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

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

Verbetersleutel

1. \(\text{28 is het kleinste gemene veelvoud van 7, 14 en 4} \\ \begin{align} & \color{blue}{28.} (\frac{x}{7}-\frac{5}{14})& = & (\frac{-3}{4}x-8) \color{blue}{.28} \\\Leftrightarrow & 4x-10& = & -21x-224 \\\Leftrightarrow & 4x \color{red}{-10} \color{blue}{+10} \color{blue}{+21x} & = & \color{red}{-21x} -224 \color{blue}{+21x} \color{blue}{+10} \\\Leftrightarrow & 25x& = & -214 \\\Leftrightarrow & \frac{25x}{ \color{red}{25} }& = & \frac{-214}{25} \\\Leftrightarrow & x = \frac{-214}{25} & & \\ & V = \left\{ \frac{-214}{25} \right\} & \\\end{align}\)
2. \(\text{210 is het kleinste gemene veelvoud van 5, 14 en 3} \\ \begin{align} & \color{blue}{210.} (\frac{x}{5}+\frac{3}{14})& = & (\frac{-2}{3}x+2) \color{blue}{.210} \\\Leftrightarrow & 42x+45& = & -140x+420 \\\Leftrightarrow & 42x \color{red}{+45} \color{blue}{-45} \color{blue}{+140x} & = & \color{red}{-140x} +420 \color{blue}{+140x} \color{blue}{-45} \\\Leftrightarrow & 182x& = & 375 \\\Leftrightarrow & \frac{182x}{ \color{red}{182} }& = & \frac{375}{182} \\\Leftrightarrow & x = \frac{375}{182} & & \\ & V = \left\{ \frac{375}{182} \right\} & \\\end{align}\)
3. \(\text{240 is het kleinste gemene veelvoud van 5, 16 en 3} \\ \begin{align} & \color{blue}{240.} (\frac{x}{5}-\frac{5}{16})& = & (\frac{2}{3}x+1) \color{blue}{.240} \\\Leftrightarrow & 48x-75& = & 160x+240 \\\Leftrightarrow & 48x \color{red}{-75} \color{blue}{+75} \color{blue}{-160x} & = & \color{red}{160x} +240 \color{blue}{-160x} \color{blue}{+75} \\\Leftrightarrow & -112x& = & 315 \\\Leftrightarrow & \frac{-112x}{ \color{red}{-112} }& = & \frac{315}{-112} \\\Leftrightarrow & x = \frac{-45}{16} & & \\ & V = \left\{ \frac{-45}{16} \right\} & \\\end{align}\)
4. \(\text{105 is het kleinste gemene veelvoud van 3, 7 en 5} \\ \begin{align} & \color{blue}{105.} (\frac{x}{3}+\frac{5}{7})& = & (\frac{-7}{5}x+1) \color{blue}{.105} \\\Leftrightarrow & 35x+75& = & -147x+105 \\\Leftrightarrow & 35x \color{red}{+75} \color{blue}{-75} \color{blue}{+147x} & = & \color{red}{-147x} +105 \color{blue}{+147x} \color{blue}{-75} \\\Leftrightarrow & 182x& = & 30 \\\Leftrightarrow & \frac{182x}{ \color{red}{182} }& = & \frac{30}{182} \\\Leftrightarrow & x = \frac{15}{91} & & \\ & V = \left\{ \frac{15}{91} \right\} & \\\end{align}\)
5. \(\text{20 is het kleinste gemene veelvoud van 5, 10 en 4} \\ \begin{align} & \color{blue}{20.} (\frac{x}{5}-\frac{3}{10})& = & (\frac{5}{4}x+6) \color{blue}{.20} \\\Leftrightarrow & 4x-6& = & 25x+120 \\\Leftrightarrow & 4x \color{red}{-6} \color{blue}{+6} \color{blue}{-25x} & = & \color{red}{25x} +120 \color{blue}{-25x} \color{blue}{+6} \\\Leftrightarrow & -21x& = & 126 \\\Leftrightarrow & \frac{-21x}{ \color{red}{-21} }& = & \frac{126}{-21} \\\Leftrightarrow & x = -6 & & \\ & V = \left\{ -6 \right\} & \\\end{align}\)
6. \(\text{90 is het kleinste gemene veelvoud van 6, 9 en 5} \\ \begin{align} & \color{blue}{90.} (\frac{x}{6}+\frac{2}{9})& = & (\frac{-7}{5}x+7) \color{blue}{.90} \\\Leftrightarrow & 15x+20& = & -126x+630 \\\Leftrightarrow & 15x \color{red}{+20} \color{blue}{-20} \color{blue}{+126x} & = & \color{red}{-126x} +630 \color{blue}{+126x} \color{blue}{-20} \\\Leftrightarrow & 141x& = & 610 \\\Leftrightarrow & \frac{141x}{ \color{red}{141} }& = & \frac{610}{141} \\\Leftrightarrow & x = \frac{610}{141} & & \\ & V = \left\{ \frac{610}{141} \right\} & \\\end{align}\)
