Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}-\frac{4}{7}& = & \frac{2}{5}x+1 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }- \frac{ 120 }{ \color{blue}{210} })& = & (\frac{84}{ \color{blue}{210} }x+\frac{210}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x-120& = & 84x+210 \\\Leftrightarrow & 35x \color{red}{-120} \color{blue}{+120} \color{blue}{-84x} & = & \color{red}{84x} +210 \color{blue}{-84x} \color{blue}{+120} \\\Leftrightarrow & -49x& = & 330 \\\Leftrightarrow & \frac{-49x}{ \color{red}{-49} }& = & \frac{330}{-49} \\\Leftrightarrow & x = \frac{-330}{49} & & \\ & V = \left\{ \frac{-330}{49} \right\} & \\\end{align}\)
2. \(\text{105 is het kleinste gemene veelvoud van 5, 7 en 3} \\ \begin{align} & \frac{x}{5}-\frac{2}{7}& = & \frac{4}{3}x-3 \\\Leftrightarrow & \color{blue}{105.} (\frac{21x}{ \color{blue}{105} }- \frac{ 30 }{ \color{blue}{105} })& = & (\frac{140}{ \color{blue}{105} }x-\frac{315}{ \color{blue}{105} }) \color{blue}{.105} \\\Leftrightarrow & 21x-30& = & 140x-315 \\\Leftrightarrow & 21x \color{red}{-30} \color{blue}{+30} \color{blue}{-140x} & = & \color{red}{140x} -315 \color{blue}{-140x} \color{blue}{+30} \\\Leftrightarrow & -119x& = & -285 \\\Leftrightarrow & \frac{-119x}{ \color{red}{-119} }& = & \frac{-285}{-119} \\\Leftrightarrow & x = \frac{285}{119} & & \\ & V = \left\{ \frac{285}{119} \right\} & \\\end{align}\)
3. \(\text{10 is het kleinste gemene veelvoud van 2, 10 en 5} \\ \begin{align} & \frac{x}{2}+\frac{3}{10}& = & \frac{-7}{5}x-8 \\\Leftrightarrow & \color{blue}{10.} (\frac{5x}{ \color{blue}{10} }+ \frac{ 3 }{ \color{blue}{10} })& = & (\frac{-14}{ \color{blue}{10} }x-\frac{80}{ \color{blue}{10} }) \color{blue}{.10} \\\Leftrightarrow & 5x+3& = & -14x-80 \\\Leftrightarrow & 5x \color{red}{+3} \color{blue}{-3} \color{blue}{+14x} & = & \color{red}{-14x} -80 \color{blue}{+14x} \color{blue}{-3} \\\Leftrightarrow & 19x& = & -83 \\\Leftrightarrow & \frac{19x}{ \color{red}{19} }& = & \frac{-83}{19} \\\Leftrightarrow & x = \frac{-83}{19} & & \\ & V = \left\{ \frac{-83}{19} \right\} & \\\end{align}\)
4. \(\text{105 is het kleinste gemene veelvoud van 5, 7 en 3} \\ \begin{align} & \frac{x}{5}-\frac{3}{7}& = & \frac{4}{3}x-5 \\\Leftrightarrow & \color{blue}{105.} (\frac{21x}{ \color{blue}{105} }- \frac{ 45 }{ \color{blue}{105} })& = & (\frac{140}{ \color{blue}{105} }x-\frac{525}{ \color{blue}{105} }) \color{blue}{.105} \\\Leftrightarrow & 21x-45& = & 140x-525 \\\Leftrightarrow & 21x \color{red}{-45} \color{blue}{+45} \color{blue}{-140x} & = & \color{red}{140x} -525 \color{blue}{-140x} \color{blue}{+45} \\\Leftrightarrow & -119x& = & -480 \\\Leftrightarrow & \frac{-119x}{ \color{red}{-119} }& = & \frac{-480}{-119} \\\Leftrightarrow & x = \frac{480}{119} & & \\ & V = \left\{ \frac{480}{119} \right\} & \\\end{align}\)
