Vgln. eerste graad (reeks 4)

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

1. \(\frac{x}{5}+\frac{3}{14}=\frac{5}{2}x+7\)
2. \(\frac{x}{4}-\frac{2}{7}=\frac{1}{3}x+5\)
3. \(\frac{x}{5}-\frac{2}{13}=\frac{5}{6}x-2\)
4. \(\frac{x}{2}+\frac{4}{9}=\frac{-2}{3}x+2\)
5. \(\frac{x}{3}-\frac{2}{9}=\frac{1}{4}x-1\)
6. \(\frac{x}{6}-\frac{2}{13}=\frac{4}{5}x-8\)
7. \(\frac{x}{7}+\frac{5}{13}=\frac{-2}{3}x-8\)
8. \(\frac{x}{2}+\frac{5}{8}=\frac{1}{3}x+1\)
9. \(\frac{x}{6}+\frac{5}{6}=\frac{-4}{5}x-2\)
10. \(\frac{x}{6}+\frac{3}{13}=\frac{1}{5}x-5\)
11. \(\frac{x}{4}+\frac{4}{9}=\frac{4}{3}x+7\)
12. \(\frac{x}{7}+\frac{4}{15}=\frac{7}{3}x-2\)

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

Verbetersleutel

1. \(\text{70 is het kleinste gemene veelvoud van 5, 14 en 2} \\ \begin{align} & \frac{x}{5}+\frac{3}{14}& = & \frac{5}{2}x+7 \\\Leftrightarrow & \color{blue}{70.} (\frac{14x}{ \color{blue}{70} }+ \frac{ 15 }{ \color{blue}{70} })& = & (\frac{175}{ \color{blue}{70} }x+\frac{490}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 14x+15& = & 175x+490 \\\Leftrightarrow & 14x \color{red}{+15} \color{blue}{-15} \color{blue}{-175x} & = & \color{red}{175x} +490 \color{blue}{-175x} \color{blue}{-15} \\\Leftrightarrow & -161x& = & 475 \\\Leftrightarrow & \frac{-161x}{ \color{red}{-161} }& = & \frac{475}{-161} \\\Leftrightarrow & x = \frac{-475}{161} & & \\ & V = \left\{ \frac{-475}{161} \right\} & \\\end{align}\)
2. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}-\frac{2}{7}& = & \frac{1}{3}x+5 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }- \frac{ 24 }{ \color{blue}{84} })& = & (\frac{28}{ \color{blue}{84} }x+\frac{420}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x-24& = & 28x+420 \\\Leftrightarrow & 21x \color{red}{-24} \color{blue}{+24} \color{blue}{-28x} & = & \color{red}{28x} +420 \color{blue}{-28x} \color{blue}{+24} \\\Leftrightarrow & -7x& = & 444 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{444}{-7} \\\Leftrightarrow & x = \frac{-444}{7} & & \\ & V = \left\{ \frac{-444}{7} \right\} & \\\end{align}\)
3. \(\text{390 is het kleinste gemene veelvoud van 5, 13 en 6} \\ \begin{align} & \frac{x}{5}-\frac{2}{13}& = & \frac{5}{6}x-2 \\\Leftrightarrow & \color{blue}{390.} (\frac{78x}{ \color{blue}{390} }- \frac{ 60 }{ \color{blue}{390} })& = & (\frac{325}{ \color{blue}{390} }x-\frac{780}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 78x-60& = & 325x-780 \\\Leftrightarrow & 78x \color{red}{-60} \color{blue}{+60} \color{blue}{-325x} & = & \color{red}{325x} -780 \color{blue}{-325x} \color{blue}{+60} \\\Leftrightarrow & -247x& = & -720 \\\Leftrightarrow & \frac{-247x}{ \color{red}{-247} }& = & \frac{-720}{-247} \\\Leftrightarrow & x = \frac{720}{247} & & \\ & V = \left\{ \frac{720}{247} \right\} & \\\end{align}\)
