Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{420 is het kleinste gemene veelvoud van 7, 12 en 5} \\ \begin{align} & \frac{x}{7}-\frac{5}{12}& = & \frac{-7}{5}x-4 \\\Leftrightarrow & \color{blue}{420.} (\frac{60x}{ \color{blue}{420} }- \frac{ 175 }{ \color{blue}{420} })& = & (\frac{-588}{ \color{blue}{420} }x-\frac{1680}{ \color{blue}{420} }) \color{blue}{.420} \\\Leftrightarrow & 60x-175& = & -588x-1680 \\\Leftrightarrow & 60x \color{red}{-175} \color{blue}{+175} \color{blue}{+588x} & = & \color{red}{-588x} -1680 \color{blue}{+588x} \color{blue}{+175} \\\Leftrightarrow & 648x& = & -1505 \\\Leftrightarrow & \frac{648x}{ \color{red}{648} }& = & \frac{-1505}{648} \\\Leftrightarrow & x = \frac{-1505}{648} & & \\ & V = \left\{ \frac{-1505}{648} \right\} & \\\end{align}\)
2. \(\text{120 is het kleinste gemene veelvoud van 6, 8 en 5} \\ \begin{align} & \frac{x}{6}+\frac{5}{8}& = & \frac{-7}{5}x+6 \\\Leftrightarrow & \color{blue}{120.} (\frac{20x}{ \color{blue}{120} }+ \frac{ 75 }{ \color{blue}{120} })& = & (\frac{-168}{ \color{blue}{120} }x+\frac{720}{ \color{blue}{120} }) \color{blue}{.120} \\\Leftrightarrow & 20x+75& = & -168x+720 \\\Leftrightarrow & 20x \color{red}{+75} \color{blue}{-75} \color{blue}{+168x} & = & \color{red}{-168x} +720 \color{blue}{+168x} \color{blue}{-75} \\\Leftrightarrow & 188x& = & 645 \\\Leftrightarrow & \frac{188x}{ \color{red}{188} }& = & \frac{645}{188} \\\Leftrightarrow & x = \frac{645}{188} & & \\ & V = \left\{ \frac{645}{188} \right\} & \\\end{align}\)
3. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}+\frac{5}{7}& = & \frac{-8}{3}x+2 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }+ \frac{ 60 }{ \color{blue}{84} })& = & (\frac{-224}{ \color{blue}{84} }x+\frac{168}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x+60& = & -224x+168 \\\Leftrightarrow & 21x \color{red}{+60} \color{blue}{-60} \color{blue}{+224x} & = & \color{red}{-224x} +168 \color{blue}{+224x} \color{blue}{-60} \\\Leftrightarrow & 245x& = & 108 \\\Leftrightarrow & \frac{245x}{ \color{red}{245} }& = & \frac{108}{245} \\\Leftrightarrow & x = \frac{108}{245} & & \\ & V = \left\{ \frac{108}{245} \right\} & \\\end{align}\)
4. \(\text{132 is het kleinste gemene veelvoud van 3, 11 en 4} \\ \begin{align} & \frac{x}{3}-\frac{5}{11}& = & \frac{1}{4}x-2 \\\Leftrightarrow & \color{blue}{132.} (\frac{44x}{ \color{blue}{132} }- \frac{ 60 }{ \color{blue}{132} })& = & (\frac{33}{ \color{blue}{132} }x-\frac{264}{ \color{blue}{132} }) \color{blue}{.132} \\\Leftrightarrow & 44x-60& = & 33x-264 \\\Leftrightarrow & 44x \color{red}{-60} \color{blue}{+60} \color{blue}{-33x} & = & \color{red}{33x} -264 \color{blue}{-33x} \color{blue}{+60} \\\Leftrightarrow & 11x& = & -204 \\\Leftrightarrow & \frac{11x}{ \color{red}{11} }& = & \frac{-204}{11} \\\Leftrightarrow & x = \frac{-204}{11} & & \\ & V = \left\{ \frac{-204}{11} \right\} & \\\end{align}\)
