Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{140 is het kleinste gemene veelvoud van 4, 7 en 5} \\ \begin{align} & \frac{x}{4}+\frac{3}{7}& = & \frac{7}{5}x-3 \\\Leftrightarrow & \color{blue}{140.} (\frac{35x}{ \color{blue}{140} }+ \frac{ 60 }{ \color{blue}{140} })& = & (\frac{196}{ \color{blue}{140} }x-\frac{420}{ \color{blue}{140} }) \color{blue}{.140} \\\Leftrightarrow & 35x+60& = & 196x-420 \\\Leftrightarrow & 35x \color{red}{+60} \color{blue}{-60} \color{blue}{-196x} & = & \color{red}{196x} -420 \color{blue}{-196x} \color{blue}{-60} \\\Leftrightarrow & -161x& = & -480 \\\Leftrightarrow & \frac{-161x}{ \color{red}{-161} }& = & \frac{-480}{-161} \\\Leftrightarrow & x = \frac{480}{161} & & \\ & V = \left\{ \frac{480}{161} \right\} & \\\end{align}\)
2. \(\text{462 is het kleinste gemene veelvoud van 7, 11 en 6} \\ \begin{align} & \frac{x}{7}-\frac{3}{11}& = & \frac{-5}{6}x+4 \\\Leftrightarrow & \color{blue}{462.} (\frac{66x}{ \color{blue}{462} }- \frac{ 126 }{ \color{blue}{462} })& = & (\frac{-385}{ \color{blue}{462} }x+\frac{1848}{ \color{blue}{462} }) \color{blue}{.462} \\\Leftrightarrow & 66x-126& = & -385x+1848 \\\Leftrightarrow & 66x \color{red}{-126} \color{blue}{+126} \color{blue}{+385x} & = & \color{red}{-385x} +1848 \color{blue}{+385x} \color{blue}{+126} \\\Leftrightarrow & 451x& = & 1974 \\\Leftrightarrow & \frac{451x}{ \color{red}{451} }& = & \frac{1974}{451} \\\Leftrightarrow & x = \frac{1974}{451} & & \\ & V = \left\{ \frac{1974}{451} \right\} & \\\end{align}\)
3. \(\text{42 is het kleinste gemene veelvoud van 2, 7 en 3} \\ \begin{align} & \frac{x}{2}-\frac{2}{7}& = & \frac{-8}{3}x+8 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }- \frac{ 12 }{ \color{blue}{42} })& = & (\frac{-112}{ \color{blue}{42} }x+\frac{336}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x-12& = & -112x+336 \\\Leftrightarrow & 21x \color{red}{-12} \color{blue}{+12} \color{blue}{+112x} & = & \color{red}{-112x} +336 \color{blue}{+112x} \color{blue}{+12} \\\Leftrightarrow & 133x& = & 348 \\\Leftrightarrow & \frac{133x}{ \color{red}{133} }& = & \frac{348}{133} \\\Leftrightarrow & x = \frac{348}{133} & & \\ & V = \left\{ \frac{348}{133} \right\} & \\\end{align}\)
4. \(\text{35 is het kleinste gemene veelvoud van 7, 7 en 5} \\ \begin{align} & \frac{x}{7}-\frac{2}{7}& = & \frac{-4}{5}x+5 \\\Leftrightarrow & \color{blue}{35.} (\frac{5x}{ \color{blue}{35} }- \frac{ 10 }{ \color{blue}{35} })& = & (\frac{-28}{ \color{blue}{35} }x+\frac{175}{ \color{blue}{35} }) \color{blue}{.35} \\\Leftrightarrow & 5x-10& = & -28x+175 \\\Leftrightarrow & 5x \color{red}{-10} \color{blue}{+10} \color{blue}{+28x} & = & \color{red}{-28x} +175 \color{blue}{+28x} \color{blue}{+10} \\\Leftrightarrow & 33x& = & 185 \\\Leftrightarrow & \frac{33x}{ \color{red}{33} }& = & \frac{185}{33} \\\Leftrightarrow & x = \frac{185}{33} & & \\ & V = \left\{ \frac{185}{33} \right\} & \\\end{align}\)
