Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{70 is het kleinste gemene veelvoud van 5, 7 en 2} \\ \begin{align} & \frac{x}{5}+\frac{2}{7}& = & \frac{7}{2}x-5 \\\Leftrightarrow & \color{blue}{70.} (\frac{14x}{ \color{blue}{70} }+ \frac{ 20 }{ \color{blue}{70} })& = & (\frac{245}{ \color{blue}{70} }x-\frac{350}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 14x+20& = & 245x-350 \\\Leftrightarrow & 14x \color{red}{+20} \color{blue}{-20} \color{blue}{-245x} & = & \color{red}{245x} -350 \color{blue}{-245x} \color{blue}{-20} \\\Leftrightarrow & -231x& = & -370 \\\Leftrightarrow & \frac{-231x}{ \color{red}{-231} }& = & \frac{-370}{-231} \\\Leftrightarrow & x = \frac{370}{231} & & \\ & V = \left\{ \frac{370}{231} \right\} & \\\end{align}\)
2. \(\text{120 is het kleinste gemene veelvoud van 6, 8 en 5} \\ \begin{align} & \frac{x}{6}+\frac{5}{8}& = & \frac{1}{5}x-2 \\\Leftrightarrow & \color{blue}{120.} (\frac{20x}{ \color{blue}{120} }+ \frac{ 75 }{ \color{blue}{120} })& = & (\frac{24}{ \color{blue}{120} }x-\frac{240}{ \color{blue}{120} }) \color{blue}{.120} \\\Leftrightarrow & 20x+75& = & 24x-240 \\\Leftrightarrow & 20x \color{red}{+75} \color{blue}{-75} \color{blue}{-24x} & = & \color{red}{24x} -240 \color{blue}{-24x} \color{blue}{-75} \\\Leftrightarrow & -4x& = & -315 \\\Leftrightarrow & \frac{-4x}{ \color{red}{-4} }& = & \frac{-315}{-4} \\\Leftrightarrow & x = \frac{315}{4} & & \\ & V = \left\{ \frac{315}{4} \right\} & \\\end{align}\)
3. \(\text{70 is het kleinste gemene veelvoud van 2, 7 en 5} \\ \begin{align} & \frac{x}{2}+\frac{4}{7}& = & \frac{3}{5}x-7 \\\Leftrightarrow & \color{blue}{70.} (\frac{35x}{ \color{blue}{70} }+ \frac{ 40 }{ \color{blue}{70} })& = & (\frac{42}{ \color{blue}{70} }x-\frac{490}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 35x+40& = & 42x-490 \\\Leftrightarrow & 35x \color{red}{+40} \color{blue}{-40} \color{blue}{-42x} & = & \color{red}{42x} -490 \color{blue}{-42x} \color{blue}{-40} \\\Leftrightarrow & -7x& = & -530 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{-530}{-7} \\\Leftrightarrow & x = \frac{530}{7} & & \\ & V = \left\{ \frac{530}{7} \right\} & \\\end{align}\)
4. \(\text{18 is het kleinste gemene veelvoud van 3, 9 en 2} \\ \begin{align} & \frac{x}{3}+\frac{4}{9}& = & \frac{1}{2}x-4 \\\Leftrightarrow & \color{blue}{18.} (\frac{6x}{ \color{blue}{18} }+ \frac{ 8 }{ \color{blue}{18} })& = & (\frac{9}{ \color{blue}{18} }x-\frac{72}{ \color{blue}{18} }) \color{blue}{.18} \\\Leftrightarrow & 6x+8& = & 9x-72 \\\Leftrightarrow & 6x \color{red}{+8} \color{blue}{-8} \color{blue}{-9x} & = & \color{red}{9x} -72 \color{blue}{-9x} \color{blue}{-8} \\\Leftrightarrow & -3x& = & -80 \\\Leftrightarrow & \frac{-3x}{ \color{red}{-3} }& = & \frac{-80}{-3} \\\Leftrightarrow & x = \frac{80}{3} & & \\ & V = \left\{ \frac{80}{3} \right\} & \\\end{align}\)