5. \(\text{42 is het kleinste gemene veelvoud van 2, 7 en 3} \\ \begin{align} & \frac{x}{2}+\frac{4}{7}& = & \frac{2}{3}x+2 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }+ \frac{ 24 }{ \color{blue}{42} })& = & (\frac{28}{ \color{blue}{42} }x+\frac{84}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x+24& = & 28x+84 \\\Leftrightarrow & 21x \color{red}{+24} \color{blue}{-24} \color{blue}{-28x} & = & \color{red}{28x} +84 \color{blue}{-28x} \color{blue}{-24} \\\Leftrightarrow & -7x& = & 60 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{60}{-7} \\\Leftrightarrow & x = \frac{-60}{7} & & \\ & V = \left\{ \frac{-60}{7} \right\} & \\\end{align}\)
6. \(\text{260 is het kleinste gemene veelvoud van 4, 13 en 5} \\ \begin{align} & \frac{x}{4}-\frac{4}{13}& = & \frac{1}{5}x-3 \\\Leftrightarrow & \color{blue}{260.} (\frac{65x}{ \color{blue}{260} }- \frac{ 80 }{ \color{blue}{260} })& = & (\frac{52}{ \color{blue}{260} }x-\frac{780}{ \color{blue}{260} }) \color{blue}{.260} \\\Leftrightarrow & 65x-80& = & 52x-780 \\\Leftrightarrow & 65x \color{red}{-80} \color{blue}{+80} \color{blue}{-52x} & = & \color{red}{52x} -780 \color{blue}{-52x} \color{blue}{+80} \\\Leftrightarrow & 13x& = & -700 \\\Leftrightarrow & \frac{13x}{ \color{red}{13} }& = & \frac{-700}{13} \\\Leftrightarrow & x = \frac{-700}{13} & & \\ & V = \left\{ \frac{-700}{13} \right\} & \\\end{align}\)
7. \(\text{12 is het kleinste gemene veelvoud van 4, 6 en 3} \\ \begin{align} & \frac{x}{4}-\frac{5}{6}& = & \frac{-5}{3}x-5 \\\Leftrightarrow & \color{blue}{12.} (\frac{3x}{ \color{blue}{12} }- \frac{ 10 }{ \color{blue}{12} })& = & (\frac{-20}{ \color{blue}{12} }x-\frac{60}{ \color{blue}{12} }) \color{blue}{.12} \\\Leftrightarrow & 3x-10& = & -20x-60 \\\Leftrightarrow & 3x \color{red}{-10} \color{blue}{+10} \color{blue}{+20x} & = & \color{red}{-20x} -60 \color{blue}{+20x} \color{blue}{+10} \\\Leftrightarrow & 23x& = & -50 \\\Leftrightarrow & \frac{23x}{ \color{red}{23} }& = & \frac{-50}{23} \\\Leftrightarrow & x = \frac{-50}{23} & & \\ & V = \left\{ \frac{-50}{23} \right\} & \\\end{align}\)
8. \(\text{30 is het kleinste gemene veelvoud van 5, 15 en 2} \\ \begin{align} & \frac{x}{5}+\frac{2}{15}& = & \frac{1}{2}x-1 \\\Leftrightarrow & \color{blue}{30.} (\frac{6x}{ \color{blue}{30} }+ \frac{ 4 }{ \color{blue}{30} })& = & (\frac{15}{ \color{blue}{30} }x-\frac{30}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 6x+4& = & 15x-30 \\\Leftrightarrow & 6x \color{red}{+4} \color{blue}{-4} \color{blue}{-15x} & = & \color{red}{15x} -30 \color{blue}{-15x} \color{blue}{-4} \\\Leftrightarrow & -9x& = & -34 \\\Leftrightarrow & \frac{-9x}{ \color{red}{-9} }& = & \frac{-34}{-9} \\\Leftrightarrow & x = \frac{34}{9} & & \\ & V = \left\{ \frac{34}{9} \right\} & \\\end{align}\)
