Vgln. eerste graad (reeks 4)

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1. \(\frac{x}{2}+\frac{4}{7}=\frac{7}{5}x-4\)
2. \(\frac{x}{2}+\frac{2}{13}=\frac{4}{5}x-6\)
3. \(\frac{x}{6}-\frac{5}{11}=\frac{1}{5}x-8\)
4. \(\frac{x}{5}-\frac{5}{16}=\frac{7}{2}x+7\)
5. \(\frac{x}{3}-\frac{2}{11}=\frac{-3}{4}x+5\)
6. \(\frac{x}{5}+\frac{3}{14}=\frac{7}{2}x-2\)
7. \(\frac{x}{4}+\frac{5}{6}=\frac{7}{5}x+8\)
8. \(\frac{x}{3}+\frac{4}{11}=\frac{6}{5}x-5\)
9. \(\frac{x}{7}-\frac{5}{11}=\frac{5}{6}x-7\)
10. \(\frac{x}{2}-\frac{3}{7}=\frac{-2}{3}x-7\)
11. \(\frac{x}{4}-\frac{2}{13}=\frac{-8}{3}x+7\)
12. \(\frac{x}{2}+\frac{5}{16}=\frac{4}{3}x-4\)

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

Verbetersleutel

1. \(\text{70 is het kleinste gemene veelvoud van 2, 7 en 5} \\ \begin{align} & \frac{x}{2}+\frac{4}{7}& = & \frac{7}{5}x-4 \\\Leftrightarrow & \color{blue}{70.} (\frac{35x}{ \color{blue}{70} }+ \frac{ 40 }{ \color{blue}{70} })& = & (\frac{98}{ \color{blue}{70} }x-\frac{280}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 35x+40& = & 98x-280 \\\Leftrightarrow & 35x \color{red}{+40} \color{blue}{-40} \color{blue}{-98x} & = & \color{red}{98x} -280 \color{blue}{-98x} \color{blue}{-40} \\\Leftrightarrow & -63x& = & -320 \\\Leftrightarrow & \frac{-63x}{ \color{red}{-63} }& = & \frac{-320}{-63} \\\Leftrightarrow & x = \frac{320}{63} & & \\ & V = \left\{ \frac{320}{63} \right\} & \\\end{align}\)
2. \(\text{130 is het kleinste gemene veelvoud van 2, 13 en 5} \\ \begin{align} & \frac{x}{2}+\frac{2}{13}& = & \frac{4}{5}x-6 \\\Leftrightarrow & \color{blue}{130.} (\frac{65x}{ \color{blue}{130} }+ \frac{ 20 }{ \color{blue}{130} })& = & (\frac{104}{ \color{blue}{130} }x-\frac{780}{ \color{blue}{130} }) \color{blue}{.130} \\\Leftrightarrow & 65x+20& = & 104x-780 \\\Leftrightarrow & 65x \color{red}{+20} \color{blue}{-20} \color{blue}{-104x} & = & \color{red}{104x} -780 \color{blue}{-104x} \color{blue}{-20} \\\Leftrightarrow & -39x& = & -800 \\\Leftrightarrow & \frac{-39x}{ \color{red}{-39} }& = & \frac{-800}{-39} \\\Leftrightarrow & x = \frac{800}{39} & & \\ & V = \left\{ \frac{800}{39} \right\} & \\\end{align}\)
3. \(\text{330 is het kleinste gemene veelvoud van 6, 11 en 5} \\ \begin{align} & \frac{x}{6}-\frac{5}{11}& = & \frac{1}{5}x-8 \\\Leftrightarrow & \color{blue}{330.} (\frac{55x}{ \color{blue}{330} }- \frac{ 150 }{ \color{blue}{330} })& = & (\frac{66}{ \color{blue}{330} }x-\frac{2640}{ \color{blue}{330} }) \color{blue}{.330} \\\Leftrightarrow & 55x-150& = & 66x-2640 \\\Leftrightarrow & 55x \color{red}{-150} \color{blue}{+150} \color{blue}{-66x} & = & \color{red}{66x} -2640 \color{blue}{-66x} \color{blue}{+150} \\\Leftrightarrow & -11x& = & -2490 \\\Leftrightarrow & \frac{-11x}{ \color{red}{-11} }& = & \frac{-2490}{-11} \\\Leftrightarrow & x = \frac{2490}{11} & & \\ & V = \left\{ \frac{2490}{11} \right\} & \\\end{align}\)
