Vgln. eerste graad (reeks 4)

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1. \(\frac{x}{2}-\frac{2}{13}=\frac{-4}{5}x+4\)
2. \(\frac{x}{7}+\frac{4}{7}=\frac{6}{5}x-1\)
3. \(\frac{x}{6}+\frac{3}{16}=\frac{-7}{5}x+8\)
4. \(\frac{x}{2}-\frac{5}{6}=\frac{1}{3}x+8\)
5. \(\frac{x}{2}-\frac{4}{15}=\frac{6}{5}x+7\)
6. \(\frac{x}{3}-\frac{5}{9}=\frac{1}{2}x-2\)
7. \(\frac{x}{6}-\frac{2}{15}=\frac{1}{5}x-5\)
8. \(\frac{x}{7}+\frac{2}{13}=\frac{5}{3}x-7\)
9. \(\frac{x}{3}+\frac{3}{7}=\frac{5}{2}x+8\)
10. \(\frac{x}{7}-\frac{4}{9}=\frac{5}{4}x+8\)
11. \(\frac{x}{6}-\frac{3}{10}=\frac{-2}{5}x+1\)
12. \(\frac{x}{5}+\frac{5}{13}=\frac{1}{2}x+2\)

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

Verbetersleutel

1. \(\text{130 is het kleinste gemene veelvoud van 2, 13 en 5} \\ \begin{align} & \frac{x}{2}-\frac{2}{13}& = & \frac{-4}{5}x+4 \\\Leftrightarrow & \color{blue}{130.} (\frac{65x}{ \color{blue}{130} }- \frac{ 20 }{ \color{blue}{130} })& = & (\frac{-104}{ \color{blue}{130} }x+\frac{520}{ \color{blue}{130} }) \color{blue}{.130} \\\Leftrightarrow & 65x-20& = & -104x+520 \\\Leftrightarrow & 65x \color{red}{-20} \color{blue}{+20} \color{blue}{+104x} & = & \color{red}{-104x} +520 \color{blue}{+104x} \color{blue}{+20} \\\Leftrightarrow & 169x& = & 540 \\\Leftrightarrow & \frac{169x}{ \color{red}{169} }& = & \frac{540}{169} \\\Leftrightarrow & x = \frac{540}{169} & & \\ & V = \left\{ \frac{540}{169} \right\} & \\\end{align}\)
2. \(\text{35 is het kleinste gemene veelvoud van 7, 7 en 5} \\ \begin{align} & \frac{x}{7}+\frac{4}{7}& = & \frac{6}{5}x-1 \\\Leftrightarrow & \color{blue}{35.} (\frac{5x}{ \color{blue}{35} }+ \frac{ 20 }{ \color{blue}{35} })& = & (\frac{42}{ \color{blue}{35} }x-\frac{35}{ \color{blue}{35} }) \color{blue}{.35} \\\Leftrightarrow & 5x+20& = & 42x-35 \\\Leftrightarrow & 5x \color{red}{+20} \color{blue}{-20} \color{blue}{-42x} & = & \color{red}{42x} -35 \color{blue}{-42x} \color{blue}{-20} \\\Leftrightarrow & -37x& = & -55 \\\Leftrightarrow & \frac{-37x}{ \color{red}{-37} }& = & \frac{-55}{-37} \\\Leftrightarrow & x = \frac{55}{37} & & \\ & V = \left\{ \frac{55}{37} \right\} & \\\end{align}\)
3. \(\text{240 is het kleinste gemene veelvoud van 6, 16 en 5} \\ \begin{align} & \frac{x}{6}+\frac{3}{16}& = & \frac{-7}{5}x+8 \\\Leftrightarrow & \color{blue}{240.} (\frac{40x}{ \color{blue}{240} }+ \frac{ 45 }{ \color{blue}{240} })& = & (\frac{-336}{ \color{blue}{240} }x+\frac{1920}{ \color{blue}{240} }) \color{blue}{.240} \\\Leftrightarrow & 40x+45& = & -336x+1920 \\\Leftrightarrow & 40x \color{red}{+45} \color{blue}{-45} \color{blue}{+336x} & = & \color{red}{-336x} +1920 \color{blue}{+336x} \color{blue}{-45} \\\Leftrightarrow & 376x& = & 1875 \\\Leftrightarrow & \frac{376x}{ \color{red}{376} }& = & \frac{1875}{376} \\\Leftrightarrow & x = \frac{1875}{376} & & \\ & V = \left\{ \frac{1875}{376} \right\} & \\\end{align}\)
