Vgln. eerste graad (reeks 4)

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1. \(\frac{x}{2}+\frac{2}{7}=\frac{1}{3}x-7\)
2. \(\frac{x}{7}-\frac{3}{16}=\frac{6}{5}x-5\)
3. \(\frac{x}{6}+\frac{5}{14}=\frac{2}{5}x+7\)
4. \(\frac{x}{6}+\frac{5}{9}=\frac{2}{5}x+2\)
5. \(\frac{x}{3}-\frac{2}{13}=\frac{7}{5}x-4\)
6. \(\frac{x}{4}+\frac{2}{11}=\frac{-8}{3}x+1\)
7. \(\frac{x}{7}+\frac{5}{11}=\frac{7}{6}x+6\)
8. \(\frac{x}{4}+\frac{4}{7}=\frac{4}{3}x-7\)
9. \(\frac{x}{6}+\frac{2}{7}=\frac{-2}{5}x-8\)
10. \(\frac{x}{7}-\frac{4}{15}=\frac{-2}{3}x-4\)
11. \(\frac{x}{2}+\frac{2}{13}=\frac{6}{5}x-3\)
12. \(\frac{x}{5}-\frac{3}{7}=\frac{-5}{6}x-1\)

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

Verbetersleutel

1. \(\text{42 is het kleinste gemene veelvoud van 2, 7 en 3} \\ \begin{align} & \frac{x}{2}+\frac{2}{7}& = & \frac{1}{3}x-7 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }+ \frac{ 12 }{ \color{blue}{42} })& = & (\frac{14}{ \color{blue}{42} }x-\frac{294}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x+12& = & 14x-294 \\\Leftrightarrow & 21x \color{red}{+12} \color{blue}{-12} \color{blue}{-14x} & = & \color{red}{14x} -294 \color{blue}{-14x} \color{blue}{-12} \\\Leftrightarrow & 7x& = & -306 \\\Leftrightarrow & \frac{7x}{ \color{red}{7} }& = & \frac{-306}{7} \\\Leftrightarrow & x = \frac{-306}{7} & & \\ & V = \left\{ \frac{-306}{7} \right\} & \\\end{align}\)
2. \(\text{560 is het kleinste gemene veelvoud van 7, 16 en 5} \\ \begin{align} & \frac{x}{7}-\frac{3}{16}& = & \frac{6}{5}x-5 \\\Leftrightarrow & \color{blue}{560.} (\frac{80x}{ \color{blue}{560} }- \frac{ 105 }{ \color{blue}{560} })& = & (\frac{672}{ \color{blue}{560} }x-\frac{2800}{ \color{blue}{560} }) \color{blue}{.560} \\\Leftrightarrow & 80x-105& = & 672x-2800 \\\Leftrightarrow & 80x \color{red}{-105} \color{blue}{+105} \color{blue}{-672x} & = & \color{red}{672x} -2800 \color{blue}{-672x} \color{blue}{+105} \\\Leftrightarrow & -592x& = & -2695 \\\Leftrightarrow & \frac{-592x}{ \color{red}{-592} }& = & \frac{-2695}{-592} \\\Leftrightarrow & x = \frac{2695}{592} & & \\ & V = \left\{ \frac{2695}{592} \right\} & \\\end{align}\)
3. \(\text{210 is het kleinste gemene veelvoud van 6, 14 en 5} \\ \begin{align} & \frac{x}{6}+\frac{5}{14}& = & \frac{2}{5}x+7 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }+ \frac{ 75 }{ \color{blue}{210} })& = & (\frac{84}{ \color{blue}{210} }x+\frac{1470}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x+75& = & 84x+1470 \\\Leftrightarrow & 35x \color{red}{+75} \color{blue}{-75} \color{blue}{-84x} & = & \color{red}{84x} +1470 \color{blue}{-84x} \color{blue}{-75} \\\Leftrightarrow & -49x& = & 1395 \\\Leftrightarrow & \frac{-49x}{ \color{red}{-49} }& = & \frac{1395}{-49} \\\Leftrightarrow & x = \frac{-1395}{49} & & \\ & V = \left\{ \frac{-1395}{49} \right\} & \\\end{align}\)
