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

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

Verbetersleutel

1. \(\text{357 is het kleinste gemene veelvoud van 7, 17 en 3} \\ \begin{align} & \color{blue}{357.} (\frac{x}{7}+\frac{4}{17})& = & (\frac{1}{3}x-1) \color{blue}{.357} \\\Leftrightarrow & 51x+84& = & 119x-357 \\\Leftrightarrow & 51x \color{red}{+84} \color{blue}{-84} \color{blue}{-119x} & = & \color{red}{119x} -357 \color{blue}{-119x} \color{blue}{-84} \\\Leftrightarrow & -68x& = & -441 \\\Leftrightarrow & \frac{-68x}{ \color{red}{-68} }& = & \frac{-441}{-68} \\\Leftrightarrow & x = \frac{441}{68} & & \\ & V = \left\{ \frac{441}{68} \right\} & \\\end{align}\)
2. \(\text{90 is het kleinste gemene veelvoud van 2, 9 en 5} \\ \begin{align} & \color{blue}{90.} (\frac{x}{2}+\frac{4}{9})& = & (\frac{-7}{5}x-2) \color{blue}{.90} \\\Leftrightarrow & 45x+40& = & -126x-180 \\\Leftrightarrow & 45x \color{red}{+40} \color{blue}{-40} \color{blue}{+126x} & = & \color{red}{-126x} -180 \color{blue}{+126x} \color{blue}{-40} \\\Leftrightarrow & 171x& = & -220 \\\Leftrightarrow & \frac{171x}{ \color{red}{171} }& = & \frac{-220}{171} \\\Leftrightarrow & x = \frac{-220}{171} & & \\ & V = \left\{ \frac{-220}{171} \right\} & \\\end{align}\)
3. \(\text{156 is het kleinste gemene veelvoud van 3, 13 en 4} \\ \begin{align} & \color{blue}{156.} (\frac{x}{3}+\frac{5}{13})& = & (\frac{1}{4}x+1) \color{blue}{.156} \\\Leftrightarrow & 52x+60& = & 39x+156 \\\Leftrightarrow & 52x \color{red}{+60} \color{blue}{-60} \color{blue}{-39x} & = & \color{red}{39x} +156 \color{blue}{-39x} \color{blue}{-60} \\\Leftrightarrow & 13x& = & 96 \\\Leftrightarrow & \frac{13x}{ \color{red}{13} }& = & \frac{96}{13} \\\Leftrightarrow & x = \frac{96}{13} & & \\ & V = \left\{ \frac{96}{13} \right\} & \\\end{align}\)
4. \(\text{42 is het kleinste gemene veelvoud van 3, 7 en 2} \\ \begin{align} & \color{blue}{42.} (\frac{x}{3}-\frac{3}{7})& = & (\frac{1}{2}x+2) \color{blue}{.42} \\\Leftrightarrow & 14x-18& = & 21x+84 \\\Leftrightarrow & 14x \color{red}{-18} \color{blue}{+18} \color{blue}{-21x} & = & \color{red}{21x} +84 \color{blue}{-21x} \color{blue}{+18} \\\Leftrightarrow & -7x& = & 102 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{102}{-7} \\\Leftrightarrow & x = \frac{-102}{7} & & \\ & V = \left\{ \frac{-102}{7} \right\} & \\\end{align}\)
5. \(\text{78 is het kleinste gemene veelvoud van 2, 13 en 3} \\ \begin{align} & \color{blue}{78.} (\frac{x}{2}-\frac{2}{13})& = & (\frac{8}{3}x+8) \color{blue}{.78} \\\Leftrightarrow & 39x-12& = & 208x+624 \\\Leftrightarrow & 39x \color{red}{-12} \color{blue}{+12} \color{blue}{-208x} & = & \color{red}{208x} +624 \color{blue}{-208x} \color{blue}{+12} \\\Leftrightarrow & -169x& = & 636 \\\Leftrightarrow & \frac{-169x}{ \color{red}{-169} }& = & \frac{636}{-169} \\\Leftrightarrow & x = \frac{-636}{169} & & \\ & V = \left\{ \frac{-636}{169} \right\} & \\\end{align}\)
6. \(\text{340 is het kleinste gemene veelvoud van 4, 17 en 5} \\ \begin{align} & \color{blue}{340.} (\frac{x}{4}-\frac{4}{17})& = & (\frac{-2}{5}x+7) \color{blue}{.340} \\\Leftrightarrow & 85x-80& = & -136x+2380 \\\Leftrightarrow & 85x \color{red}{-80} \color{blue}{+80} \color{blue}{+136x} & = & \color{red}{-136x} +2380 \color{blue}{+136x} \color{blue}{+80} \\\Leftrightarrow & 221x& = & 2460 \\\Leftrightarrow & \frac{221x}{ \color{red}{221} }& = & \frac{2460}{221} \\\Leftrightarrow & x = \frac{2460}{221} & & \\ & V = \left\{ \frac{2460}{221} \right\} & \\\end{align}\)
