Vgln. eerste graad (reeks 4)

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1. \(\frac{x}{5}-\frac{4}{7}=\frac{1}{2}x+7\)
2. \(\frac{x}{3}+\frac{5}{12}=\frac{1}{2}x+3\)
3. \(\frac{x}{7}-\frac{2}{11}=\frac{1}{6}x+3\)
4. \(\frac{x}{2}+\frac{5}{13}=\frac{2}{5}x-7\)
5. \(\frac{x}{7}-\frac{3}{13}=\frac{1}{6}x+8\)
6. \(\frac{x}{7}+\frac{5}{6}=\frac{2}{5}x-4\)
7. \(\frac{x}{2}+\frac{3}{7}=\frac{1}{5}x-7\)
8. \(\frac{x}{5}+\frac{3}{14}=\frac{5}{6}x+2\)
9. \(\frac{x}{5}+\frac{2}{13}=\frac{-5}{6}x-5\)
10. \(\frac{x}{3}-\frac{2}{9}=\frac{-3}{4}x+8\)
11. \(\frac{x}{2}-\frac{3}{16}=\frac{7}{3}x-4\)
12. \(\frac{x}{4}+\frac{3}{8}=\frac{-8}{3}x-8\)

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

Verbetersleutel

1. \(\text{70 is het kleinste gemene veelvoud van 5, 7 en 2} \\ \begin{align} & \frac{x}{5}-\frac{4}{7}& = & \frac{1}{2}x+7 \\\Leftrightarrow & \color{blue}{70.} (\frac{14x}{ \color{blue}{70} }- \frac{ 40 }{ \color{blue}{70} })& = & (\frac{35}{ \color{blue}{70} }x+\frac{490}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 14x-40& = & 35x+490 \\\Leftrightarrow & 14x \color{red}{-40} \color{blue}{+40} \color{blue}{-35x} & = & \color{red}{35x} +490 \color{blue}{-35x} \color{blue}{+40} \\\Leftrightarrow & -21x& = & 530 \\\Leftrightarrow & \frac{-21x}{ \color{red}{-21} }& = & \frac{530}{-21} \\\Leftrightarrow & x = \frac{-530}{21} & & \\ & V = \left\{ \frac{-530}{21} \right\} & \\\end{align}\)
2. \(\text{12 is het kleinste gemene veelvoud van 3, 12 en 2} \\ \begin{align} & \frac{x}{3}+\frac{5}{12}& = & \frac{1}{2}x+3 \\\Leftrightarrow & \color{blue}{12.} (\frac{4x}{ \color{blue}{12} }+ \frac{ 5 }{ \color{blue}{12} })& = & (\frac{6}{ \color{blue}{12} }x+\frac{36}{ \color{blue}{12} }) \color{blue}{.12} \\\Leftrightarrow & 4x+5& = & 6x+36 \\\Leftrightarrow & 4x \color{red}{+5} \color{blue}{-5} \color{blue}{-6x} & = & \color{red}{6x} +36 \color{blue}{-6x} \color{blue}{-5} \\\Leftrightarrow & -2x& = & 31 \\\Leftrightarrow & \frac{-2x}{ \color{red}{-2} }& = & \frac{31}{-2} \\\Leftrightarrow & x = \frac{-31}{2} & & \\ & V = \left\{ \frac{-31}{2} \right\} & \\\end{align}\)
3. \(\text{462 is het kleinste gemene veelvoud van 7, 11 en 6} \\ \begin{align} & \frac{x}{7}-\frac{2}{11}& = & \frac{1}{6}x+3 \\\Leftrightarrow & \color{blue}{462.} (\frac{66x}{ \color{blue}{462} }- \frac{ 84 }{ \color{blue}{462} })& = & (\frac{77}{ \color{blue}{462} }x+\frac{1386}{ \color{blue}{462} }) \color{blue}{.462} \\\Leftrightarrow & 66x-84& = & 77x+1386 \\\Leftrightarrow & 66x \color{red}{-84} \color{blue}{+84} \color{blue}{-77x} & = & \color{red}{77x} +1386 \color{blue}{-77x} \color{blue}{+84} \\\Leftrightarrow & -11x& = & 1470 \\\Leftrightarrow & \frac{-11x}{ \color{red}{-11} }& = & \frac{1470}{-11} \\\Leftrightarrow & x = \frac{-1470}{11} & & \\ & V = \left\{ \frac{-1470}{11} \right\} & \\\end{align}\)