9. \(\text{240 is het kleinste gemene veelvoud van 5, 16 en 6} \\ \begin{align} & \frac{x}{5}+\frac{3}{16}& = & \frac{-5}{6}x+7 \\\Leftrightarrow & \color{blue}{240.} (\frac{48x}{ \color{blue}{240} }+ \frac{ 45 }{ \color{blue}{240} })& = & (\frac{-200}{ \color{blue}{240} }x+\frac{1680}{ \color{blue}{240} }) \color{blue}{.240} \\\Leftrightarrow & 48x+45& = & -200x+1680 \\\Leftrightarrow & 48x \color{red}{+45} \color{blue}{-45} \color{blue}{+200x} & = & \color{red}{-200x} +1680 \color{blue}{+200x} \color{blue}{-45} \\\Leftrightarrow & 248x& = & 1635 \\\Leftrightarrow & \frac{248x}{ \color{red}{248} }& = & \frac{1635}{248} \\\Leftrightarrow & x = \frac{1635}{248} & & \\ & V = \left\{ \frac{1635}{248} \right\} & \\\end{align}\)
10. \(\text{390 is het kleinste gemene veelvoud van 5, 13 en 6} \\ \begin{align} & \frac{x}{5}-\frac{2}{13}& = & \frac{7}{6}x+5 \\\Leftrightarrow & \color{blue}{390.} (\frac{78x}{ \color{blue}{390} }- \frac{ 60 }{ \color{blue}{390} })& = & (\frac{455}{ \color{blue}{390} }x+\frac{1950}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 78x-60& = & 455x+1950 \\\Leftrightarrow & 78x \color{red}{-60} \color{blue}{+60} \color{blue}{-455x} & = & \color{red}{455x} +1950 \color{blue}{-455x} \color{blue}{+60} \\\Leftrightarrow & -377x& = & 2010 \\\Leftrightarrow & \frac{-377x}{ \color{red}{-377} }& = & \frac{2010}{-377} \\\Leftrightarrow & x = \frac{-2010}{377} & & \\ & V = \left\{ \frac{-2010}{377} \right\} & \\\end{align}\)
11. \(\text{30 is het kleinste gemene veelvoud van 2, 15 en 3} \\ \begin{align} & \frac{x}{2}+\frac{2}{15}& = & \frac{7}{3}x+6 \\\Leftrightarrow & \color{blue}{30.} (\frac{15x}{ \color{blue}{30} }+ \frac{ 4 }{ \color{blue}{30} })& = & (\frac{70}{ \color{blue}{30} }x+\frac{180}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 15x+4& = & 70x+180 \\\Leftrightarrow & 15x \color{red}{+4} \color{blue}{-4} \color{blue}{-70x} & = & \color{red}{70x} +180 \color{blue}{-70x} \color{blue}{-4} \\\Leftrightarrow & -55x& = & 176 \\\Leftrightarrow & \frac{-55x}{ \color{red}{-55} }& = & \frac{176}{-55} \\\Leftrightarrow & x = \frac{-16}{5} & & \\ & V = \left\{ \frac{-16}{5} \right\} & \\\end{align}\)
12. \(\text{130 is het kleinste gemene veelvoud van 5, 13 en 2} \\ \begin{align} & \frac{x}{5}-\frac{3}{13}& = & \frac{5}{2}x+7 \\\Leftrightarrow & \color{blue}{130.} (\frac{26x}{ \color{blue}{130} }- \frac{ 30 }{ \color{blue}{130} })& = & (\frac{325}{ \color{blue}{130} }x+\frac{910}{ \color{blue}{130} }) \color{blue}{.130} \\\Leftrightarrow & 26x-30& = & 325x+910 \\\Leftrightarrow & 26x \color{red}{-30} \color{blue}{+30} \color{blue}{-325x} & = & \color{red}{325x} +910 \color{blue}{-325x} \color{blue}{+30} \\\Leftrightarrow & -299x& = & 940 \\\Leftrightarrow & \frac{-299x}{ \color{red}{-299} }& = & \frac{940}{-299} \\\Leftrightarrow & x = \frac{-940}{299} & & \\ & V = \left\{ \frac{-940}{299} \right\} & \\\end{align}\)