5. \(\text{30 is het kleinste gemene veelvoud van 2, 15 en 5} \\ \begin{align} & \frac{x}{2}-\frac{2}{15}& = & \frac{1}{5}x-8 \\\Leftrightarrow & \color{blue}{30.} (\frac{15x}{ \color{blue}{30} }- \frac{ 4 }{ \color{blue}{30} })& = & (\frac{6}{ \color{blue}{30} }x-\frac{240}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 15x-4& = & 6x-240 \\\Leftrightarrow & 15x \color{red}{-4} \color{blue}{+4} \color{blue}{-6x} & = & \color{red}{6x} -240 \color{blue}{-6x} \color{blue}{+4} \\\Leftrightarrow & 9x& = & -236 \\\Leftrightarrow & \frac{9x}{ \color{red}{9} }& = & \frac{-236}{9} \\\Leftrightarrow & x = \frac{-236}{9} & & \\ & V = \left\{ \frac{-236}{9} \right\} & \\\end{align}\)
6. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}-\frac{2}{7}& = & \frac{1}{3}x-6 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }- \frac{ 24 }{ \color{blue}{84} })& = & (\frac{28}{ \color{blue}{84} }x-\frac{504}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x-24& = & 28x-504 \\\Leftrightarrow & 21x \color{red}{-24} \color{blue}{+24} \color{blue}{-28x} & = & \color{red}{28x} -504 \color{blue}{-28x} \color{blue}{+24} \\\Leftrightarrow & -7x& = & -480 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{-480}{-7} \\\Leftrightarrow & x = \frac{480}{7} & & \\ & V = \left\{ \frac{480}{7} \right\} & \\\end{align}\)
7. \(\text{42 is het kleinste gemene veelvoud van 2, 7 en 3} \\ \begin{align} & \frac{x}{2}-\frac{4}{7}& = & \frac{4}{3}x-3 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }- \frac{ 24 }{ \color{blue}{42} })& = & (\frac{56}{ \color{blue}{42} }x-\frac{126}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x-24& = & 56x-126 \\\Leftrightarrow & 21x \color{red}{-24} \color{blue}{+24} \color{blue}{-56x} & = & \color{red}{56x} -126 \color{blue}{-56x} \color{blue}{+24} \\\Leftrightarrow & -35x& = & -102 \\\Leftrightarrow & \frac{-35x}{ \color{red}{-35} }& = & \frac{-102}{-35} \\\Leftrightarrow & x = \frac{102}{35} & & \\ & V = \left\{ \frac{102}{35} \right\} & \\\end{align}\)
8. \(\text{132 is het kleinste gemene veelvoud van 4, 11 en 3} \\ \begin{align} & \frac{x}{4}-\frac{4}{11}& = & \frac{4}{3}x-3 \\\Leftrightarrow & \color{blue}{132.} (\frac{33x}{ \color{blue}{132} }- \frac{ 48 }{ \color{blue}{132} })& = & (\frac{176}{ \color{blue}{132} }x-\frac{396}{ \color{blue}{132} }) \color{blue}{.132} \\\Leftrightarrow & 33x-48& = & 176x-396 \\\Leftrightarrow & 33x \color{red}{-48} \color{blue}{+48} \color{blue}{-176x} & = & \color{red}{176x} -396 \color{blue}{-176x} \color{blue}{+48} \\\Leftrightarrow & -143x& = & -348 \\\Leftrightarrow & \frac{-143x}{ \color{red}{-143} }& = & \frac{-348}{-143} \\\Leftrightarrow & x = \frac{348}{143} & & \\ & V = \left\{ \frac{348}{143} \right\} & \\\end{align}\)
