Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{105 is het kleinste gemene veelvoud van 7, 15 en 5} \\ \begin{align} & \frac{x}{7}+\frac{2}{15}& = & \frac{-7}{5}x+1 \\\Leftrightarrow & \color{blue}{105.} (\frac{15x}{ \color{blue}{105} }+ \frac{ 14 }{ \color{blue}{105} })& = & (\frac{-147}{ \color{blue}{105} }x+\frac{105}{ \color{blue}{105} }) \color{blue}{.105} \\\Leftrightarrow & 15x+14& = & -147x+105 \\\Leftrightarrow & 15x \color{red}{+14} \color{blue}{-14} \color{blue}{+147x} & = & \color{red}{-147x} +105 \color{blue}{+147x} \color{blue}{-14} \\\Leftrightarrow & 162x& = & 91 \\\Leftrightarrow & \frac{162x}{ \color{red}{162} }& = & \frac{91}{162} \\\Leftrightarrow & x = \frac{91}{162} & & \\ & V = \left\{ \frac{91}{162} \right\} & \\\end{align}\)
2. \(\text{462 is het kleinste gemene veelvoud van 7, 11 en 6} \\ \begin{align} & \frac{x}{7}+\frac{2}{11}& = & \frac{5}{6}x+7 \\\Leftrightarrow & \color{blue}{462.} (\frac{66x}{ \color{blue}{462} }+ \frac{ 84 }{ \color{blue}{462} })& = & (\frac{385}{ \color{blue}{462} }x+\frac{3234}{ \color{blue}{462} }) \color{blue}{.462} \\\Leftrightarrow & 66x+84& = & 385x+3234 \\\Leftrightarrow & 66x \color{red}{+84} \color{blue}{-84} \color{blue}{-385x} & = & \color{red}{385x} +3234 \color{blue}{-385x} \color{blue}{-84} \\\Leftrightarrow & -319x& = & 3150 \\\Leftrightarrow & \frac{-319x}{ \color{red}{-319} }& = & \frac{3150}{-319} \\\Leftrightarrow & x = \frac{-3150}{319} & & \\ & V = \left\{ \frac{-3150}{319} \right\} & \\\end{align}\)
3. \(\text{10 is het kleinste gemene veelvoud van 2, 10 en 5} \\ \begin{align} & \frac{x}{2}-\frac{3}{10}& = & \frac{-4}{5}x-5 \\\Leftrightarrow & \color{blue}{10.} (\frac{5x}{ \color{blue}{10} }- \frac{ 3 }{ \color{blue}{10} })& = & (\frac{-8}{ \color{blue}{10} }x-\frac{50}{ \color{blue}{10} }) \color{blue}{.10} \\\Leftrightarrow & 5x-3& = & -8x-50 \\\Leftrightarrow & 5x \color{red}{-3} \color{blue}{+3} \color{blue}{+8x} & = & \color{red}{-8x} -50 \color{blue}{+8x} \color{blue}{+3} \\\Leftrightarrow & 13x& = & -47 \\\Leftrightarrow & \frac{13x}{ \color{red}{13} }& = & \frac{-47}{13} \\\Leftrightarrow & x = \frac{-47}{13} & & \\ & V = \left\{ \frac{-47}{13} \right\} & \\\end{align}\)
4. \(\text{165 is het kleinste gemene veelvoud van 3, 11 en 5} \\ \begin{align} & \frac{x}{3}+\frac{2}{11}& = & \frac{-4}{5}x-3 \\\Leftrightarrow & \color{blue}{165.} (\frac{55x}{ \color{blue}{165} }+ \frac{ 30 }{ \color{blue}{165} })& = & (\frac{-132}{ \color{blue}{165} }x-\frac{495}{ \color{blue}{165} }) \color{blue}{.165} \\\Leftrightarrow & 55x+30& = & -132x-495 \\\Leftrightarrow & 55x \color{red}{+30} \color{blue}{-30} \color{blue}{+132x} & = & \color{red}{-132x} -495 \color{blue}{+132x} \color{blue}{-30} \\\Leftrightarrow & 187x& = & -525 \\\Leftrightarrow & \frac{187x}{ \color{red}{187} }& = & \frac{-525}{187} \\\Leftrightarrow & x = \frac{-525}{187} & & \\ & V = \left\{ \frac{-525}{187} \right\} & \\\end{align}\)
