Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{140 is het kleinste gemene veelvoud van 4, 7 en 5} \\ \begin{align} & \frac{x}{4}+\frac{3}{7}& = & \frac{1}{5}x+2 \\\Leftrightarrow & \color{blue}{140.} (\frac{35x}{ \color{blue}{140} }+ \frac{ 60 }{ \color{blue}{140} })& = & (\frac{28}{ \color{blue}{140} }x+\frac{280}{ \color{blue}{140} }) \color{blue}{.140} \\\Leftrightarrow & 35x+60& = & 28x+280 \\\Leftrightarrow & 35x \color{red}{+60} \color{blue}{-60} \color{blue}{-28x} & = & \color{red}{28x} +280 \color{blue}{-28x} \color{blue}{-60} \\\Leftrightarrow & 7x& = & 220 \\\Leftrightarrow & \frac{7x}{ \color{red}{7} }& = & \frac{220}{7} \\\Leftrightarrow & x = \frac{220}{7} & & \\ & V = \left\{ \frac{220}{7} \right\} & \\\end{align}\)
2. \(\text{546 is het kleinste gemene veelvoud van 7, 13 en 6} \\ \begin{align} & \frac{x}{7}-\frac{5}{13}& = & \frac{5}{6}x-1 \\\Leftrightarrow & \color{blue}{546.} (\frac{78x}{ \color{blue}{546} }- \frac{ 210 }{ \color{blue}{546} })& = & (\frac{455}{ \color{blue}{546} }x-\frac{546}{ \color{blue}{546} }) \color{blue}{.546} \\\Leftrightarrow & 78x-210& = & 455x-546 \\\Leftrightarrow & 78x \color{red}{-210} \color{blue}{+210} \color{blue}{-455x} & = & \color{red}{455x} -546 \color{blue}{-455x} \color{blue}{+210} \\\Leftrightarrow & -377x& = & -336 \\\Leftrightarrow & \frac{-377x}{ \color{red}{-377} }& = & \frac{-336}{-377} \\\Leftrightarrow & x = \frac{336}{377} & & \\ & V = \left\{ \frac{336}{377} \right\} & \\\end{align}\)
3. \(\text{6 is het kleinste gemene veelvoud van 2, 6 en 3} \\ \begin{align} & \frac{x}{2}+\frac{5}{6}& = & \frac{4}{3}x-1 \\\Leftrightarrow & \color{blue}{6.} (\frac{3x}{ \color{blue}{6} }+ \frac{ 5 }{ \color{blue}{6} })& = & (\frac{8}{ \color{blue}{6} }x-\frac{6}{ \color{blue}{6} }) \color{blue}{.6} \\\Leftrightarrow & 3x+5& = & 8x-6 \\\Leftrightarrow & 3x \color{red}{+5} \color{blue}{-5} \color{blue}{-8x} & = & \color{red}{8x} -6 \color{blue}{-8x} \color{blue}{-5} \\\Leftrightarrow & -5x& = & -11 \\\Leftrightarrow & \frac{-5x}{ \color{red}{-5} }& = & \frac{-11}{-5} \\\Leftrightarrow & x = \frac{11}{5} & & \\ & V = \left\{ \frac{11}{5} \right\} & \\\end{align}\)
4. \(\text{110 is het kleinste gemene veelvoud van 5, 11 en 2} \\ \begin{align} & \frac{x}{5}-\frac{5}{11}& = & \frac{3}{2}x+6 \\\Leftrightarrow & \color{blue}{110.} (\frac{22x}{ \color{blue}{110} }- \frac{ 50 }{ \color{blue}{110} })& = & (\frac{165}{ \color{blue}{110} }x+\frac{660}{ \color{blue}{110} }) \color{blue}{.110} \\\Leftrightarrow & 22x-50& = & 165x+660 \\\Leftrightarrow & 22x \color{red}{-50} \color{blue}{+50} \color{blue}{-165x} & = & \color{red}{165x} +660 \color{blue}{-165x} \color{blue}{+50} \\\Leftrightarrow & -143x& = & 710 \\\Leftrightarrow & \frac{-143x}{ \color{red}{-143} }& = & \frac{710}{-143} \\\Leftrightarrow & x = \frac{-710}{143} & & \\ & V = \left\{ \frac{-710}{143} \right\} & \\\end{align}\)
