Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{140 is het kleinste gemene veelvoud van 4, 7 en 5} \\ \begin{align} & \frac{x}{4}+\frac{2}{7}& = & \frac{6}{5}x-2 \\\Leftrightarrow & \color{blue}{140.} (\frac{35x}{ \color{blue}{140} }+ \frac{ 40 }{ \color{blue}{140} })& = & (\frac{168}{ \color{blue}{140} }x-\frac{280}{ \color{blue}{140} }) \color{blue}{.140} \\\Leftrightarrow & 35x+40& = & 168x-280 \\\Leftrightarrow & 35x \color{red}{+40} \color{blue}{-40} \color{blue}{-168x} & = & \color{red}{168x} -280 \color{blue}{-168x} \color{blue}{-40} \\\Leftrightarrow & -133x& = & -320 \\\Leftrightarrow & \frac{-133x}{ \color{red}{-133} }& = & \frac{-320}{-133} \\\Leftrightarrow & x = \frac{320}{133} & & \\ & V = \left\{ \frac{320}{133} \right\} & \\\end{align}\)
2. \(\text{390 is het kleinste gemene veelvoud van 6, 13 en 5} \\ \begin{align} & \frac{x}{6}-\frac{4}{13}& = & \frac{-7}{5}x+6 \\\Leftrightarrow & \color{blue}{390.} (\frac{65x}{ \color{blue}{390} }- \frac{ 120 }{ \color{blue}{390} })& = & (\frac{-546}{ \color{blue}{390} }x+\frac{2340}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 65x-120& = & -546x+2340 \\\Leftrightarrow & 65x \color{red}{-120} \color{blue}{+120} \color{blue}{+546x} & = & \color{red}{-546x} +2340 \color{blue}{+546x} \color{blue}{+120} \\\Leftrightarrow & 611x& = & 2460 \\\Leftrightarrow & \frac{611x}{ \color{red}{611} }& = & \frac{2460}{611} \\\Leftrightarrow & x = \frac{2460}{611} & & \\ & V = \left\{ \frac{2460}{611} \right\} & \\\end{align}\)
3. \(\text{70 is het kleinste gemene veelvoud van 5, 14 en 2} \\ \begin{align} & \frac{x}{5}-\frac{3}{14}& = & \frac{1}{2}x+4 \\\Leftrightarrow & \color{blue}{70.} (\frac{14x}{ \color{blue}{70} }- \frac{ 15 }{ \color{blue}{70} })& = & (\frac{35}{ \color{blue}{70} }x+\frac{280}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 14x-15& = & 35x+280 \\\Leftrightarrow & 14x \color{red}{-15} \color{blue}{+15} \color{blue}{-35x} & = & \color{red}{35x} +280 \color{blue}{-35x} \color{blue}{+15} \\\Leftrightarrow & -21x& = & 295 \\\Leftrightarrow & \frac{-21x}{ \color{red}{-21} }& = & \frac{295}{-21} \\\Leftrightarrow & x = \frac{-295}{21} & & \\ & V = \left\{ \frac{-295}{21} \right\} & \\\end{align}\)
4. \(\text{30 is het kleinste gemene veelvoud van 3, 15 en 2} \\ \begin{align} & \frac{x}{3}-\frac{4}{15}& = & \frac{1}{2}x-2 \\\Leftrightarrow & \color{blue}{30.} (\frac{10x}{ \color{blue}{30} }- \frac{ 8 }{ \color{blue}{30} })& = & (\frac{15}{ \color{blue}{30} }x-\frac{60}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 10x-8& = & 15x-60 \\\Leftrightarrow & 10x \color{red}{-8} \color{blue}{+8} \color{blue}{-15x} & = & \color{red}{15x} -60 \color{blue}{-15x} \color{blue}{+8} \\\Leftrightarrow & -5x& = & -52 \\\Leftrightarrow & \frac{-5x}{ \color{red}{-5} }& = & \frac{-52}{-5} \\\Leftrightarrow & x = \frac{52}{5} & & \\ & V = \left\{ \frac{52}{5} \right\} & \\\end{align}\)
5. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}-\frac{2}{7}& = & \frac{1}{5}x-2 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }- \frac{ 60 }{ \color{blue}{210} })& = & (\frac{42}{ \color{blue}{210} }x-\frac{420}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x-60& = & 42x-420 \\\Leftrightarrow & 35x \color{red}{-60} \color{blue}{+60} \color{blue}{-42x} & = & \color{red}{42x} -420 \color{blue}{-42x} \color{blue}{+60} \\\Leftrightarrow & -7x& = & -360 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{-360}{-7} \\\Leftrightarrow & x = \frac{360}{7} & & \\ & V = \left\{ \frac{360}{7} \right\} & \\\end{align}\)
6. \(\text{84 is het kleinste gemene veelvoud van 3, 7 en 4} \\ \begin{align} & \frac{x}{3}+\frac{5}{7}& = & \frac{1}{4}x+2 \\\Leftrightarrow & \color{blue}{84.} (\frac{28x}{ \color{blue}{84} }+ \frac{ 60 }{ \color{blue}{84} })& = & (\frac{21}{ \color{blue}{84} }x+\frac{168}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 28x+60& = & 21x+168 \\\Leftrightarrow & 28x \color{red}{+60} \color{blue}{-60} \color{blue}{-21x} & = & \color{red}{21x} +168 \color{blue}{-21x} \color{blue}{-60} \\\Leftrightarrow & 7x& = & 108 \\\Leftrightarrow & \frac{7x}{ \color{red}{7} }& = & \frac{108}{7} \\\Leftrightarrow & x = \frac{108}{7} & & \\ & V = \left\{ \frac{108}{7} \right\} & \\\end{align}\)
7. \(\text{84 is het kleinste gemene veelvoud van 3, 14 en 4} \\ \begin{align} & \frac{x}{3}-\frac{3}{14}& = & \frac{-7}{4}x+2 \\\Leftrightarrow & \color{blue}{84.} (\frac{28x}{ \color{blue}{84} }- \frac{ 18 }{ \color{blue}{84} })& = & (\frac{-147}{ \color{blue}{84} }x+\frac{168}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 28x-18& = & -147x+168 \\\Leftrightarrow & 28x \color{red}{-18} \color{blue}{+18} \color{blue}{+147x} & = & \color{red}{-147x} +168 \color{blue}{+147x} \color{blue}{+18} \\\Leftrightarrow & 175x& = & 186 \\\Leftrightarrow & \frac{175x}{ \color{red}{175} }& = & \frac{186}{175} \\\Leftrightarrow & x = \frac{186}{175} & & \\ & V = \left\{ \frac{186}{175} \right\} & \\\end{align}\)
8. \(\text{390 is het kleinste gemene veelvoud van 5, 13 en 6} \\ \begin{align} & \frac{x}{5}-\frac{5}{13}& = & \frac{-5}{6}x-4 \\\Leftrightarrow & \color{blue}{390.} (\frac{78x}{ \color{blue}{390} }- \frac{ 150 }{ \color{blue}{390} })& = & (\frac{-325}{ \color{blue}{390} }x-\frac{1560}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 78x-150& = & -325x-1560 \\\Leftrightarrow & 78x \color{red}{-150} \color{blue}{+150} \color{blue}{+325x} & = & \color{red}{-325x} -1560 \color{blue}{+325x} \color{blue}{+150} \\\Leftrightarrow & 403x& = & -1410 \\\Leftrightarrow & \frac{403x}{ \color{red}{403} }& = & \frac{-1410}{403} \\\Leftrightarrow & x = \frac{-1410}{403} & & \\ & V = \left\{ \frac{-1410}{403} \right\} & \\\end{align}\)