9. \(\text{36 is het kleinste gemene veelvoud van 4, 9 en 3} \\ \begin{align} & \frac{x}{4}+\frac{2}{9}& = & \frac{-5}{3}x-7 \\\Leftrightarrow & \color{blue}{36.} (\frac{9x}{ \color{blue}{36} }+ \frac{ 8 }{ \color{blue}{36} })& = & (\frac{-60}{ \color{blue}{36} }x-\frac{252}{ \color{blue}{36} }) \color{blue}{.36} \\\Leftrightarrow & 9x+8& = & -60x-252 \\\Leftrightarrow & 9x \color{red}{+8} \color{blue}{-8} \color{blue}{+60x} & = & \color{red}{-60x} -252 \color{blue}{+60x} \color{blue}{-8} \\\Leftrightarrow & 69x& = & -260 \\\Leftrightarrow & \frac{69x}{ \color{red}{69} }& = & \frac{-260}{69} \\\Leftrightarrow & x = \frac{-260}{69} & & \\ & V = \left\{ \frac{-260}{69} \right\} & \\\end{align}\)
10. \(\text{130 is het kleinste gemene veelvoud van 5, 13 en 2} \\ \begin{align} & \frac{x}{5}+\frac{4}{13}& = & \frac{1}{2}x-1 \\\Leftrightarrow & \color{blue}{130.} (\frac{26x}{ \color{blue}{130} }+ \frac{ 40 }{ \color{blue}{130} })& = & (\frac{65}{ \color{blue}{130} }x-\frac{130}{ \color{blue}{130} }) \color{blue}{.130} \\\Leftrightarrow & 26x+40& = & 65x-130 \\\Leftrightarrow & 26x \color{red}{+40} \color{blue}{-40} \color{blue}{-65x} & = & \color{red}{65x} -130 \color{blue}{-65x} \color{blue}{-40} \\\Leftrightarrow & -39x& = & -170 \\\Leftrightarrow & \frac{-39x}{ \color{red}{-39} }& = & \frac{-170}{-39} \\\Leftrightarrow & x = \frac{170}{39} & & \\ & V = \left\{ \frac{170}{39} \right\} & \\\end{align}\)
11. \(\text{420 is het kleinste gemene veelvoud van 7, 15 en 4} \\ \begin{align} & \frac{x}{7}-\frac{2}{15}& = & \frac{-3}{4}x-7 \\\Leftrightarrow & \color{blue}{420.} (\frac{60x}{ \color{blue}{420} }- \frac{ 56 }{ \color{blue}{420} })& = & (\frac{-315}{ \color{blue}{420} }x-\frac{2940}{ \color{blue}{420} }) \color{blue}{.420} \\\Leftrightarrow & 60x-56& = & -315x-2940 \\\Leftrightarrow & 60x \color{red}{-56} \color{blue}{+56} \color{blue}{+315x} & = & \color{red}{-315x} -2940 \color{blue}{+315x} \color{blue}{+56} \\\Leftrightarrow & 375x& = & -2884 \\\Leftrightarrow & \frac{375x}{ \color{red}{375} }& = & \frac{-2884}{375} \\\Leftrightarrow & x = \frac{-2884}{375} & & \\ & V = \left\{ \frac{-2884}{375} \right\} & \\\end{align}\)
12. \(\text{56 is het kleinste gemene veelvoud van 7, 8 en 4} \\ \begin{align} & \frac{x}{7}-\frac{3}{8}& = & \frac{-3}{4}x+2 \\\Leftrightarrow & \color{blue}{56.} (\frac{8x}{ \color{blue}{56} }- \frac{ 21 }{ \color{blue}{56} })& = & (\frac{-42}{ \color{blue}{56} }x+\frac{112}{ \color{blue}{56} }) \color{blue}{.56} \\\Leftrightarrow & 8x-21& = & -42x+112 \\\Leftrightarrow & 8x \color{red}{-21} \color{blue}{+21} \color{blue}{+42x} & = & \color{red}{-42x} +112 \color{blue}{+42x} \color{blue}{+21} \\\Leftrightarrow & 50x& = & 133 \\\Leftrightarrow & \frac{50x}{ \color{red}{50} }& = & \frac{133}{50} \\\Leftrightarrow & x = \frac{133}{50} & & \\ & V = \left\{ \frac{133}{50} \right\} & \\\end{align}\)