7. \(\text{195 is het kleinste gemene veelvoud van 3, 13 en 5} \\ \begin{align} & \color{blue}{195.} (\frac{x}{3}+\frac{4}{13})& = & (\frac{-2}{5}x-2) \color{blue}{.195} \\\Leftrightarrow & 65x+60& = & -78x-390 \\\Leftrightarrow & 65x \color{red}{+60} \color{blue}{-60} \color{blue}{+78x} & = & \color{red}{-78x} -390 \color{blue}{+78x} \color{blue}{-60} \\\Leftrightarrow & 143x& = & -450 \\\Leftrightarrow & \frac{143x}{ \color{red}{143} }& = & \frac{-450}{143} \\\Leftrightarrow & x = \frac{-450}{143} & & \\ & V = \left\{ \frac{-450}{143} \right\} & \\\end{align}\)
8. \(\text{340 is het kleinste gemene veelvoud van 5, 17 en 4} \\ \begin{align} & \color{blue}{340.} (\frac{x}{5}+\frac{2}{17})& = & (\frac{-7}{4}x+4) \color{blue}{.340} \\\Leftrightarrow & 68x+40& = & -595x+1360 \\\Leftrightarrow & 68x \color{red}{+40} \color{blue}{-40} \color{blue}{+595x} & = & \color{red}{-595x} +1360 \color{blue}{+595x} \color{blue}{-40} \\\Leftrightarrow & 663x& = & 1320 \\\Leftrightarrow & \frac{663x}{ \color{red}{663} }& = & \frac{1320}{663} \\\Leftrightarrow & x = \frac{440}{221} & & \\ & V = \left\{ \frac{440}{221} \right\} & \\\end{align}\)
9. \(\text{30 is het kleinste gemene veelvoud van 6, 10 en 5} \\ \begin{align} & \color{blue}{30.} (\frac{x}{6}-\frac{3}{10})& = & (\frac{-7}{5}x+1) \color{blue}{.30} \\\Leftrightarrow & 5x-9& = & -42x+30 \\\Leftrightarrow & 5x \color{red}{-9} \color{blue}{+9} \color{blue}{+42x} & = & \color{red}{-42x} +30 \color{blue}{+42x} \color{blue}{+9} \\\Leftrightarrow & 47x& = & 39 \\\Leftrightarrow & \frac{47x}{ \color{red}{47} }& = & \frac{39}{47} \\\Leftrightarrow & x = \frac{39}{47} & & \\ & V = \left\{ \frac{39}{47} \right\} & \\\end{align}\)
10. \(\text{560 is het kleinste gemene veelvoud van 5, 16 en 7} \\ \begin{align} & \color{blue}{560.} (\frac{x}{5}+\frac{3}{16})& = & (\frac{1}{7}x-5) \color{blue}{.560} \\\Leftrightarrow & 112x+105& = & 80x-2800 \\\Leftrightarrow & 112x \color{red}{+105} \color{blue}{-105} \color{blue}{-80x} & = & \color{red}{80x} -2800 \color{blue}{-80x} \color{blue}{-105} \\\Leftrightarrow & 32x& = & -2905 \\\Leftrightarrow & \frac{32x}{ \color{red}{32} }& = & \frac{-2905}{32} \\\Leftrightarrow & x = \frac{-2905}{32} & & \\ & V = \left\{ \frac{-2905}{32} \right\} & \\\end{align}\)
11. \(\text{21 is het kleinste gemene veelvoud van 7, 7 en 3} \\ \begin{align} & \color{blue}{21.} (\frac{x}{7}-\frac{5}{7})& = & (\frac{-2}{3}x-2) \color{blue}{.21} \\\Leftrightarrow & 3x-15& = & -14x-42 \\\Leftrightarrow & 3x \color{red}{-15} \color{blue}{+15} \color{blue}{+14x} & = & \color{red}{-14x} -42 \color{blue}{+14x} \color{blue}{+15} \\\Leftrightarrow & 17x& = & -27 \\\Leftrightarrow & \frac{17x}{ \color{red}{17} }& = & \frac{-27}{17} \\\Leftrightarrow & x = \frac{-27}{17} & & \\ & V = \left\{ \frac{-27}{17} \right\} & \\\end{align}\)
12. \(\text{56 is het kleinste gemene veelvoud van 4, 8 en 7} \\ \begin{align} & \color{blue}{56.} (\frac{x}{4}-\frac{3}{8})& = & (\frac{-3}{7}x-8) \color{blue}{.56} \\\Leftrightarrow & 14x-21& = & -24x-448 \\\Leftrightarrow & 14x \color{red}{-21} \color{blue}{+21} \color{blue}{+24x} & = & \color{red}{-24x} -448 \color{blue}{+24x} \color{blue}{+21} \\\Leftrightarrow & 38x& = & -427 \\\Leftrightarrow & \frac{38x}{ \color{red}{38} }& = & \frac{-427}{38} \\\Leftrightarrow & x = \frac{-427}{38} & & \\ & V = \left\{ \frac{-427}{38} \right\} & \\\end{align}\)