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. \(\frac{x}{3}-\frac{4}{17}=\frac{-7}{4}x-3\)
2. \(\frac{x}{4}-\frac{2}{7}=\frac{5}{3}x-5\)
3. \(\frac{x}{2}-\frac{2}{11}=\frac{-6}{7}x+8\)
4. \(\frac{x}{5}-\frac{5}{13}=\frac{3}{2}x+3\)
5. \(\frac{x}{7}+\frac{5}{16}=\frac{3}{2}x+2\)
6. \(\frac{x}{5}-\frac{5}{12}=\frac{5}{4}x+3\)
7. \(\frac{x}{7}-\frac{3}{16}=\frac{7}{6}x-4\)
8. \(\frac{x}{5}-\frac{3}{16}=\frac{7}{6}x+4\)
9. \(\frac{x}{7}+\frac{2}{7}=\frac{3}{4}x+5\)
10. \(\frac{x}{3}-\frac{5}{8}=\frac{7}{4}x-6\)
11. \(\frac{x}{5}-\frac{5}{14}=\frac{3}{7}x-1\)
12. \(\frac{x}{7}-\frac{4}{11}=\frac{-7}{4}x-3\)

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

Verbetersleutel

1. \(\text{204 is het kleinste gemene veelvoud van 3, 17 en 4} \\ \begin{align} & \color{blue}{204.} (\frac{x}{3}-\frac{4}{17})& = & (\frac{-7}{4}x-3) \color{blue}{.204} \\\Leftrightarrow & 68x-48& = & -357x-612 \\\Leftrightarrow & 68x \color{red}{-48} \color{blue}{+48} \color{blue}{+357x} & = & \color{red}{-357x} -612 \color{blue}{+357x} \color{blue}{+48} \\\Leftrightarrow & 425x& = & -564 \\\Leftrightarrow & \frac{425x}{ \color{red}{425} }& = & \frac{-564}{425} \\\Leftrightarrow & x = \frac{-564}{425} & & \\ & V = \left\{ \frac{-564}{425} \right\} & \\\end{align}\)
2. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \color{blue}{84.} (\frac{x}{4}-\frac{2}{7})& = & (\frac{5}{3}x-5) \color{blue}{.84} \\\Leftrightarrow & 21x-24& = & 140x-420 \\\Leftrightarrow & 21x \color{red}{-24} \color{blue}{+24} \color{blue}{-140x} & = & \color{red}{140x} -420 \color{blue}{-140x} \color{blue}{+24} \\\Leftrightarrow & -119x& = & -396 \\\Leftrightarrow & \frac{-119x}{ \color{red}{-119} }& = & \frac{-396}{-119} \\\Leftrightarrow & x = \frac{396}{119} & & \\ & V = \left\{ \frac{396}{119} \right\} & \\\end{align}\)
3. \(\text{154 is het kleinste gemene veelvoud van 2, 11 en 7} \\ \begin{align} & \color{blue}{154.} (\frac{x}{2}-\frac{2}{11})& = & (\frac{-6}{7}x+8) \color{blue}{.154} \\\Leftrightarrow & 77x-28& = & -132x+1232 \\\Leftrightarrow & 77x \color{red}{-28} \color{blue}{+28} \color{blue}{+132x} & = & \color{red}{-132x} +1232 \color{blue}{+132x} \color{blue}{+28} \\\Leftrightarrow & 209x& = & 1260 \\\Leftrightarrow & \frac{209x}{ \color{red}{209} }& = & \frac{1260}{209} \\\Leftrightarrow & x = \frac{1260}{209} & & \\ & V = \left\{ \frac{1260}{209} \right\} & \\\end{align}\)
4. \(\text{130 is het kleinste gemene veelvoud van 5, 13 en 2} \\ \begin{align} & \color{blue}{130.} (\frac{x}{5}-\frac{5}{13})& = & (\frac{3}{2}x+3) \color{blue}{.130} \\\Leftrightarrow & 26x-50& = & 195x+390 \\\Leftrightarrow & 26x \color{red}{-50} \color{blue}{+50} \color{blue}{-195x} & = & \color{red}{195x} +390 \color{blue}{-195x} \color{blue}{+50} \\\Leftrightarrow & -169x& = & 440 \\\Leftrightarrow & \frac{-169x}{ \color{red}{-169} }& = & \frac{440}{-169} \\\Leftrightarrow & x = \frac{-440}{169} & & \\ & V = \left\{ \frac{-440}{169} \right\} & \\\end{align}\)
5. \(\text{112 is het kleinste gemene veelvoud van 7, 16 en 2} \\ \begin{align} & \color{blue}{112.} (\frac{x}{7}+\frac{5}{16})& = & (\frac{3}{2}x+2) \color{blue}{.112} \\\Leftrightarrow & 16x+35& = & 168x+224 \\\Leftrightarrow & 16x \color{red}{+35} \color{blue}{-35} \color{blue}{-168x} & = & \color{red}{168x} +224 \color{blue}{-168x} \color{blue}{-35} \\\Leftrightarrow & -152x& = & 189 \\\Leftrightarrow & \frac{-152x}{ \color{red}{-152} }& = & \frac{189}{-152} \\\Leftrightarrow & x = \frac{-189}{152} & & \\ & V = \left\{ \frac{-189}{152} \right\} & \\\end{align}\)
