Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{330 is het kleinste gemene veelvoud van 6, 11 en 5} \\ \begin{align} & \frac{x}{6}-\frac{5}{11}& = & \frac{1}{5}x+3 \\\Leftrightarrow & \color{blue}{330.} (\frac{55x}{ \color{blue}{330} }- \frac{ 150 }{ \color{blue}{330} })& = & (\frac{66}{ \color{blue}{330} }x+\frac{990}{ \color{blue}{330} }) \color{blue}{.330} \\\Leftrightarrow & 55x-150& = & 66x+990 \\\Leftrightarrow & 55x \color{red}{-150} \color{blue}{+150} \color{blue}{-66x} & = & \color{red}{66x} +990 \color{blue}{-66x} \color{blue}{+150} \\\Leftrightarrow & -11x& = & 1140 \\\Leftrightarrow & \frac{-11x}{ \color{red}{-11} }& = & \frac{1140}{-11} \\\Leftrightarrow & x = \frac{-1140}{11} & & \\ & V = \left\{ \frac{-1140}{11} \right\} & \\\end{align}\)
2. \(\text{10 is het kleinste gemene veelvoud van 5, 10 en 2} \\ \begin{align} & \frac{x}{5}+\frac{3}{10}& = & \frac{3}{2}x+2 \\\Leftrightarrow & \color{blue}{10.} (\frac{2x}{ \color{blue}{10} }+ \frac{ 3 }{ \color{blue}{10} })& = & (\frac{15}{ \color{blue}{10} }x+\frac{20}{ \color{blue}{10} }) \color{blue}{.10} \\\Leftrightarrow & 2x+3& = & 15x+20 \\\Leftrightarrow & 2x \color{red}{+3} \color{blue}{-3} \color{blue}{-15x} & = & \color{red}{15x} +20 \color{blue}{-15x} \color{blue}{-3} \\\Leftrightarrow & -13x& = & 17 \\\Leftrightarrow & \frac{-13x}{ \color{red}{-13} }& = & \frac{17}{-13} \\\Leftrightarrow & x = \frac{-17}{13} & & \\ & V = \left\{ \frac{-17}{13} \right\} & \\\end{align}\)
3. \(\text{240 is het kleinste gemene veelvoud van 6, 16 en 5} \\ \begin{align} & \frac{x}{6}+\frac{3}{16}& = & \frac{-7}{5}x-6 \\\Leftrightarrow & \color{blue}{240.} (\frac{40x}{ \color{blue}{240} }+ \frac{ 45 }{ \color{blue}{240} })& = & (\frac{-336}{ \color{blue}{240} }x-\frac{1440}{ \color{blue}{240} }) \color{blue}{.240} \\\Leftrightarrow & 40x+45& = & -336x-1440 \\\Leftrightarrow & 40x \color{red}{+45} \color{blue}{-45} \color{blue}{+336x} & = & \color{red}{-336x} -1440 \color{blue}{+336x} \color{blue}{-45} \\\Leftrightarrow & 376x& = & -1485 \\\Leftrightarrow & \frac{376x}{ \color{red}{376} }& = & \frac{-1485}{376} \\\Leftrightarrow & x = \frac{-1485}{376} & & \\ & V = \left\{ \frac{-1485}{376} \right\} & \\\end{align}\)
4. \(\text{308 is het kleinste gemene veelvoud van 7, 11 en 4} \\ \begin{align} & \frac{x}{7}+\frac{5}{11}& = & \frac{-7}{4}x-1 \\\Leftrightarrow & \color{blue}{308.} (\frac{44x}{ \color{blue}{308} }+ \frac{ 140 }{ \color{blue}{308} })& = & (\frac{-539}{ \color{blue}{308} }x-\frac{308}{ \color{blue}{308} }) \color{blue}{.308} \\\Leftrightarrow & 44x+140& = & -539x-308 \\\Leftrightarrow & 44x \color{red}{+140} \color{blue}{-140} \color{blue}{+539x} & = & \color{red}{-539x} -308 \color{blue}{+539x} \color{blue}{-140} \\\Leftrightarrow & 583x& = & -448 \\\Leftrightarrow & \frac{583x}{ \color{red}{583} }& = & \frac{-448}{583} \\\Leftrightarrow & x = \frac{-448}{583} & & \\ & V = \left\{ \frac{-448}{583} \right\} & \\\end{align}\)
