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

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

Verbetersleutel

1. \(\text{80 is het kleinste gemene veelvoud van 2, 16 en 5} \\ \begin{align} & \frac{x}{2}-\frac{5}{16}& = & \frac{6}{5}x-8 \\\Leftrightarrow & \color{blue}{80.} (\frac{40x}{ \color{blue}{80} }- \frac{ 25 }{ \color{blue}{80} })& = & (\frac{96}{ \color{blue}{80} }x-\frac{640}{ \color{blue}{80} }) \color{blue}{.80} \\\Leftrightarrow & 40x-25& = & 96x-640 \\\Leftrightarrow & 40x \color{red}{-25} \color{blue}{+25} \color{blue}{-96x} & = & \color{red}{96x} -640 \color{blue}{-96x} \color{blue}{+25} \\\Leftrightarrow & -56x& = & -615 \\\Leftrightarrow & \frac{-56x}{ \color{red}{-56} }& = & \frac{-615}{-56} \\\Leftrightarrow & x = \frac{615}{56} & & \\ & V = \left\{ \frac{615}{56} \right\} & \\\end{align}\)
2. \(\text{336 is het kleinste gemene veelvoud van 7, 16 en 6} \\ \begin{align} & \frac{x}{7}+\frac{3}{16}& = & \frac{7}{6}x+6 \\\Leftrightarrow & \color{blue}{336.} (\frac{48x}{ \color{blue}{336} }+ \frac{ 63 }{ \color{blue}{336} })& = & (\frac{392}{ \color{blue}{336} }x+\frac{2016}{ \color{blue}{336} }) \color{blue}{.336} \\\Leftrightarrow & 48x+63& = & 392x+2016 \\\Leftrightarrow & 48x \color{red}{+63} \color{blue}{-63} \color{blue}{-392x} & = & \color{red}{392x} +2016 \color{blue}{-392x} \color{blue}{-63} \\\Leftrightarrow & -344x& = & 1953 \\\Leftrightarrow & \frac{-344x}{ \color{red}{-344} }& = & \frac{1953}{-344} \\\Leftrightarrow & x = \frac{-1953}{344} & & \\ & V = \left\{ \frac{-1953}{344} \right\} & \\\end{align}\)
3. \(\text{78 is het kleinste gemene veelvoud van 2, 13 en 3} \\ \begin{align} & \frac{x}{2}-\frac{2}{13}& = & \frac{1}{3}x-1 \\\Leftrightarrow & \color{blue}{78.} (\frac{39x}{ \color{blue}{78} }- \frac{ 12 }{ \color{blue}{78} })& = & (\frac{26}{ \color{blue}{78} }x-\frac{78}{ \color{blue}{78} }) \color{blue}{.78} \\\Leftrightarrow & 39x-12& = & 26x-78 \\\Leftrightarrow & 39x \color{red}{-12} \color{blue}{+12} \color{blue}{-26x} & = & \color{red}{26x} -78 \color{blue}{-26x} \color{blue}{+12} \\\Leftrightarrow & 13x& = & -66 \\\Leftrightarrow & \frac{13x}{ \color{red}{13} }& = & \frac{-66}{13} \\\Leftrightarrow & x = \frac{-66}{13} & & \\ & V = \left\{ \frac{-66}{13} \right\} & \\\end{align}\)
4. \(\text{60 is het kleinste gemene veelvoud van 4, 15 en 5} \\ \begin{align} & \frac{x}{4}-\frac{2}{15}& = & \frac{6}{5}x+4 \\\Leftrightarrow & \color{blue}{60.} (\frac{15x}{ \color{blue}{60} }- \frac{ 8 }{ \color{blue}{60} })& = & (\frac{72}{ \color{blue}{60} }x+\frac{240}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 15x-8& = & 72x+240 \\\Leftrightarrow & 15x \color{red}{-8} \color{blue}{+8} \color{blue}{-72x} & = & \color{red}{72x} +240 \color{blue}{-72x} \color{blue}{+8} \\\Leftrightarrow & -57x& = & 248 \\\Leftrightarrow & \frac{-57x}{ \color{red}{-57} }& = & \frac{248}{-57} \\\Leftrightarrow & x = \frac{-248}{57} & & \\ & V = \left\{ \frac{-248}{57} \right\} & \\\end{align}\)
