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. \(\frac{x}{6}+\frac{5}{17}=\frac{2}{5}x+5\)
2. \(\frac{x}{4}-\frac{4}{11}=\frac{-8}{3}x-4\)
3. \(\frac{x}{2}-\frac{5}{12}=\frac{5}{3}x-5\)
4. \(\frac{x}{7}-\frac{2}{17}=\frac{8}{3}x+8\)
5. \(\frac{x}{4}-\frac{2}{15}=\frac{1}{5}x+7\)
6. \(\frac{x}{4}-\frac{2}{7}=\frac{-5}{3}x-8\)
7. \(\frac{x}{7}-\frac{3}{8}=\frac{7}{2}x-8\)
8. \(\frac{x}{2}+\frac{4}{15}=\frac{-2}{3}x+6\)
9. \(\frac{x}{3}+\frac{3}{8}=\frac{1}{7}x+1\)
10. \(\frac{x}{3}-\frac{5}{6}=\frac{3}{4}x+1\)
11. \(\frac{x}{5}+\frac{2}{7}=\frac{7}{6}x-7\)
12. \(\frac{x}{3}+\frac{3}{17}=\frac{7}{4}x-8\)

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

Verbetersleutel

1. \(\text{510 is het kleinste gemene veelvoud van 6, 17 en 5} \\ \begin{align} & \color{blue}{510.} (\frac{x}{6}+\frac{5}{17})& = & (\frac{2}{5}x+5) \color{blue}{.510} \\\Leftrightarrow & 85x+150& = & 204x+2550 \\\Leftrightarrow & 85x \color{red}{+150} \color{blue}{-150} \color{blue}{-204x} & = & \color{red}{204x} +2550 \color{blue}{-204x} \color{blue}{-150} \\\Leftrightarrow & -119x& = & 2400 \\\Leftrightarrow & \frac{-119x}{ \color{red}{-119} }& = & \frac{2400}{-119} \\\Leftrightarrow & x = \frac{-2400}{119} & & \\ & V = \left\{ \frac{-2400}{119} \right\} & \\\end{align}\)
2. \(\text{132 is het kleinste gemene veelvoud van 4, 11 en 3} \\ \begin{align} & \color{blue}{132.} (\frac{x}{4}-\frac{4}{11})& = & (\frac{-8}{3}x-4) \color{blue}{.132} \\\Leftrightarrow & 33x-48& = & -352x-528 \\\Leftrightarrow & 33x \color{red}{-48} \color{blue}{+48} \color{blue}{+352x} & = & \color{red}{-352x} -528 \color{blue}{+352x} \color{blue}{+48} \\\Leftrightarrow & 385x& = & -480 \\\Leftrightarrow & \frac{385x}{ \color{red}{385} }& = & \frac{-480}{385} \\\Leftrightarrow & x = \frac{-96}{77} & & \\ & V = \left\{ \frac{-96}{77} \right\} & \\\end{align}\)
3. \(\text{12 is het kleinste gemene veelvoud van 2, 12 en 3} \\ \begin{align} & \color{blue}{12.} (\frac{x}{2}-\frac{5}{12})& = & (\frac{5}{3}x-5) \color{blue}{.12} \\\Leftrightarrow & 6x-5& = & 20x-60 \\\Leftrightarrow & 6x \color{red}{-5} \color{blue}{+5} \color{blue}{-20x} & = & \color{red}{20x} -60 \color{blue}{-20x} \color{blue}{+5} \\\Leftrightarrow & -14x& = & -55 \\\Leftrightarrow & \frac{-14x}{ \color{red}{-14} }& = & \frac{-55}{-14} \\\Leftrightarrow & x = \frac{55}{14} & & \\ & V = \left\{ \frac{55}{14} \right\} & \\\end{align}\)
4. \(\text{357 is het kleinste gemene veelvoud van 7, 17 en 3} \\ \begin{align} & \color{blue}{357.} (\frac{x}{7}-\frac{2}{17})& = & (\frac{8}{3}x+8) \color{blue}{.357} \\\Leftrightarrow & 51x-42& = & 952x+2856 \\\Leftrightarrow & 51x \color{red}{-42} \color{blue}{+42} \color{blue}{-952x} & = & \color{red}{952x} +2856 \color{blue}{-952x} \color{blue}{+42} \\\Leftrightarrow & -901x& = & 2898 \\\Leftrightarrow & \frac{-901x}{ \color{red}{-901} }& = & \frac{2898}{-901} \\\Leftrightarrow & x = \frac{-2898}{901} & & \\ & V = \left\{ \frac{-2898}{901} \right\} & \\\end{align}\)
5. \(\text{60 is het kleinste gemene veelvoud van 4, 15 en 5} \\ \begin{align} & \color{blue}{60.} (\frac{x}{4}-\frac{2}{15})& = & (\frac{1}{5}x+7) \color{blue}{.60} \\\Leftrightarrow & 15x-8& = & 12x+420 \\\Leftrightarrow & 15x \color{red}{-8} \color{blue}{+8} \color{blue}{-12x} & = & \color{red}{12x} +420 \color{blue}{-12x} \color{blue}{+8} \\\Leftrightarrow & 3x& = & 428 \\\Leftrightarrow & \frac{3x}{ \color{red}{3} }& = & \frac{428}{3} \\\Leftrightarrow & x = \frac{428}{3} & & \\ & V = \left\{ \frac{428}{3} \right\} & \\\end{align}\)
6. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \color{blue}{84.} (\frac{x}{4}-\frac{2}{7})& = & (\frac{-5}{3}x-8) \color{blue}{.84} \\\Leftrightarrow & 21x-24& = & -140x-672 \\\Leftrightarrow & 21x \color{red}{-24} \color{blue}{+24} \color{blue}{+140x} & = & \color{red}{-140x} -672 \color{blue}{+140x} \color{blue}{+24} \\\Leftrightarrow & 161x& = & -648 \\\Leftrightarrow & \frac{161x}{ \color{red}{161} }& = & \frac{-648}{161} \\\Leftrightarrow & x = \frac{-648}{161} & & \\ & V = \left\{ \frac{-648}{161} \right\} & \\\end{align}\)
7. \(\text{56 is het kleinste gemene veelvoud van 7, 8 en 2} \\ \begin{align} & \color{blue}{56.} (\frac{x}{7}-\frac{3}{8})& = & (\frac{7}{2}x-8) \color{blue}{.56} \\\Leftrightarrow & 8x-21& = & 196x-448 \\\Leftrightarrow & 8x \color{red}{-21} \color{blue}{+21} \color{blue}{-196x} & = & \color{red}{196x} -448 \color{blue}{-196x} \color{blue}{+21} \\\Leftrightarrow & -188x& = & -427 \\\Leftrightarrow & \frac{-188x}{ \color{red}{-188} }& = & \frac{-427}{-188} \\\Leftrightarrow & x = \frac{427}{188} & & \\ & V = \left\{ \frac{427}{188} \right\} & \\\end{align}\)
8. \(\text{30 is het kleinste gemene veelvoud van 2, 15 en 3} \\ \begin{align} & \color{blue}{30.} (\frac{x}{2}+\frac{4}{15})& = & (\frac{-2}{3}x+6) \color{blue}{.30} \\\Leftrightarrow & 15x+8& = & -20x+180 \\\Leftrightarrow & 15x \color{red}{+8} \color{blue}{-8} \color{blue}{+20x} & = & \color{red}{-20x} +180 \color{blue}{+20x} \color{blue}{-8} \\\Leftrightarrow & 35x& = & 172 \\\Leftrightarrow & \frac{35x}{ \color{red}{35} }& = & \frac{172}{35} \\\Leftrightarrow & x = \frac{172}{35} & & \\ & V = \left\{ \frac{172}{35} \right\} & \\\end{align}\)
9. \(\text{168 is het kleinste gemene veelvoud van 3, 8 en 7} \\ \begin{align} & \color{blue}{168.} (\frac{x}{3}+\frac{3}{8})& = & (\frac{1}{7}x+1) \color{blue}{.168} \\\Leftrightarrow & 56x+63& = & 24x+168 \\\Leftrightarrow & 56x \color{red}{+63} \color{blue}{-63} \color{blue}{-24x} & = & \color{red}{24x} +168 \color{blue}{-24x} \color{blue}{-63} \\\Leftrightarrow & 32x& = & 105 \\\Leftrightarrow & \frac{32x}{ \color{red}{32} }& = & \frac{105}{32} \\\Leftrightarrow & x = \frac{105}{32} & & \\ & V = \left\{ \frac{105}{32} \right\} & \\\end{align}\)
10. \(\text{12 is het kleinste gemene veelvoud van 3, 6 en 4} \\ \begin{align} & \color{blue}{12.} (\frac{x}{3}-\frac{5}{6})& = & (\frac{3}{4}x+1) \color{blue}{.12} \\\Leftrightarrow & 4x-10& = & 9x+12 \\\Leftrightarrow & 4x \color{red}{-10} \color{blue}{+10} \color{blue}{-9x} & = & \color{red}{9x} +12 \color{blue}{-9x} \color{blue}{+10} \\\Leftrightarrow & -5x& = & 22 \\\Leftrightarrow & \frac{-5x}{ \color{red}{-5} }& = & \frac{22}{-5} \\\Leftrightarrow & x = \frac{-22}{5} & & \\ & V = \left\{ \frac{-22}{5} \right\} & \\\end{align}\)
11. \(\text{210 is het kleinste gemene veelvoud van 5, 7 en 6} \\ \begin{align} & \color{blue}{210.} (\frac{x}{5}+\frac{2}{7})& = & (\frac{7}{6}x-7) \color{blue}{.210} \\\Leftrightarrow & 42x+60& = & 245x-1470 \\\Leftrightarrow & 42x \color{red}{+60} \color{blue}{-60} \color{blue}{-245x} & = & \color{red}{245x} -1470 \color{blue}{-245x} \color{blue}{-60} \\\Leftrightarrow & -203x& = & -1530 \\\Leftrightarrow & \frac{-203x}{ \color{red}{-203} }& = & \frac{-1530}{-203} \\\Leftrightarrow & x = \frac{1530}{203} & & \\ & V = \left\{ \frac{1530}{203} \right\} & \\\end{align}\)
12. \(\text{204 is het kleinste gemene veelvoud van 3, 17 en 4} \\ \begin{align} & \color{blue}{204.} (\frac{x}{3}+\frac{3}{17})& = & (\frac{7}{4}x-8) \color{blue}{.204} \\\Leftrightarrow & 68x+36& = & 357x-1632 \\\Leftrightarrow & 68x \color{red}{+36} \color{blue}{-36} \color{blue}{-357x} & = & \color{red}{357x} -1632 \color{blue}{-357x} \color{blue}{-36} \\\Leftrightarrow & -289x& = & -1668 \\\Leftrightarrow & \frac{-289x}{ \color{red}{-289} }& = & \frac{-1668}{-289} \\\Leftrightarrow & x = \frac{1668}{289} & & \\ & V = \left\{ \frac{1668}{289} \right\} & \\\end{align}\)