Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{210 is het kleinste gemene veelvoud van 5, 7 en 6} \\ \begin{align} & \frac{x}{5}-\frac{4}{7}& = & \frac{1}{6}x+8 \\\Leftrightarrow & \color{blue}{210.} (\frac{42x}{ \color{blue}{210} }- \frac{ 120 }{ \color{blue}{210} })& = & (\frac{35}{ \color{blue}{210} }x+\frac{1680}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 42x-120& = & 35x+1680 \\\Leftrightarrow & 42x \color{red}{-120} \color{blue}{+120} \color{blue}{-35x} & = & \color{red}{35x} +1680 \color{blue}{-35x} \color{blue}{+120} \\\Leftrightarrow & 7x& = & 1800 \\\Leftrightarrow & \frac{7x}{ \color{red}{7} }& = & \frac{1800}{7} \\\Leftrightarrow & x = \frac{1800}{7} & & \\ & V = \left\{ \frac{1800}{7} \right\} & \\\end{align}\)
2. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}-\frac{2}{7}& = & \frac{5}{3}x+1 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }- \frac{ 24 }{ \color{blue}{84} })& = & (\frac{140}{ \color{blue}{84} }x+\frac{84}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x-24& = & 140x+84 \\\Leftrightarrow & 21x \color{red}{-24} \color{blue}{+24} \color{blue}{-140x} & = & \color{red}{140x} +84 \color{blue}{-140x} \color{blue}{+24} \\\Leftrightarrow & -119x& = & 108 \\\Leftrightarrow & \frac{-119x}{ \color{red}{-119} }& = & \frac{108}{-119} \\\Leftrightarrow & x = \frac{-108}{119} & & \\ & V = \left\{ \frac{-108}{119} \right\} & \\\end{align}\)
3. \(\text{42 is het kleinste gemene veelvoud van 2, 7 en 3} \\ \begin{align} & \frac{x}{2}+\frac{4}{7}& = & \frac{-8}{3}x-6 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }+ \frac{ 24 }{ \color{blue}{42} })& = & (\frac{-112}{ \color{blue}{42} }x-\frac{252}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x+24& = & -112x-252 \\\Leftrightarrow & 21x \color{red}{+24} \color{blue}{-24} \color{blue}{+112x} & = & \color{red}{-112x} -252 \color{blue}{+112x} \color{blue}{-24} \\\Leftrightarrow & 133x& = & -276 \\\Leftrightarrow & \frac{133x}{ \color{red}{133} }& = & \frac{-276}{133} \\\Leftrightarrow & x = \frac{-276}{133} & & \\ & V = \left\{ \frac{-276}{133} \right\} & \\\end{align}\)
4. \(\text{110 is het kleinste gemene veelvoud van 5, 11 en 2} \\ \begin{align} & \frac{x}{5}+\frac{4}{11}& = & \frac{1}{2}x-2 \\\Leftrightarrow & \color{blue}{110.} (\frac{22x}{ \color{blue}{110} }+ \frac{ 40 }{ \color{blue}{110} })& = & (\frac{55}{ \color{blue}{110} }x-\frac{220}{ \color{blue}{110} }) \color{blue}{.110} \\\Leftrightarrow & 22x+40& = & 55x-220 \\\Leftrightarrow & 22x \color{red}{+40} \color{blue}{-40} \color{blue}{-55x} & = & \color{red}{55x} -220 \color{blue}{-55x} \color{blue}{-40} \\\Leftrightarrow & -33x& = & -260 \\\Leftrightarrow & \frac{-33x}{ \color{red}{-33} }& = & \frac{-260}{-33} \\\Leftrightarrow & x = \frac{260}{33} & & \\ & V = \left\{ \frac{260}{33} \right\} & \\\end{align}\)
