Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}-\frac{2}{7}& = & \frac{2}{3}x+7 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }- \frac{ 24 }{ \color{blue}{84} })& = & (\frac{56}{ \color{blue}{84} }x+\frac{588}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x-24& = & 56x+588 \\\Leftrightarrow & 21x \color{red}{-24} \color{blue}{+24} \color{blue}{-56x} & = & \color{red}{56x} +588 \color{blue}{-56x} \color{blue}{+24} \\\Leftrightarrow & -35x& = & 612 \\\Leftrightarrow & \frac{-35x}{ \color{red}{-35} }& = & \frac{612}{-35} \\\Leftrightarrow & x = \frac{-612}{35} & & \\ & V = \left\{ \frac{-612}{35} \right\} & \\\end{align}\)
2. \(\text{110 is het kleinste gemene veelvoud van 5, 11 en 2} \\ \begin{align} & \frac{x}{5}+\frac{4}{11}& = & \frac{3}{2}x-7 \\\Leftrightarrow & \color{blue}{110.} (\frac{22x}{ \color{blue}{110} }+ \frac{ 40 }{ \color{blue}{110} })& = & (\frac{165}{ \color{blue}{110} }x-\frac{770}{ \color{blue}{110} }) \color{blue}{.110} \\\Leftrightarrow & 22x+40& = & 165x-770 \\\Leftrightarrow & 22x \color{red}{+40} \color{blue}{-40} \color{blue}{-165x} & = & \color{red}{165x} -770 \color{blue}{-165x} \color{blue}{-40} \\\Leftrightarrow & -143x& = & -810 \\\Leftrightarrow & \frac{-143x}{ \color{red}{-143} }& = & \frac{-810}{-143} \\\Leftrightarrow & x = \frac{810}{143} & & \\ & V = \left\{ \frac{810}{143} \right\} & \\\end{align}\)
3. \(\text{60 is het kleinste gemene veelvoud van 3, 15 en 4} \\ \begin{align} & \frac{x}{3}+\frac{2}{15}& = & \frac{1}{4}x-3 \\\Leftrightarrow & \color{blue}{60.} (\frac{20x}{ \color{blue}{60} }+ \frac{ 8 }{ \color{blue}{60} })& = & (\frac{15}{ \color{blue}{60} }x-\frac{180}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 20x+8& = & 15x-180 \\\Leftrightarrow & 20x \color{red}{+8} \color{blue}{-8} \color{blue}{-15x} & = & \color{red}{15x} -180 \color{blue}{-15x} \color{blue}{-8} \\\Leftrightarrow & 5x& = & -188 \\\Leftrightarrow & \frac{5x}{ \color{red}{5} }& = & \frac{-188}{5} \\\Leftrightarrow & x = \frac{-188}{5} & & \\ & V = \left\{ \frac{-188}{5} \right\} & \\\end{align}\)
4. \(\text{240 is het kleinste gemene veelvoud van 3, 16 en 5} \\ \begin{align} & \frac{x}{3}+\frac{5}{16}& = & \frac{4}{5}x-4 \\\Leftrightarrow & \color{blue}{240.} (\frac{80x}{ \color{blue}{240} }+ \frac{ 75 }{ \color{blue}{240} })& = & (\frac{192}{ \color{blue}{240} }x-\frac{960}{ \color{blue}{240} }) \color{blue}{.240} \\\Leftrightarrow & 80x+75& = & 192x-960 \\\Leftrightarrow & 80x \color{red}{+75} \color{blue}{-75} \color{blue}{-192x} & = & \color{red}{192x} -960 \color{blue}{-192x} \color{blue}{-75} \\\Leftrightarrow & -112x& = & -1035 \\\Leftrightarrow & \frac{-112x}{ \color{red}{-112} }& = & \frac{-1035}{-112} \\\Leftrightarrow & x = \frac{1035}{112} & & \\ & V = \left\{ \frac{1035}{112} \right\} & \\\end{align}\)