4. \(\text{6 is het kleinste gemene veelvoud van 2, 6 en 3} \\ \begin{align} & \frac{x}{2}-\frac{5}{6}& = & \frac{1}{3}x+8 \\\Leftrightarrow & \color{blue}{6.} (\frac{3x}{ \color{blue}{6} }- \frac{ 5 }{ \color{blue}{6} })& = & (\frac{2}{ \color{blue}{6} }x+\frac{48}{ \color{blue}{6} }) \color{blue}{.6} \\\Leftrightarrow & 3x-5& = & 2x+48 \\\Leftrightarrow & 3x \color{red}{-5} \color{blue}{+5} \color{blue}{-2x} & = & \color{red}{2x} +48 \color{blue}{-2x} \color{blue}{+5} \\\Leftrightarrow & x& = & 53 \\ & V = \left\{ 53 \right\} & \\\end{align}\)
5. \(\text{30 is het kleinste gemene veelvoud van 2, 15 en 5} \\ \begin{align} & \frac{x}{2}-\frac{4}{15}& = & \frac{6}{5}x+7 \\\Leftrightarrow & \color{blue}{30.} (\frac{15x}{ \color{blue}{30} }- \frac{ 8 }{ \color{blue}{30} })& = & (\frac{36}{ \color{blue}{30} }x+\frac{210}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 15x-8& = & 36x+210 \\\Leftrightarrow & 15x \color{red}{-8} \color{blue}{+8} \color{blue}{-36x} & = & \color{red}{36x} +210 \color{blue}{-36x} \color{blue}{+8} \\\Leftrightarrow & -21x& = & 218 \\\Leftrightarrow & \frac{-21x}{ \color{red}{-21} }& = & \frac{218}{-21} \\\Leftrightarrow & x = \frac{-218}{21} & & \\ & V = \left\{ \frac{-218}{21} \right\} & \\\end{align}\)
6. \(\text{18 is het kleinste gemene veelvoud van 3, 9 en 2} \\ \begin{align} & \frac{x}{3}-\frac{5}{9}& = & \frac{1}{2}x-2 \\\Leftrightarrow & \color{blue}{18.} (\frac{6x}{ \color{blue}{18} }- \frac{ 10 }{ \color{blue}{18} })& = & (\frac{9}{ \color{blue}{18} }x-\frac{36}{ \color{blue}{18} }) \color{blue}{.18} \\\Leftrightarrow & 6x-10& = & 9x-36 \\\Leftrightarrow & 6x \color{red}{-10} \color{blue}{+10} \color{blue}{-9x} & = & \color{red}{9x} -36 \color{blue}{-9x} \color{blue}{+10} \\\Leftrightarrow & -3x& = & -26 \\\Leftrightarrow & \frac{-3x}{ \color{red}{-3} }& = & \frac{-26}{-3} \\\Leftrightarrow & x = \frac{26}{3} & & \\ & V = \left\{ \frac{26}{3} \right\} & \\\end{align}\)
7. \(\text{30 is het kleinste gemene veelvoud van 6, 15 en 5} \\ \begin{align} & \frac{x}{6}-\frac{2}{15}& = & \frac{1}{5}x-5 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }- \frac{ 4 }{ \color{blue}{30} })& = & (\frac{6}{ \color{blue}{30} }x-\frac{150}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x-4& = & 6x-150 \\\Leftrightarrow & 5x \color{red}{-4} \color{blue}{+4} \color{blue}{-6x} & = & \color{red}{6x} -150 \color{blue}{-6x} \color{blue}{+4} \\\Leftrightarrow & -x& = & -146 \\\Leftrightarrow & \frac{-x}{ \color{red}{-1} }& = & \frac{-146}{-1} \\\Leftrightarrow & x = 146 & & \\ & V = \left\{ 146 \right\} & \\\end{align}\)
8. \(\text{273 is het kleinste gemene veelvoud van 7, 13 en 3} \\ \begin{align} & \frac{x}{7}+\frac{2}{13}& = & \frac{5}{3}x-7 \\\Leftrightarrow & \color{blue}{273.} (\frac{39x}{ \color{blue}{273} }+ \frac{ 42 }{ \color{blue}{273} })& = & (\frac{455}{ \color{blue}{273} }x-\frac{1911}{ \color{blue}{273} }) \color{blue}{.273} \\\Leftrightarrow & 39x+42& = & 455x-1911 \\\Leftrightarrow & 39x \color{red}{+42} \color{blue}{-42} \color{blue}{-455x} & = & \color{red}{455x} -1911 \color{blue}{-455x} \color{blue}{-42} \\\Leftrightarrow & -416x& = & -1953 \\\Leftrightarrow & \frac{-416x}{ \color{red}{-416} }& = & \frac{-1953}{-416} \\\Leftrightarrow & x = \frac{1953}{416} & & \\ & V = \left\{ \frac{1953}{416} \right\} & \\\end{align}\)