6. \(\text{60 is het kleinste gemene veelvoud van 5, 12 en 4} \\ \begin{align} & \color{blue}{60.} (\frac{x}{5}-\frac{5}{12})& = & (\frac{5}{4}x+3) \color{blue}{.60} \\\Leftrightarrow & 12x-25& = & 75x+180 \\\Leftrightarrow & 12x \color{red}{-25} \color{blue}{+25} \color{blue}{-75x} & = & \color{red}{75x} +180 \color{blue}{-75x} \color{blue}{+25} \\\Leftrightarrow & -63x& = & 205 \\\Leftrightarrow & \frac{-63x}{ \color{red}{-63} }& = & \frac{205}{-63} \\\Leftrightarrow & x = \frac{-205}{63} & & \\ & V = \left\{ \frac{-205}{63} \right\} & \\\end{align}\)
7. \(\text{336 is het kleinste gemene veelvoud van 7, 16 en 6} \\ \begin{align} & \color{blue}{336.} (\frac{x}{7}-\frac{3}{16})& = & (\frac{7}{6}x-4) \color{blue}{.336} \\\Leftrightarrow & 48x-63& = & 392x-1344 \\\Leftrightarrow & 48x \color{red}{-63} \color{blue}{+63} \color{blue}{-392x} & = & \color{red}{392x} -1344 \color{blue}{-392x} \color{blue}{+63} \\\Leftrightarrow & -344x& = & -1281 \\\Leftrightarrow & \frac{-344x}{ \color{red}{-344} }& = & \frac{-1281}{-344} \\\Leftrightarrow & x = \frac{1281}{344} & & \\ & V = \left\{ \frac{1281}{344} \right\} & \\\end{align}\)
8. \(\text{240 is het kleinste gemene veelvoud van 5, 16 en 6} \\ \begin{align} & \color{blue}{240.} (\frac{x}{5}-\frac{3}{16})& = & (\frac{7}{6}x+4) \color{blue}{.240} \\\Leftrightarrow & 48x-45& = & 280x+960 \\\Leftrightarrow & 48x \color{red}{-45} \color{blue}{+45} \color{blue}{-280x} & = & \color{red}{280x} +960 \color{blue}{-280x} \color{blue}{+45} \\\Leftrightarrow & -232x& = & 1005 \\\Leftrightarrow & \frac{-232x}{ \color{red}{-232} }& = & \frac{1005}{-232} \\\Leftrightarrow & x = \frac{-1005}{232} & & \\ & V = \left\{ \frac{-1005}{232} \right\} & \\\end{align}\)
9. \(\text{28 is het kleinste gemene veelvoud van 7, 7 en 4} \\ \begin{align} & \color{blue}{28.} (\frac{x}{7}+\frac{2}{7})& = & (\frac{3}{4}x+5) \color{blue}{.28} \\\Leftrightarrow & 4x+8& = & 21x+140 \\\Leftrightarrow & 4x \color{red}{+8} \color{blue}{-8} \color{blue}{-21x} & = & \color{red}{21x} +140 \color{blue}{-21x} \color{blue}{-8} \\\Leftrightarrow & -17x& = & 132 \\\Leftrightarrow & \frac{-17x}{ \color{red}{-17} }& = & \frac{132}{-17} \\\Leftrightarrow & x = \frac{-132}{17} & & \\ & V = \left\{ \frac{-132}{17} \right\} & \\\end{align}\)
10. \(\text{24 is het kleinste gemene veelvoud van 3, 8 en 4} \\ \begin{align} & \color{blue}{24.} (\frac{x}{3}-\frac{5}{8})& = & (\frac{7}{4}x-6) \color{blue}{.24} \\\Leftrightarrow & 8x-15& = & 42x-144 \\\Leftrightarrow & 8x \color{red}{-15} \color{blue}{+15} \color{blue}{-42x} & = & \color{red}{42x} -144 \color{blue}{-42x} \color{blue}{+15} \\\Leftrightarrow & -34x& = & -129 \\\Leftrightarrow & \frac{-34x}{ \color{red}{-34} }& = & \frac{-129}{-34} \\\Leftrightarrow & x = \frac{129}{34} & & \\ & V = \left\{ \frac{129}{34} \right\} & \\\end{align}\)
11. \(\text{70 is het kleinste gemene veelvoud van 5, 14 en 7} \\ \begin{align} & \color{blue}{70.} (\frac{x}{5}-\frac{5}{14})& = & (\frac{3}{7}x-1) \color{blue}{.70} \\\Leftrightarrow & 14x-25& = & 30x-70 \\\Leftrightarrow & 14x \color{red}{-25} \color{blue}{+25} \color{blue}{-30x} & = & \color{red}{30x} -70 \color{blue}{-30x} \color{blue}{+25} \\\Leftrightarrow & -16x& = & -45 \\\Leftrightarrow & \frac{-16x}{ \color{red}{-16} }& = & \frac{-45}{-16} \\\Leftrightarrow & x = \frac{45}{16} & & \\ & V = \left\{ \frac{45}{16} \right\} & \\\end{align}\)
12. \(\text{308 is het kleinste gemene veelvoud van 7, 11 en 4} \\ \begin{align} & \color{blue}{308.} (\frac{x}{7}-\frac{4}{11})& = & (\frac{-7}{4}x-3) \color{blue}{.308} \\\Leftrightarrow & 44x-112& = & -539x-924 \\\Leftrightarrow & 44x \color{red}{-112} \color{blue}{+112} \color{blue}{+539x} & = & \color{red}{-539x} -924 \color{blue}{+539x} \color{blue}{+112} \\\Leftrightarrow & 583x& = & -812 \\\Leftrightarrow & \frac{583x}{ \color{red}{583} }& = & \frac{-812}{583} \\\Leftrightarrow & x = \frac{-812}{583} & & \\ & V = \left\{ \frac{-812}{583} \right\} & \\\end{align}\)