5. \(\text{21 is het kleinste gemene veelvoud van 7, 7 en 3} \\ \begin{align} & \frac{x}{7}+\frac{4}{7}& = & \frac{1}{3}x-3 \\\Leftrightarrow & \color{blue}{21.} (\frac{3x}{ \color{blue}{21} }+ \frac{ 12 }{ \color{blue}{21} })& = & (\frac{7}{ \color{blue}{21} }x-\frac{63}{ \color{blue}{21} }) \color{blue}{.21} \\\Leftrightarrow & 3x+12& = & 7x-63 \\\Leftrightarrow & 3x \color{red}{+12} \color{blue}{-12} \color{blue}{-7x} & = & \color{red}{7x} -63 \color{blue}{-7x} \color{blue}{-12} \\\Leftrightarrow & -4x& = & -75 \\\Leftrightarrow & \frac{-4x}{ \color{red}{-4} }& = & \frac{-75}{-4} \\\Leftrightarrow & x = \frac{75}{4} & & \\ & V = \left\{ \frac{75}{4} \right\} & \\\end{align}\)
6. \(\text{165 is het kleinste gemene veelvoud van 5, 11 en 3} \\ \begin{align} & \frac{x}{5}+\frac{4}{11}& = & \frac{-5}{3}x+2 \\\Leftrightarrow & \color{blue}{165.} (\frac{33x}{ \color{blue}{165} }+ \frac{ 60 }{ \color{blue}{165} })& = & (\frac{-275}{ \color{blue}{165} }x+\frac{330}{ \color{blue}{165} }) \color{blue}{.165} \\\Leftrightarrow & 33x+60& = & -275x+330 \\\Leftrightarrow & 33x \color{red}{+60} \color{blue}{-60} \color{blue}{+275x} & = & \color{red}{-275x} +330 \color{blue}{+275x} \color{blue}{-60} \\\Leftrightarrow & 308x& = & 270 \\\Leftrightarrow & \frac{308x}{ \color{red}{308} }& = & \frac{270}{308} \\\Leftrightarrow & x = \frac{135}{154} & & \\ & V = \left\{ \frac{135}{154} \right\} & \\\end{align}\)
7. \(\text{130 is het kleinste gemene veelvoud van 5, 13 en 2} \\ \begin{align} & \frac{x}{5}-\frac{2}{13}& = & \frac{1}{2}x-3 \\\Leftrightarrow & \color{blue}{130.} (\frac{26x}{ \color{blue}{130} }- \frac{ 20 }{ \color{blue}{130} })& = & (\frac{65}{ \color{blue}{130} }x-\frac{390}{ \color{blue}{130} }) \color{blue}{.130} \\\Leftrightarrow & 26x-20& = & 65x-390 \\\Leftrightarrow & 26x \color{red}{-20} \color{blue}{+20} \color{blue}{-65x} & = & \color{red}{65x} -390 \color{blue}{-65x} \color{blue}{+20} \\\Leftrightarrow & -39x& = & -370 \\\Leftrightarrow & \frac{-39x}{ \color{red}{-39} }& = & \frac{-370}{-39} \\\Leftrightarrow & x = \frac{370}{39} & & \\ & V = \left\{ \frac{370}{39} \right\} & \\\end{align}\)
8. \(\text{210 is het kleinste gemene veelvoud van 7, 10 en 3} \\ \begin{align} & \frac{x}{7}+\frac{3}{10}& = & \frac{-2}{3}x-3 \\\Leftrightarrow & \color{blue}{210.} (\frac{30x}{ \color{blue}{210} }+ \frac{ 63 }{ \color{blue}{210} })& = & (\frac{-140}{ \color{blue}{210} }x-\frac{630}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 30x+63& = & -140x-630 \\\Leftrightarrow & 30x \color{red}{+63} \color{blue}{-63} \color{blue}{+140x} & = & \color{red}{-140x} -630 \color{blue}{+140x} \color{blue}{-63} \\\Leftrightarrow & 170x& = & -693 \\\Leftrightarrow & \frac{170x}{ \color{red}{170} }& = & \frac{-693}{170} \\\Leftrightarrow & x = \frac{-693}{170} & & \\ & V = \left\{ \frac{-693}{170} \right\} & \\\end{align}\)