4. \(\text{80 is het kleinste gemene veelvoud van 5, 16 en 2} \\ \begin{align} & \frac{x}{5}-\frac{5}{16}& = & \frac{7}{2}x+7 \\\Leftrightarrow & \color{blue}{80.} (\frac{16x}{ \color{blue}{80} }- \frac{ 25 }{ \color{blue}{80} })& = & (\frac{280}{ \color{blue}{80} }x+\frac{560}{ \color{blue}{80} }) \color{blue}{.80} \\\Leftrightarrow & 16x-25& = & 280x+560 \\\Leftrightarrow & 16x \color{red}{-25} \color{blue}{+25} \color{blue}{-280x} & = & \color{red}{280x} +560 \color{blue}{-280x} \color{blue}{+25} \\\Leftrightarrow & -264x& = & 585 \\\Leftrightarrow & \frac{-264x}{ \color{red}{-264} }& = & \frac{585}{-264} \\\Leftrightarrow & x = \frac{-195}{88} & & \\ & V = \left\{ \frac{-195}{88} \right\} & \\\end{align}\)
5. \(\text{132 is het kleinste gemene veelvoud van 3, 11 en 4} \\ \begin{align} & \frac{x}{3}-\frac{2}{11}& = & \frac{-3}{4}x+5 \\\Leftrightarrow & \color{blue}{132.} (\frac{44x}{ \color{blue}{132} }- \frac{ 24 }{ \color{blue}{132} })& = & (\frac{-99}{ \color{blue}{132} }x+\frac{660}{ \color{blue}{132} }) \color{blue}{.132} \\\Leftrightarrow & 44x-24& = & -99x+660 \\\Leftrightarrow & 44x \color{red}{-24} \color{blue}{+24} \color{blue}{+99x} & = & \color{red}{-99x} +660 \color{blue}{+99x} \color{blue}{+24} \\\Leftrightarrow & 143x& = & 684 \\\Leftrightarrow & \frac{143x}{ \color{red}{143} }& = & \frac{684}{143} \\\Leftrightarrow & x = \frac{684}{143} & & \\ & V = \left\{ \frac{684}{143} \right\} & \\\end{align}\)
6. \(\text{70 is het kleinste gemene veelvoud van 5, 14 en 2} \\ \begin{align} & \frac{x}{5}+\frac{3}{14}& = & \frac{7}{2}x-2 \\\Leftrightarrow & \color{blue}{70.} (\frac{14x}{ \color{blue}{70} }+ \frac{ 15 }{ \color{blue}{70} })& = & (\frac{245}{ \color{blue}{70} }x-\frac{140}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 14x+15& = & 245x-140 \\\Leftrightarrow & 14x \color{red}{+15} \color{blue}{-15} \color{blue}{-245x} & = & \color{red}{245x} -140 \color{blue}{-245x} \color{blue}{-15} \\\Leftrightarrow & -231x& = & -155 \\\Leftrightarrow & \frac{-231x}{ \color{red}{-231} }& = & \frac{-155}{-231} \\\Leftrightarrow & x = \frac{155}{231} & & \\ & V = \left\{ \frac{155}{231} \right\} & \\\end{align}\)
7. \(\text{60 is het kleinste gemene veelvoud van 4, 6 en 5} \\ \begin{align} & \frac{x}{4}+\frac{5}{6}& = & \frac{7}{5}x+8 \\\Leftrightarrow & \color{blue}{60.} (\frac{15x}{ \color{blue}{60} }+ \frac{ 50 }{ \color{blue}{60} })& = & (\frac{84}{ \color{blue}{60} }x+\frac{480}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 15x+50& = & 84x+480 \\\Leftrightarrow & 15x \color{red}{+50} \color{blue}{-50} \color{blue}{-84x} & = & \color{red}{84x} +480 \color{blue}{-84x} \color{blue}{-50} \\\Leftrightarrow & -69x& = & 430 \\\Leftrightarrow & \frac{-69x}{ \color{red}{-69} }& = & \frac{430}{-69} \\\Leftrightarrow & x = \frac{-430}{69} & & \\ & V = \left\{ \frac{-430}{69} \right\} & \\\end{align}\)
8. \(\text{165 is het kleinste gemene veelvoud van 3, 11 en 5} \\ \begin{align} & \frac{x}{3}+\frac{4}{11}& = & \frac{6}{5}x-5 \\\Leftrightarrow & \color{blue}{165.} (\frac{55x}{ \color{blue}{165} }+ \frac{ 60 }{ \color{blue}{165} })& = & (\frac{198}{ \color{blue}{165} }x-\frac{825}{ \color{blue}{165} }) \color{blue}{.165} \\\Leftrightarrow & 55x+60& = & 198x-825 \\\Leftrightarrow & 55x \color{red}{+60} \color{blue}{-60} \color{blue}{-198x} & = & \color{red}{198x} -825 \color{blue}{-198x} \color{blue}{-60} \\\Leftrightarrow & -143x& = & -885 \\\Leftrightarrow & \frac{-143x}{ \color{red}{-143} }& = & \frac{-885}{-143} \\\Leftrightarrow & x = \frac{885}{143} & & \\ & V = \left\{ \frac{885}{143} \right\} & \\\end{align}\)
