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

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

Verbetersleutel

1. \(\text{14 is het kleinste gemene veelvoud van 7, 7 en 2} \\ \begin{align} & \frac{x}{7}-\frac{3}{7}& = & \frac{5}{2}x-6 \\\Leftrightarrow & \color{blue}{14.} (\frac{2x}{ \color{blue}{14} }- \frac{ 6 }{ \color{blue}{14} })& = & (\frac{35}{ \color{blue}{14} }x-\frac{84}{ \color{blue}{14} }) \color{blue}{.14} \\\Leftrightarrow & 2x-6& = & 35x-84 \\\Leftrightarrow & 2x \color{red}{-6} \color{blue}{+6} \color{blue}{-35x} & = & \color{red}{35x} -84 \color{blue}{-35x} \color{blue}{+6} \\\Leftrightarrow & -33x& = & -78 \\\Leftrightarrow & \frac{-33x}{ \color{red}{-33} }& = & \frac{-78}{-33} \\\Leftrightarrow & x = \frac{26}{11} & & \\ & V = \left\{ \frac{26}{11} \right\} & \\\end{align}\)
2. \(\text{90 is het kleinste gemene veelvoud van 5, 9 en 2} \\ \begin{align} & \frac{x}{5}-\frac{2}{9}& = & \frac{1}{2}x+6 \\\Leftrightarrow & \color{blue}{90.} (\frac{18x}{ \color{blue}{90} }- \frac{ 20 }{ \color{blue}{90} })& = & (\frac{45}{ \color{blue}{90} }x+\frac{540}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 18x-20& = & 45x+540 \\\Leftrightarrow & 18x \color{red}{-20} \color{blue}{+20} \color{blue}{-45x} & = & \color{red}{45x} +540 \color{blue}{-45x} \color{blue}{+20} \\\Leftrightarrow & -27x& = & 560 \\\Leftrightarrow & \frac{-27x}{ \color{red}{-27} }& = & \frac{560}{-27} \\\Leftrightarrow & x = \frac{-560}{27} & & \\ & V = \left\{ \frac{-560}{27} \right\} & \\\end{align}\)
3. \(\text{105 is het kleinste gemene veelvoud van 3, 7 en 5} \\ \begin{align} & \frac{x}{3}-\frac{2}{7}& = & \frac{-4}{5}x-1 \\\Leftrightarrow & \color{blue}{105.} (\frac{35x}{ \color{blue}{105} }- \frac{ 30 }{ \color{blue}{105} })& = & (\frac{-84}{ \color{blue}{105} }x-\frac{105}{ \color{blue}{105} }) \color{blue}{.105} \\\Leftrightarrow & 35x-30& = & -84x-105 \\\Leftrightarrow & 35x \color{red}{-30} \color{blue}{+30} \color{blue}{+84x} & = & \color{red}{-84x} -105 \color{blue}{+84x} \color{blue}{+30} \\\Leftrightarrow & 119x& = & -75 \\\Leftrightarrow & \frac{119x}{ \color{red}{119} }& = & \frac{-75}{119} \\\Leftrightarrow & x = \frac{-75}{119} & & \\ & V = \left\{ \frac{-75}{119} \right\} & \\\end{align}\)
4. \(\text{36 is het kleinste gemene veelvoud van 4, 9 en 3} \\ \begin{align} & \frac{x}{4}-\frac{4}{9}& = & \frac{7}{3}x-5 \\\Leftrightarrow & \color{blue}{36.} (\frac{9x}{ \color{blue}{36} }- \frac{ 16 }{ \color{blue}{36} })& = & (\frac{84}{ \color{blue}{36} }x-\frac{180}{ \color{blue}{36} }) \color{blue}{.36} \\\Leftrightarrow & 9x-16& = & 84x-180 \\\Leftrightarrow & 9x \color{red}{-16} \color{blue}{+16} \color{blue}{-84x} & = & \color{red}{84x} -180 \color{blue}{-84x} \color{blue}{+16} \\\Leftrightarrow & -75x& = & -164 \\\Leftrightarrow & \frac{-75x}{ \color{red}{-75} }& = & \frac{-164}{-75} \\\Leftrightarrow & x = \frac{164}{75} & & \\ & V = \left\{ \frac{164}{75} \right\} & \\\end{align}\)