5. \(\text{280 is het kleinste gemene veelvoud van 7, 8 en 5} \\ \begin{align} & \frac{x}{7}+\frac{3}{8}& = & \frac{1}{5}x+6 \\\Leftrightarrow & \color{blue}{280.} (\frac{40x}{ \color{blue}{280} }+ \frac{ 105 }{ \color{blue}{280} })& = & (\frac{56}{ \color{blue}{280} }x+\frac{1680}{ \color{blue}{280} }) \color{blue}{.280} \\\Leftrightarrow & 40x+105& = & 56x+1680 \\\Leftrightarrow & 40x \color{red}{+105} \color{blue}{-105} \color{blue}{-56x} & = & \color{red}{56x} +1680 \color{blue}{-56x} \color{blue}{-105} \\\Leftrightarrow & -16x& = & 1575 \\\Leftrightarrow & \frac{-16x}{ \color{red}{-16} }& = & \frac{1575}{-16} \\\Leftrightarrow & x = \frac{-1575}{16} & & \\ & V = \left\{ \frac{-1575}{16} \right\} & \\\end{align}\)
6. \(\text{90 is het kleinste gemene veelvoud van 6, 9 en 5} \\ \begin{align} & \frac{x}{6}+\frac{2}{9}& = & \frac{1}{5}x+2 \\\Leftrightarrow & \color{blue}{90.} (\frac{15x}{ \color{blue}{90} }+ \frac{ 20 }{ \color{blue}{90} })& = & (\frac{18}{ \color{blue}{90} }x+\frac{180}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 15x+20& = & 18x+180 \\\Leftrightarrow & 15x \color{red}{+20} \color{blue}{-20} \color{blue}{-18x} & = & \color{red}{18x} +180 \color{blue}{-18x} \color{blue}{-20} \\\Leftrightarrow & -3x& = & 160 \\\Leftrightarrow & \frac{-3x}{ \color{red}{-3} }& = & \frac{160}{-3} \\\Leftrightarrow & x = \frac{-160}{3} & & \\ & V = \left\{ \frac{-160}{3} \right\} & \\\end{align}\)
7. \(\text{14 is het kleinste gemene veelvoud van 7, 7 en 2} \\ \begin{align} & \frac{x}{7}-\frac{4}{7}& = & \frac{1}{2}x-2 \\\Leftrightarrow & \color{blue}{14.} (\frac{2x}{ \color{blue}{14} }- \frac{ 8 }{ \color{blue}{14} })& = & (\frac{7}{ \color{blue}{14} }x-\frac{28}{ \color{blue}{14} }) \color{blue}{.14} \\\Leftrightarrow & 2x-8& = & 7x-28 \\\Leftrightarrow & 2x \color{red}{-8} \color{blue}{+8} \color{blue}{-7x} & = & \color{red}{7x} -28 \color{blue}{-7x} \color{blue}{+8} \\\Leftrightarrow & -5x& = & -20 \\\Leftrightarrow & \frac{-5x}{ \color{red}{-5} }& = & \frac{-20}{-5} \\\Leftrightarrow & x = 4 & & \\ & V = \left\{ 4 \right\} & \\\end{align}\)
8. \(\text{12 is het kleinste gemene veelvoud van 4, 6 en 3} \\ \begin{align} & \frac{x}{4}-\frac{5}{6}& = & \frac{-2}{3}x+3 \\\Leftrightarrow & \color{blue}{12.} (\frac{3x}{ \color{blue}{12} }- \frac{ 10 }{ \color{blue}{12} })& = & (\frac{-8}{ \color{blue}{12} }x+\frac{36}{ \color{blue}{12} }) \color{blue}{.12} \\\Leftrightarrow & 3x-10& = & -8x+36 \\\Leftrightarrow & 3x \color{red}{-10} \color{blue}{+10} \color{blue}{+8x} & = & \color{red}{-8x} +36 \color{blue}{+8x} \color{blue}{+10} \\\Leftrightarrow & 11x& = & 46 \\\Leftrightarrow & \frac{11x}{ \color{red}{11} }& = & \frac{46}{11} \\\Leftrightarrow & x = \frac{46}{11} & & \\ & V = \left\{ \frac{46}{11} \right\} & \\\end{align}\)