9. \(\text{30 is het kleinste gemene veelvoud van 2, 10 en 3} \\ \begin{align} & \frac{x}{2}+\frac{3}{10}& = & \frac{2}{3}x-8 \\\Leftrightarrow & \color{blue}{30.} (\frac{15x}{ \color{blue}{30} }+ \frac{ 9 }{ \color{blue}{30} })& = & (\frac{20}{ \color{blue}{30} }x-\frac{240}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 15x+9& = & 20x-240 \\\Leftrightarrow & 15x \color{red}{+9} \color{blue}{-9} \color{blue}{-20x} & = & \color{red}{20x} -240 \color{blue}{-20x} \color{blue}{-9} \\\Leftrightarrow & -5x& = & -249 \\\Leftrightarrow & \frac{-5x}{ \color{red}{-5} }& = & \frac{-249}{-5} \\\Leftrightarrow & x = \frac{249}{5} & & \\ & V = \left\{ \frac{249}{5} \right\} & \\\end{align}\)
10. \(\text{120 is het kleinste gemene veelvoud van 6, 8 en 5} \\ \begin{align} & \frac{x}{6}+\frac{5}{8}& = & \frac{-7}{5}x+8 \\\Leftrightarrow & \color{blue}{120.} (\frac{20x}{ \color{blue}{120} }+ \frac{ 75 }{ \color{blue}{120} })& = & (\frac{-168}{ \color{blue}{120} }x+\frac{960}{ \color{blue}{120} }) \color{blue}{.120} \\\Leftrightarrow & 20x+75& = & -168x+960 \\\Leftrightarrow & 20x \color{red}{+75} \color{blue}{-75} \color{blue}{+168x} & = & \color{red}{-168x} +960 \color{blue}{+168x} \color{blue}{-75} \\\Leftrightarrow & 188x& = & 885 \\\Leftrightarrow & \frac{188x}{ \color{red}{188} }& = & \frac{885}{188} \\\Leftrightarrow & x = \frac{885}{188} & & \\ & V = \left\{ \frac{885}{188} \right\} & \\\end{align}\)
11. \(\text{66 is het kleinste gemene veelvoud van 3, 11 en 2} \\ \begin{align} & \frac{x}{3}-\frac{4}{11}& = & \frac{1}{2}x-1 \\\Leftrightarrow & \color{blue}{66.} (\frac{22x}{ \color{blue}{66} }- \frac{ 24 }{ \color{blue}{66} })& = & (\frac{33}{ \color{blue}{66} }x-\frac{66}{ \color{blue}{66} }) \color{blue}{.66} \\\Leftrightarrow & 22x-24& = & 33x-66 \\\Leftrightarrow & 22x \color{red}{-24} \color{blue}{+24} \color{blue}{-33x} & = & \color{red}{33x} -66 \color{blue}{-33x} \color{blue}{+24} \\\Leftrightarrow & -11x& = & -42 \\\Leftrightarrow & \frac{-11x}{ \color{red}{-11} }& = & \frac{-42}{-11} \\\Leftrightarrow & x = \frac{42}{11} & & \\ & V = \left\{ \frac{42}{11} \right\} & \\\end{align}\)
12. \(\text{130 is het kleinste gemene veelvoud van 2, 13 en 5} \\ \begin{align} & \frac{x}{2}+\frac{4}{13}& = & \frac{-4}{5}x-3 \\\Leftrightarrow & \color{blue}{130.} (\frac{65x}{ \color{blue}{130} }+ \frac{ 40 }{ \color{blue}{130} })& = & (\frac{-104}{ \color{blue}{130} }x-\frac{390}{ \color{blue}{130} }) \color{blue}{.130} \\\Leftrightarrow & 65x+40& = & -104x-390 \\\Leftrightarrow & 65x \color{red}{+40} \color{blue}{-40} \color{blue}{+104x} & = & \color{red}{-104x} -390 \color{blue}{+104x} \color{blue}{-40} \\\Leftrightarrow & 169x& = & -430 \\\Leftrightarrow & \frac{169x}{ \color{red}{169} }& = & \frac{-430}{169} \\\Leftrightarrow & x = \frac{-430}{169} & & \\ & V = \left\{ \frac{-430}{169} \right\} & \\\end{align}\)