Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{336 is het kleinste gemene veelvoud van 7, 16 en 3} \\ \begin{align} & \frac{x}{7}+\frac{5}{16}& = & \frac{1}{3}x-5 \\\Leftrightarrow & \color{blue}{336.} (\frac{48x}{ \color{blue}{336} }+ \frac{ 105 }{ \color{blue}{336} })& = & (\frac{112}{ \color{blue}{336} }x-\frac{1680}{ \color{blue}{336} }) \color{blue}{.336} \\\Leftrightarrow & 48x+105& = & 112x-1680 \\\Leftrightarrow & 48x \color{red}{+105} \color{blue}{-105} \color{blue}{-112x} & = & \color{red}{112x} -1680 \color{blue}{-112x} \color{blue}{-105} \\\Leftrightarrow & -64x& = & -1785 \\\Leftrightarrow & \frac{-64x}{ \color{red}{-64} }& = & \frac{-1785}{-64} \\\Leftrightarrow & x = \frac{1785}{64} & & \\ & V = \left\{ \frac{1785}{64} \right\} & \\\end{align}\)
2. \(\text{130 is het kleinste gemene veelvoud van 2, 13 en 5} \\ \begin{align} & \frac{x}{2}+\frac{4}{13}& = & \frac{3}{5}x+2 \\\Leftrightarrow & \color{blue}{130.} (\frac{65x}{ \color{blue}{130} }+ \frac{ 40 }{ \color{blue}{130} })& = & (\frac{78}{ \color{blue}{130} }x+\frac{260}{ \color{blue}{130} }) \color{blue}{.130} \\\Leftrightarrow & 65x+40& = & 78x+260 \\\Leftrightarrow & 65x \color{red}{+40} \color{blue}{-40} \color{blue}{-78x} & = & \color{red}{78x} +260 \color{blue}{-78x} \color{blue}{-40} \\\Leftrightarrow & -13x& = & 220 \\\Leftrightarrow & \frac{-13x}{ \color{red}{-13} }& = & \frac{220}{-13} \\\Leftrightarrow & x = \frac{-220}{13} & & \\ & V = \left\{ \frac{-220}{13} \right\} & \\\end{align}\)
3. \(\text{90 is het kleinste gemene veelvoud van 6, 9 en 5} \\ \begin{align} & \frac{x}{6}-\frac{4}{9}& = & \frac{-4}{5}x-6 \\\Leftrightarrow & \color{blue}{90.} (\frac{15x}{ \color{blue}{90} }- \frac{ 40 }{ \color{blue}{90} })& = & (\frac{-72}{ \color{blue}{90} }x-\frac{540}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 15x-40& = & -72x-540 \\\Leftrightarrow & 15x \color{red}{-40} \color{blue}{+40} \color{blue}{+72x} & = & \color{red}{-72x} -540 \color{blue}{+72x} \color{blue}{+40} \\\Leftrightarrow & 87x& = & -500 \\\Leftrightarrow & \frac{87x}{ \color{red}{87} }& = & \frac{-500}{87} \\\Leftrightarrow & x = \frac{-500}{87} & & \\ & V = \left\{ \frac{-500}{87} \right\} & \\\end{align}\)
4. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}+\frac{5}{7}& = & \frac{-4}{5}x-8 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }+ \frac{ 150 }{ \color{blue}{210} })& = & (\frac{-168}{ \color{blue}{210} }x-\frac{1680}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x+150& = & -168x-1680 \\\Leftrightarrow & 35x \color{red}{+150} \color{blue}{-150} \color{blue}{+168x} & = & \color{red}{-168x} -1680 \color{blue}{+168x} \color{blue}{-150} \\\Leftrightarrow & 203x& = & -1830 \\\Leftrightarrow & \frac{203x}{ \color{red}{203} }& = & \frac{-1830}{203} \\\Leftrightarrow & x = \frac{-1830}{203} & & \\ & V = \left\{ \frac{-1830}{203} \right\} & \\\end{align}\)
