Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{30 is het kleinste gemene veelvoud van 3, 10 en 5} \\ \begin{align} & \frac{x}{3}+\frac{3}{10}& = & \frac{3}{5}x+5 \\\Leftrightarrow & \color{blue}{30.} (\frac{10x}{ \color{blue}{30} }+ \frac{ 9 }{ \color{blue}{30} })& = & (\frac{18}{ \color{blue}{30} }x+\frac{150}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 10x+9& = & 18x+150 \\\Leftrightarrow & 10x \color{red}{+9} \color{blue}{-9} \color{blue}{-18x} & = & \color{red}{18x} +150 \color{blue}{-18x} \color{blue}{-9} \\\Leftrightarrow & -8x& = & 141 \\\Leftrightarrow & \frac{-8x}{ \color{red}{-8} }& = & \frac{141}{-8} \\\Leftrightarrow & x = \frac{-141}{8} & & \\ & V = \left\{ \frac{-141}{8} \right\} & \\\end{align}\)
2. \(\text{15 is het kleinste gemene veelvoud van 5, 15 en 3} \\ \begin{align} & \frac{x}{5}+\frac{4}{15}& = & \frac{4}{3}x-6 \\\Leftrightarrow & \color{blue}{15.} (\frac{3x}{ \color{blue}{15} }+ \frac{ 4 }{ \color{blue}{15} })& = & (\frac{20}{ \color{blue}{15} }x-\frac{90}{ \color{blue}{15} }) \color{blue}{.15} \\\Leftrightarrow & 3x+4& = & 20x-90 \\\Leftrightarrow & 3x \color{red}{+4} \color{blue}{-4} \color{blue}{-20x} & = & \color{red}{20x} -90 \color{blue}{-20x} \color{blue}{-4} \\\Leftrightarrow & -17x& = & -94 \\\Leftrightarrow & \frac{-17x}{ \color{red}{-17} }& = & \frac{-94}{-17} \\\Leftrightarrow & x = \frac{94}{17} & & \\ & V = \left\{ \frac{94}{17} \right\} & \\\end{align}\)
3. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}-\frac{4}{7}& = & \frac{7}{5}x+7 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }- \frac{ 120 }{ \color{blue}{210} })& = & (\frac{294}{ \color{blue}{210} }x+\frac{1470}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x-120& = & 294x+1470 \\\Leftrightarrow & 35x \color{red}{-120} \color{blue}{+120} \color{blue}{-294x} & = & \color{red}{294x} +1470 \color{blue}{-294x} \color{blue}{+120} \\\Leftrightarrow & -259x& = & 1590 \\\Leftrightarrow & \frac{-259x}{ \color{red}{-259} }& = & \frac{1590}{-259} \\\Leftrightarrow & x = \frac{-1590}{259} & & \\ & V = \left\{ \frac{-1590}{259} \right\} & \\\end{align}\)
4. \(\text{70 is het kleinste gemene veelvoud van 7, 10 en 2} \\ \begin{align} & \frac{x}{7}+\frac{3}{10}& = & \frac{1}{2}x-7 \\\Leftrightarrow & \color{blue}{70.} (\frac{10x}{ \color{blue}{70} }+ \frac{ 21 }{ \color{blue}{70} })& = & (\frac{35}{ \color{blue}{70} }x-\frac{490}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 10x+21& = & 35x-490 \\\Leftrightarrow & 10x \color{red}{+21} \color{blue}{-21} \color{blue}{-35x} & = & \color{red}{35x} -490 \color{blue}{-35x} \color{blue}{-21} \\\Leftrightarrow & -25x& = & -511 \\\Leftrightarrow & \frac{-25x}{ \color{red}{-25} }& = & \frac{-511}{-25} \\\Leftrightarrow & x = \frac{511}{25} & & \\ & V = \left\{ \frac{511}{25} \right\} & \\\end{align}\)