9. \(\text{48 is het kleinste gemene veelvoud van 2, 16 en 3} \\ \begin{align} & \frac{x}{2}+\frac{5}{16}& = & \frac{5}{3}x-5 \\\Leftrightarrow & \color{blue}{48.} (\frac{24x}{ \color{blue}{48} }+ \frac{ 15 }{ \color{blue}{48} })& = & (\frac{80}{ \color{blue}{48} }x-\frac{240}{ \color{blue}{48} }) \color{blue}{.48} \\\Leftrightarrow & 24x+15& = & 80x-240 \\\Leftrightarrow & 24x \color{red}{+15} \color{blue}{-15} \color{blue}{-80x} & = & \color{red}{80x} -240 \color{blue}{-80x} \color{blue}{-15} \\\Leftrightarrow & -56x& = & -255 \\\Leftrightarrow & \frac{-56x}{ \color{red}{-56} }& = & \frac{-255}{-56} \\\Leftrightarrow & x = \frac{255}{56} & & \\ & V = \left\{ \frac{255}{56} \right\} & \\\end{align}\)
10. \(\text{6 is het kleinste gemene veelvoud van 2, 6 en 3} \\ \begin{align} & \frac{x}{2}-\frac{5}{6}& = & \frac{1}{3}x-7 \\\Leftrightarrow & \color{blue}{6.} (\frac{3x}{ \color{blue}{6} }- \frac{ 5 }{ \color{blue}{6} })& = & (\frac{2}{ \color{blue}{6} }x-\frac{42}{ \color{blue}{6} }) \color{blue}{.6} \\\Leftrightarrow & 3x-5& = & 2x-42 \\\Leftrightarrow & 3x \color{red}{-5} \color{blue}{+5} \color{blue}{-2x} & = & \color{red}{2x} -42 \color{blue}{-2x} \color{blue}{+5} \\\Leftrightarrow & x& = & -37 \\ & V = \left\{ -37 \right\} & \\\end{align}\)
11. \(\text{462 is het kleinste gemene veelvoud van 7, 11 en 6} \\ \begin{align} & \frac{x}{7}-\frac{2}{11}& = & \frac{5}{6}x+8 \\\Leftrightarrow & \color{blue}{462.} (\frac{66x}{ \color{blue}{462} }- \frac{ 84 }{ \color{blue}{462} })& = & (\frac{385}{ \color{blue}{462} }x+\frac{3696}{ \color{blue}{462} }) \color{blue}{.462} \\\Leftrightarrow & 66x-84& = & 385x+3696 \\\Leftrightarrow & 66x \color{red}{-84} \color{blue}{+84} \color{blue}{-385x} & = & \color{red}{385x} +3696 \color{blue}{-385x} \color{blue}{+84} \\\Leftrightarrow & -319x& = & 3780 \\\Leftrightarrow & \frac{-319x}{ \color{red}{-319} }& = & \frac{3780}{-319} \\\Leftrightarrow & x = \frac{-3780}{319} & & \\ & V = \left\{ \frac{-3780}{319} \right\} & \\\end{align}\)
12. \(\text{66 is het kleinste gemene veelvoud van 2, 11 en 3} \\ \begin{align} & \frac{x}{2}-\frac{2}{11}& = & \frac{4}{3}x+4 \\\Leftrightarrow & \color{blue}{66.} (\frac{33x}{ \color{blue}{66} }- \frac{ 12 }{ \color{blue}{66} })& = & (\frac{88}{ \color{blue}{66} }x+\frac{264}{ \color{blue}{66} }) \color{blue}{.66} \\\Leftrightarrow & 33x-12& = & 88x+264 \\\Leftrightarrow & 33x \color{red}{-12} \color{blue}{+12} \color{blue}{-88x} & = & \color{red}{88x} +264 \color{blue}{-88x} \color{blue}{+12} \\\Leftrightarrow & -55x& = & 276 \\\Leftrightarrow & \frac{-55x}{ \color{red}{-55} }& = & \frac{276}{-55} \\\Leftrightarrow & x = \frac{-276}{55} & & \\ & V = \left\{ \frac{-276}{55} \right\} & \\\end{align}\)