4. \(\text{130 is het kleinste gemene veelvoud van 2, 13 en 5} \\ \begin{align} & \frac{x}{2}+\frac{5}{13}& = & \frac{2}{5}x-7 \\\Leftrightarrow & \color{blue}{130.} (\frac{65x}{ \color{blue}{130} }+ \frac{ 50 }{ \color{blue}{130} })& = & (\frac{52}{ \color{blue}{130} }x-\frac{910}{ \color{blue}{130} }) \color{blue}{.130} \\\Leftrightarrow & 65x+50& = & 52x-910 \\\Leftrightarrow & 65x \color{red}{+50} \color{blue}{-50} \color{blue}{-52x} & = & \color{red}{52x} -910 \color{blue}{-52x} \color{blue}{-50} \\\Leftrightarrow & 13x& = & -960 \\\Leftrightarrow & \frac{13x}{ \color{red}{13} }& = & \frac{-960}{13} \\\Leftrightarrow & x = \frac{-960}{13} & & \\ & V = \left\{ \frac{-960}{13} \right\} & \\\end{align}\)
5. \(\text{546 is het kleinste gemene veelvoud van 7, 13 en 6} \\ \begin{align} & \frac{x}{7}-\frac{3}{13}& = & \frac{1}{6}x+8 \\\Leftrightarrow & \color{blue}{546.} (\frac{78x}{ \color{blue}{546} }- \frac{ 126 }{ \color{blue}{546} })& = & (\frac{91}{ \color{blue}{546} }x+\frac{4368}{ \color{blue}{546} }) \color{blue}{.546} \\\Leftrightarrow & 78x-126& = & 91x+4368 \\\Leftrightarrow & 78x \color{red}{-126} \color{blue}{+126} \color{blue}{-91x} & = & \color{red}{91x} +4368 \color{blue}{-91x} \color{blue}{+126} \\\Leftrightarrow & -13x& = & 4494 \\\Leftrightarrow & \frac{-13x}{ \color{red}{-13} }& = & \frac{4494}{-13} \\\Leftrightarrow & x = \frac{-4494}{13} & & \\ & V = \left\{ \frac{-4494}{13} \right\} & \\\end{align}\)
6. \(\text{210 is het kleinste gemene veelvoud van 7, 6 en 5} \\ \begin{align} & \frac{x}{7}+\frac{5}{6}& = & \frac{2}{5}x-4 \\\Leftrightarrow & \color{blue}{210.} (\frac{30x}{ \color{blue}{210} }+ \frac{ 175 }{ \color{blue}{210} })& = & (\frac{84}{ \color{blue}{210} }x-\frac{840}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 30x+175& = & 84x-840 \\\Leftrightarrow & 30x \color{red}{+175} \color{blue}{-175} \color{blue}{-84x} & = & \color{red}{84x} -840 \color{blue}{-84x} \color{blue}{-175} \\\Leftrightarrow & -54x& = & -1015 \\\Leftrightarrow & \frac{-54x}{ \color{red}{-54} }& = & \frac{-1015}{-54} \\\Leftrightarrow & x = \frac{1015}{54} & & \\ & V = \left\{ \frac{1015}{54} \right\} & \\\end{align}\)
7. \(\text{70 is het kleinste gemene veelvoud van 2, 7 en 5} \\ \begin{align} & \frac{x}{2}+\frac{3}{7}& = & \frac{1}{5}x-7 \\\Leftrightarrow & \color{blue}{70.} (\frac{35x}{ \color{blue}{70} }+ \frac{ 30 }{ \color{blue}{70} })& = & (\frac{14}{ \color{blue}{70} }x-\frac{490}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 35x+30& = & 14x-490 \\\Leftrightarrow & 35x \color{red}{+30} \color{blue}{-30} \color{blue}{-14x} & = & \color{red}{14x} -490 \color{blue}{-14x} \color{blue}{-30} \\\Leftrightarrow & 21x& = & -520 \\\Leftrightarrow & \frac{21x}{ \color{red}{21} }& = & \frac{-520}{21} \\\Leftrightarrow & x = \frac{-520}{21} & & \\ & V = \left\{ \frac{-520}{21} \right\} & \\\end{align}\)
8. \(\text{210 is het kleinste gemene veelvoud van 5, 14 en 6} \\ \begin{align} & \frac{x}{5}+\frac{3}{14}& = & \frac{5}{6}x+2 \\\Leftrightarrow & \color{blue}{210.} (\frac{42x}{ \color{blue}{210} }+ \frac{ 45 }{ \color{blue}{210} })& = & (\frac{175}{ \color{blue}{210} }x+\frac{420}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 42x+45& = & 175x+420 \\\Leftrightarrow & 42x \color{red}{+45} \color{blue}{-45} \color{blue}{-175x} & = & \color{red}{175x} +420 \color{blue}{-175x} \color{blue}{-45} \\\Leftrightarrow & -133x& = & 375 \\\Leftrightarrow & \frac{-133x}{ \color{red}{-133} }& = & \frac{375}{-133} \\\Leftrightarrow & x = \frac{-375}{133} & & \\ & V = \left\{ \frac{-375}{133} \right\} & \\\end{align}\)