5. \(\text{18 is het kleinste gemene veelvoud van 2, 9 en 3} \\ \begin{align} & \frac{x}{2}-\frac{4}{9}& = & \frac{-5}{3}x+4 \\\Leftrightarrow & \color{blue}{18.} (\frac{9x}{ \color{blue}{18} }- \frac{ 8 }{ \color{blue}{18} })& = & (\frac{-30}{ \color{blue}{18} }x+\frac{72}{ \color{blue}{18} }) \color{blue}{.18} \\\Leftrightarrow & 9x-8& = & -30x+72 \\\Leftrightarrow & 9x \color{red}{-8} \color{blue}{+8} \color{blue}{+30x} & = & \color{red}{-30x} +72 \color{blue}{+30x} \color{blue}{+8} \\\Leftrightarrow & 39x& = & 80 \\\Leftrightarrow & \frac{39x}{ \color{red}{39} }& = & \frac{80}{39} \\\Leftrightarrow & x = \frac{80}{39} & & \\ & V = \left\{ \frac{80}{39} \right\} & \\\end{align}\)
6. \(\text{110 is het kleinste gemene veelvoud van 2, 11 en 5} \\ \begin{align} & \frac{x}{2}+\frac{4}{11}& = & \frac{-2}{5}x-1 \\\Leftrightarrow & \color{blue}{110.} (\frac{55x}{ \color{blue}{110} }+ \frac{ 40 }{ \color{blue}{110} })& = & (\frac{-44}{ \color{blue}{110} }x-\frac{110}{ \color{blue}{110} }) \color{blue}{.110} \\\Leftrightarrow & 55x+40& = & -44x-110 \\\Leftrightarrow & 55x \color{red}{+40} \color{blue}{-40} \color{blue}{+44x} & = & \color{red}{-44x} -110 \color{blue}{+44x} \color{blue}{-40} \\\Leftrightarrow & 99x& = & -150 \\\Leftrightarrow & \frac{99x}{ \color{red}{99} }& = & \frac{-150}{99} \\\Leftrightarrow & x = \frac{-50}{33} & & \\ & V = \left\{ \frac{-50}{33} \right\} & \\\end{align}\)
7. \(\text{42 is het kleinste gemene veelvoud van 2, 7 en 3} \\ \begin{align} & \frac{x}{2}-\frac{2}{7}& = & \frac{-5}{3}x-7 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }- \frac{ 12 }{ \color{blue}{42} })& = & (\frac{-70}{ \color{blue}{42} }x-\frac{294}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x-12& = & -70x-294 \\\Leftrightarrow & 21x \color{red}{-12} \color{blue}{+12} \color{blue}{+70x} & = & \color{red}{-70x} -294 \color{blue}{+70x} \color{blue}{+12} \\\Leftrightarrow & 91x& = & -282 \\\Leftrightarrow & \frac{91x}{ \color{red}{91} }& = & \frac{-282}{91} \\\Leftrightarrow & x = \frac{-282}{91} & & \\ & V = \left\{ \frac{-282}{91} \right\} & \\\end{align}\)
8. \(\text{156 is het kleinste gemene veelvoud van 4, 13 en 3} \\ \begin{align} & \frac{x}{4}-\frac{3}{13}& = & \frac{1}{3}x-5 \\\Leftrightarrow & \color{blue}{156.} (\frac{39x}{ \color{blue}{156} }- \frac{ 36 }{ \color{blue}{156} })& = & (\frac{52}{ \color{blue}{156} }x-\frac{780}{ \color{blue}{156} }) \color{blue}{.156} \\\Leftrightarrow & 39x-36& = & 52x-780 \\\Leftrightarrow & 39x \color{red}{-36} \color{blue}{+36} \color{blue}{-52x} & = & \color{red}{52x} -780 \color{blue}{-52x} \color{blue}{+36} \\\Leftrightarrow & -13x& = & -744 \\\Leftrightarrow & \frac{-13x}{ \color{red}{-13} }& = & \frac{-744}{-13} \\\Leftrightarrow & x = \frac{744}{13} & & \\ & V = \left\{ \frac{744}{13} \right\} & \\\end{align}\)