9. \(\text{70 is het kleinste gemene veelvoud van 2, 14 en 5} \\ \begin{align} & \frac{x}{2}-\frac{5}{14}& = & \frac{1}{5}x-7 \\\Leftrightarrow & \color{blue}{70.} (\frac{35x}{ \color{blue}{70} }- \frac{ 25 }{ \color{blue}{70} })& = & (\frac{14}{ \color{blue}{70} }x-\frac{490}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 35x-25& = & 14x-490 \\\Leftrightarrow & 35x \color{red}{-25} \color{blue}{+25} \color{blue}{-14x} & = & \color{red}{14x} -490 \color{blue}{-14x} \color{blue}{+25} \\\Leftrightarrow & 21x& = & -465 \\\Leftrightarrow & \frac{21x}{ \color{red}{21} }& = & \frac{-465}{21} \\\Leftrightarrow & x = \frac{-155}{7} & & \\ & V = \left\{ \frac{-155}{7} \right\} & \\\end{align}\)
10. \(\text{132 is het kleinste gemene veelvoud van 4, 11 en 3} \\ \begin{align} & \frac{x}{4}+\frac{5}{11}& = & \frac{-2}{3}x-1 \\\Leftrightarrow & \color{blue}{132.} (\frac{33x}{ \color{blue}{132} }+ \frac{ 60 }{ \color{blue}{132} })& = & (\frac{-88}{ \color{blue}{132} }x-\frac{132}{ \color{blue}{132} }) \color{blue}{.132} \\\Leftrightarrow & 33x+60& = & -88x-132 \\\Leftrightarrow & 33x \color{red}{+60} \color{blue}{-60} \color{blue}{+88x} & = & \color{red}{-88x} -132 \color{blue}{+88x} \color{blue}{-60} \\\Leftrightarrow & 121x& = & -192 \\\Leftrightarrow & \frac{121x}{ \color{red}{121} }& = & \frac{-192}{121} \\\Leftrightarrow & x = \frac{-192}{121} & & \\ & V = \left\{ \frac{-192}{121} \right\} & \\\end{align}\)
11. \(\text{330 is het kleinste gemene veelvoud van 6, 11 en 5} \\ \begin{align} & \frac{x}{6}-\frac{5}{11}& = & \frac{1}{5}x-8 \\\Leftrightarrow & \color{blue}{330.} (\frac{55x}{ \color{blue}{330} }- \frac{ 150 }{ \color{blue}{330} })& = & (\frac{66}{ \color{blue}{330} }x-\frac{2640}{ \color{blue}{330} }) \color{blue}{.330} \\\Leftrightarrow & 55x-150& = & 66x-2640 \\\Leftrightarrow & 55x \color{red}{-150} \color{blue}{+150} \color{blue}{-66x} & = & \color{red}{66x} -2640 \color{blue}{-66x} \color{blue}{+150} \\\Leftrightarrow & -11x& = & -2490 \\\Leftrightarrow & \frac{-11x}{ \color{red}{-11} }& = & \frac{-2490}{-11} \\\Leftrightarrow & x = \frac{2490}{11} & & \\ & V = \left\{ \frac{2490}{11} \right\} & \\\end{align}\)
12. \(\text{14 is het kleinste gemene veelvoud van 7, 7 en 2} \\ \begin{align} & \frac{x}{7}-\frac{3}{7}& = & \frac{1}{2}x+4 \\\Leftrightarrow & \color{blue}{14.} (\frac{2x}{ \color{blue}{14} }- \frac{ 6 }{ \color{blue}{14} })& = & (\frac{7}{ \color{blue}{14} }x+\frac{56}{ \color{blue}{14} }) \color{blue}{.14} \\\Leftrightarrow & 2x-6& = & 7x+56 \\\Leftrightarrow & 2x \color{red}{-6} \color{blue}{+6} \color{blue}{-7x} & = & \color{red}{7x} +56 \color{blue}{-7x} \color{blue}{+6} \\\Leftrightarrow & -5x& = & 62 \\\Leftrightarrow & \frac{-5x}{ \color{red}{-5} }& = & \frac{62}{-5} \\\Leftrightarrow & x = \frac{-62}{5} & & \\ & V = \left\{ \frac{-62}{5} \right\} & \\\end{align}\)