5. \(\text{30 is het kleinste gemene veelvoud van 2, 15 en 5} \\ \begin{align} & \frac{x}{2}-\frac{4}{15}& = & \frac{-7}{5}x+5 \\\Leftrightarrow & \color{blue}{30.} (\frac{15x}{ \color{blue}{30} }- \frac{ 8 }{ \color{blue}{30} })& = & (\frac{-42}{ \color{blue}{30} }x+\frac{150}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 15x-8& = & -42x+150 \\\Leftrightarrow & 15x \color{red}{-8} \color{blue}{+8} \color{blue}{+42x} & = & \color{red}{-42x} +150 \color{blue}{+42x} \color{blue}{+8} \\\Leftrightarrow & 57x& = & 158 \\\Leftrightarrow & \frac{57x}{ \color{red}{57} }& = & \frac{158}{57} \\\Leftrightarrow & x = \frac{158}{57} & & \\ & V = \left\{ \frac{158}{57} \right\} & \\\end{align}\)
6. \(\text{30 is het kleinste gemene veelvoud van 6, 10 en 5} \\ \begin{align} & \frac{x}{6}-\frac{3}{10}& = & \frac{-7}{5}x-2 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }- \frac{ 9 }{ \color{blue}{30} })& = & (\frac{-42}{ \color{blue}{30} }x-\frac{60}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x-9& = & -42x-60 \\\Leftrightarrow & 5x \color{red}{-9} \color{blue}{+9} \color{blue}{+42x} & = & \color{red}{-42x} -60 \color{blue}{+42x} \color{blue}{+9} \\\Leftrightarrow & 47x& = & -51 \\\Leftrightarrow & \frac{47x}{ \color{red}{47} }& = & \frac{-51}{47} \\\Leftrightarrow & x = \frac{-51}{47} & & \\ & V = \left\{ \frac{-51}{47} \right\} & \\\end{align}\)
7. \(\text{18 is het kleinste gemene veelvoud van 2, 9 en 3} \\ \begin{align} & \frac{x}{2}-\frac{5}{9}& = & \frac{4}{3}x+7 \\\Leftrightarrow & \color{blue}{18.} (\frac{9x}{ \color{blue}{18} }- \frac{ 10 }{ \color{blue}{18} })& = & (\frac{24}{ \color{blue}{18} }x+\frac{126}{ \color{blue}{18} }) \color{blue}{.18} \\\Leftrightarrow & 9x-10& = & 24x+126 \\\Leftrightarrow & 9x \color{red}{-10} \color{blue}{+10} \color{blue}{-24x} & = & \color{red}{24x} +126 \color{blue}{-24x} \color{blue}{+10} \\\Leftrightarrow & -15x& = & 136 \\\Leftrightarrow & \frac{-15x}{ \color{red}{-15} }& = & \frac{136}{-15} \\\Leftrightarrow & x = \frac{-136}{15} & & \\ & V = \left\{ \frac{-136}{15} \right\} & \\\end{align}\)
8. \(\text{70 is het kleinste gemene veelvoud van 2, 7 en 5} \\ \begin{align} & \frac{x}{2}+\frac{4}{7}& = & \frac{7}{5}x+7 \\\Leftrightarrow & \color{blue}{70.} (\frac{35x}{ \color{blue}{70} }+ \frac{ 40 }{ \color{blue}{70} })& = & (\frac{98}{ \color{blue}{70} }x+\frac{490}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 35x+40& = & 98x+490 \\\Leftrightarrow & 35x \color{red}{+40} \color{blue}{-40} \color{blue}{-98x} & = & \color{red}{98x} +490 \color{blue}{-98x} \color{blue}{-40} \\\Leftrightarrow & -63x& = & 450 \\\Leftrightarrow & \frac{-63x}{ \color{red}{-63} }& = & \frac{450}{-63} \\\Leftrightarrow & x = \frac{-50}{7} & & \\ & V = \left\{ \frac{-50}{7} \right\} & \\\end{align}\)