5. \(\text{36 is het kleinste gemene veelvoud van 4, 9 en 3} \\ \begin{align} & \frac{x}{4}-\frac{5}{9}& = & \frac{-2}{3}x-4 \\\Leftrightarrow & \color{blue}{36.} (\frac{9x}{ \color{blue}{36} }- \frac{ 20 }{ \color{blue}{36} })& = & (\frac{-24}{ \color{blue}{36} }x-\frac{144}{ \color{blue}{36} }) \color{blue}{.36} \\\Leftrightarrow & 9x-20& = & -24x-144 \\\Leftrightarrow & 9x \color{red}{-20} \color{blue}{+20} \color{blue}{+24x} & = & \color{red}{-24x} -144 \color{blue}{+24x} \color{blue}{+20} \\\Leftrightarrow & 33x& = & -124 \\\Leftrightarrow & \frac{33x}{ \color{red}{33} }& = & \frac{-124}{33} \\\Leftrightarrow & x = \frac{-124}{33} & & \\ & V = \left\{ \frac{-124}{33} \right\} & \\\end{align}\)
6. \(\text{90 is het kleinste gemene veelvoud van 5, 9 en 6} \\ \begin{align} & \frac{x}{5}-\frac{4}{9}& = & \frac{-5}{6}x-1 \\\Leftrightarrow & \color{blue}{90.} (\frac{18x}{ \color{blue}{90} }- \frac{ 40 }{ \color{blue}{90} })& = & (\frac{-75}{ \color{blue}{90} }x-\frac{90}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 18x-40& = & -75x-90 \\\Leftrightarrow & 18x \color{red}{-40} \color{blue}{+40} \color{blue}{+75x} & = & \color{red}{-75x} -90 \color{blue}{+75x} \color{blue}{+40} \\\Leftrightarrow & 93x& = & -50 \\\Leftrightarrow & \frac{93x}{ \color{red}{93} }& = & \frac{-50}{93} \\\Leftrightarrow & x = \frac{-50}{93} & & \\ & V = \left\{ \frac{-50}{93} \right\} & \\\end{align}\)
7. \(\text{42 is het kleinste gemene veelvoud van 2, 7 en 3} \\ \begin{align} & \frac{x}{2}-\frac{5}{7}& = & \frac{7}{3}x-1 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }- \frac{ 30 }{ \color{blue}{42} })& = & (\frac{98}{ \color{blue}{42} }x-\frac{42}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x-30& = & 98x-42 \\\Leftrightarrow & 21x \color{red}{-30} \color{blue}{+30} \color{blue}{-98x} & = & \color{red}{98x} -42 \color{blue}{-98x} \color{blue}{+30} \\\Leftrightarrow & -77x& = & -12 \\\Leftrightarrow & \frac{-77x}{ \color{red}{-77} }& = & \frac{-12}{-77} \\\Leftrightarrow & x = \frac{12}{77} & & \\ & V = \left\{ \frac{12}{77} \right\} & \\\end{align}\)
8. \(\text{14 is het kleinste gemene veelvoud van 7, 7 en 2} \\ \begin{align} & \frac{x}{7}-\frac{3}{7}& = & \frac{3}{2}x+5 \\\Leftrightarrow & \color{blue}{14.} (\frac{2x}{ \color{blue}{14} }- \frac{ 6 }{ \color{blue}{14} })& = & (\frac{21}{ \color{blue}{14} }x+\frac{70}{ \color{blue}{14} }) \color{blue}{.14} \\\Leftrightarrow & 2x-6& = & 21x+70 \\\Leftrightarrow & 2x \color{red}{-6} \color{blue}{+6} \color{blue}{-21x} & = & \color{red}{21x} +70 \color{blue}{-21x} \color{blue}{+6} \\\Leftrightarrow & -19x& = & 76 \\\Leftrightarrow & \frac{-19x}{ \color{red}{-19} }& = & \frac{76}{-19} \\\Leftrightarrow & x = -4 & & \\ & V = \left\{ -4 \right\} & \\\end{align}\)