5. \(\text{110 is het kleinste gemene veelvoud van 5, 11 en 2} \\ \begin{align} & \frac{x}{5}-\frac{5}{11}& = & \frac{1}{2}x+3 \\\Leftrightarrow & \color{blue}{110.} (\frac{22x}{ \color{blue}{110} }- \frac{ 50 }{ \color{blue}{110} })& = & (\frac{55}{ \color{blue}{110} }x+\frac{330}{ \color{blue}{110} }) \color{blue}{.110} \\\Leftrightarrow & 22x-50& = & 55x+330 \\\Leftrightarrow & 22x \color{red}{-50} \color{blue}{+50} \color{blue}{-55x} & = & \color{red}{55x} +330 \color{blue}{-55x} \color{blue}{+50} \\\Leftrightarrow & -33x& = & 380 \\\Leftrightarrow & \frac{-33x}{ \color{red}{-33} }& = & \frac{380}{-33} \\\Leftrightarrow & x = \frac{-380}{33} & & \\ & V = \left\{ \frac{-380}{33} \right\} & \\\end{align}\)
6. \(\text{60 is het kleinste gemene veelvoud van 4, 10 en 3} \\ \begin{align} & \frac{x}{4}-\frac{3}{10}& = & \frac{-5}{3}x+7 \\\Leftrightarrow & \color{blue}{60.} (\frac{15x}{ \color{blue}{60} }- \frac{ 18 }{ \color{blue}{60} })& = & (\frac{-100}{ \color{blue}{60} }x+\frac{420}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 15x-18& = & -100x+420 \\\Leftrightarrow & 15x \color{red}{-18} \color{blue}{+18} \color{blue}{+100x} & = & \color{red}{-100x} +420 \color{blue}{+100x} \color{blue}{+18} \\\Leftrightarrow & 115x& = & 438 \\\Leftrightarrow & \frac{115x}{ \color{red}{115} }& = & \frac{438}{115} \\\Leftrightarrow & x = \frac{438}{115} & & \\ & V = \left\{ \frac{438}{115} \right\} & \\\end{align}\)
7. \(\text{42 is het kleinste gemene veelvoud van 7, 7 en 6} \\ \begin{align} & \frac{x}{7}+\frac{2}{7}& = & \frac{5}{6}x-3 \\\Leftrightarrow & \color{blue}{42.} (\frac{6x}{ \color{blue}{42} }+ \frac{ 12 }{ \color{blue}{42} })& = & (\frac{35}{ \color{blue}{42} }x-\frac{126}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 6x+12& = & 35x-126 \\\Leftrightarrow & 6x \color{red}{+12} \color{blue}{-12} \color{blue}{-35x} & = & \color{red}{35x} -126 \color{blue}{-35x} \color{blue}{-12} \\\Leftrightarrow & -29x& = & -138 \\\Leftrightarrow & \frac{-29x}{ \color{red}{-29} }& = & \frac{-138}{-29} \\\Leftrightarrow & x = \frac{138}{29} & & \\ & V = \left\{ \frac{138}{29} \right\} & \\\end{align}\)
8. \(\text{20 is het kleinste gemene veelvoud van 5, 10 en 4} \\ \begin{align} & \frac{x}{5}-\frac{3}{10}& = & \frac{-3}{4}x-3 \\\Leftrightarrow & \color{blue}{20.} (\frac{4x}{ \color{blue}{20} }- \frac{ 6 }{ \color{blue}{20} })& = & (\frac{-15}{ \color{blue}{20} }x-\frac{60}{ \color{blue}{20} }) \color{blue}{.20} \\\Leftrightarrow & 4x-6& = & -15x-60 \\\Leftrightarrow & 4x \color{red}{-6} \color{blue}{+6} \color{blue}{+15x} & = & \color{red}{-15x} -60 \color{blue}{+15x} \color{blue}{+6} \\\Leftrightarrow & 19x& = & -54 \\\Leftrightarrow & \frac{19x}{ \color{red}{19} }& = & \frac{-54}{19} \\\Leftrightarrow & x = \frac{-54}{19} & & \\ & V = \left\{ \frac{-54}{19} \right\} & \\\end{align}\)