5. \(\text{126 is het kleinste gemene veelvoud van 7, 9 en 2} \\ \begin{align} & \frac{x}{7}-\frac{4}{9}& = & \frac{1}{2}x+4 \\\Leftrightarrow & \color{blue}{126.} (\frac{18x}{ \color{blue}{126} }- \frac{ 56 }{ \color{blue}{126} })& = & (\frac{63}{ \color{blue}{126} }x+\frac{504}{ \color{blue}{126} }) \color{blue}{.126} \\\Leftrightarrow & 18x-56& = & 63x+504 \\\Leftrightarrow & 18x \color{red}{-56} \color{blue}{+56} \color{blue}{-63x} & = & \color{red}{63x} +504 \color{blue}{-63x} \color{blue}{+56} \\\Leftrightarrow & -45x& = & 560 \\\Leftrightarrow & \frac{-45x}{ \color{red}{-45} }& = & \frac{560}{-45} \\\Leftrightarrow & x = \frac{-112}{9} & & \\ & V = \left\{ \frac{-112}{9} \right\} & \\\end{align}\)
6. \(\text{36 is het kleinste gemene veelvoud van 3, 9 en 4} \\ \begin{align} & \frac{x}{3}+\frac{4}{9}& = & \frac{1}{4}x-4 \\\Leftrightarrow & \color{blue}{36.} (\frac{12x}{ \color{blue}{36} }+ \frac{ 16 }{ \color{blue}{36} })& = & (\frac{9}{ \color{blue}{36} }x-\frac{144}{ \color{blue}{36} }) \color{blue}{.36} \\\Leftrightarrow & 12x+16& = & 9x-144 \\\Leftrightarrow & 12x \color{red}{+16} \color{blue}{-16} \color{blue}{-9x} & = & \color{red}{9x} -144 \color{blue}{-9x} \color{blue}{-16} \\\Leftrightarrow & 3x& = & -160 \\\Leftrightarrow & \frac{3x}{ \color{red}{3} }& = & \frac{-160}{3} \\\Leftrightarrow & x = \frac{-160}{3} & & \\ & V = \left\{ \frac{-160}{3} \right\} & \\\end{align}\)
7. \(\text{330 is het kleinste gemene veelvoud van 5, 11 en 6} \\ \begin{align} & \frac{x}{5}-\frac{5}{11}& = & \frac{5}{6}x-5 \\\Leftrightarrow & \color{blue}{330.} (\frac{66x}{ \color{blue}{330} }- \frac{ 150 }{ \color{blue}{330} })& = & (\frac{275}{ \color{blue}{330} }x-\frac{1650}{ \color{blue}{330} }) \color{blue}{.330} \\\Leftrightarrow & 66x-150& = & 275x-1650 \\\Leftrightarrow & 66x \color{red}{-150} \color{blue}{+150} \color{blue}{-275x} & = & \color{red}{275x} -1650 \color{blue}{-275x} \color{blue}{+150} \\\Leftrightarrow & -209x& = & -1500 \\\Leftrightarrow & \frac{-209x}{ \color{red}{-209} }& = & \frac{-1500}{-209} \\\Leftrightarrow & x = \frac{1500}{209} & & \\ & V = \left\{ \frac{1500}{209} \right\} & \\\end{align}\)
8. \(\text{330 is het kleinste gemene veelvoud van 6, 11 en 5} \\ \begin{align} & \frac{x}{6}-\frac{4}{11}& = & \frac{-2}{5}x-5 \\\Leftrightarrow & \color{blue}{330.} (\frac{55x}{ \color{blue}{330} }- \frac{ 120 }{ \color{blue}{330} })& = & (\frac{-132}{ \color{blue}{330} }x-\frac{1650}{ \color{blue}{330} }) \color{blue}{.330} \\\Leftrightarrow & 55x-120& = & -132x-1650 \\\Leftrightarrow & 55x \color{red}{-120} \color{blue}{+120} \color{blue}{+132x} & = & \color{red}{-132x} -1650 \color{blue}{+132x} \color{blue}{+120} \\\Leftrightarrow & 187x& = & -1530 \\\Leftrightarrow & \frac{187x}{ \color{red}{187} }& = & \frac{-1530}{187} \\\Leftrightarrow & x = \frac{-90}{11} & & \\ & V = \left\{ \frac{-90}{11} \right\} & \\\end{align}\)