9. \(\text{132 is het kleinste gemene veelvoud van 3, 11 en 4} \\ \begin{align} & \frac{x}{3}+\frac{5}{11}& = & \frac{-3}{4}x+7 \\\Leftrightarrow & \color{blue}{132.} (\frac{44x}{ \color{blue}{132} }+ \frac{ 60 }{ \color{blue}{132} })& = & (\frac{-99}{ \color{blue}{132} }x+\frac{924}{ \color{blue}{132} }) \color{blue}{.132} \\\Leftrightarrow & 44x+60& = & -99x+924 \\\Leftrightarrow & 44x \color{red}{+60} \color{blue}{-60} \color{blue}{+99x} & = & \color{red}{-99x} +924 \color{blue}{+99x} \color{blue}{-60} \\\Leftrightarrow & 143x& = & 864 \\\Leftrightarrow & \frac{143x}{ \color{red}{143} }& = & \frac{864}{143} \\\Leftrightarrow & x = \frac{864}{143} & & \\ & V = \left\{ \frac{864}{143} \right\} & \\\end{align}\)
10. \(\text{220 is het kleinste gemene veelvoud van 4, 11 en 5} \\ \begin{align} & \frac{x}{4}-\frac{2}{11}& = & \frac{-2}{5}x+3 \\\Leftrightarrow & \color{blue}{220.} (\frac{55x}{ \color{blue}{220} }- \frac{ 40 }{ \color{blue}{220} })& = & (\frac{-88}{ \color{blue}{220} }x+\frac{660}{ \color{blue}{220} }) \color{blue}{.220} \\\Leftrightarrow & 55x-40& = & -88x+660 \\\Leftrightarrow & 55x \color{red}{-40} \color{blue}{+40} \color{blue}{+88x} & = & \color{red}{-88x} +660 \color{blue}{+88x} \color{blue}{+40} \\\Leftrightarrow & 143x& = & 700 \\\Leftrightarrow & \frac{143x}{ \color{red}{143} }& = & \frac{700}{143} \\\Leftrightarrow & x = \frac{700}{143} & & \\ & V = \left\{ \frac{700}{143} \right\} & \\\end{align}\)
11. \(\text{24 is het kleinste gemene veelvoud van 2, 8 en 3} \\ \begin{align} & \frac{x}{2}+\frac{3}{8}& = & \frac{7}{3}x-6 \\\Leftrightarrow & \color{blue}{24.} (\frac{12x}{ \color{blue}{24} }+ \frac{ 9 }{ \color{blue}{24} })& = & (\frac{56}{ \color{blue}{24} }x-\frac{144}{ \color{blue}{24} }) \color{blue}{.24} \\\Leftrightarrow & 12x+9& = & 56x-144 \\\Leftrightarrow & 12x \color{red}{+9} \color{blue}{-9} \color{blue}{-56x} & = & \color{red}{56x} -144 \color{blue}{-56x} \color{blue}{-9} \\\Leftrightarrow & -44x& = & -153 \\\Leftrightarrow & \frac{-44x}{ \color{red}{-44} }& = & \frac{-153}{-44} \\\Leftrightarrow & x = \frac{153}{44} & & \\ & V = \left\{ \frac{153}{44} \right\} & \\\end{align}\)
12. \(\text{14 is het kleinste gemene veelvoud van 7, 7 en 2} \\ \begin{align} & \frac{x}{7}-\frac{4}{7}& = & \frac{1}{2}x-5 \\\Leftrightarrow & \color{blue}{14.} (\frac{2x}{ \color{blue}{14} }- \frac{ 8 }{ \color{blue}{14} })& = & (\frac{7}{ \color{blue}{14} }x-\frac{70}{ \color{blue}{14} }) \color{blue}{.14} \\\Leftrightarrow & 2x-8& = & 7x-70 \\\Leftrightarrow & 2x \color{red}{-8} \color{blue}{+8} \color{blue}{-7x} & = & \color{red}{7x} -70 \color{blue}{-7x} \color{blue}{+8} \\\Leftrightarrow & -5x& = & -62 \\\Leftrightarrow & \frac{-5x}{ \color{red}{-5} }& = & \frac{-62}{-5} \\\Leftrightarrow & x = \frac{62}{5} & & \\ & V = \left\{ \frac{62}{5} \right\} & \\\end{align}\)