9. \(\text{390 is het kleinste gemene veelvoud van 5, 13 en 6} \\ \begin{align} & \frac{x}{5}+\frac{2}{13}& = & \frac{-5}{6}x-5 \\\Leftrightarrow & \color{blue}{390.} (\frac{78x}{ \color{blue}{390} }+ \frac{ 60 }{ \color{blue}{390} })& = & (\frac{-325}{ \color{blue}{390} }x-\frac{1950}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 78x+60& = & -325x-1950 \\\Leftrightarrow & 78x \color{red}{+60} \color{blue}{-60} \color{blue}{+325x} & = & \color{red}{-325x} -1950 \color{blue}{+325x} \color{blue}{-60} \\\Leftrightarrow & 403x& = & -2010 \\\Leftrightarrow & \frac{403x}{ \color{red}{403} }& = & \frac{-2010}{403} \\\Leftrightarrow & x = \frac{-2010}{403} & & \\ & V = \left\{ \frac{-2010}{403} \right\} & \\\end{align}\)
10. \(\text{36 is het kleinste gemene veelvoud van 3, 9 en 4} \\ \begin{align} & \frac{x}{3}-\frac{2}{9}& = & \frac{-3}{4}x+8 \\\Leftrightarrow & \color{blue}{36.} (\frac{12x}{ \color{blue}{36} }- \frac{ 8 }{ \color{blue}{36} })& = & (\frac{-27}{ \color{blue}{36} }x+\frac{288}{ \color{blue}{36} }) \color{blue}{.36} \\\Leftrightarrow & 12x-8& = & -27x+288 \\\Leftrightarrow & 12x \color{red}{-8} \color{blue}{+8} \color{blue}{+27x} & = & \color{red}{-27x} +288 \color{blue}{+27x} \color{blue}{+8} \\\Leftrightarrow & 39x& = & 296 \\\Leftrightarrow & \frac{39x}{ \color{red}{39} }& = & \frac{296}{39} \\\Leftrightarrow & x = \frac{296}{39} & & \\ & V = \left\{ \frac{296}{39} \right\} & \\\end{align}\)
11. \(\text{48 is het kleinste gemene veelvoud van 2, 16 en 3} \\ \begin{align} & \frac{x}{2}-\frac{3}{16}& = & \frac{7}{3}x-4 \\\Leftrightarrow & \color{blue}{48.} (\frac{24x}{ \color{blue}{48} }- \frac{ 9 }{ \color{blue}{48} })& = & (\frac{112}{ \color{blue}{48} }x-\frac{192}{ \color{blue}{48} }) \color{blue}{.48} \\\Leftrightarrow & 24x-9& = & 112x-192 \\\Leftrightarrow & 24x \color{red}{-9} \color{blue}{+9} \color{blue}{-112x} & = & \color{red}{112x} -192 \color{blue}{-112x} \color{blue}{+9} \\\Leftrightarrow & -88x& = & -183 \\\Leftrightarrow & \frac{-88x}{ \color{red}{-88} }& = & \frac{-183}{-88} \\\Leftrightarrow & x = \frac{183}{88} & & \\ & V = \left\{ \frac{183}{88} \right\} & \\\end{align}\)
12. \(\text{24 is het kleinste gemene veelvoud van 4, 8 en 3} \\ \begin{align} & \frac{x}{4}+\frac{3}{8}& = & \frac{-8}{3}x-8 \\\Leftrightarrow & \color{blue}{24.} (\frac{6x}{ \color{blue}{24} }+ \frac{ 9 }{ \color{blue}{24} })& = & (\frac{-64}{ \color{blue}{24} }x-\frac{192}{ \color{blue}{24} }) \color{blue}{.24} \\\Leftrightarrow & 6x+9& = & -64x-192 \\\Leftrightarrow & 6x \color{red}{+9} \color{blue}{-9} \color{blue}{+64x} & = & \color{red}{-64x} -192 \color{blue}{+64x} \color{blue}{-9} \\\Leftrightarrow & 70x& = & -201 \\\Leftrightarrow & \frac{70x}{ \color{red}{70} }& = & \frac{-201}{70} \\\Leftrightarrow & x = \frac{-201}{70} & & \\ & V = \left\{ \frac{-201}{70} \right\} & \\\end{align}\)