5. \(\text{390 is het kleinste gemene veelvoud van 6, 13 en 5} \\ \begin{align} & \frac{x}{6}-\frac{2}{13}& = & \frac{1}{5}x-5 \\\Leftrightarrow & \color{blue}{390.} (\frac{65x}{ \color{blue}{390} }- \frac{ 60 }{ \color{blue}{390} })& = & (\frac{78}{ \color{blue}{390} }x-\frac{1950}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 65x-60& = & 78x-1950 \\\Leftrightarrow & 65x \color{red}{-60} \color{blue}{+60} \color{blue}{-78x} & = & \color{red}{78x} -1950 \color{blue}{-78x} \color{blue}{+60} \\\Leftrightarrow & -13x& = & -1890 \\\Leftrightarrow & \frac{-13x}{ \color{red}{-13} }& = & \frac{-1890}{-13} \\\Leftrightarrow & x = \frac{1890}{13} & & \\ & V = \left\{ \frac{1890}{13} \right\} & \\\end{align}\)
6. \(\text{105 is het kleinste gemene veelvoud van 3, 7 en 5} \\ \begin{align} & \frac{x}{3}-\frac{4}{7}& = & \frac{3}{5}x+7 \\\Leftrightarrow & \color{blue}{105.} (\frac{35x}{ \color{blue}{105} }- \frac{ 60 }{ \color{blue}{105} })& = & (\frac{63}{ \color{blue}{105} }x+\frac{735}{ \color{blue}{105} }) \color{blue}{.105} \\\Leftrightarrow & 35x-60& = & 63x+735 \\\Leftrightarrow & 35x \color{red}{-60} \color{blue}{+60} \color{blue}{-63x} & = & \color{red}{63x} +735 \color{blue}{-63x} \color{blue}{+60} \\\Leftrightarrow & -28x& = & 795 \\\Leftrightarrow & \frac{-28x}{ \color{red}{-28} }& = & \frac{795}{-28} \\\Leftrightarrow & x = \frac{-795}{28} & & \\ & V = \left\{ \frac{-795}{28} \right\} & \\\end{align}\)
7. \(\text{66 is het kleinste gemene veelvoud van 3, 11 en 2} \\ \begin{align} & \frac{x}{3}+\frac{3}{11}& = & \frac{5}{2}x+2 \\\Leftrightarrow & \color{blue}{66.} (\frac{22x}{ \color{blue}{66} }+ \frac{ 18 }{ \color{blue}{66} })& = & (\frac{165}{ \color{blue}{66} }x+\frac{132}{ \color{blue}{66} }) \color{blue}{.66} \\\Leftrightarrow & 22x+18& = & 165x+132 \\\Leftrightarrow & 22x \color{red}{+18} \color{blue}{-18} \color{blue}{-165x} & = & \color{red}{165x} +132 \color{blue}{-165x} \color{blue}{-18} \\\Leftrightarrow & -143x& = & 114 \\\Leftrightarrow & \frac{-143x}{ \color{red}{-143} }& = & \frac{114}{-143} \\\Leftrightarrow & x = \frac{-114}{143} & & \\ & V = \left\{ \frac{-114}{143} \right\} & \\\end{align}\)
8. \(\text{24 is het kleinste gemene veelvoud van 4, 8 en 3} \\ \begin{align} & \frac{x}{4}+\frac{5}{8}& = & \frac{-5}{3}x+8 \\\Leftrightarrow & \color{blue}{24.} (\frac{6x}{ \color{blue}{24} }+ \frac{ 15 }{ \color{blue}{24} })& = & (\frac{-40}{ \color{blue}{24} }x+\frac{192}{ \color{blue}{24} }) \color{blue}{.24} \\\Leftrightarrow & 6x+15& = & -40x+192 \\\Leftrightarrow & 6x \color{red}{+15} \color{blue}{-15} \color{blue}{+40x} & = & \color{red}{-40x} +192 \color{blue}{+40x} \color{blue}{-15} \\\Leftrightarrow & 46x& = & 177 \\\Leftrightarrow & \frac{46x}{ \color{red}{46} }& = & \frac{177}{46} \\\Leftrightarrow & x = \frac{177}{46} & & \\ & V = \left\{ \frac{177}{46} \right\} & \\\end{align}\)
9. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}-\frac{2}{7}& = & \frac{2}{3}x+4 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }- \frac{ 24 }{ \color{blue}{84} })& = & (\frac{56}{ \color{blue}{84} }x+\frac{336}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x-24& = & 56x+336 \\\Leftrightarrow & 21x \color{red}{-24} \color{blue}{+24} \color{blue}{-56x} & = & \color{red}{56x} +336 \color{blue}{-56x} \color{blue}{+24} \\\Leftrightarrow & -35x& = & 360 \\\Leftrightarrow & \frac{-35x}{ \color{red}{-35} }& = & \frac{360}{-35} \\\Leftrightarrow & x = \frac{-72}{7} & & \\ & V = \left\{ \frac{-72}{7} \right\} & \\\end{align}\)
10. \(\text{80 is het kleinste gemene veelvoud van 5, 16 en 2} \\ \begin{align} & \frac{x}{5}+\frac{5}{16}& = & \frac{3}{2}x-7 \\\Leftrightarrow & \color{blue}{80.} (\frac{16x}{ \color{blue}{80} }+ \frac{ 25 }{ \color{blue}{80} })& = & (\frac{120}{ \color{blue}{80} }x-\frac{560}{ \color{blue}{80} }) \color{blue}{.80} \\\Leftrightarrow & 16x+25& = & 120x-560 \\\Leftrightarrow & 16x \color{red}{+25} \color{blue}{-25} \color{blue}{-120x} & = & \color{red}{120x} -560 \color{blue}{-120x} \color{blue}{-25} \\\Leftrightarrow & -104x& = & -585 \\\Leftrightarrow & \frac{-104x}{ \color{red}{-104} }& = & \frac{-585}{-104} \\\Leftrightarrow & x = \frac{45}{8} & & \\ & V = \left\{ \frac{45}{8} \right\} & \\\end{align}\)
11. \(\text{132 is het kleinste gemene veelvoud van 4, 11 en 3} \\ \begin{align} & \frac{x}{4}-\frac{4}{11}& = & \frac{-5}{3}x-5 \\\Leftrightarrow & \color{blue}{132.} (\frac{33x}{ \color{blue}{132} }- \frac{ 48 }{ \color{blue}{132} })& = & (\frac{-220}{ \color{blue}{132} }x-\frac{660}{ \color{blue}{132} }) \color{blue}{.132} \\\Leftrightarrow & 33x-48& = & -220x-660 \\\Leftrightarrow & 33x \color{red}{-48} \color{blue}{+48} \color{blue}{+220x} & = & \color{red}{-220x} -660 \color{blue}{+220x} \color{blue}{+48} \\\Leftrightarrow & 253x& = & -612 \\\Leftrightarrow & \frac{253x}{ \color{red}{253} }& = & \frac{-612}{253} \\\Leftrightarrow & x = \frac{-612}{253} & & \\ & V = \left\{ \frac{-612}{253} \right\} & \\\end{align}\)
12. \(\text{240 is het kleinste gemene veelvoud van 6, 16 en 5} \\ \begin{align} & \frac{x}{6}-\frac{3}{16}& = & \frac{-4}{5}x+3 \\\Leftrightarrow & \color{blue}{240.} (\frac{40x}{ \color{blue}{240} }- \frac{ 45 }{ \color{blue}{240} })& = & (\frac{-192}{ \color{blue}{240} }x+\frac{720}{ \color{blue}{240} }) \color{blue}{.240} \\\Leftrightarrow & 40x-45& = & -192x+720 \\\Leftrightarrow & 40x \color{red}{-45} \color{blue}{+45} \color{blue}{+192x} & = & \color{red}{-192x} +720 \color{blue}{+192x} \color{blue}{+45} \\\Leftrightarrow & 232x& = & 765 \\\Leftrightarrow & \frac{232x}{ \color{red}{232} }& = & \frac{765}{232} \\\Leftrightarrow & x = \frac{765}{232} & & \\ & V = \left\{ \frac{765}{232} \right\} & \\\end{align}\)