Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{70 is het kleinste gemene veelvoud van 2, 7 en 5} \\ \begin{align} & \frac{x}{2}+\frac{5}{7}& = & \frac{-7}{5}x-3 \\\Leftrightarrow & \color{blue}{70.} (\frac{35x}{ \color{blue}{70} }+ \frac{ 50 }{ \color{blue}{70} })& = & (\frac{-98}{ \color{blue}{70} }x-\frac{210}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 35x+50& = & -98x-210 \\\Leftrightarrow & 35x \color{red}{+50} \color{blue}{-50} \color{blue}{+98x} & = & \color{red}{-98x} -210 \color{blue}{+98x} \color{blue}{-50} \\\Leftrightarrow & 133x& = & -260 \\\Leftrightarrow & \frac{133x}{ \color{red}{133} }& = & \frac{-260}{133} \\\Leftrightarrow & x = \frac{-260}{133} & & \\ & V = \left\{ \frac{-260}{133} \right\} & \\\end{align}\)
2. \(\text{390 is het kleinste gemene veelvoud van 6, 13 en 5} \\ \begin{align} & \frac{x}{6}-\frac{4}{13}& = & \frac{1}{5}x-4 \\\Leftrightarrow & \color{blue}{390.} (\frac{65x}{ \color{blue}{390} }- \frac{ 120 }{ \color{blue}{390} })& = & (\frac{78}{ \color{blue}{390} }x-\frac{1560}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 65x-120& = & 78x-1560 \\\Leftrightarrow & 65x \color{red}{-120} \color{blue}{+120} \color{blue}{-78x} & = & \color{red}{78x} -1560 \color{blue}{-78x} \color{blue}{+120} \\\Leftrightarrow & -13x& = & -1440 \\\Leftrightarrow & \frac{-13x}{ \color{red}{-13} }& = & \frac{-1440}{-13} \\\Leftrightarrow & x = \frac{1440}{13} & & \\ & V = \left\{ \frac{1440}{13} \right\} & \\\end{align}\)
3. \(\text{42 is het kleinste gemene veelvoud van 3, 7 en 2} \\ \begin{align} & \frac{x}{3}+\frac{4}{7}& = & \frac{1}{2}x+5 \\\Leftrightarrow & \color{blue}{42.} (\frac{14x}{ \color{blue}{42} }+ \frac{ 24 }{ \color{blue}{42} })& = & (\frac{21}{ \color{blue}{42} }x+\frac{210}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 14x+24& = & 21x+210 \\\Leftrightarrow & 14x \color{red}{+24} \color{blue}{-24} \color{blue}{-21x} & = & \color{red}{21x} +210 \color{blue}{-21x} \color{blue}{-24} \\\Leftrightarrow & -7x& = & 186 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{186}{-7} \\\Leftrightarrow & x = \frac{-186}{7} & & \\ & V = \left\{ \frac{-186}{7} \right\} & \\\end{align}\)
4. \(\text{110 is het kleinste gemene veelvoud van 2, 11 en 5} \\ \begin{align} & \frac{x}{2}-\frac{2}{11}& = & \frac{1}{5}x+8 \\\Leftrightarrow & \color{blue}{110.} (\frac{55x}{ \color{blue}{110} }- \frac{ 20 }{ \color{blue}{110} })& = & (\frac{22}{ \color{blue}{110} }x+\frac{880}{ \color{blue}{110} }) \color{blue}{.110} \\\Leftrightarrow & 55x-20& = & 22x+880 \\\Leftrightarrow & 55x \color{red}{-20} \color{blue}{+20} \color{blue}{-22x} & = & \color{red}{22x} +880 \color{blue}{-22x} \color{blue}{+20} \\\Leftrightarrow & 33x& = & 900 \\\Leftrightarrow & \frac{33x}{ \color{red}{33} }& = & \frac{900}{33} \\\Leftrightarrow & x = \frac{300}{11} & & \\ & V = \left\{ \frac{300}{11} \right\} & \\\end{align}\)