5. \(\text{240 is het kleinste gemene veelvoud van 6, 16 en 5} \\ \begin{align} & \frac{x}{6}+\frac{5}{16}& = & \frac{-7}{5}x-7 \\\Leftrightarrow & \color{blue}{240.} (\frac{40x}{ \color{blue}{240} }+ \frac{ 75 }{ \color{blue}{240} })& = & (\frac{-336}{ \color{blue}{240} }x-\frac{1680}{ \color{blue}{240} }) \color{blue}{.240} \\\Leftrightarrow & 40x+75& = & -336x-1680 \\\Leftrightarrow & 40x \color{red}{+75} \color{blue}{-75} \color{blue}{+336x} & = & \color{red}{-336x} -1680 \color{blue}{+336x} \color{blue}{-75} \\\Leftrightarrow & 376x& = & -1755 \\\Leftrightarrow & \frac{376x}{ \color{red}{376} }& = & \frac{-1755}{376} \\\Leftrightarrow & x = \frac{-1755}{376} & & \\ & V = \left\{ \frac{-1755}{376} \right\} & \\\end{align}\)
6. \(\text{70 is het kleinste gemene veelvoud van 2, 7 en 5} \\ \begin{align} & \frac{x}{2}-\frac{3}{7}& = & \frac{-7}{5}x-7 \\\Leftrightarrow & \color{blue}{70.} (\frac{35x}{ \color{blue}{70} }- \frac{ 30 }{ \color{blue}{70} })& = & (\frac{-98}{ \color{blue}{70} }x-\frac{490}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 35x-30& = & -98x-490 \\\Leftrightarrow & 35x \color{red}{-30} \color{blue}{+30} \color{blue}{+98x} & = & \color{red}{-98x} -490 \color{blue}{+98x} \color{blue}{+30} \\\Leftrightarrow & 133x& = & -460 \\\Leftrightarrow & \frac{133x}{ \color{red}{133} }& = & \frac{-460}{133} \\\Leftrightarrow & x = \frac{-460}{133} & & \\ & V = \left\{ \frac{-460}{133} \right\} & \\\end{align}\)
7. \(\text{156 is het kleinste gemene veelvoud van 4, 13 en 3} \\ \begin{align} & \frac{x}{4}-\frac{4}{13}& = & \frac{2}{3}x-3 \\\Leftrightarrow & \color{blue}{156.} (\frac{39x}{ \color{blue}{156} }- \frac{ 48 }{ \color{blue}{156} })& = & (\frac{104}{ \color{blue}{156} }x-\frac{468}{ \color{blue}{156} }) \color{blue}{.156} \\\Leftrightarrow & 39x-48& = & 104x-468 \\\Leftrightarrow & 39x \color{red}{-48} \color{blue}{+48} \color{blue}{-104x} & = & \color{red}{104x} -468 \color{blue}{-104x} \color{blue}{+48} \\\Leftrightarrow & -65x& = & -420 \\\Leftrightarrow & \frac{-65x}{ \color{red}{-65} }& = & \frac{-420}{-65} \\\Leftrightarrow & x = \frac{84}{13} & & \\ & V = \left\{ \frac{84}{13} \right\} & \\\end{align}\)
8. \(\text{78 is het kleinste gemene veelvoud van 3, 13 en 2} \\ \begin{align} & \frac{x}{3}+\frac{3}{13}& = & \frac{7}{2}x-5 \\\Leftrightarrow & \color{blue}{78.} (\frac{26x}{ \color{blue}{78} }+ \frac{ 18 }{ \color{blue}{78} })& = & (\frac{273}{ \color{blue}{78} }x-\frac{390}{ \color{blue}{78} }) \color{blue}{.78} \\\Leftrightarrow & 26x+18& = & 273x-390 \\\Leftrightarrow & 26x \color{red}{+18} \color{blue}{-18} \color{blue}{-273x} & = & \color{red}{273x} -390 \color{blue}{-273x} \color{blue}{-18} \\\Leftrightarrow & -247x& = & -408 \\\Leftrightarrow & \frac{-247x}{ \color{red}{-247} }& = & \frac{-408}{-247} \\\Leftrightarrow & x = \frac{408}{247} & & \\ & V = \left\{ \frac{408}{247} \right\} & \\\end{align}\)
