Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}-\frac{5}{7}& = & \frac{-4}{5}x-1 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }- \frac{ 150 }{ \color{blue}{210} })& = & (\frac{-168}{ \color{blue}{210} }x-\frac{210}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x-150& = & -168x-210 \\\Leftrightarrow & 35x \color{red}{-150} \color{blue}{+150} \color{blue}{+168x} & = & \color{red}{-168x} -210 \color{blue}{+168x} \color{blue}{+150} \\\Leftrightarrow & 203x& = & -60 \\\Leftrightarrow & \frac{203x}{ \color{red}{203} }& = & \frac{-60}{203} \\\Leftrightarrow & x = \frac{-60}{203} & & \\ & V = \left\{ \frac{-60}{203} \right\} & \\\end{align}\)
2. \(\text{260 is het kleinste gemene veelvoud van 5, 13 en 4} \\ \begin{align} & \frac{x}{5}-\frac{4}{13}& = & \frac{-3}{4}x-2 \\\Leftrightarrow & \color{blue}{260.} (\frac{52x}{ \color{blue}{260} }- \frac{ 80 }{ \color{blue}{260} })& = & (\frac{-195}{ \color{blue}{260} }x-\frac{520}{ \color{blue}{260} }) \color{blue}{.260} \\\Leftrightarrow & 52x-80& = & -195x-520 \\\Leftrightarrow & 52x \color{red}{-80} \color{blue}{+80} \color{blue}{+195x} & = & \color{red}{-195x} -520 \color{blue}{+195x} \color{blue}{+80} \\\Leftrightarrow & 247x& = & -440 \\\Leftrightarrow & \frac{247x}{ \color{red}{247} }& = & \frac{-440}{247} \\\Leftrightarrow & x = \frac{-440}{247} & & \\ & V = \left\{ \frac{-440}{247} \right\} & \\\end{align}\)
3. \(\text{36 is het kleinste gemene veelvoud van 4, 9 en 3} \\ \begin{align} & \frac{x}{4}+\frac{4}{9}& = & \frac{-5}{3}x-8 \\\Leftrightarrow & \color{blue}{36.} (\frac{9x}{ \color{blue}{36} }+ \frac{ 16 }{ \color{blue}{36} })& = & (\frac{-60}{ \color{blue}{36} }x-\frac{288}{ \color{blue}{36} }) \color{blue}{.36} \\\Leftrightarrow & 9x+16& = & -60x-288 \\\Leftrightarrow & 9x \color{red}{+16} \color{blue}{-16} \color{blue}{+60x} & = & \color{red}{-60x} -288 \color{blue}{+60x} \color{blue}{-16} \\\Leftrightarrow & 69x& = & -304 \\\Leftrightarrow & \frac{69x}{ \color{red}{69} }& = & \frac{-304}{69} \\\Leftrightarrow & x = \frac{-304}{69} & & \\ & V = \left\{ \frac{-304}{69} \right\} & \\\end{align}\)
4. \(\text{78 is het kleinste gemene veelvoud van 3, 13 en 2} \\ \begin{align} & \frac{x}{3}-\frac{2}{13}& = & \frac{5}{2}x+5 \\\Leftrightarrow & \color{blue}{78.} (\frac{26x}{ \color{blue}{78} }- \frac{ 12 }{ \color{blue}{78} })& = & (\frac{195}{ \color{blue}{78} }x+\frac{390}{ \color{blue}{78} }) \color{blue}{.78} \\\Leftrightarrow & 26x-12& = & 195x+390 \\\Leftrightarrow & 26x \color{red}{-12} \color{blue}{+12} \color{blue}{-195x} & = & \color{red}{195x} +390 \color{blue}{-195x} \color{blue}{+12} \\\Leftrightarrow & -169x& = & 402 \\\Leftrightarrow & \frac{-169x}{ \color{red}{-169} }& = & \frac{402}{-169} \\\Leftrightarrow & x = \frac{-402}{169} & & \\ & V = \left\{ \frac{-402}{169} \right\} & \\\end{align}\)
