kleinste gemene veelvoud van 5, 13 en 3} \\ \begin{align} & \frac{x}{5}-\frac{4}{13}& = & \frac{7}{3}x+1 \\\Leftrightarrow & \color{blue}{195.} (\frac{39x}{ \color{blue}{195} }- \frac{ 60 }{ \color{blue}{195} })& = & (\frac{455}{ \color{blue}{195} }x+\frac{195}{ \color{blue}{195} }) \color{blue}{.195} \\\Leftrightarrow & 39x-60& = & 455x+195 \\\Leftrightarrow & 39x \color{red}{-60} \color{blue}{+60} \color{blue}{-455x} & = & \color{red}{455x} +195 \color{blue}{-455x} \color{blue}{+60} \\\Leftrightarrow & -416x& = & 255 \\\Leftrightarrow & \frac{-416x}{ \color{red}{-416} }& = & \frac{255}{-416} \\\Leftrightarrow & x = \frac{-255}{416} & & \\ & V = \left\{ \frac{-255}{416} \right\} & \\\end{align}\)
11. \(\text{240 is het kleinste gemene veelvoud van 6, 16 en 5} \\ \begin{align} & \frac{x}{6}-\frac{3}{16}& = & \frac{6}{5}x+8 \\\Leftrightarrow & \color{blue}{240.} (\frac{40x}{ \color{blue}{240} }- \frac{ 45 }{ \color{blue}{240} })& = & (\frac{288}{ \color{blue}{240} }x+\frac{1920}{ \color{blue}{240} }) \color{blue}{.240} \\\Leftrightarrow & 40x-45& = & 288x+1920 \\\Leftrightarrow & 40x \color{red}{-45} \color{blue}{+45} \color{blue}{-288x} & = & \color{red}{288x} +1920 \color{blue}{-288x} \color{blue}{+45} \\\Leftrightarrow & -248x& = & 1965 \\\Leftrightarrow & \frac{-248x}{ \color{red}{-248} }& = & \frac{1965}{-248} \\\Leftrightarrow & x = \frac{-1965}{248} & & \\ & V = \left\{ \frac{-1965}{248} \right\} & \\\end{align}\)
12. \(\text{90 is het kleinste gemene veelvoud van 5, 9 en 2} \\ \begin{align} & \frac{x}{5}+\frac{2}{9}& = & \frac{5}{2}x-8 \\\Leftrightarrow & \color{blue}{90.} (\frac{18x}{ \color{blue}{90} }+ \frac{ 20 }{ \color{blue}{90} })& = & (\frac{225}{ \color{blue}{90} }x-\frac{720}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 18x+20& = & 225x-720 \\\Leftrightarrow & 18x \color{red}{+20} \color{blue}{-20} \color{blue}{-225x} & = & \color{red}{225x} -720 \color{blue}{-225x} \color{blue}{-20} \\\Leftrightarrow & -207x& = & -740 \\\Leftrightarrow & \frac{-207x}{ \color{red}{-207} }& = & \frac{-740}{-207} \\\Leftrightarrow & x = \frac{740}{207} & & \\ & V = \left\{ \frac{740}{207} \right\} & \\\end{align}\)