5. \(\text{66 is het kleinste gemene veelvoud van 3, 11 en 2} \\ \begin{align} & \frac{x}{3}+\frac{2}{11}& = & \frac{1}{2}x-5 \\\Leftrightarrow & \color{blue}{66.} (\frac{22x}{ \color{blue}{66} }+ \frac{ 12 }{ \color{blue}{66} })& = & (\frac{33}{ \color{blue}{66} }x-\frac{330}{ \color{blue}{66} }) \color{blue}{.66} \\\Leftrightarrow & 22x+12& = & 33x-330 \\\Leftrightarrow & 22x \color{red}{+12} \color{blue}{-12} \color{blue}{-33x} & = & \color{red}{33x} -330 \color{blue}{-33x} \color{blue}{-12} \\\Leftrightarrow & -11x& = & -342 \\\Leftrightarrow & \frac{-11x}{ \color{red}{-11} }& = & \frac{-342}{-11} \\\Leftrightarrow & x = \frac{342}{11} & & \\ & V = \left\{ \frac{342}{11} \right\} & \\\end{align}\)
6. \(\text{120 is het kleinste gemene veelvoud van 5, 8 en 6} \\ \begin{align} & \frac{x}{5}-\frac{3}{8}& = & \frac{1}{6}x+8 \\\Leftrightarrow & \color{blue}{120.} (\frac{24x}{ \color{blue}{120} }- \frac{ 45 }{ \color{blue}{120} })& = & (\frac{20}{ \color{blue}{120} }x+\frac{960}{ \color{blue}{120} }) \color{blue}{.120} \\\Leftrightarrow & 24x-45& = & 20x+960 \\\Leftrightarrow & 24x \color{red}{-45} \color{blue}{+45} \color{blue}{-20x} & = & \color{red}{20x} +960 \color{blue}{-20x} \color{blue}{+45} \\\Leftrightarrow & 4x& = & 1005 \\\Leftrightarrow & \frac{4x}{ \color{red}{4} }& = & \frac{1005}{4} \\\Leftrightarrow & x = \frac{1005}{4} & & \\ & V = \left\{ \frac{1005}{4} \right\} & \\\end{align}\)
7. \(\text{390 is het kleinste gemene veelvoud van 5, 13 en 6} \\ \begin{align} & \frac{x}{5}+\frac{3}{13}& = & \frac{-5}{6}x-7 \\\Leftrightarrow & \color{blue}{390.} (\frac{78x}{ \color{blue}{390} }+ \frac{ 90 }{ \color{blue}{390} })& = & (\frac{-325}{ \color{blue}{390} }x-\frac{2730}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 78x+90& = & -325x-2730 \\\Leftrightarrow & 78x \color{red}{+90} \color{blue}{-90} \color{blue}{+325x} & = & \color{red}{-325x} -2730 \color{blue}{+325x} \color{blue}{-90} \\\Leftrightarrow & 403x& = & -2820 \\\Leftrightarrow & \frac{403x}{ \color{red}{403} }& = & \frac{-2820}{403} \\\Leftrightarrow & x = \frac{-2820}{403} & & \\ & V = \left\{ \frac{-2820}{403} \right\} & \\\end{align}\)
8. \(\text{180 is het kleinste gemene veelvoud van 4, 9 en 5} \\ \begin{align} & \frac{x}{4}+\frac{4}{9}& = & \frac{2}{5}x-5 \\\Leftrightarrow & \color{blue}{180.} (\frac{45x}{ \color{blue}{180} }+ \frac{ 80 }{ \color{blue}{180} })& = & (\frac{72}{ \color{blue}{180} }x-\frac{900}{ \color{blue}{180} }) \color{blue}{.180} \\\Leftrightarrow & 45x+80& = & 72x-900 \\\Leftrightarrow & 45x \color{red}{+80} \color{blue}{-80} \color{blue}{-72x} & = & \color{red}{72x} -900 \color{blue}{-72x} \color{blue}{-80} \\\Leftrightarrow & -27x& = & -980 \\\Leftrightarrow & \frac{-27x}{ \color{red}{-27} }& = & \frac{-980}{-27} \\\Leftrightarrow & x = \frac{980}{27} & & \\ & V = \left\{ \frac{980}{27} \right\} & \\\end{align}\)
