Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{330 is het kleinste gemene veelvoud van 5, 11 en 6} \\ \begin{align} & \frac{x}{5}-\frac{5}{11}& = & \frac{1}{6}x-7 \\\Leftrightarrow & \color{blue}{330.} (\frac{66x}{ \color{blue}{330} }- \frac{ 150 }{ \color{blue}{330} })& = & (\frac{55}{ \color{blue}{330} }x-\frac{2310}{ \color{blue}{330} }) \color{blue}{.330} \\\Leftrightarrow & 66x-150& = & 55x-2310 \\\Leftrightarrow & 66x \color{red}{-150} \color{blue}{+150} \color{blue}{-55x} & = & \color{red}{55x} -2310 \color{blue}{-55x} \color{blue}{+150} \\\Leftrightarrow & 11x& = & -2160 \\\Leftrightarrow & \frac{11x}{ \color{red}{11} }& = & \frac{-2160}{11} \\\Leftrightarrow & x = \frac{-2160}{11} & & \\ & V = \left\{ \frac{-2160}{11} \right\} & \\\end{align}\)
2. \(\text{15 is het kleinste gemene veelvoud van 3, 15 en 5} \\ \begin{align} & \frac{x}{3}+\frac{2}{15}& = & \frac{-4}{5}x+5 \\\Leftrightarrow & \color{blue}{15.} (\frac{5x}{ \color{blue}{15} }+ \frac{ 2 }{ \color{blue}{15} })& = & (\frac{-12}{ \color{blue}{15} }x+\frac{75}{ \color{blue}{15} }) \color{blue}{.15} \\\Leftrightarrow & 5x+2& = & -12x+75 \\\Leftrightarrow & 5x \color{red}{+2} \color{blue}{-2} \color{blue}{+12x} & = & \color{red}{-12x} +75 \color{blue}{+12x} \color{blue}{-2} \\\Leftrightarrow & 17x& = & 73 \\\Leftrightarrow & \frac{17x}{ \color{red}{17} }& = & \frac{73}{17} \\\Leftrightarrow & x = \frac{73}{17} & & \\ & V = \left\{ \frac{73}{17} \right\} & \\\end{align}\)
3. \(\text{220 is het kleinste gemene veelvoud van 4, 11 en 5} \\ \begin{align} & \frac{x}{4}+\frac{5}{11}& = & \frac{7}{5}x+1 \\\Leftrightarrow & \color{blue}{220.} (\frac{55x}{ \color{blue}{220} }+ \frac{ 100 }{ \color{blue}{220} })& = & (\frac{308}{ \color{blue}{220} }x+\frac{220}{ \color{blue}{220} }) \color{blue}{.220} \\\Leftrightarrow & 55x+100& = & 308x+220 \\\Leftrightarrow & 55x \color{red}{+100} \color{blue}{-100} \color{blue}{-308x} & = & \color{red}{308x} +220 \color{blue}{-308x} \color{blue}{-100} \\\Leftrightarrow & -253x& = & 120 \\\Leftrightarrow & \frac{-253x}{ \color{red}{-253} }& = & \frac{120}{-253} \\\Leftrightarrow & x = \frac{-120}{253} & & \\ & V = \left\{ \frac{-120}{253} \right\} & \\\end{align}\)
4. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}-\frac{4}{7}& = & \frac{-2}{3}x+6 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }- \frac{ 48 }{ \color{blue}{84} })& = & (\frac{-56}{ \color{blue}{84} }x+\frac{504}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x-48& = & -56x+504 \\\Leftrightarrow & 21x \color{red}{-48} \color{blue}{+48} \color{blue}{+56x} & = & \color{red}{-56x} +504 \color{blue}{+56x} \color{blue}{+48} \\\Leftrightarrow & 77x& = & 552 \\\Leftrightarrow & \frac{77x}{ \color{red}{77} }& = & \frac{552}{77} \\\Leftrightarrow & x = \frac{552}{77} & & \\ & V = \left\{ \frac{552}{77} \right\} & \\\end{align}\)