5. \(\text{48 is het kleinste gemene veelvoud van 3, 16 en 2} \\ \begin{align} & \frac{x}{3}-\frac{5}{16}& = & \frac{3}{2}x-3 \\\Leftrightarrow & \color{blue}{48.} (\frac{16x}{ \color{blue}{48} }- \frac{ 15 }{ \color{blue}{48} })& = & (\frac{72}{ \color{blue}{48} }x-\frac{144}{ \color{blue}{48} }) \color{blue}{.48} \\\Leftrightarrow & 16x-15& = & 72x-144 \\\Leftrightarrow & 16x \color{red}{-15} \color{blue}{+15} \color{blue}{-72x} & = & \color{red}{72x} -144 \color{blue}{-72x} \color{blue}{+15} \\\Leftrightarrow & -56x& = & -129 \\\Leftrightarrow & \frac{-56x}{ \color{red}{-56} }& = & \frac{-129}{-56} \\\Leftrightarrow & x = \frac{129}{56} & & \\ & V = \left\{ \frac{129}{56} \right\} & \\\end{align}\)
6. \(\text{66 is het kleinste gemene veelvoud van 3, 11 en 2} \\ \begin{align} & \frac{x}{3}-\frac{2}{11}& = & \frac{5}{2}x+3 \\\Leftrightarrow & \color{blue}{66.} (\frac{22x}{ \color{blue}{66} }- \frac{ 12 }{ \color{blue}{66} })& = & (\frac{165}{ \color{blue}{66} }x+\frac{198}{ \color{blue}{66} }) \color{blue}{.66} \\\Leftrightarrow & 22x-12& = & 165x+198 \\\Leftrightarrow & 22x \color{red}{-12} \color{blue}{+12} \color{blue}{-165x} & = & \color{red}{165x} +198 \color{blue}{-165x} \color{blue}{+12} \\\Leftrightarrow & -143x& = & 210 \\\Leftrightarrow & \frac{-143x}{ \color{red}{-143} }& = & \frac{210}{-143} \\\Leftrightarrow & x = \frac{-210}{143} & & \\ & V = \left\{ \frac{-210}{143} \right\} & \\\end{align}\)
7. \(\text{210 is het kleinste gemene veelvoud van 5, 7 en 6} \\ \begin{align} & \frac{x}{5}+\frac{2}{7}& = & \frac{1}{6}x-4 \\\Leftrightarrow & \color{blue}{210.} (\frac{42x}{ \color{blue}{210} }+ \frac{ 60 }{ \color{blue}{210} })& = & (\frac{35}{ \color{blue}{210} }x-\frac{840}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 42x+60& = & 35x-840 \\\Leftrightarrow & 42x \color{red}{+60} \color{blue}{-60} \color{blue}{-35x} & = & \color{red}{35x} -840 \color{blue}{-35x} \color{blue}{-60} \\\Leftrightarrow & 7x& = & -900 \\\Leftrightarrow & \frac{7x}{ \color{red}{7} }& = & \frac{-900}{7} \\\Leftrightarrow & x = \frac{-900}{7} & & \\ & V = \left\{ \frac{-900}{7} \right\} & \\\end{align}\)
8. \(\text{462 is het kleinste gemene veelvoud van 7, 11 en 6} \\ \begin{align} & \frac{x}{7}+\frac{2}{11}& = & \frac{7}{6}x+6 \\\Leftrightarrow & \color{blue}{462.} (\frac{66x}{ \color{blue}{462} }+ \frac{ 84 }{ \color{blue}{462} })& = & (\frac{539}{ \color{blue}{462} }x+\frac{2772}{ \color{blue}{462} }) \color{blue}{.462} \\\Leftrightarrow & 66x+84& = & 539x+2772 \\\Leftrightarrow & 66x \color{red}{+84} \color{blue}{-84} \color{blue}{-539x} & = & \color{red}{539x} +2772 \color{blue}{-539x} \color{blue}{-84} \\\Leftrightarrow & -473x& = & 2688 \\\Leftrightarrow & \frac{-473x}{ \color{red}{-473} }& = & \frac{2688}{-473} \\\Leftrightarrow & x = \frac{-2688}{473} & & \\ & V = \left\{ \frac{-2688}{473} \right\} & \\\end{align}\)