5. \(\text{110 is het kleinste gemene veelvoud van 5, 11 en 2} \\ \begin{align} & \frac{x}{5}-\frac{5}{11}& = & \frac{1}{2}x-7 \\\Leftrightarrow & \color{blue}{110.} (\frac{22x}{ \color{blue}{110} }- \frac{ 50 }{ \color{blue}{110} })& = & (\frac{55}{ \color{blue}{110} }x-\frac{770}{ \color{blue}{110} }) \color{blue}{.110} \\\Leftrightarrow & 22x-50& = & 55x-770 \\\Leftrightarrow & 22x \color{red}{-50} \color{blue}{+50} \color{blue}{-55x} & = & \color{red}{55x} -770 \color{blue}{-55x} \color{blue}{+50} \\\Leftrightarrow & -33x& = & -720 \\\Leftrightarrow & \frac{-33x}{ \color{red}{-33} }& = & \frac{-720}{-33} \\\Leftrightarrow & x = \frac{240}{11} & & \\ & V = \left\{ \frac{240}{11} \right\} & \\\end{align}\)
6. \(\text{330 is het kleinste gemene veelvoud van 6, 11 en 5} \\ \begin{align} & \frac{x}{6}-\frac{5}{11}& = & \frac{-4}{5}x-2 \\\Leftrightarrow & \color{blue}{330.} (\frac{55x}{ \color{blue}{330} }- \frac{ 150 }{ \color{blue}{330} })& = & (\frac{-264}{ \color{blue}{330} }x-\frac{660}{ \color{blue}{330} }) \color{blue}{.330} \\\Leftrightarrow & 55x-150& = & -264x-660 \\\Leftrightarrow & 55x \color{red}{-150} \color{blue}{+150} \color{blue}{+264x} & = & \color{red}{-264x} -660 \color{blue}{+264x} \color{blue}{+150} \\\Leftrightarrow & 319x& = & -510 \\\Leftrightarrow & \frac{319x}{ \color{red}{319} }& = & \frac{-510}{319} \\\Leftrightarrow & x = \frac{-510}{319} & & \\ & V = \left\{ \frac{-510}{319} \right\} & \\\end{align}\)
7. \(\text{30 is het kleinste gemene veelvoud van 5, 10 en 3} \\ \begin{align} & \frac{x}{5}-\frac{3}{10}& = & \frac{1}{3}x+6 \\\Leftrightarrow & \color{blue}{30.} (\frac{6x}{ \color{blue}{30} }- \frac{ 9 }{ \color{blue}{30} })& = & (\frac{10}{ \color{blue}{30} }x+\frac{180}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 6x-9& = & 10x+180 \\\Leftrightarrow & 6x \color{red}{-9} \color{blue}{+9} \color{blue}{-10x} & = & \color{red}{10x} +180 \color{blue}{-10x} \color{blue}{+9} \\\Leftrightarrow & -4x& = & 189 \\\Leftrightarrow & \frac{-4x}{ \color{red}{-4} }& = & \frac{189}{-4} \\\Leftrightarrow & x = \frac{-189}{4} & & \\ & V = \left\{ \frac{-189}{4} \right\} & \\\end{align}\)
8. \(\text{30 is het kleinste gemene veelvoud van 6, 15 en 5} \\ \begin{align} & \frac{x}{6}-\frac{4}{15}& = & \frac{-4}{5}x-8 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }- \frac{ 8 }{ \color{blue}{30} })& = & (\frac{-24}{ \color{blue}{30} }x-\frac{240}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x-8& = & -24x-240 \\\Leftrightarrow & 5x \color{red}{-8} \color{blue}{+8} \color{blue}{+24x} & = & \color{red}{-24x} -240 \color{blue}{+24x} \color{blue}{+8} \\\Leftrightarrow & 29x& = & -232 \\\Leftrightarrow & \frac{29x}{ \color{red}{29} }& = & \frac{-232}{29} \\\Leftrightarrow & x = -8 & & \\ & V = \left\{ -8 \right\} & \\\end{align}\)