7. \(\text{18 is het kleinste gemene veelvoud van 2, 9 en 3} \\ \begin{align} & \color{blue}{18.} (\frac{x}{2}+\frac{2}{9})& = & (\frac{2}{3}x-4) \color{blue}{.18} \\\Leftrightarrow & 9x+4& = & 12x-72 \\\Leftrightarrow & 9x \color{red}{+4} \color{blue}{-4} \color{blue}{-12x} & = & \color{red}{12x} -72 \color{blue}{-12x} \color{blue}{-4} \\\Leftrightarrow & -3x& = & -76 \\\Leftrightarrow & \frac{-3x}{ \color{red}{-3} }& = & \frac{-76}{-3} \\\Leftrightarrow & x = \frac{76}{3} & & \\ & V = \left\{ \frac{76}{3} \right\} & \\\end{align}\)
8. \(\text{140 is het kleinste gemene veelvoud van 5, 7 en 4} \\ \begin{align} & \color{blue}{140.} (\frac{x}{5}-\frac{5}{7})& = & (\frac{3}{4}x-4) \color{blue}{.140} \\\Leftrightarrow & 28x-100& = & 105x-560 \\\Leftrightarrow & 28x \color{red}{-100} \color{blue}{+100} \color{blue}{-105x} & = & \color{red}{105x} -560 \color{blue}{-105x} \color{blue}{+100} \\\Leftrightarrow & -77x& = & -460 \\\Leftrightarrow & \frac{-77x}{ \color{red}{-77} }& = & \frac{-460}{-77} \\\Leftrightarrow & x = \frac{460}{77} & & \\ & V = \left\{ \frac{460}{77} \right\} & \\\end{align}\)
9. \(\text{385 is het kleinste gemene veelvoud van 5, 11 en 7} \\ \begin{align} & \color{blue}{385.} (\frac{x}{5}-\frac{2}{11})& = & (\frac{6}{7}x+5) \color{blue}{.385} \\\Leftrightarrow & 77x-70& = & 330x+1925 \\\Leftrightarrow & 77x \color{red}{-70} \color{blue}{+70} \color{blue}{-330x} & = & \color{red}{330x} +1925 \color{blue}{-330x} \color{blue}{+70} \\\Leftrightarrow & -253x& = & 1995 \\\Leftrightarrow & \frac{-253x}{ \color{red}{-253} }& = & \frac{1995}{-253} \\\Leftrightarrow & x = \frac{-1995}{253} & & \\ & V = \left\{ \frac{-1995}{253} \right\} & \\\end{align}\)
10. \(\text{63 is het kleinste gemene veelvoud van 7, 9 en 3} \\ \begin{align} & \color{blue}{63.} (\frac{x}{7}+\frac{5}{9})& = & (\frac{-2}{3}x-2) \color{blue}{.63} \\\Leftrightarrow & 9x+35& = & -42x-126 \\\Leftrightarrow & 9x \color{red}{+35} \color{blue}{-35} \color{blue}{+42x} & = & \color{red}{-42x} -126 \color{blue}{+42x} \color{blue}{-35} \\\Leftrightarrow & 51x& = & -161 \\\Leftrightarrow & \frac{51x}{ \color{red}{51} }& = & \frac{-161}{51} \\\Leftrightarrow & x = \frac{-161}{51} & & \\ & V = \left\{ \frac{-161}{51} \right\} & \\\end{align}\)
11. \(\text{385 is het kleinste gemene veelvoud van 7, 11 en 5} \\ \begin{align} & \color{blue}{385.} (\frac{x}{7}+\frac{5}{11})& = & (\frac{7}{5}x+8) \color{blue}{.385} \\\Leftrightarrow & 55x+175& = & 539x+3080 \\\Leftrightarrow & 55x \color{red}{+175} \color{blue}{-175} \color{blue}{-539x} & = & \color{red}{539x} +3080 \color{blue}{-539x} \color{blue}{-175} \\\Leftrightarrow & -484x& = & 2905 \\\Leftrightarrow & \frac{-484x}{ \color{red}{-484} }& = & \frac{2905}{-484} \\\Leftrightarrow & x = \frac{-2905}{484} & & \\ & V = \left\{ \frac{-2905}{484} \right\} & \\\end{align}\)
12. \(\text{78 is het kleinste gemene veelvoud van 3, 13 en 2} \\ \begin{align} & \color{blue}{78.} (\frac{x}{3}-\frac{3}{13})& = & (\frac{5}{2}x+4) \color{blue}{.78} \\\Leftrightarrow & 26x-18& = & 195x+312 \\\Leftrightarrow & 26x \color{red}{-18} \color{blue}{+18} \color{blue}{-195x} & = & \color{red}{195x} +312 \color{blue}{-195x} \color{blue}{+18} \\\Leftrightarrow & -169x& = & 330 \\\Leftrightarrow & \frac{-169x}{ \color{red}{-169} }& = & \frac{330}{-169} \\\Leftrightarrow & x = \frac{-330}{169} & & \\ & V = \left\{ \frac{-330}{169} \right\} & \\\end{align}\)