Vgln. eerste graad (reeks 4)

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

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

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

Verbetersleutel

1. \(\text{60 is het kleinste gemene veelvoud van 4, 15 en 5} \\ \begin{align} & \frac{x}{4}+\frac{4}{15}& = & \frac{-7}{5}x+4 \\\Leftrightarrow & \color{blue}{60.} (\frac{15x}{ \color{blue}{60} }+ \frac{ 16 }{ \color{blue}{60} })& = & (\frac{-84}{ \color{blue}{60} }x+\frac{240}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 15x+16& = & -84x+240 \\\Leftrightarrow & 15x \color{red}{+16} \color{blue}{-16} \color{blue}{+84x} & = & \color{red}{-84x} +240 \color{blue}{+84x} \color{blue}{-16} \\\Leftrightarrow & 99x& = & 224 \\\Leftrightarrow & \frac{99x}{ \color{red}{99} }& = & \frac{224}{99} \\\Leftrightarrow & x = \frac{224}{99} & & \\ & V = \left\{ \frac{224}{99} \right\} & \\\end{align}\)
2. \(\text{66 is het kleinste gemene veelvoud van 2, 11 en 3} \\ \begin{align} & \frac{x}{2}+\frac{5}{11}& = & \frac{-8}{3}x+6 \\\Leftrightarrow & \color{blue}{66.} (\frac{33x}{ \color{blue}{66} }+ \frac{ 30 }{ \color{blue}{66} })& = & (\frac{-176}{ \color{blue}{66} }x+\frac{396}{ \color{blue}{66} }) \color{blue}{.66} \\\Leftrightarrow & 33x+30& = & -176x+396 \\\Leftrightarrow & 33x \color{red}{+30} \color{blue}{-30} \color{blue}{+176x} & = & \color{red}{-176x} +396 \color{blue}{+176x} \color{blue}{-30} \\\Leftrightarrow & 209x& = & 366 \\\Leftrightarrow & \frac{209x}{ \color{red}{209} }& = & \frac{366}{209} \\\Leftrightarrow & x = \frac{366}{209} & & \\ & V = \left\{ \frac{366}{209} \right\} & \\\end{align}\)
3. \(\text{390 is het kleinste gemene veelvoud van 5, 13 en 6} \\ \begin{align} & \frac{x}{5}-\frac{5}{13}& = & \frac{-5}{6}x-4 \\\Leftrightarrow & \color{blue}{390.} (\frac{78x}{ \color{blue}{390} }- \frac{ 150 }{ \color{blue}{390} })& = & (\frac{-325}{ \color{blue}{390} }x-\frac{1560}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 78x-150& = & -325x-1560 \\\Leftrightarrow & 78x \color{red}{-150} \color{blue}{+150} \color{blue}{+325x} & = & \color{red}{-325x} -1560 \color{blue}{+325x} \color{blue}{+150} \\\Leftrightarrow & 403x& = & -1410 \\\Leftrightarrow & \frac{403x}{ \color{red}{403} }& = & \frac{-1410}{403} \\\Leftrightarrow & x = \frac{-1410}{403} & & \\ & V = \left\{ \frac{-1410}{403} \right\} & \\\end{align}\)
4. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}-\frac{4}{7}& = & \frac{7}{3}x-1 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }- \frac{ 48 }{ \color{blue}{84} })& = & (\frac{196}{ \color{blue}{84} }x-\frac{84}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x-48& = & 196x-84 \\\Leftrightarrow & 21x \color{red}{-48} \color{blue}{+48} \color{blue}{-196x} & = & \color{red}{196x} -84 \color{blue}{-196x} \color{blue}{+48} \\\Leftrightarrow & -175x& = & -36 \\\Leftrightarrow & \frac{-175x}{ \color{red}{-175} }& = & \frac{-36}{-175} \\\Leftrightarrow & x = \frac{36}{175} & & \\ & V = \left\{ \frac{36}{175} \right\} & \\\end{align}\)