5. \(\text{90 is het kleinste gemene veelvoud van 2, 9 en 5} \\ \begin{align} & \frac{x}{2}-\frac{4}{9}& = & \frac{-4}{5}x+6 \\\Leftrightarrow & \color{blue}{90.} (\frac{45x}{ \color{blue}{90} }- \frac{ 40 }{ \color{blue}{90} })& = & (\frac{-72}{ \color{blue}{90} }x+\frac{540}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 45x-40& = & -72x+540 \\\Leftrightarrow & 45x \color{red}{-40} \color{blue}{+40} \color{blue}{+72x} & = & \color{red}{-72x} +540 \color{blue}{+72x} \color{blue}{+40} \\\Leftrightarrow & 117x& = & 580 \\\Leftrightarrow & \frac{117x}{ \color{red}{117} }& = & \frac{580}{117} \\\Leftrightarrow & x = \frac{580}{117} & & \\ & V = \left\{ \frac{580}{117} \right\} & \\\end{align}\)
6. \(\text{18 is het kleinste gemene veelvoud van 3, 9 en 2} \\ \begin{align} & \frac{x}{3}+\frac{2}{9}& = & \frac{1}{2}x-7 \\\Leftrightarrow & \color{blue}{18.} (\frac{6x}{ \color{blue}{18} }+ \frac{ 4 }{ \color{blue}{18} })& = & (\frac{9}{ \color{blue}{18} }x-\frac{126}{ \color{blue}{18} }) \color{blue}{.18} \\\Leftrightarrow & 6x+4& = & 9x-126 \\\Leftrightarrow & 6x \color{red}{+4} \color{blue}{-4} \color{blue}{-9x} & = & \color{red}{9x} -126 \color{blue}{-9x} \color{blue}{-4} \\\Leftrightarrow & -3x& = & -130 \\\Leftrightarrow & \frac{-3x}{ \color{red}{-3} }& = & \frac{-130}{-3} \\\Leftrightarrow & x = \frac{130}{3} & & \\ & V = \left\{ \frac{130}{3} \right\} & \\\end{align}\)
7. \(\text{132 is het kleinste gemene veelvoud van 3, 11 en 4} \\ \begin{align} & \frac{x}{3}-\frac{4}{11}& = & \frac{1}{4}x+5 \\\Leftrightarrow & \color{blue}{132.} (\frac{44x}{ \color{blue}{132} }- \frac{ 48 }{ \color{blue}{132} })& = & (\frac{33}{ \color{blue}{132} }x+\frac{660}{ \color{blue}{132} }) \color{blue}{.132} \\\Leftrightarrow & 44x-48& = & 33x+660 \\\Leftrightarrow & 44x \color{red}{-48} \color{blue}{+48} \color{blue}{-33x} & = & \color{red}{33x} +660 \color{blue}{-33x} \color{blue}{+48} \\\Leftrightarrow & 11x& = & 708 \\\Leftrightarrow & \frac{11x}{ \color{red}{11} }& = & \frac{708}{11} \\\Leftrightarrow & x = \frac{708}{11} & & \\ & V = \left\{ \frac{708}{11} \right\} & \\\end{align}\)
8. \(\text{120 is het kleinste gemene veelvoud van 5, 8 en 6} \\ \begin{align} & \frac{x}{5}-\frac{3}{8}& = & \frac{1}{6}x-1 \\\Leftrightarrow & \color{blue}{120.} (\frac{24x}{ \color{blue}{120} }- \frac{ 45 }{ \color{blue}{120} })& = & (\frac{20}{ \color{blue}{120} }x-\frac{120}{ \color{blue}{120} }) \color{blue}{.120} \\\Leftrightarrow & 24x-45& = & 20x-120 \\\Leftrightarrow & 24x \color{red}{-45} \color{blue}{+45} \color{blue}{-20x} & = & \color{red}{20x} -120 \color{blue}{-20x} \color{blue}{+45} \\\Leftrightarrow & 4x& = & -75 \\\Leftrightarrow & \frac{4x}{ \color{red}{4} }& = & \frac{-75}{4} \\\Leftrightarrow & x = \frac{-75}{4} & & \\ & V = \left\{ \frac{-75}{4} \right\} & \\\end{align}\)