9. \(\text{120 is het kleinste gemene veelvoud van 5, 8 en 6} \\ \begin{align} & \frac{x}{5}+\frac{5}{8}& = & \frac{1}{6}x+6 \\\Leftrightarrow & \color{blue}{120.} (\frac{24x}{ \color{blue}{120} }+ \frac{ 75 }{ \color{blue}{120} })& = & (\frac{20}{ \color{blue}{120} }x+\frac{720}{ \color{blue}{120} }) \color{blue}{.120} \\\Leftrightarrow & 24x+75& = & 20x+720 \\\Leftrightarrow & 24x \color{red}{+75} \color{blue}{-75} \color{blue}{-20x} & = & \color{red}{20x} +720 \color{blue}{-20x} \color{blue}{-75} \\\Leftrightarrow & 4x& = & 645 \\\Leftrightarrow & \frac{4x}{ \color{red}{4} }& = & \frac{645}{4} \\\Leftrightarrow & x = \frac{645}{4} & & \\ & V = \left\{ \frac{645}{4} \right\} & \\\end{align}\)
10. \(\text{36 is het kleinste gemene veelvoud van 4, 9 en 3} \\ \begin{align} & \frac{x}{4}+\frac{2}{9}& = & \frac{-2}{3}x-8 \\\Leftrightarrow & \color{blue}{36.} (\frac{9x}{ \color{blue}{36} }+ \frac{ 8 }{ \color{blue}{36} })& = & (\frac{-24}{ \color{blue}{36} }x-\frac{288}{ \color{blue}{36} }) \color{blue}{.36} \\\Leftrightarrow & 9x+8& = & -24x-288 \\\Leftrightarrow & 9x \color{red}{+8} \color{blue}{-8} \color{blue}{+24x} & = & \color{red}{-24x} -288 \color{blue}{+24x} \color{blue}{-8} \\\Leftrightarrow & 33x& = & -296 \\\Leftrightarrow & \frac{33x}{ \color{red}{33} }& = & \frac{-296}{33} \\\Leftrightarrow & x = \frac{-296}{33} & & \\ & V = \left\{ \frac{-296}{33} \right\} & \\\end{align}\)
11. \(\text{42 is het kleinste gemene veelvoud van 3, 7 en 2} \\ \begin{align} & \frac{x}{3}+\frac{2}{7}& = & \frac{3}{2}x+1 \\\Leftrightarrow & \color{blue}{42.} (\frac{14x}{ \color{blue}{42} }+ \frac{ 12 }{ \color{blue}{42} })& = & (\frac{63}{ \color{blue}{42} }x+\frac{42}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 14x+12& = & 63x+42 \\\Leftrightarrow & 14x \color{red}{+12} \color{blue}{-12} \color{blue}{-63x} & = & \color{red}{63x} +42 \color{blue}{-63x} \color{blue}{-12} \\\Leftrightarrow & -49x& = & 30 \\\Leftrightarrow & \frac{-49x}{ \color{red}{-49} }& = & \frac{30}{-49} \\\Leftrightarrow & x = \frac{-30}{49} & & \\ & V = \left\{ \frac{-30}{49} \right\} & \\\end{align}\)
12. \(\text{78 is het kleinste gemene veelvoud van 2, 13 en 3} \\ \begin{align} & \frac{x}{2}-\frac{4}{13}& = & \frac{1}{3}x-6 \\\Leftrightarrow & \color{blue}{78.} (\frac{39x}{ \color{blue}{78} }- \frac{ 24 }{ \color{blue}{78} })& = & (\frac{26}{ \color{blue}{78} }x-\frac{468}{ \color{blue}{78} }) \color{blue}{.78} \\\Leftrightarrow & 39x-24& = & 26x-468 \\\Leftrightarrow & 39x \color{red}{-24} \color{blue}{+24} \color{blue}{-26x} & = & \color{red}{26x} -468 \color{blue}{-26x} \color{blue}{+24} \\\Leftrightarrow & 13x& = & -444 \\\Leftrightarrow & \frac{13x}{ \color{red}{13} }& = & \frac{-444}{13} \\\Leftrightarrow & x = \frac{-444}{13} & & \\ & V = \left\{ \frac{-444}{13} \right\} & \\\end{align}\)