5. \(\text{330 is het kleinste gemene veelvoud van 6, 11 en 5} \\ \begin{align} & \frac{x}{6}-\frac{2}{11}& = & \frac{-4}{5}x-7 \\\Leftrightarrow & \color{blue}{330.} (\frac{55x}{ \color{blue}{330} }- \frac{ 60 }{ \color{blue}{330} })& = & (\frac{-264}{ \color{blue}{330} }x-\frac{2310}{ \color{blue}{330} }) \color{blue}{.330} \\\Leftrightarrow & 55x-60& = & -264x-2310 \\\Leftrightarrow & 55x \color{red}{-60} \color{blue}{+60} \color{blue}{+264x} & = & \color{red}{-264x} -2310 \color{blue}{+264x} \color{blue}{+60} \\\Leftrightarrow & 319x& = & -2250 \\\Leftrightarrow & \frac{319x}{ \color{red}{319} }& = & \frac{-2250}{319} \\\Leftrightarrow & x = \frac{-2250}{319} & & \\ & V = \left\{ \frac{-2250}{319} \right\} & \\\end{align}\)
6. \(\text{120 is het kleinste gemene veelvoud van 5, 8 en 6} \\ \begin{align} & \frac{x}{5}+\frac{5}{8}& = & \frac{1}{6}x-7 \\\Leftrightarrow & \color{blue}{120.} (\frac{24x}{ \color{blue}{120} }+ \frac{ 75 }{ \color{blue}{120} })& = & (\frac{20}{ \color{blue}{120} }x-\frac{840}{ \color{blue}{120} }) \color{blue}{.120} \\\Leftrightarrow & 24x+75& = & 20x-840 \\\Leftrightarrow & 24x \color{red}{+75} \color{blue}{-75} \color{blue}{-20x} & = & \color{red}{20x} -840 \color{blue}{-20x} \color{blue}{-75} \\\Leftrightarrow & 4x& = & -915 \\\Leftrightarrow & \frac{4x}{ \color{red}{4} }& = & \frac{-915}{4} \\\Leftrightarrow & x = \frac{-915}{4} & & \\ & V = \left\{ \frac{-915}{4} \right\} & \\\end{align}\)
7. \(\text{240 is het kleinste gemene veelvoud van 6, 16 en 5} \\ \begin{align} & \frac{x}{6}-\frac{5}{16}& = & \frac{3}{5}x-4 \\\Leftrightarrow & \color{blue}{240.} (\frac{40x}{ \color{blue}{240} }- \frac{ 75 }{ \color{blue}{240} })& = & (\frac{144}{ \color{blue}{240} }x-\frac{960}{ \color{blue}{240} }) \color{blue}{.240} \\\Leftrightarrow & 40x-75& = & 144x-960 \\\Leftrightarrow & 40x \color{red}{-75} \color{blue}{+75} \color{blue}{-144x} & = & \color{red}{144x} -960 \color{blue}{-144x} \color{blue}{+75} \\\Leftrightarrow & -104x& = & -885 \\\Leftrightarrow & \frac{-104x}{ \color{red}{-104} }& = & \frac{-885}{-104} \\\Leftrightarrow & x = \frac{885}{104} & & \\ & V = \left\{ \frac{885}{104} \right\} & \\\end{align}\)
8. \(\text{560 is het kleinste gemene veelvoud van 7, 16 en 5} \\ \begin{align} & \frac{x}{7}-\frac{3}{16}& = & \frac{2}{5}x+1 \\\Leftrightarrow & \color{blue}{560.} (\frac{80x}{ \color{blue}{560} }- \frac{ 105 }{ \color{blue}{560} })& = & (\frac{224}{ \color{blue}{560} }x+\frac{560}{ \color{blue}{560} }) \color{blue}{.560} \\\Leftrightarrow & 80x-105& = & 224x+560 \\\Leftrightarrow & 80x \color{red}{-105} \color{blue}{+105} \color{blue}{-224x} & = & \color{red}{224x} +560 \color{blue}{-224x} \color{blue}{+105} \\\Leftrightarrow & -144x& = & 665 \\\Leftrightarrow & \frac{-144x}{ \color{red}{-144} }& = & \frac{665}{-144} \\\Leftrightarrow & x = \frac{-665}{144} & & \\ & V = \left\{ \frac{-665}{144} \right\} & \\\end{align}\)