5. \(\text{30 is het kleinste gemene veelvoud van 6, 6 en 5} \\ \begin{align} & \frac{x}{6}+\frac{5}{6}& = & \frac{-4}{5}x+7 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }+ \frac{ 25 }{ \color{blue}{30} })& = & (\frac{-24}{ \color{blue}{30} }x+\frac{210}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x+25& = & -24x+210 \\\Leftrightarrow & 5x \color{red}{+25} \color{blue}{-25} \color{blue}{+24x} & = & \color{red}{-24x} +210 \color{blue}{+24x} \color{blue}{-25} \\\Leftrightarrow & 29x& = & 185 \\\Leftrightarrow & \frac{29x}{ \color{red}{29} }& = & \frac{185}{29} \\\Leftrightarrow & x = \frac{185}{29} & & \\ & V = \left\{ \frac{185}{29} \right\} & \\\end{align}\)
6. \(\text{156 is het kleinste gemene veelvoud van 3, 13 en 4} \\ \begin{align} & \frac{x}{3}-\frac{4}{13}& = & \frac{5}{4}x+6 \\\Leftrightarrow & \color{blue}{156.} (\frac{52x}{ \color{blue}{156} }- \frac{ 48 }{ \color{blue}{156} })& = & (\frac{195}{ \color{blue}{156} }x+\frac{936}{ \color{blue}{156} }) \color{blue}{.156} \\\Leftrightarrow & 52x-48& = & 195x+936 \\\Leftrightarrow & 52x \color{red}{-48} \color{blue}{+48} \color{blue}{-195x} & = & \color{red}{195x} +936 \color{blue}{-195x} \color{blue}{+48} \\\Leftrightarrow & -143x& = & 984 \\\Leftrightarrow & \frac{-143x}{ \color{red}{-143} }& = & \frac{984}{-143} \\\Leftrightarrow & x = \frac{-984}{143} & & \\ & V = \left\{ \frac{-984}{143} \right\} & \\\end{align}\)
7. \(\text{42 is het kleinste gemene veelvoud van 3, 7 en 2} \\ \begin{align} & \frac{x}{3}+\frac{3}{7}& = & \frac{7}{2}x-7 \\\Leftrightarrow & \color{blue}{42.} (\frac{14x}{ \color{blue}{42} }+ \frac{ 18 }{ \color{blue}{42} })& = & (\frac{147}{ \color{blue}{42} }x-\frac{294}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 14x+18& = & 147x-294 \\\Leftrightarrow & 14x \color{red}{+18} \color{blue}{-18} \color{blue}{-147x} & = & \color{red}{147x} -294 \color{blue}{-147x} \color{blue}{-18} \\\Leftrightarrow & -133x& = & -312 \\\Leftrightarrow & \frac{-133x}{ \color{red}{-133} }& = & \frac{-312}{-133} \\\Leftrightarrow & x = \frac{312}{133} & & \\ & V = \left\{ \frac{312}{133} \right\} & \\\end{align}\)
8. \(\text{24 is het kleinste gemene veelvoud van 3, 8 en 2} \\ \begin{align} & \frac{x}{3}+\frac{5}{8}& = & \frac{5}{2}x-4 \\\Leftrightarrow & \color{blue}{24.} (\frac{8x}{ \color{blue}{24} }+ \frac{ 15 }{ \color{blue}{24} })& = & (\frac{60}{ \color{blue}{24} }x-\frac{96}{ \color{blue}{24} }) \color{blue}{.24} \\\Leftrightarrow & 8x+15& = & 60x-96 \\\Leftrightarrow & 8x \color{red}{+15} \color{blue}{-15} \color{blue}{-60x} & = & \color{red}{60x} -96 \color{blue}{-60x} \color{blue}{-15} \\\Leftrightarrow & -52x& = & -111 \\\Leftrightarrow & \frac{-52x}{ \color{red}{-52} }& = & \frac{-111}{-52} \\\Leftrightarrow & x = \frac{111}{52} & & \\ & V = \left\{ \frac{111}{52} \right\} & \\\end{align}\)