9. \(\text{240 is het kleinste gemene veelvoud van 5, 16 en 6} \\ \begin{align} & \frac{x}{5}-\frac{3}{16}& = & \frac{7}{6}x-2 \\\Leftrightarrow & \color{blue}{240.} (\frac{48x}{ \color{blue}{240} }- \frac{ 45 }{ \color{blue}{240} })& = & (\frac{280}{ \color{blue}{240} }x-\frac{480}{ \color{blue}{240} }) \color{blue}{.240} \\\Leftrightarrow & 48x-45& = & 280x-480 \\\Leftrightarrow & 48x \color{red}{-45} \color{blue}{+45} \color{blue}{-280x} & = & \color{red}{280x} -480 \color{blue}{-280x} \color{blue}{+45} \\\Leftrightarrow & -232x& = & -435 \\\Leftrightarrow & \frac{-232x}{ \color{red}{-232} }& = & \frac{-435}{-232} \\\Leftrightarrow & x = \frac{15}{8} & & \\ & V = \left\{ \frac{15}{8} \right\} & \\\end{align}\)
10. \(\text{42 is het kleinste gemene veelvoud van 7, 6 en 2} \\ \begin{align} & \frac{x}{7}+\frac{5}{6}& = & \frac{1}{2}x-2 \\\Leftrightarrow & \color{blue}{42.} (\frac{6x}{ \color{blue}{42} }+ \frac{ 35 }{ \color{blue}{42} })& = & (\frac{21}{ \color{blue}{42} }x-\frac{84}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 6x+35& = & 21x-84 \\\Leftrightarrow & 6x \color{red}{+35} \color{blue}{-35} \color{blue}{-21x} & = & \color{red}{21x} -84 \color{blue}{-21x} \color{blue}{-35} \\\Leftrightarrow & -15x& = & -119 \\\Leftrightarrow & \frac{-15x}{ \color{red}{-15} }& = & \frac{-119}{-15} \\\Leftrightarrow & x = \frac{119}{15} & & \\ & V = \left\{ \frac{119}{15} \right\} & \\\end{align}\)
11. \(\text{90 is het kleinste gemene veelvoud van 6, 9 en 5} \\ \begin{align} & \frac{x}{6}+\frac{4}{9}& = & \frac{1}{5}x+6 \\\Leftrightarrow & \color{blue}{90.} (\frac{15x}{ \color{blue}{90} }+ \frac{ 40 }{ \color{blue}{90} })& = & (\frac{18}{ \color{blue}{90} }x+\frac{540}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 15x+40& = & 18x+540 \\\Leftrightarrow & 15x \color{red}{+40} \color{blue}{-40} \color{blue}{-18x} & = & \color{red}{18x} +540 \color{blue}{-18x} \color{blue}{-40} \\\Leftrightarrow & -3x& = & 500 \\\Leftrightarrow & \frac{-3x}{ \color{red}{-3} }& = & \frac{500}{-3} \\\Leftrightarrow & x = \frac{-500}{3} & & \\ & V = \left\{ \frac{-500}{3} \right\} & \\\end{align}\)
12. \(\text{48 is het kleinste gemene veelvoud van 3, 16 en 2} \\ \begin{align} & \frac{x}{3}+\frac{3}{16}& = & \frac{1}{2}x-1 \\\Leftrightarrow & \color{blue}{48.} (\frac{16x}{ \color{blue}{48} }+ \frac{ 9 }{ \color{blue}{48} })& = & (\frac{24}{ \color{blue}{48} }x-\frac{48}{ \color{blue}{48} }) \color{blue}{.48} \\\Leftrightarrow & 16x+9& = & 24x-48 \\\Leftrightarrow & 16x \color{red}{+9} \color{blue}{-9} \color{blue}{-24x} & = & \color{red}{24x} -48 \color{blue}{-24x} \color{blue}{-9} \\\Leftrightarrow & -8x& = & -57 \\\Leftrightarrow & \frac{-8x}{ \color{red}{-8} }& = & \frac{-57}{-8} \\\Leftrightarrow & x = \frac{57}{8} & & \\ & V = \left\{ \frac{57}{8} \right\} & \\\end{align}\)