9. \(\text{240 is het kleinste gemene veelvoud van 5, 16 en 6} \\ \begin{align} & \frac{x}{5}-\frac{5}{16}& = & \frac{-5}{6}x+3 \\\Leftrightarrow & \color{blue}{240.} (\frac{48x}{ \color{blue}{240} }- \frac{ 75 }{ \color{blue}{240} })& = & (\frac{-200}{ \color{blue}{240} }x+\frac{720}{ \color{blue}{240} }) \color{blue}{.240} \\\Leftrightarrow & 48x-75& = & -200x+720 \\\Leftrightarrow & 48x \color{red}{-75} \color{blue}{+75} \color{blue}{+200x} & = & \color{red}{-200x} +720 \color{blue}{+200x} \color{blue}{+75} \\\Leftrightarrow & 248x& = & 795 \\\Leftrightarrow & \frac{248x}{ \color{red}{248} }& = & \frac{795}{248} \\\Leftrightarrow & x = \frac{795}{248} & & \\ & V = \left\{ \frac{795}{248} \right\} & \\\end{align}\)
10. \(\text{60 is het kleinste gemene veelvoud van 4, 10 en 3} \\ \begin{align} & \frac{x}{4}-\frac{3}{10}& = & \frac{7}{3}x-4 \\\Leftrightarrow & \color{blue}{60.} (\frac{15x}{ \color{blue}{60} }- \frac{ 18 }{ \color{blue}{60} })& = & (\frac{140}{ \color{blue}{60} }x-\frac{240}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 15x-18& = & 140x-240 \\\Leftrightarrow & 15x \color{red}{-18} \color{blue}{+18} \color{blue}{-140x} & = & \color{red}{140x} -240 \color{blue}{-140x} \color{blue}{+18} \\\Leftrightarrow & -125x& = & -222 \\\Leftrightarrow & \frac{-125x}{ \color{red}{-125} }& = & \frac{-222}{-125} \\\Leftrightarrow & x = \frac{222}{125} & & \\ & V = \left\{ \frac{222}{125} \right\} & \\\end{align}\)
11. \(\text{60 is het kleinste gemene veelvoud van 5, 12 en 6} \\ \begin{align} & \frac{x}{5}-\frac{5}{12}& = & \frac{-5}{6}x-3 \\\Leftrightarrow & \color{blue}{60.} (\frac{12x}{ \color{blue}{60} }- \frac{ 25 }{ \color{blue}{60} })& = & (\frac{-50}{ \color{blue}{60} }x-\frac{180}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 12x-25& = & -50x-180 \\\Leftrightarrow & 12x \color{red}{-25} \color{blue}{+25} \color{blue}{+50x} & = & \color{red}{-50x} -180 \color{blue}{+50x} \color{blue}{+25} \\\Leftrightarrow & 62x& = & -155 \\\Leftrightarrow & \frac{62x}{ \color{red}{62} }& = & \frac{-155}{62} \\\Leftrightarrow & x = \frac{-5}{2} & & \\ & V = \left\{ \frac{-5}{2} \right\} & \\\end{align}\)
12. \(\text{105 is het kleinste gemene veelvoud van 7, 15 en 3} \\ \begin{align} & \frac{x}{7}+\frac{2}{15}& = & \frac{-2}{3}x+2 \\\Leftrightarrow & \color{blue}{105.} (\frac{15x}{ \color{blue}{105} }+ \frac{ 14 }{ \color{blue}{105} })& = & (\frac{-70}{ \color{blue}{105} }x+\frac{210}{ \color{blue}{105} }) \color{blue}{.105} \\\Leftrightarrow & 15x+14& = & -70x+210 \\\Leftrightarrow & 15x \color{red}{+14} \color{blue}{-14} \color{blue}{+70x} & = & \color{red}{-70x} +210 \color{blue}{+70x} \color{blue}{-14} \\\Leftrightarrow & 85x& = & 196 \\\Leftrightarrow & \frac{85x}{ \color{red}{85} }& = & \frac{196}{85} \\\Leftrightarrow & x = \frac{196}{85} & & \\ & V = \left\{ \frac{196}{85} \right\} & \\\end{align}\)