9. \(\text{15 is het kleinste gemene veelvoud van 3, 15 en 5} \\ \begin{align} & \frac{x}{3}-\frac{4}{15}& = & \frac{4}{5}x-8 \\\Leftrightarrow & \color{blue}{15.} (\frac{5x}{ \color{blue}{15} }- \frac{ 4 }{ \color{blue}{15} })& = & (\frac{12}{ \color{blue}{15} }x-\frac{120}{ \color{blue}{15} }) \color{blue}{.15} \\\Leftrightarrow & 5x-4& = & 12x-120 \\\Leftrightarrow & 5x \color{red}{-4} \color{blue}{+4} \color{blue}{-12x} & = & \color{red}{12x} -120 \color{blue}{-12x} \color{blue}{+4} \\\Leftrightarrow & -7x& = & -116 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{-116}{-7} \\\Leftrightarrow & x = \frac{116}{7} & & \\ & V = \left\{ \frac{116}{7} \right\} & \\\end{align}\)
10. \(\text{12 is het kleinste gemene veelvoud van 2, 12 en 3} \\ \begin{align} & \frac{x}{2}-\frac{5}{12}& = & \frac{-8}{3}x-8 \\\Leftrightarrow & \color{blue}{12.} (\frac{6x}{ \color{blue}{12} }- \frac{ 5 }{ \color{blue}{12} })& = & (\frac{-32}{ \color{blue}{12} }x-\frac{96}{ \color{blue}{12} }) \color{blue}{.12} \\\Leftrightarrow & 6x-5& = & -32x-96 \\\Leftrightarrow & 6x \color{red}{-5} \color{blue}{+5} \color{blue}{+32x} & = & \color{red}{-32x} -96 \color{blue}{+32x} \color{blue}{+5} \\\Leftrightarrow & 38x& = & -91 \\\Leftrightarrow & \frac{38x}{ \color{red}{38} }& = & \frac{-91}{38} \\\Leftrightarrow & x = \frac{-91}{38} & & \\ & V = \left\{ \frac{-91}{38} \right\} & \\\end{align}\)
11. \(\text{42 is het kleinste gemene veelvoud van 3, 7 en 2} \\ \begin{align} & \frac{x}{3}-\frac{3}{7}& = & \frac{3}{2}x-3 \\\Leftrightarrow & \color{blue}{42.} (\frac{14x}{ \color{blue}{42} }- \frac{ 18 }{ \color{blue}{42} })& = & (\frac{63}{ \color{blue}{42} }x-\frac{126}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 14x-18& = & 63x-126 \\\Leftrightarrow & 14x \color{red}{-18} \color{blue}{+18} \color{blue}{-63x} & = & \color{red}{63x} -126 \color{blue}{-63x} \color{blue}{+18} \\\Leftrightarrow & -49x& = & -108 \\\Leftrightarrow & \frac{-49x}{ \color{red}{-49} }& = & \frac{-108}{-49} \\\Leftrightarrow & x = \frac{108}{49} & & \\ & V = \left\{ \frac{108}{49} \right\} & \\\end{align}\)
12. \(\text{30 is het kleinste gemene veelvoud van 2, 15 en 5} \\ \begin{align} & \frac{x}{2}-\frac{4}{15}& = & \frac{3}{5}x+7 \\\Leftrightarrow & \color{blue}{30.} (\frac{15x}{ \color{blue}{30} }- \frac{ 8 }{ \color{blue}{30} })& = & (\frac{18}{ \color{blue}{30} }x+\frac{210}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 15x-8& = & 18x+210 \\\Leftrightarrow & 15x \color{red}{-8} \color{blue}{+8} \color{blue}{-18x} & = & \color{red}{18x} +210 \color{blue}{-18x} \color{blue}{+8} \\\Leftrightarrow & -3x& = & 218 \\\Leftrightarrow & \frac{-3x}{ \color{red}{-3} }& = & \frac{218}{-3} \\\Leftrightarrow & x = \frac{-218}{3} & & \\ & V = \left\{ \frac{-218}{3} \right\} & \\\end{align}\)