9. \(\text{14 is het kleinste gemene veelvoud van 7, 14 en 2} \\ \begin{align} & \frac{x}{7}+\frac{3}{14}& = & \frac{3}{2}x+1 \\\Leftrightarrow & \color{blue}{14.} (\frac{2x}{ \color{blue}{14} }+ \frac{ 3 }{ \color{blue}{14} })& = & (\frac{21}{ \color{blue}{14} }x+\frac{14}{ \color{blue}{14} }) \color{blue}{.14} \\\Leftrightarrow & 2x+3& = & 21x+14 \\\Leftrightarrow & 2x \color{red}{+3} \color{blue}{-3} \color{blue}{-21x} & = & \color{red}{21x} +14 \color{blue}{-21x} \color{blue}{-3} \\\Leftrightarrow & -19x& = & 11 \\\Leftrightarrow & \frac{-19x}{ \color{red}{-19} }& = & \frac{11}{-19} \\\Leftrightarrow & x = \frac{-11}{19} & & \\ & V = \left\{ \frac{-11}{19} \right\} & \\\end{align}\)
10. \(\text{132 is het kleinste gemene veelvoud van 4, 11 en 3} \\ \begin{align} & \frac{x}{4}+\frac{2}{11}& = & \frac{-5}{3}x+3 \\\Leftrightarrow & \color{blue}{132.} (\frac{33x}{ \color{blue}{132} }+ \frac{ 24 }{ \color{blue}{132} })& = & (\frac{-220}{ \color{blue}{132} }x+\frac{396}{ \color{blue}{132} }) \color{blue}{.132} \\\Leftrightarrow & 33x+24& = & -220x+396 \\\Leftrightarrow & 33x \color{red}{+24} \color{blue}{-24} \color{blue}{+220x} & = & \color{red}{-220x} +396 \color{blue}{+220x} \color{blue}{-24} \\\Leftrightarrow & 253x& = & 372 \\\Leftrightarrow & \frac{253x}{ \color{red}{253} }& = & \frac{372}{253} \\\Leftrightarrow & x = \frac{372}{253} & & \\ & V = \left\{ \frac{372}{253} \right\} & \\\end{align}\)
11. \(\text{45 is het kleinste gemene veelvoud van 5, 9 en 3} \\ \begin{align} & \frac{x}{5}+\frac{5}{9}& = & \frac{-5}{3}x+1 \\\Leftrightarrow & \color{blue}{45.} (\frac{9x}{ \color{blue}{45} }+ \frac{ 25 }{ \color{blue}{45} })& = & (\frac{-75}{ \color{blue}{45} }x+\frac{45}{ \color{blue}{45} }) \color{blue}{.45} \\\Leftrightarrow & 9x+25& = & -75x+45 \\\Leftrightarrow & 9x \color{red}{+25} \color{blue}{-25} \color{blue}{+75x} & = & \color{red}{-75x} +45 \color{blue}{+75x} \color{blue}{-25} \\\Leftrightarrow & 84x& = & 20 \\\Leftrightarrow & \frac{84x}{ \color{red}{84} }& = & \frac{20}{84} \\\Leftrightarrow & x = \frac{5}{21} & & \\ & V = \left\{ \frac{5}{21} \right\} & \\\end{align}\)
12. \(\text{60 is het kleinste gemene veelvoud van 5, 15 en 4} \\ \begin{align} & \frac{x}{5}-\frac{4}{15}& = & \frac{3}{4}x-4 \\\Leftrightarrow & \color{blue}{60.} (\frac{12x}{ \color{blue}{60} }- \frac{ 16 }{ \color{blue}{60} })& = & (\frac{45}{ \color{blue}{60} }x-\frac{240}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 12x-16& = & 45x-240 \\\Leftrightarrow & 12x \color{red}{-16} \color{blue}{+16} \color{blue}{-45x} & = & \color{red}{45x} -240 \color{blue}{-45x} \color{blue}{+16} \\\Leftrightarrow & -33x& = & -224 \\\Leftrightarrow & \frac{-33x}{ \color{red}{-33} }& = & \frac{-224}{-33} \\\Leftrightarrow & x = \frac{224}{33} & & \\ & V = \left\{ \frac{224}{33} \right\} & \\\end{align}\)