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. \(\frac{x}{5}-\frac{4}{9}=\frac{8}{3}x-5\)
2. \(\frac{x}{4}-\frac{5}{8}=\frac{-2}{5}x+2\)
3. \(\frac{x}{2}+\frac{3}{10}=\frac{1}{7}x+2\)
4. \(\frac{x}{2}+\frac{4}{13}=\frac{3}{7}x+3\)
5. \(\frac{x}{7}+\frac{5}{14}=\frac{-7}{4}x-2\)
6. \(\frac{x}{2}-\frac{2}{11}=\frac{5}{3}x-7\)
7. \(\frac{x}{7}-\frac{4}{11}=\frac{7}{6}x+4\)
8. \(\frac{x}{7}+\frac{5}{8}=\frac{7}{6}x-7\)
9. \(\frac{x}{2}-\frac{4}{17}=\frac{6}{7}x+6\)
10. \(\frac{x}{6}+\frac{2}{11}=\frac{-4}{5}x-2\)
11. \(\frac{x}{5}+\frac{2}{17}=\frac{5}{7}x+8\)
12. \(\frac{x}{4}+\frac{5}{9}=\frac{4}{3}x+2\)

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

Verbetersleutel

1. \(\text{45 is het kleinste gemene veelvoud van 5, 9 en 3} \\ \begin{align} & \color{blue}{45.} (\frac{x}{5}-\frac{4}{9})& = & (\frac{8}{3}x-5) \color{blue}{.45} \\\Leftrightarrow & 9x-20& = & 120x-225 \\\Leftrightarrow & 9x \color{red}{-20} \color{blue}{+20} \color{blue}{-120x} & = & \color{red}{120x} -225 \color{blue}{-120x} \color{blue}{+20} \\\Leftrightarrow & -111x& = & -205 \\\Leftrightarrow & \frac{-111x}{ \color{red}{-111} }& = & \frac{-205}{-111} \\\Leftrightarrow & x = \frac{205}{111} & & \\ & V = \left\{ \frac{205}{111} \right\} & \\\end{align}\)
2. \(\text{40 is het kleinste gemene veelvoud van 4, 8 en 5} \\ \begin{align} & \color{blue}{40.} (\frac{x}{4}-\frac{5}{8})& = & (\frac{-2}{5}x+2) \color{blue}{.40} \\\Leftrightarrow & 10x-25& = & -16x+80 \\\Leftrightarrow & 10x \color{red}{-25} \color{blue}{+25} \color{blue}{+16x} & = & \color{red}{-16x} +80 \color{blue}{+16x} \color{blue}{+25} \\\Leftrightarrow & 26x& = & 105 \\\Leftrightarrow & \frac{26x}{ \color{red}{26} }& = & \frac{105}{26} \\\Leftrightarrow & x = \frac{105}{26} & & \\ & V = \left\{ \frac{105}{26} \right\} & \\\end{align}\)
3. \(\text{70 is het kleinste gemene veelvoud van 2, 10 en 7} \\ \begin{align} & \color{blue}{70.} (\frac{x}{2}+\frac{3}{10})& = & (\frac{1}{7}x+2) \color{blue}{.70} \\\Leftrightarrow & 35x+21& = & 10x+140 \\\Leftrightarrow & 35x \color{red}{+21} \color{blue}{-21} \color{blue}{-10x} & = & \color{red}{10x} +140 \color{blue}{-10x} \color{blue}{-21} \\\Leftrightarrow & 25x& = & 119 \\\Leftrightarrow & \frac{25x}{ \color{red}{25} }& = & \frac{119}{25} \\\Leftrightarrow & x = \frac{119}{25} & & \\ & V = \left\{ \frac{119}{25} \right\} & \\\end{align}\)
4. \(\text{182 is het kleinste gemene veelvoud van 2, 13 en 7} \\ \begin{align} & \color{blue}{182.} (\frac{x}{2}+\frac{4}{13})& = & (\frac{3}{7}x+3) \color{blue}{.182} \\\Leftrightarrow & 91x+56& = & 78x+546 \\\Leftrightarrow & 91x \color{red}{+56} \color{blue}{-56} \color{blue}{-78x} & = & \color{red}{78x} +546 \color{blue}{-78x} \color{blue}{-56} \\\Leftrightarrow & 13x& = & 490 \\\Leftrightarrow & \frac{13x}{ \color{red}{13} }& = & \frac{490}{13} \\\Leftrightarrow & x = \frac{490}{13} & & \\ & V = \left\{ \frac{490}{13} \right\} & \\\end{align}\)
5. \(\text{28 is het kleinste gemene veelvoud van 7, 14 en 4} \\ \begin{align} & \color{blue}{28.} (\frac{x}{7}+\frac{5}{14})& = & (\frac{-7}{4}x-2) \color{blue}{.28} \\\Leftrightarrow & 4x+10& = & -49x-56 \\\Leftrightarrow & 4x \color{red}{+10} \color{blue}{-10} \color{blue}{+49x} & = & \color{red}{-49x} -56 \color{blue}{+49x} \color{blue}{-10} \\\Leftrightarrow & 53x& = & -66 \\\Leftrightarrow & \frac{53x}{ \color{red}{53} }& = & \frac{-66}{53} \\\Leftrightarrow & x = \frac{-66}{53} & & \\ & V = \left\{ \frac{-66}{53} \right\} & \\\end{align}\)