9. \(\text{70 is het kleinste gemene veelvoud van 2, 7 en 5} \\ \begin{align} & \frac{x}{2}-\frac{4}{7}& = & \frac{2}{5}x+4 \\\Leftrightarrow & \color{blue}{70.} (\frac{35x}{ \color{blue}{70} }- \frac{ 40 }{ \color{blue}{70} })& = & (\frac{28}{ \color{blue}{70} }x+\frac{280}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 35x-40& = & 28x+280 \\\Leftrightarrow & 35x \color{red}{-40} \color{blue}{+40} \color{blue}{-28x} & = & \color{red}{28x} +280 \color{blue}{-28x} \color{blue}{+40} \\\Leftrightarrow & 7x& = & 320 \\\Leftrightarrow & \frac{7x}{ \color{red}{7} }& = & \frac{320}{7} \\\Leftrightarrow & x = \frac{320}{7} & & \\ & V = \left\{ \frac{320}{7} \right\} & \\\end{align}\)
10. \(\text{90 is het kleinste gemene veelvoud van 2, 9 en 5} \\ \begin{align} & \frac{x}{2}-\frac{2}{9}& = & \frac{6}{5}x-7 \\\Leftrightarrow & \color{blue}{90.} (\frac{45x}{ \color{blue}{90} }- \frac{ 20 }{ \color{blue}{90} })& = & (\frac{108}{ \color{blue}{90} }x-\frac{630}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 45x-20& = & 108x-630 \\\Leftrightarrow & 45x \color{red}{-20} \color{blue}{+20} \color{blue}{-108x} & = & \color{red}{108x} -630 \color{blue}{-108x} \color{blue}{+20} \\\Leftrightarrow & -63x& = & -610 \\\Leftrightarrow & \frac{-63x}{ \color{red}{-63} }& = & \frac{-610}{-63} \\\Leftrightarrow & x = \frac{610}{63} & & \\ & V = \left\{ \frac{610}{63} \right\} & \\\end{align}\)
11. \(\text{45 is het kleinste gemene veelvoud van 5, 9 en 3} \\ \begin{align} & \frac{x}{5}-\frac{4}{9}& = & \frac{-8}{3}x+5 \\\Leftrightarrow & \color{blue}{45.} (\frac{9x}{ \color{blue}{45} }- \frac{ 20 }{ \color{blue}{45} })& = & (\frac{-120}{ \color{blue}{45} }x+\frac{225}{ \color{blue}{45} }) \color{blue}{.45} \\\Leftrightarrow & 9x-20& = & -120x+225 \\\Leftrightarrow & 9x \color{red}{-20} \color{blue}{+20} \color{blue}{+120x} & = & \color{red}{-120x} +225 \color{blue}{+120x} \color{blue}{+20} \\\Leftrightarrow & 129x& = & 245 \\\Leftrightarrow & \frac{129x}{ \color{red}{129} }& = & \frac{245}{129} \\\Leftrightarrow & x = \frac{245}{129} & & \\ & V = \left\{ \frac{245}{129} \right\} & \\\end{align}\)
12. \(\text{260 is het kleinste gemene veelvoud van 5, 13 en 4} \\ \begin{align} & \frac{x}{5}-\frac{3}{13}& = & \frac{-3}{4}x+7 \\\Leftrightarrow & \color{blue}{260.} (\frac{52x}{ \color{blue}{260} }- \frac{ 60 }{ \color{blue}{260} })& = & (\frac{-195}{ \color{blue}{260} }x+\frac{1820}{ \color{blue}{260} }) \color{blue}{.260} \\\Leftrightarrow & 52x-60& = & -195x+1820 \\\Leftrightarrow & 52x \color{red}{-60} \color{blue}{+60} \color{blue}{+195x} & = & \color{red}{-195x} +1820 \color{blue}{+195x} \color{blue}{+60} \\\Leftrightarrow & 247x& = & 1880 \\\Leftrightarrow & \frac{247x}{ \color{red}{247} }& = & \frac{1880}{247} \\\Leftrightarrow & x = \frac{1880}{247} & & \\ & V = \left\{ \frac{1880}{247} \right\} & \\\end{align}\)