9. \(\text{30 is het kleinste gemene veelvoud van 5, 6 en 6} \\ \begin{align} & \frac{x}{5}+\frac{5}{6}& = & \frac{-5}{6}x-7 \\\Leftrightarrow & \color{blue}{30.} (\frac{6x}{ \color{blue}{30} }+ \frac{ 25 }{ \color{blue}{30} })& = & (\frac{-25}{ \color{blue}{30} }x-\frac{210}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 6x+25& = & -25x-210 \\\Leftrightarrow & 6x \color{red}{+25} \color{blue}{-25} \color{blue}{+25x} & = & \color{red}{-25x} -210 \color{blue}{+25x} \color{blue}{-25} \\\Leftrightarrow & 31x& = & -235 \\\Leftrightarrow & \frac{31x}{ \color{red}{31} }& = & \frac{-235}{31} \\\Leftrightarrow & x = \frac{-235}{31} & & \\ & V = \left\{ \frac{-235}{31} \right\} & \\\end{align}\)
10. \(\text{330 is het kleinste gemene veelvoud van 6, 11 en 5} \\ \begin{align} & \frac{x}{6}+\frac{5}{11}& = & \frac{-7}{5}x-7 \\\Leftrightarrow & \color{blue}{330.} (\frac{55x}{ \color{blue}{330} }+ \frac{ 150 }{ \color{blue}{330} })& = & (\frac{-462}{ \color{blue}{330} }x-\frac{2310}{ \color{blue}{330} }) \color{blue}{.330} \\\Leftrightarrow & 55x+150& = & -462x-2310 \\\Leftrightarrow & 55x \color{red}{+150} \color{blue}{-150} \color{blue}{+462x} & = & \color{red}{-462x} -2310 \color{blue}{+462x} \color{blue}{-150} \\\Leftrightarrow & 517x& = & -2460 \\\Leftrightarrow & \frac{517x}{ \color{red}{517} }& = & \frac{-2460}{517} \\\Leftrightarrow & x = \frac{-2460}{517} & & \\ & V = \left\{ \frac{-2460}{517} \right\} & \\\end{align}\)
11. \(\text{385 is het kleinste gemene veelvoud van 7, 11 en 5} \\ \begin{align} & \frac{x}{7}-\frac{2}{11}& = & \frac{6}{5}x-8 \\\Leftrightarrow & \color{blue}{385.} (\frac{55x}{ \color{blue}{385} }- \frac{ 70 }{ \color{blue}{385} })& = & (\frac{462}{ \color{blue}{385} }x-\frac{3080}{ \color{blue}{385} }) \color{blue}{.385} \\\Leftrightarrow & 55x-70& = & 462x-3080 \\\Leftrightarrow & 55x \color{red}{-70} \color{blue}{+70} \color{blue}{-462x} & = & \color{red}{462x} -3080 \color{blue}{-462x} \color{blue}{+70} \\\Leftrightarrow & -407x& = & -3010 \\\Leftrightarrow & \frac{-407x}{ \color{red}{-407} }& = & \frac{-3010}{-407} \\\Leftrightarrow & x = \frac{3010}{407} & & \\ & V = \left\{ \frac{3010}{407} \right\} & \\\end{align}\)
12. \(\text{42 is het kleinste gemene veelvoud van 2, 7 en 3} \\ \begin{align} & \frac{x}{2}+\frac{3}{7}& = & \frac{7}{3}x-5 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }+ \frac{ 18 }{ \color{blue}{42} })& = & (\frac{98}{ \color{blue}{42} }x-\frac{210}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x+18& = & 98x-210 \\\Leftrightarrow & 21x \color{red}{+18} \color{blue}{-18} \color{blue}{-98x} & = & \color{red}{98x} -210 \color{blue}{-98x} \color{blue}{-18} \\\Leftrightarrow & -77x& = & -228 \\\Leftrightarrow & \frac{-77x}{ \color{red}{-77} }& = & \frac{-228}{-77} \\\Leftrightarrow & x = \frac{228}{77} & & \\ & V = \left\{ \frac{228}{77} \right\} & \\\end{align}\)