Vgln. eerste graad (reeks 4)

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1. \(\frac{x}{2}-\frac{3}{11}=\frac{-2}{3}x+5\)
2. \(\frac{x}{5}-\frac{5}{16}=\frac{7}{3}x-4\)
3. \(\frac{x}{4}+\frac{3}{10}=\frac{-7}{5}x+1\)
4. \(\frac{x}{7}+\frac{4}{7}=\frac{-8}{3}x-4\)
5. \(\frac{x}{7}-\frac{3}{11}=\frac{-5}{6}x+2\)
6. \(\frac{x}{7}+\frac{2}{15}=\frac{1}{2}x-2\)
7. \(\frac{x}{4}-\frac{4}{15}=\frac{-8}{3}x+4\)
8. \(\frac{x}{2}-\frac{2}{7}=\frac{-4}{5}x+3\)
9. \(\frac{x}{6}-\frac{5}{8}=\frac{2}{5}x+8\)
10. \(\frac{x}{2}+\frac{3}{8}=\frac{7}{5}x-4\)
11. \(\frac{x}{7}+\frac{5}{11}=\frac{4}{5}x-8\)
12. \(\frac{x}{6}+\frac{4}{15}=\frac{-4}{5}x+2\)

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

Verbetersleutel

1. \(\text{66 is het kleinste gemene veelvoud van 2, 11 en 3} \\ \begin{align} & \frac{x}{2}-\frac{3}{11}& = & \frac{-2}{3}x+5 \\\Leftrightarrow & \color{blue}{66.} (\frac{33x}{ \color{blue}{66} }- \frac{ 18 }{ \color{blue}{66} })& = & (\frac{-44}{ \color{blue}{66} }x+\frac{330}{ \color{blue}{66} }) \color{blue}{.66} \\\Leftrightarrow & 33x-18& = & -44x+330 \\\Leftrightarrow & 33x \color{red}{-18} \color{blue}{+18} \color{blue}{+44x} & = & \color{red}{-44x} +330 \color{blue}{+44x} \color{blue}{+18} \\\Leftrightarrow & 77x& = & 348 \\\Leftrightarrow & \frac{77x}{ \color{red}{77} }& = & \frac{348}{77} \\\Leftrightarrow & x = \frac{348}{77} & & \\ & V = \left\{ \frac{348}{77} \right\} & \\\end{align}\)
2. \(\text{240 is het kleinste gemene veelvoud van 5, 16 en 3} \\ \begin{align} & \frac{x}{5}-\frac{5}{16}& = & \frac{7}{3}x-4 \\\Leftrightarrow & \color{blue}{240.} (\frac{48x}{ \color{blue}{240} }- \frac{ 75 }{ \color{blue}{240} })& = & (\frac{560}{ \color{blue}{240} }x-\frac{960}{ \color{blue}{240} }) \color{blue}{.240} \\\Leftrightarrow & 48x-75& = & 560x-960 \\\Leftrightarrow & 48x \color{red}{-75} \color{blue}{+75} \color{blue}{-560x} & = & \color{red}{560x} -960 \color{blue}{-560x} \color{blue}{+75} \\\Leftrightarrow & -512x& = & -885 \\\Leftrightarrow & \frac{-512x}{ \color{red}{-512} }& = & \frac{-885}{-512} \\\Leftrightarrow & x = \frac{885}{512} & & \\ & V = \left\{ \frac{885}{512} \right\} & \\\end{align}\)
3. \(\text{20 is het kleinste gemene veelvoud van 4, 10 en 5} \\ \begin{align} & \frac{x}{4}+\frac{3}{10}& = & \frac{-7}{5}x+1 \\\Leftrightarrow & \color{blue}{20.} (\frac{5x}{ \color{blue}{20} }+ \frac{ 6 }{ \color{blue}{20} })& = & (\frac{-28}{ \color{blue}{20} }x+\frac{20}{ \color{blue}{20} }) \color{blue}{.20} \\\Leftrightarrow & 5x+6& = & -28x+20 \\\Leftrightarrow & 5x \color{red}{+6} \color{blue}{-6} \color{blue}{+28x} & = & \color{red}{-28x} +20 \color{blue}{+28x} \color{blue}{-6} \\\Leftrightarrow & 33x& = & 14 \\\Leftrightarrow & \frac{33x}{ \color{red}{33} }& = & \frac{14}{33} \\\Leftrightarrow & x = \frac{14}{33} & & \\ & V = \left\{ \frac{14}{33} \right\} & \\\end{align}\)