9. \(\text{42 is het kleinste gemene veelvoud van 3, 7 en 2} \\ \begin{align} & \frac{x}{3}+\frac{3}{7}& = & \frac{5}{2}x+8 \\\Leftrightarrow & \color{blue}{42.} (\frac{14x}{ \color{blue}{42} }+ \frac{ 18 }{ \color{blue}{42} })& = & (\frac{105}{ \color{blue}{42} }x+\frac{336}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 14x+18& = & 105x+336 \\\Leftrightarrow & 14x \color{red}{+18} \color{blue}{-18} \color{blue}{-105x} & = & \color{red}{105x} +336 \color{blue}{-105x} \color{blue}{-18} \\\Leftrightarrow & -91x& = & 318 \\\Leftrightarrow & \frac{-91x}{ \color{red}{-91} }& = & \frac{318}{-91} \\\Leftrightarrow & x = \frac{-318}{91} & & \\ & V = \left\{ \frac{-318}{91} \right\} & \\\end{align}\)
10. \(\text{252 is het kleinste gemene veelvoud van 7, 9 en 4} \\ \begin{align} & \frac{x}{7}-\frac{4}{9}& = & \frac{5}{4}x+8 \\\Leftrightarrow & \color{blue}{252.} (\frac{36x}{ \color{blue}{252} }- \frac{ 112 }{ \color{blue}{252} })& = & (\frac{315}{ \color{blue}{252} }x+\frac{2016}{ \color{blue}{252} }) \color{blue}{.252} \\\Leftrightarrow & 36x-112& = & 315x+2016 \\\Leftrightarrow & 36x \color{red}{-112} \color{blue}{+112} \color{blue}{-315x} & = & \color{red}{315x} +2016 \color{blue}{-315x} \color{blue}{+112} \\\Leftrightarrow & -279x& = & 2128 \\\Leftrightarrow & \frac{-279x}{ \color{red}{-279} }& = & \frac{2128}{-279} \\\Leftrightarrow & x = \frac{-2128}{279} & & \\ & V = \left\{ \frac{-2128}{279} \right\} & \\\end{align}\)
11. \(\text{30 is het kleinste gemene veelvoud van 6, 10 en 5} \\ \begin{align} & \frac{x}{6}-\frac{3}{10}& = & \frac{-2}{5}x+1 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }- \frac{ 9 }{ \color{blue}{30} })& = & (\frac{-12}{ \color{blue}{30} }x+\frac{30}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x-9& = & -12x+30 \\\Leftrightarrow & 5x \color{red}{-9} \color{blue}{+9} \color{blue}{+12x} & = & \color{red}{-12x} +30 \color{blue}{+12x} \color{blue}{+9} \\\Leftrightarrow & 17x& = & 39 \\\Leftrightarrow & \frac{17x}{ \color{red}{17} }& = & \frac{39}{17} \\\Leftrightarrow & x = \frac{39}{17} & & \\ & V = \left\{ \frac{39}{17} \right\} & \\\end{align}\)
12. \(\text{130 is het kleinste gemene veelvoud van 5, 13 en 2} \\ \begin{align} & \frac{x}{5}+\frac{5}{13}& = & \frac{1}{2}x+2 \\\Leftrightarrow & \color{blue}{130.} (\frac{26x}{ \color{blue}{130} }+ \frac{ 50 }{ \color{blue}{130} })& = & (\frac{65}{ \color{blue}{130} }x+\frac{260}{ \color{blue}{130} }) \color{blue}{.130} \\\Leftrightarrow & 26x+50& = & 65x+260 \\\Leftrightarrow & 26x \color{red}{+50} \color{blue}{-50} \color{blue}{-65x} & = & \color{red}{65x} +260 \color{blue}{-65x} \color{blue}{-50} \\\Leftrightarrow & -39x& = & 210 \\\Leftrightarrow & \frac{-39x}{ \color{red}{-39} }& = & \frac{210}{-39} \\\Leftrightarrow & x = \frac{-70}{13} & & \\ & V = \left\{ \frac{-70}{13} \right\} & \\\end{align}\)