9. \(\text{126 is het kleinste gemene veelvoud van 7, 9 en 6} \\ \begin{align} & \frac{x}{7}-\frac{2}{9}& = & \frac{-5}{6}x+8 \\\Leftrightarrow & \color{blue}{126.} (\frac{18x}{ \color{blue}{126} }- \frac{ 28 }{ \color{blue}{126} })& = & (\frac{-105}{ \color{blue}{126} }x+\frac{1008}{ \color{blue}{126} }) \color{blue}{.126} \\\Leftrightarrow & 18x-28& = & -105x+1008 \\\Leftrightarrow & 18x \color{red}{-28} \color{blue}{+28} \color{blue}{+105x} & = & \color{red}{-105x} +1008 \color{blue}{+105x} \color{blue}{+28} \\\Leftrightarrow & 123x& = & 1036 \\\Leftrightarrow & \frac{123x}{ \color{red}{123} }& = & \frac{1036}{123} \\\Leftrightarrow & x = \frac{1036}{123} & & \\ & V = \left\{ \frac{1036}{123} \right\} & \\\end{align}\)
10. \(\text{70 is het kleinste gemene veelvoud van 5, 7 en 2} \\ \begin{align} & \frac{x}{5}-\frac{4}{7}& = & \frac{1}{2}x-7 \\\Leftrightarrow & \color{blue}{70.} (\frac{14x}{ \color{blue}{70} }- \frac{ 40 }{ \color{blue}{70} })& = & (\frac{35}{ \color{blue}{70} }x-\frac{490}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 14x-40& = & 35x-490 \\\Leftrightarrow & 14x \color{red}{-40} \color{blue}{+40} \color{blue}{-35x} & = & \color{red}{35x} -490 \color{blue}{-35x} \color{blue}{+40} \\\Leftrightarrow & -21x& = & -450 \\\Leftrightarrow & \frac{-21x}{ \color{red}{-21} }& = & \frac{-450}{-21} \\\Leftrightarrow & x = \frac{150}{7} & & \\ & V = \left\{ \frac{150}{7} \right\} & \\\end{align}\)
11. \(\text{66 is het kleinste gemene veelvoud van 3, 11 en 2} \\ \begin{align} & \frac{x}{3}-\frac{3}{11}& = & \frac{1}{2}x+2 \\\Leftrightarrow & \color{blue}{66.} (\frac{22x}{ \color{blue}{66} }- \frac{ 18 }{ \color{blue}{66} })& = & (\frac{33}{ \color{blue}{66} }x+\frac{132}{ \color{blue}{66} }) \color{blue}{.66} \\\Leftrightarrow & 22x-18& = & 33x+132 \\\Leftrightarrow & 22x \color{red}{-18} \color{blue}{+18} \color{blue}{-33x} & = & \color{red}{33x} +132 \color{blue}{-33x} \color{blue}{+18} \\\Leftrightarrow & -11x& = & 150 \\\Leftrightarrow & \frac{-11x}{ \color{red}{-11} }& = & \frac{150}{-11} \\\Leftrightarrow & x = \frac{-150}{11} & & \\ & V = \left\{ \frac{-150}{11} \right\} & \\\end{align}\)
12. \(\text{156 is het kleinste gemene veelvoud van 3, 13 en 4} \\ \begin{align} & \frac{x}{3}-\frac{2}{13}& = & \frac{7}{4}x+1 \\\Leftrightarrow & \color{blue}{156.} (\frac{52x}{ \color{blue}{156} }- \frac{ 24 }{ \color{blue}{156} })& = & (\frac{273}{ \color{blue}{156} }x+\frac{156}{ \color{blue}{156} }) \color{blue}{.156} \\\Leftrightarrow & 52x-24& = & 273x+156 \\\Leftrightarrow & 52x \color{red}{-24} \color{blue}{+24} \color{blue}{-273x} & = & \color{red}{273x} +156 \color{blue}{-273x} \color{blue}{+24} \\\Leftrightarrow & -221x& = & 180 \\\Leftrightarrow & \frac{-221x}{ \color{red}{-221} }& = & \frac{180}{-221} \\\Leftrightarrow & x = \frac{-180}{221} & & \\ & V = \left\{ \frac{-180}{221} \right\} & \\\end{align}\)