9. \(\text{66 is het kleinste gemene veelvoud van 3, 11 en 2} \\ \begin{align} & \frac{x}{3}-\frac{2}{11}& = & \frac{5}{2}x-4 \\\Leftrightarrow & \color{blue}{66.} (\frac{22x}{ \color{blue}{66} }- \frac{ 12 }{ \color{blue}{66} })& = & (\frac{165}{ \color{blue}{66} }x-\frac{264}{ \color{blue}{66} }) \color{blue}{.66} \\\Leftrightarrow & 22x-12& = & 165x-264 \\\Leftrightarrow & 22x \color{red}{-12} \color{blue}{+12} \color{blue}{-165x} & = & \color{red}{165x} -264 \color{blue}{-165x} \color{blue}{+12} \\\Leftrightarrow & -143x& = & -252 \\\Leftrightarrow & \frac{-143x}{ \color{red}{-143} }& = & \frac{-252}{-143} \\\Leftrightarrow & x = \frac{252}{143} & & \\ & V = \left\{ \frac{252}{143} \right\} & \\\end{align}\)
10. \(\text{14 is het kleinste gemene veelvoud van 7, 7 en 2} \\ \begin{align} & \frac{x}{7}-\frac{3}{7}& = & \frac{1}{2}x+1 \\\Leftrightarrow & \color{blue}{14.} (\frac{2x}{ \color{blue}{14} }- \frac{ 6 }{ \color{blue}{14} })& = & (\frac{7}{ \color{blue}{14} }x+\frac{14}{ \color{blue}{14} }) \color{blue}{.14} \\\Leftrightarrow & 2x-6& = & 7x+14 \\\Leftrightarrow & 2x \color{red}{-6} \color{blue}{+6} \color{blue}{-7x} & = & \color{red}{7x} +14 \color{blue}{-7x} \color{blue}{+6} \\\Leftrightarrow & -5x& = & 20 \\\Leftrightarrow & \frac{-5x}{ \color{red}{-5} }& = & \frac{20}{-5} \\\Leftrightarrow & x = -4 & & \\ & V = \left\{ -4 \right\} & \\\end{align}\)
11. \(\text{60 is het kleinste gemene veelvoud van 4, 15 en 3} \\ \begin{align} & \frac{x}{4}+\frac{4}{15}& = & \frac{-5}{3}x-2 \\\Leftrightarrow & \color{blue}{60.} (\frac{15x}{ \color{blue}{60} }+ \frac{ 16 }{ \color{blue}{60} })& = & (\frac{-100}{ \color{blue}{60} }x-\frac{120}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 15x+16& = & -100x-120 \\\Leftrightarrow & 15x \color{red}{+16} \color{blue}{-16} \color{blue}{+100x} & = & \color{red}{-100x} -120 \color{blue}{+100x} \color{blue}{-16} \\\Leftrightarrow & 115x& = & -136 \\\Leftrightarrow & \frac{115x}{ \color{red}{115} }& = & \frac{-136}{115} \\\Leftrightarrow & x = \frac{-136}{115} & & \\ & V = \left\{ \frac{-136}{115} \right\} & \\\end{align}\)
12. \(\text{30 is het kleinste gemene veelvoud van 6, 10 en 5} \\ \begin{align} & \frac{x}{6}+\frac{3}{10}& = & \frac{-4}{5}x+4 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }+ \frac{ 9 }{ \color{blue}{30} })& = & (\frac{-24}{ \color{blue}{30} }x+\frac{120}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x+9& = & -24x+120 \\\Leftrightarrow & 5x \color{red}{+9} \color{blue}{-9} \color{blue}{+24x} & = & \color{red}{-24x} +120 \color{blue}{+24x} \color{blue}{-9} \\\Leftrightarrow & 29x& = & 111 \\\Leftrightarrow & \frac{29x}{ \color{red}{29} }& = & \frac{111}{29} \\\Leftrightarrow & x = \frac{111}{29} & & \\ & V = \left\{ \frac{111}{29} \right\} & \\\end{align}\)