6. \(\text{66 is het kleinste gemene veelvoud van 2, 11 en 3} \\ \begin{align} & \color{blue}{66.} (\frac{x}{2}-\frac{2}{11})& = & (\frac{5}{3}x-7) \color{blue}{.66} \\\Leftrightarrow & 33x-12& = & 110x-462 \\\Leftrightarrow & 33x \color{red}{-12} \color{blue}{+12} \color{blue}{-110x} & = & \color{red}{110x} -462 \color{blue}{-110x} \color{blue}{+12} \\\Leftrightarrow & -77x& = & -450 \\\Leftrightarrow & \frac{-77x}{ \color{red}{-77} }& = & \frac{-450}{-77} \\\Leftrightarrow & x = \frac{450}{77} & & \\ & V = \left\{ \frac{450}{77} \right\} & \\\end{align}\)
7. \(\text{462 is het kleinste gemene veelvoud van 7, 11 en 6} \\ \begin{align} & \color{blue}{462.} (\frac{x}{7}-\frac{4}{11})& = & (\frac{7}{6}x+4) \color{blue}{.462} \\\Leftrightarrow & 66x-168& = & 539x+1848 \\\Leftrightarrow & 66x \color{red}{-168} \color{blue}{+168} \color{blue}{-539x} & = & \color{red}{539x} +1848 \color{blue}{-539x} \color{blue}{+168} \\\Leftrightarrow & -473x& = & 2016 \\\Leftrightarrow & \frac{-473x}{ \color{red}{-473} }& = & \frac{2016}{-473} \\\Leftrightarrow & x = \frac{-2016}{473} & & \\ & V = \left\{ \frac{-2016}{473} \right\} & \\\end{align}\)
8. \(\text{168 is het kleinste gemene veelvoud van 7, 8 en 6} \\ \begin{align} & \color{blue}{168.} (\frac{x}{7}+\frac{5}{8})& = & (\frac{7}{6}x-7) \color{blue}{.168} \\\Leftrightarrow & 24x+105& = & 196x-1176 \\\Leftrightarrow & 24x \color{red}{+105} \color{blue}{-105} \color{blue}{-196x} & = & \color{red}{196x} -1176 \color{blue}{-196x} \color{blue}{-105} \\\Leftrightarrow & -172x& = & -1281 \\\Leftrightarrow & \frac{-172x}{ \color{red}{-172} }& = & \frac{-1281}{-172} \\\Leftrightarrow & x = \frac{1281}{172} & & \\ & V = \left\{ \frac{1281}{172} \right\} & \\\end{align}\)
9. \(\text{238 is het kleinste gemene veelvoud van 2, 17 en 7} \\ \begin{align} & \color{blue}{238.} (\frac{x}{2}-\frac{4}{17})& = & (\frac{6}{7}x+6) \color{blue}{.238} \\\Leftrightarrow & 119x-56& = & 204x+1428 \\\Leftrightarrow & 119x \color{red}{-56} \color{blue}{+56} \color{blue}{-204x} & = & \color{red}{204x} +1428 \color{blue}{-204x} \color{blue}{+56} \\\Leftrightarrow & -85x& = & 1484 \\\Leftrightarrow & \frac{-85x}{ \color{red}{-85} }& = & \frac{1484}{-85} \\\Leftrightarrow & x = \frac{-1484}{85} & & \\ & V = \left\{ \frac{-1484}{85} \right\} & \\\end{align}\)
10. \(\text{330 is het kleinste gemene veelvoud van 6, 11 en 5} \\ \begin{align} & \color{blue}{330.} (\frac{x}{6}+\frac{2}{11})& = & (\frac{-4}{5}x-2) \color{blue}{.330} \\\Leftrightarrow & 55x+60& = & -264x-660 \\\Leftrightarrow & 55x \color{red}{+60} \color{blue}{-60} \color{blue}{+264x} & = & \color{red}{-264x} -660 \color{blue}{+264x} \color{blue}{-60} \\\Leftrightarrow & 319x& = & -720 \\\Leftrightarrow & \frac{319x}{ \color{red}{319} }& = & \frac{-720}{319} \\\Leftrightarrow & x = \frac{-720}{319} & & \\ & V = \left\{ \frac{-720}{319} \right\} & \\\end{align}\)
11. \(\text{595 is het kleinste gemene veelvoud van 5, 17 en 7} \\ \begin{align} & \color{blue}{595.} (\frac{x}{5}+\frac{2}{17})& = & (\frac{5}{7}x+8) \color{blue}{.595} \\\Leftrightarrow & 119x+70& = & 425x+4760 \\\Leftrightarrow & 119x \color{red}{+70} \color{blue}{-70} \color{blue}{-425x} & = & \color{red}{425x} +4760 \color{blue}{-425x} \color{blue}{-70} \\\Leftrightarrow & -306x& = & 4690 \\\Leftrightarrow & \frac{-306x}{ \color{red}{-306} }& = & \frac{4690}{-306} \\\Leftrightarrow & x = \frac{-2345}{153} & & \\ & V = \left\{ \frac{-2345}{153} \right\} & \\\end{align}\)
12. \(\text{36 is het kleinste gemene veelvoud van 4, 9 en 3} \\ \begin{align} & \color{blue}{36.} (\frac{x}{4}+\frac{5}{9})& = & (\frac{4}{3}x+2) \color{blue}{.36} \\\Leftrightarrow & 9x+20& = & 48x+72 \\\Leftrightarrow & 9x \color{red}{+20} \color{blue}{-20} \color{blue}{-48x} & = & \color{red}{48x} +72 \color{blue}{-48x} \color{blue}{-20} \\\Leftrightarrow & -39x& = & 52 \\\Leftrightarrow & \frac{-39x}{ \color{red}{-39} }& = & \frac{52}{-39} \\\Leftrightarrow & x = \frac{-4}{3} & & \\ & V = \left\{ \frac{-4}{3} \right\} & \\\end{align}\)