4. \(\text{18 is het kleinste gemene veelvoud van 2, 9 en 3} \\ \begin{align} & \frac{x}{2}+\frac{4}{9}& = & \frac{-2}{3}x+2 \\\Leftrightarrow & \color{blue}{18.} (\frac{9x}{ \color{blue}{18} }+ \frac{ 8 }{ \color{blue}{18} })& = & (\frac{-12}{ \color{blue}{18} }x+\frac{36}{ \color{blue}{18} }) \color{blue}{.18} \\\Leftrightarrow & 9x+8& = & -12x+36 \\\Leftrightarrow & 9x \color{red}{+8} \color{blue}{-8} \color{blue}{+12x} & = & \color{red}{-12x} +36 \color{blue}{+12x} \color{blue}{-8} \\\Leftrightarrow & 21x& = & 28 \\\Leftrightarrow & \frac{21x}{ \color{red}{21} }& = & \frac{28}{21} \\\Leftrightarrow & x = \frac{4}{3} & & \\ & V = \left\{ \frac{4}{3} \right\} & \\\end{align}\)
5. \(\text{36 is het kleinste gemene veelvoud van 3, 9 en 4} \\ \begin{align} & \frac{x}{3}-\frac{2}{9}& = & \frac{1}{4}x-1 \\\Leftrightarrow & \color{blue}{36.} (\frac{12x}{ \color{blue}{36} }- \frac{ 8 }{ \color{blue}{36} })& = & (\frac{9}{ \color{blue}{36} }x-\frac{36}{ \color{blue}{36} }) \color{blue}{.36} \\\Leftrightarrow & 12x-8& = & 9x-36 \\\Leftrightarrow & 12x \color{red}{-8} \color{blue}{+8} \color{blue}{-9x} & = & \color{red}{9x} -36 \color{blue}{-9x} \color{blue}{+8} \\\Leftrightarrow & 3x& = & -28 \\\Leftrightarrow & \frac{3x}{ \color{red}{3} }& = & \frac{-28}{3} \\\Leftrightarrow & x = \frac{-28}{3} & & \\ & V = \left\{ \frac{-28}{3} \right\} & \\\end{align}\)
6. \(\text{390 is het kleinste gemene veelvoud van 6, 13 en 5} \\ \begin{align} & \frac{x}{6}-\frac{2}{13}& = & \frac{4}{5}x-8 \\\Leftrightarrow & \color{blue}{390.} (\frac{65x}{ \color{blue}{390} }- \frac{ 60 }{ \color{blue}{390} })& = & (\frac{312}{ \color{blue}{390} }x-\frac{3120}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 65x-60& = & 312x-3120 \\\Leftrightarrow & 65x \color{red}{-60} \color{blue}{+60} \color{blue}{-312x} & = & \color{red}{312x} -3120 \color{blue}{-312x} \color{blue}{+60} \\\Leftrightarrow & -247x& = & -3060 \\\Leftrightarrow & \frac{-247x}{ \color{red}{-247} }& = & \frac{-3060}{-247} \\\Leftrightarrow & x = \frac{3060}{247} & & \\ & V = \left\{ \frac{3060}{247} \right\} & \\\end{align}\)
7. \(\text{273 is het kleinste gemene veelvoud van 7, 13 en 3} \\ \begin{align} & \frac{x}{7}+\frac{5}{13}& = & \frac{-2}{3}x-8 \\\Leftrightarrow & \color{blue}{273.} (\frac{39x}{ \color{blue}{273} }+ \frac{ 105 }{ \color{blue}{273} })& = & (\frac{-182}{ \color{blue}{273} }x-\frac{2184}{ \color{blue}{273} }) \color{blue}{.273} \\\Leftrightarrow & 39x+105& = & -182x-2184 \\\Leftrightarrow & 39x \color{red}{+105} \color{blue}{-105} \color{blue}{+182x} & = & \color{red}{-182x} -2184 \color{blue}{+182x} \color{blue}{-105} \\\Leftrightarrow & 221x& = & -2289 \\\Leftrightarrow & \frac{221x}{ \color{red}{221} }& = & \frac{-2289}{221} \\\Leftrightarrow & x = \frac{-2289}{221} & & \\ & V = \left\{ \frac{-2289}{221} \right\} & \\\end{align}\)
8. \(\text{24 is het kleinste gemene veelvoud van 2, 8 en 3} \\ \begin{align} & \frac{x}{2}+\frac{5}{8}& = & \frac{1}{3}x+1 \\\Leftrightarrow & \color{blue}{24.} (\frac{12x}{ \color{blue}{24} }+ \frac{ 15 }{ \color{blue}{24} })& = & (\frac{8}{ \color{blue}{24} }x+\frac{24}{ \color{blue}{24} }) \color{blue}{.24} \\\Leftrightarrow & 12x+15& = & 8x+24 \\\Leftrightarrow & 12x \color{red}{+15} \color{blue}{-15} \color{blue}{-8x} & = & \color{red}{8x} +24 \color{blue}{-8x} \color{blue}{-15} \\\Leftrightarrow & 4x& = & 9 \\\Leftrightarrow & \frac{4x}{ \color{red}{4} }& = & \frac{9}{4} \\\Leftrightarrow & x = \frac{9}{4} & & \\ & V = \left\{ \frac{9}{4} \right\} & \\\end{align}\)