9. \(\text{462 is het kleinste gemene veelvoud van 7, 11 en 6} \\ \begin{align} & \frac{x}{7}-\frac{5}{11}& = & \frac{5}{6}x-7 \\\Leftrightarrow & \color{blue}{462.} (\frac{66x}{ \color{blue}{462} }- \frac{ 210 }{ \color{blue}{462} })& = & (\frac{385}{ \color{blue}{462} }x-\frac{3234}{ \color{blue}{462} }) \color{blue}{.462} \\\Leftrightarrow & 66x-210& = & 385x-3234 \\\Leftrightarrow & 66x \color{red}{-210} \color{blue}{+210} \color{blue}{-385x} & = & \color{red}{385x} -3234 \color{blue}{-385x} \color{blue}{+210} \\\Leftrightarrow & -319x& = & -3024 \\\Leftrightarrow & \frac{-319x}{ \color{red}{-319} }& = & \frac{-3024}{-319} \\\Leftrightarrow & x = \frac{3024}{319} & & \\ & V = \left\{ \frac{3024}{319} \right\} & \\\end{align}\)
10. \(\text{42 is het kleinste gemene veelvoud van 2, 7 en 3} \\ \begin{align} & \frac{x}{2}-\frac{3}{7}& = & \frac{-2}{3}x-7 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }- \frac{ 18 }{ \color{blue}{42} })& = & (\frac{-28}{ \color{blue}{42} }x-\frac{294}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x-18& = & -28x-294 \\\Leftrightarrow & 21x \color{red}{-18} \color{blue}{+18} \color{blue}{+28x} & = & \color{red}{-28x} -294 \color{blue}{+28x} \color{blue}{+18} \\\Leftrightarrow & 49x& = & -276 \\\Leftrightarrow & \frac{49x}{ \color{red}{49} }& = & \frac{-276}{49} \\\Leftrightarrow & x = \frac{-276}{49} & & \\ & V = \left\{ \frac{-276}{49} \right\} & \\\end{align}\)
11. \(\text{156 is het kleinste gemene veelvoud van 4, 13 en 3} \\ \begin{align} & \frac{x}{4}-\frac{2}{13}& = & \frac{-8}{3}x+7 \\\Leftrightarrow & \color{blue}{156.} (\frac{39x}{ \color{blue}{156} }- \frac{ 24 }{ \color{blue}{156} })& = & (\frac{-416}{ \color{blue}{156} }x+\frac{1092}{ \color{blue}{156} }) \color{blue}{.156} \\\Leftrightarrow & 39x-24& = & -416x+1092 \\\Leftrightarrow & 39x \color{red}{-24} \color{blue}{+24} \color{blue}{+416x} & = & \color{red}{-416x} +1092 \color{blue}{+416x} \color{blue}{+24} \\\Leftrightarrow & 455x& = & 1116 \\\Leftrightarrow & \frac{455x}{ \color{red}{455} }& = & \frac{1116}{455} \\\Leftrightarrow & x = \frac{1116}{455} & & \\ & V = \left\{ \frac{1116}{455} \right\} & \\\end{align}\)
12. \(\text{48 is het kleinste gemene veelvoud van 2, 16 en 3} \\ \begin{align} & \frac{x}{2}+\frac{5}{16}& = & \frac{4}{3}x-4 \\\Leftrightarrow & \color{blue}{48.} (\frac{24x}{ \color{blue}{48} }+ \frac{ 15 }{ \color{blue}{48} })& = & (\frac{64}{ \color{blue}{48} }x-\frac{192}{ \color{blue}{48} }) \color{blue}{.48} \\\Leftrightarrow & 24x+15& = & 64x-192 \\\Leftrightarrow & 24x \color{red}{+15} \color{blue}{-15} \color{blue}{-64x} & = & \color{red}{64x} -192 \color{blue}{-64x} \color{blue}{-15} \\\Leftrightarrow & -40x& = & -207 \\\Leftrightarrow & \frac{-40x}{ \color{red}{-40} }& = & \frac{-207}{-40} \\\Leftrightarrow & x = \frac{207}{40} & & \\ & V = \left\{ \frac{207}{40} \right\} & \\\end{align}\)