9. \(\text{130 is het kleinste gemene veelvoud van 2, 13 en 5} \\ \begin{align} & \frac{x}{2}+\frac{2}{13}& = & \frac{-4}{5}x-5 \\\Leftrightarrow & \color{blue}{130.} (\frac{65x}{ \color{blue}{130} }+ \frac{ 20 }{ \color{blue}{130} })& = & (\frac{-104}{ \color{blue}{130} }x-\frac{650}{ \color{blue}{130} }) \color{blue}{.130} \\\Leftrightarrow & 65x+20& = & -104x-650 \\\Leftrightarrow & 65x \color{red}{+20} \color{blue}{-20} \color{blue}{+104x} & = & \color{red}{-104x} -650 \color{blue}{+104x} \color{blue}{-20} \\\Leftrightarrow & 169x& = & -670 \\\Leftrightarrow & \frac{169x}{ \color{red}{169} }& = & \frac{-670}{169} \\\Leftrightarrow & x = \frac{-670}{169} & & \\ & V = \left\{ \frac{-670}{169} \right\} & \\\end{align}\)
10. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}-\frac{2}{7}& = & \frac{1}{3}x+7 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }- \frac{ 24 }{ \color{blue}{84} })& = & (\frac{28}{ \color{blue}{84} }x+\frac{588}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x-24& = & 28x+588 \\\Leftrightarrow & 21x \color{red}{-24} \color{blue}{+24} \color{blue}{-28x} & = & \color{red}{28x} +588 \color{blue}{-28x} \color{blue}{+24} \\\Leftrightarrow & -7x& = & 612 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{612}{-7} \\\Leftrightarrow & x = \frac{-612}{7} & & \\ & V = \left\{ \frac{-612}{7} \right\} & \\\end{align}\)
11. \(\text{70 is het kleinste gemene veelvoud van 5, 7 en 2} \\ \begin{align} & \frac{x}{5}-\frac{4}{7}& = & \frac{5}{2}x-7 \\\Leftrightarrow & \color{blue}{70.} (\frac{14x}{ \color{blue}{70} }- \frac{ 40 }{ \color{blue}{70} })& = & (\frac{175}{ \color{blue}{70} }x-\frac{490}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 14x-40& = & 175x-490 \\\Leftrightarrow & 14x \color{red}{-40} \color{blue}{+40} \color{blue}{-175x} & = & \color{red}{175x} -490 \color{blue}{-175x} \color{blue}{+40} \\\Leftrightarrow & -161x& = & -450 \\\Leftrightarrow & \frac{-161x}{ \color{red}{-161} }& = & \frac{-450}{-161} \\\Leftrightarrow & x = \frac{450}{161} & & \\ & V = \left\{ \frac{450}{161} \right\} & \\\end{align}\)
12. \(\text{15 is het kleinste gemene veelvoud van 3, 15 en 5} \\ \begin{align} & \frac{x}{3}+\frac{2}{15}& = & \frac{2}{5}x+7 \\\Leftrightarrow & \color{blue}{15.} (\frac{5x}{ \color{blue}{15} }+ \frac{ 2 }{ \color{blue}{15} })& = & (\frac{6}{ \color{blue}{15} }x+\frac{105}{ \color{blue}{15} }) \color{blue}{.15} \\\Leftrightarrow & 5x+2& = & 6x+105 \\\Leftrightarrow & 5x \color{red}{+2} \color{blue}{-2} \color{blue}{-6x} & = & \color{red}{6x} +105 \color{blue}{-6x} \color{blue}{-2} \\\Leftrightarrow & -x& = & 103 \\\Leftrightarrow & \frac{-x}{ \color{red}{-1} }& = & \frac{103}{-1} \\\Leftrightarrow & x = -103 & & \\ & V = \left\{ -103 \right\} & \\\end{align}\)