9. \(\text{30 is het kleinste gemene veelvoud van 6, 6 en 5} \\ \begin{align} & \frac{x}{6}+\frac{5}{6}& = & \frac{-4}{5}x-2 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }+ \frac{ 25 }{ \color{blue}{30} })& = & (\frac{-24}{ \color{blue}{30} }x-\frac{60}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x+25& = & -24x-60 \\\Leftrightarrow & 5x \color{red}{+25} \color{blue}{-25} \color{blue}{+24x} & = & \color{red}{-24x} -60 \color{blue}{+24x} \color{blue}{-25} \\\Leftrightarrow & 29x& = & -85 \\\Leftrightarrow & \frac{29x}{ \color{red}{29} }& = & \frac{-85}{29} \\\Leftrightarrow & x = \frac{-85}{29} & & \\ & V = \left\{ \frac{-85}{29} \right\} & \\\end{align}\)
10. \(\text{390 is het kleinste gemene veelvoud van 6, 13 en 5} \\ \begin{align} & \frac{x}{6}+\frac{3}{13}& = & \frac{1}{5}x-5 \\\Leftrightarrow & \color{blue}{390.} (\frac{65x}{ \color{blue}{390} }+ \frac{ 90 }{ \color{blue}{390} })& = & (\frac{78}{ \color{blue}{390} }x-\frac{1950}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 65x+90& = & 78x-1950 \\\Leftrightarrow & 65x \color{red}{+90} \color{blue}{-90} \color{blue}{-78x} & = & \color{red}{78x} -1950 \color{blue}{-78x} \color{blue}{-90} \\\Leftrightarrow & -13x& = & -2040 \\\Leftrightarrow & \frac{-13x}{ \color{red}{-13} }& = & \frac{-2040}{-13} \\\Leftrightarrow & x = \frac{2040}{13} & & \\ & V = \left\{ \frac{2040}{13} \right\} & \\\end{align}\)
11. \(\text{36 is het kleinste gemene veelvoud van 4, 9 en 3} \\ \begin{align} & \frac{x}{4}+\frac{4}{9}& = & \frac{4}{3}x+7 \\\Leftrightarrow & \color{blue}{36.} (\frac{9x}{ \color{blue}{36} }+ \frac{ 16 }{ \color{blue}{36} })& = & (\frac{48}{ \color{blue}{36} }x+\frac{252}{ \color{blue}{36} }) \color{blue}{.36} \\\Leftrightarrow & 9x+16& = & 48x+252 \\\Leftrightarrow & 9x \color{red}{+16} \color{blue}{-16} \color{blue}{-48x} & = & \color{red}{48x} +252 \color{blue}{-48x} \color{blue}{-16} \\\Leftrightarrow & -39x& = & 236 \\\Leftrightarrow & \frac{-39x}{ \color{red}{-39} }& = & \frac{236}{-39} \\\Leftrightarrow & x = \frac{-236}{39} & & \\ & V = \left\{ \frac{-236}{39} \right\} & \\\end{align}\)
12. \(\text{105 is het kleinste gemene veelvoud van 7, 15 en 3} \\ \begin{align} & \frac{x}{7}+\frac{4}{15}& = & \frac{7}{3}x-2 \\\Leftrightarrow & \color{blue}{105.} (\frac{15x}{ \color{blue}{105} }+ \frac{ 28 }{ \color{blue}{105} })& = & (\frac{245}{ \color{blue}{105} }x-\frac{210}{ \color{blue}{105} }) \color{blue}{.105} \\\Leftrightarrow & 15x+28& = & 245x-210 \\\Leftrightarrow & 15x \color{red}{+28} \color{blue}{-28} \color{blue}{-245x} & = & \color{red}{245x} -210 \color{blue}{-245x} \color{blue}{-28} \\\Leftrightarrow & -230x& = & -238 \\\Leftrightarrow & \frac{-230x}{ \color{red}{-230} }& = & \frac{-238}{-230} \\\Leftrightarrow & x = \frac{119}{115} & & \\ & V = \left\{ \frac{119}{115} \right\} & \\\end{align}\)