4. \(\text{21 is het kleinste gemene veelvoud van 7, 7 en 3} \\ \begin{align} & \frac{x}{7}+\frac{4}{7}& = & \frac{-8}{3}x-4 \\\Leftrightarrow & \color{blue}{21.} (\frac{3x}{ \color{blue}{21} }+ \frac{ 12 }{ \color{blue}{21} })& = & (\frac{-56}{ \color{blue}{21} }x-\frac{84}{ \color{blue}{21} }) \color{blue}{.21} \\\Leftrightarrow & 3x+12& = & -56x-84 \\\Leftrightarrow & 3x \color{red}{+12} \color{blue}{-12} \color{blue}{+56x} & = & \color{red}{-56x} -84 \color{blue}{+56x} \color{blue}{-12} \\\Leftrightarrow & 59x& = & -96 \\\Leftrightarrow & \frac{59x}{ \color{red}{59} }& = & \frac{-96}{59} \\\Leftrightarrow & x = \frac{-96}{59} & & \\ & V = \left\{ \frac{-96}{59} \right\} & \\\end{align}\)
5. \(\text{462 is het kleinste gemene veelvoud van 7, 11 en 6} \\ \begin{align} & \frac{x}{7}-\frac{3}{11}& = & \frac{-5}{6}x+2 \\\Leftrightarrow & \color{blue}{462.} (\frac{66x}{ \color{blue}{462} }- \frac{ 126 }{ \color{blue}{462} })& = & (\frac{-385}{ \color{blue}{462} }x+\frac{924}{ \color{blue}{462} }) \color{blue}{.462} \\\Leftrightarrow & 66x-126& = & -385x+924 \\\Leftrightarrow & 66x \color{red}{-126} \color{blue}{+126} \color{blue}{+385x} & = & \color{red}{-385x} +924 \color{blue}{+385x} \color{blue}{+126} \\\Leftrightarrow & 451x& = & 1050 \\\Leftrightarrow & \frac{451x}{ \color{red}{451} }& = & \frac{1050}{451} \\\Leftrightarrow & x = \frac{1050}{451} & & \\ & V = \left\{ \frac{1050}{451} \right\} & \\\end{align}\)
6. \(\text{210 is het kleinste gemene veelvoud van 7, 15 en 2} \\ \begin{align} & \frac{x}{7}+\frac{2}{15}& = & \frac{1}{2}x-2 \\\Leftrightarrow & \color{blue}{210.} (\frac{30x}{ \color{blue}{210} }+ \frac{ 28 }{ \color{blue}{210} })& = & (\frac{105}{ \color{blue}{210} }x-\frac{420}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 30x+28& = & 105x-420 \\\Leftrightarrow & 30x \color{red}{+28} \color{blue}{-28} \color{blue}{-105x} & = & \color{red}{105x} -420 \color{blue}{-105x} \color{blue}{-28} \\\Leftrightarrow & -75x& = & -448 \\\Leftrightarrow & \frac{-75x}{ \color{red}{-75} }& = & \frac{-448}{-75} \\\Leftrightarrow & x = \frac{448}{75} & & \\ & V = \left\{ \frac{448}{75} \right\} & \\\end{align}\)
7. \(\text{60 is het kleinste gemene veelvoud van 4, 15 en 3} \\ \begin{align} & \frac{x}{4}-\frac{4}{15}& = & \frac{-8}{3}x+4 \\\Leftrightarrow & \color{blue}{60.} (\frac{15x}{ \color{blue}{60} }- \frac{ 16 }{ \color{blue}{60} })& = & (\frac{-160}{ \color{blue}{60} }x+\frac{240}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 15x-16& = & -160x+240 \\\Leftrightarrow & 15x \color{red}{-16} \color{blue}{+16} \color{blue}{+160x} & = & \color{red}{-160x} +240 \color{blue}{+160x} \color{blue}{+16} \\\Leftrightarrow & 175x& = & 256 \\\Leftrightarrow & \frac{175x}{ \color{red}{175} }& = & \frac{256}{175} \\\Leftrightarrow & x = \frac{256}{175} & & \\ & V = \left\{ \frac{256}{175} \right\} & \\\end{align}\)
8. \(\text{70 is het kleinste gemene veelvoud van 2, 7 en 5} \\ \begin{align} & \frac{x}{2}-\frac{2}{7}& = & \frac{-4}{5}x+3 \\\Leftrightarrow & \color{blue}{70.} (\frac{35x}{ \color{blue}{70} }- \frac{ 20 }{ \color{blue}{70} })& = & (\frac{-56}{ \color{blue}{70} }x+\frac{210}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 35x-20& = & -56x+210 \\\Leftrightarrow & 35x \color{red}{-20} \color{blue}{+20} \color{blue}{+56x} & = & \color{red}{-56x} +210 \color{blue}{+56x} \color{blue}{+20} \\\Leftrightarrow & 91x& = & 230 \\\Leftrightarrow & \frac{91x}{ \color{red}{91} }& = & \frac{230}{91} \\\Leftrightarrow & x = \frac{230}{91} & & \\ & V = \left\{ \frac{230}{91} \right\} & \\\end{align}\)