4. \(\text{90 is het kleinste gemene veelvoud van 6, 9 en 5} \\ \begin{align} & \frac{x}{6}+\frac{5}{9}& = & \frac{2}{5}x+2 \\\Leftrightarrow & \color{blue}{90.} (\frac{15x}{ \color{blue}{90} }+ \frac{ 50 }{ \color{blue}{90} })& = & (\frac{36}{ \color{blue}{90} }x+\frac{180}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 15x+50& = & 36x+180 \\\Leftrightarrow & 15x \color{red}{+50} \color{blue}{-50} \color{blue}{-36x} & = & \color{red}{36x} +180 \color{blue}{-36x} \color{blue}{-50} \\\Leftrightarrow & -21x& = & 130 \\\Leftrightarrow & \frac{-21x}{ \color{red}{-21} }& = & \frac{130}{-21} \\\Leftrightarrow & x = \frac{-130}{21} & & \\ & V = \left\{ \frac{-130}{21} \right\} & \\\end{align}\)
5. \(\text{195 is het kleinste gemene veelvoud van 3, 13 en 5} \\ \begin{align} & \frac{x}{3}-\frac{2}{13}& = & \frac{7}{5}x-4 \\\Leftrightarrow & \color{blue}{195.} (\frac{65x}{ \color{blue}{195} }- \frac{ 30 }{ \color{blue}{195} })& = & (\frac{273}{ \color{blue}{195} }x-\frac{780}{ \color{blue}{195} }) \color{blue}{.195} \\\Leftrightarrow & 65x-30& = & 273x-780 \\\Leftrightarrow & 65x \color{red}{-30} \color{blue}{+30} \color{blue}{-273x} & = & \color{red}{273x} -780 \color{blue}{-273x} \color{blue}{+30} \\\Leftrightarrow & -208x& = & -750 \\\Leftrightarrow & \frac{-208x}{ \color{red}{-208} }& = & \frac{-750}{-208} \\\Leftrightarrow & x = \frac{375}{104} & & \\ & V = \left\{ \frac{375}{104} \right\} & \\\end{align}\)
6. \(\text{132 is het kleinste gemene veelvoud van 4, 11 en 3} \\ \begin{align} & \frac{x}{4}+\frac{2}{11}& = & \frac{-8}{3}x+1 \\\Leftrightarrow & \color{blue}{132.} (\frac{33x}{ \color{blue}{132} }+ \frac{ 24 }{ \color{blue}{132} })& = & (\frac{-352}{ \color{blue}{132} }x+\frac{132}{ \color{blue}{132} }) \color{blue}{.132} \\\Leftrightarrow & 33x+24& = & -352x+132 \\\Leftrightarrow & 33x \color{red}{+24} \color{blue}{-24} \color{blue}{+352x} & = & \color{red}{-352x} +132 \color{blue}{+352x} \color{blue}{-24} \\\Leftrightarrow & 385x& = & 108 \\\Leftrightarrow & \frac{385x}{ \color{red}{385} }& = & \frac{108}{385} \\\Leftrightarrow & x = \frac{108}{385} & & \\ & V = \left\{ \frac{108}{385} \right\} & \\\end{align}\)
7. \(\text{462 is het kleinste gemene veelvoud van 7, 11 en 6} \\ \begin{align} & \frac{x}{7}+\frac{5}{11}& = & \frac{7}{6}x+6 \\\Leftrightarrow & \color{blue}{462.} (\frac{66x}{ \color{blue}{462} }+ \frac{ 210 }{ \color{blue}{462} })& = & (\frac{539}{ \color{blue}{462} }x+\frac{2772}{ \color{blue}{462} }) \color{blue}{.462} \\\Leftrightarrow & 66x+210& = & 539x+2772 \\\Leftrightarrow & 66x \color{red}{+210} \color{blue}{-210} \color{blue}{-539x} & = & \color{red}{539x} +2772 \color{blue}{-539x} \color{blue}{-210} \\\Leftrightarrow & -473x& = & 2562 \\\Leftrightarrow & \frac{-473x}{ \color{red}{-473} }& = & \frac{2562}{-473} \\\Leftrightarrow & x = \frac{-2562}{473} & & \\ & V = \left\{ \frac{-2562}{473} \right\} & \\\end{align}\)
8. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}+\frac{4}{7}& = & \frac{4}{3}x-7 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }+ \frac{ 48 }{ \color{blue}{84} })& = & (\frac{112}{ \color{blue}{84} }x-\frac{588}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x+48& = & 112x-588 \\\Leftrightarrow & 21x \color{red}{+48} \color{blue}{-48} \color{blue}{-112x} & = & \color{red}{112x} -588 \color{blue}{-112x} \color{blue}{-48} \\\Leftrightarrow & -91x& = & -636 \\\Leftrightarrow & \frac{-91x}{ \color{red}{-91} }& = & \frac{-636}{-91} \\\Leftrightarrow & x = \frac{636}{91} & & \\ & V = \left\{ \frac{636}{91} \right\} & \\\end{align}\)
9. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}+\frac{2}{7}& = & \frac{-2}{5}x-8 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }+ \frac{ 60 }{ \color{blue}{210} })& = & (\frac{-84}{ \color{blue}{210} }x-\frac{1680}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x+60& = & -84x-1680 \\\Leftrightarrow & 35x \color{red}{+60} \color{blue}{-60} \color{blue}{+84x} & = & \color{red}{-84x} -1680 \color{blue}{+84x} \color{blue}{-60} \\\Leftrightarrow & 119x& = & -1740 \\\Leftrightarrow & \frac{119x}{ \color{red}{119} }& = & \frac{-1740}{119} \\\Leftrightarrow & x = \frac{-1740}{119} & & \\ & V = \left\{ \frac{-1740}{119} \right\} & \\\end{align}\)
10. \(\text{105 is het kleinste gemene veelvoud van 7, 15 en 3} \\ \begin{align} & \frac{x}{7}-\frac{4}{15}& = & \frac{-2}{3}x-4 \\\Leftrightarrow & \color{blue}{105.} (\frac{15x}{ \color{blue}{105} }- \frac{ 28 }{ \color{blue}{105} })& = & (\frac{-70}{ \color{blue}{105} }x-\frac{420}{ \color{blue}{105} }) \color{blue}{.105} \\\Leftrightarrow & 15x-28& = & -70x-420 \\\Leftrightarrow & 15x \color{red}{-28} \color{blue}{+28} \color{blue}{+70x} & = & \color{red}{-70x} -420 \color{blue}{+70x} \color{blue}{+28} \\\Leftrightarrow & 85x& = & -392 \\\Leftrightarrow & \frac{85x}{ \color{red}{85} }& = & \frac{-392}{85} \\\Leftrightarrow & x = \frac{-392}{85} & & \\ & V = \left\{ \frac{-392}{85} \right\} & \\\end{align}\)
11. \(\text{130 is het kleinste gemene veelvoud van 2, 13 en 5} \\ \begin{align} & \frac{x}{2}+\frac{2}{13}& = & \frac{6}{5}x-3 \\\Leftrightarrow & \color{blue}{130.} (\frac{65x}{ \color{blue}{130} }+ \frac{ 20 }{ \color{blue}{130} })& = & (\frac{156}{ \color{blue}{130} }x-\frac{390}{ \color{blue}{130} }) \color{blue}{.130} \\\Leftrightarrow & 65x+20& = & 156x-390 \\\Leftrightarrow & 65x \color{red}{+20} \color{blue}{-20} \color{blue}{-156x} & = & \color{red}{156x} -390 \color{blue}{-156x} \color{blue}{-20} \\\Leftrightarrow & -91x& = & -410 \\\Leftrightarrow & \frac{-91x}{ \color{red}{-91} }& = & \frac{-410}{-91} \\\Leftrightarrow & x = \frac{410}{91} & & \\ & V = \left\{ \frac{410}{91} \right\} & \\\end{align}\)
12. \(\text{210 is het kleinste gemene veelvoud van 5, 7 en 6} \\ \begin{align} & \frac{x}{5}-\frac{3}{7}& = & \frac{-5}{6}x-1 \\\Leftrightarrow & \color{blue}{210.} (\frac{42x}{ \color{blue}{210} }- \frac{ 90 }{ \color{blue}{210} })& = & (\frac{-175}{ \color{blue}{210} }x-\frac{210}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 42x-90& = & -175x-210 \\\Leftrightarrow & 42x \color{red}{-90} \color{blue}{+90} \color{blue}{+175x} & = & \color{red}{-175x} -210 \color{blue}{+175x} \color{blue}{+90} \\\Leftrightarrow & 217x& = & -120 \\\Leftrightarrow & \frac{217x}{ \color{red}{217} }& = & \frac{-120}{217} \\\Leftrightarrow & x = \frac{-120}{217} & & \\ & V = \left\{ \frac{-120}{217} \right\} & \\\end{align}\)