9. \(\text{42 is het kleinste gemene veelvoud van 2, 7 en 3} \\ \begin{align} & \frac{x}{2}+\frac{2}{7}& = & \frac{7}{3}x+4 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }+ \frac{ 12 }{ \color{blue}{42} })& = & (\frac{98}{ \color{blue}{42} }x+\frac{168}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x+12& = & 98x+168 \\\Leftrightarrow & 21x \color{red}{+12} \color{blue}{-12} \color{blue}{-98x} & = & \color{red}{98x} +168 \color{blue}{-98x} \color{blue}{-12} \\\Leftrightarrow & -77x& = & 156 \\\Leftrightarrow & \frac{-77x}{ \color{red}{-77} }& = & \frac{156}{-77} \\\Leftrightarrow & x = \frac{-156}{77} & & \\ & V = \left\{ \frac{-156}{77} \right\} & \\\end{align}\)
10. \(\text{90 is het kleinste gemene veelvoud van 5, 9 en 6} \\ \begin{align} & \frac{x}{5}-\frac{4}{9}& = & \frac{7}{6}x+1 \\\Leftrightarrow & \color{blue}{90.} (\frac{18x}{ \color{blue}{90} }- \frac{ 40 }{ \color{blue}{90} })& = & (\frac{105}{ \color{blue}{90} }x+\frac{90}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 18x-40& = & 105x+90 \\\Leftrightarrow & 18x \color{red}{-40} \color{blue}{+40} \color{blue}{-105x} & = & \color{red}{105x} +90 \color{blue}{-105x} \color{blue}{+40} \\\Leftrightarrow & -87x& = & 130 \\\Leftrightarrow & \frac{-87x}{ \color{red}{-87} }& = & \frac{130}{-87} \\\Leftrightarrow & x = \frac{-130}{87} & & \\ & V = \left\{ \frac{-130}{87} \right\} & \\\end{align}\)
11. \(\text{20 is het kleinste gemene veelvoud van 4, 10 en 5} \\ \begin{align} & \frac{x}{4}-\frac{3}{10}& = & \frac{1}{5}x-2 \\\Leftrightarrow & \color{blue}{20.} (\frac{5x}{ \color{blue}{20} }- \frac{ 6 }{ \color{blue}{20} })& = & (\frac{4}{ \color{blue}{20} }x-\frac{40}{ \color{blue}{20} }) \color{blue}{.20} \\\Leftrightarrow & 5x-6& = & 4x-40 \\\Leftrightarrow & 5x \color{red}{-6} \color{blue}{+6} \color{blue}{-4x} & = & \color{red}{4x} -40 \color{blue}{-4x} \color{blue}{+6} \\\Leftrightarrow & x& = & -34 \\ & V = \left\{ -34 \right\} & \\\end{align}\)
12. \(\text{140 is het kleinste gemene veelvoud van 5, 7 en 4} \\ \begin{align} & \frac{x}{5}+\frac{3}{7}& = & \frac{-3}{4}x-8 \\\Leftrightarrow & \color{blue}{140.} (\frac{28x}{ \color{blue}{140} }+ \frac{ 60 }{ \color{blue}{140} })& = & (\frac{-105}{ \color{blue}{140} }x-\frac{1120}{ \color{blue}{140} }) \color{blue}{.140} \\\Leftrightarrow & 28x+60& = & -105x-1120 \\\Leftrightarrow & 28x \color{red}{+60} \color{blue}{-60} \color{blue}{+105x} & = & \color{red}{-105x} -1120 \color{blue}{+105x} \color{blue}{-60} \\\Leftrightarrow & 133x& = & -1180 \\\Leftrightarrow & \frac{133x}{ \color{red}{133} }& = & \frac{-1180}{133} \\\Leftrightarrow & x = \frac{-1180}{133} & & \\ & V = \left\{ \frac{-1180}{133} \right\} & \\\end{align}\)