9. \(\text{120 is het kleinste gemene veelvoud van 6, 8 en 5} \\ \begin{align} & \frac{x}{6}-\frac{5}{8}& = & \frac{2}{5}x+8 \\\Leftrightarrow & \color{blue}{120.} (\frac{20x}{ \color{blue}{120} }- \frac{ 75 }{ \color{blue}{120} })& = & (\frac{48}{ \color{blue}{120} }x+\frac{960}{ \color{blue}{120} }) \color{blue}{.120} \\\Leftrightarrow & 20x-75& = & 48x+960 \\\Leftrightarrow & 20x \color{red}{-75} \color{blue}{+75} \color{blue}{-48x} & = & \color{red}{48x} +960 \color{blue}{-48x} \color{blue}{+75} \\\Leftrightarrow & -28x& = & 1035 \\\Leftrightarrow & \frac{-28x}{ \color{red}{-28} }& = & \frac{1035}{-28} \\\Leftrightarrow & x = \frac{-1035}{28} & & \\ & V = \left\{ \frac{-1035}{28} \right\} & \\\end{align}\)
10. \(\text{40 is het kleinste gemene veelvoud van 2, 8 en 5} \\ \begin{align} & \frac{x}{2}+\frac{3}{8}& = & \frac{7}{5}x-4 \\\Leftrightarrow & \color{blue}{40.} (\frac{20x}{ \color{blue}{40} }+ \frac{ 15 }{ \color{blue}{40} })& = & (\frac{56}{ \color{blue}{40} }x-\frac{160}{ \color{blue}{40} }) \color{blue}{.40} \\\Leftrightarrow & 20x+15& = & 56x-160 \\\Leftrightarrow & 20x \color{red}{+15} \color{blue}{-15} \color{blue}{-56x} & = & \color{red}{56x} -160 \color{blue}{-56x} \color{blue}{-15} \\\Leftrightarrow & -36x& = & -175 \\\Leftrightarrow & \frac{-36x}{ \color{red}{-36} }& = & \frac{-175}{-36} \\\Leftrightarrow & x = \frac{175}{36} & & \\ & V = \left\{ \frac{175}{36} \right\} & \\\end{align}\)
11. \(\text{385 is het kleinste gemene veelvoud van 7, 11 en 5} \\ \begin{align} & \frac{x}{7}+\frac{5}{11}& = & \frac{4}{5}x-8 \\\Leftrightarrow & \color{blue}{385.} (\frac{55x}{ \color{blue}{385} }+ \frac{ 175 }{ \color{blue}{385} })& = & (\frac{308}{ \color{blue}{385} }x-\frac{3080}{ \color{blue}{385} }) \color{blue}{.385} \\\Leftrightarrow & 55x+175& = & 308x-3080 \\\Leftrightarrow & 55x \color{red}{+175} \color{blue}{-175} \color{blue}{-308x} & = & \color{red}{308x} -3080 \color{blue}{-308x} \color{blue}{-175} \\\Leftrightarrow & -253x& = & -3255 \\\Leftrightarrow & \frac{-253x}{ \color{red}{-253} }& = & \frac{-3255}{-253} \\\Leftrightarrow & x = \frac{3255}{253} & & \\ & V = \left\{ \frac{3255}{253} \right\} & \\\end{align}\)
12. \(\text{30 is het kleinste gemene veelvoud van 6, 15 en 5} \\ \begin{align} & \frac{x}{6}+\frac{4}{15}& = & \frac{-4}{5}x+2 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }+ \frac{ 8 }{ \color{blue}{30} })& = & (\frac{-24}{ \color{blue}{30} }x+\frac{60}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x+8& = & -24x+60 \\\Leftrightarrow & 5x \color{red}{+8} \color{blue}{-8} \color{blue}{+24x} & = & \color{red}{-24x} +60 \color{blue}{+24x} \color{blue}{-8} \\\Leftrightarrow & 29x& = & 52 \\\Leftrightarrow & \frac{29x}{ \color{red}{29} }& = & \frac{52}{29} \\\Leftrightarrow & x = \frac{52}{29} & & \\ & V = \left\{ \frac{52}{29} \right\} & \\\end{align}\)