Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{28 is het kleinste gemene veelvoud van 7, 7 en 4} \\ \begin{align} & \frac{x}{7}+\frac{4}{7}& = & \frac{1}{4}x-3 \\\Leftrightarrow & \color{blue}{28.} (\frac{4x}{ \color{blue}{28} }+ \frac{ 16 }{ \color{blue}{28} })& = & (\frac{7}{ \color{blue}{28} }x-\frac{84}{ \color{blue}{28} }) \color{blue}{.28} \\\Leftrightarrow & 4x+16& = & 7x-84 \\\Leftrightarrow & 4x \color{red}{+16} \color{blue}{-16} \color{blue}{-7x} & = & \color{red}{7x} -84 \color{blue}{-7x} \color{blue}{-16} \\\Leftrightarrow & -3x& = & -100 \\\Leftrightarrow & \frac{-3x}{ \color{red}{-3} }& = & \frac{-100}{-3} \\\Leftrightarrow & x = \frac{100}{3} & & \\ & V = \left\{ \frac{100}{3} \right\} & \\\end{align}\)
2. \(\text{42 is het kleinste gemene veelvoud van 2, 7 en 3} \\ \begin{align} & \frac{x}{2}+\frac{4}{7}& = & \frac{4}{3}x+5 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }+ \frac{ 24 }{ \color{blue}{42} })& = & (\frac{56}{ \color{blue}{42} }x+\frac{210}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x+24& = & 56x+210 \\\Leftrightarrow & 21x \color{red}{+24} \color{blue}{-24} \color{blue}{-56x} & = & \color{red}{56x} +210 \color{blue}{-56x} \color{blue}{-24} \\\Leftrightarrow & -35x& = & 186 \\\Leftrightarrow & \frac{-35x}{ \color{red}{-35} }& = & \frac{186}{-35} \\\Leftrightarrow & x = \frac{-186}{35} & & \\ & V = \left\{ \frac{-186}{35} \right\} & \\\end{align}\)
3. \(\text{70 is het kleinste gemene veelvoud van 7, 14 en 5} \\ \begin{align} & \frac{x}{7}-\frac{5}{14}& = & \frac{6}{5}x+5 \\\Leftrightarrow & \color{blue}{70.} (\frac{10x}{ \color{blue}{70} }- \frac{ 25 }{ \color{blue}{70} })& = & (\frac{84}{ \color{blue}{70} }x+\frac{350}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 10x-25& = & 84x+350 \\\Leftrightarrow & 10x \color{red}{-25} \color{blue}{+25} \color{blue}{-84x} & = & \color{red}{84x} +350 \color{blue}{-84x} \color{blue}{+25} \\\Leftrightarrow & -74x& = & 375 \\\Leftrightarrow & \frac{-74x}{ \color{red}{-74} }& = & \frac{375}{-74} \\\Leftrightarrow & x = \frac{-375}{74} & & \\ & V = \left\{ \frac{-375}{74} \right\} & \\\end{align}\)
4. \(\text{195 is het kleinste gemene veelvoud van 3, 13 en 5} \\ \begin{align} & \frac{x}{3}+\frac{2}{13}& = & \frac{1}{5}x-5 \\\Leftrightarrow & \color{blue}{195.} (\frac{65x}{ \color{blue}{195} }+ \frac{ 30 }{ \color{blue}{195} })& = & (\frac{39}{ \color{blue}{195} }x-\frac{975}{ \color{blue}{195} }) \color{blue}{.195} \\\Leftrightarrow & 65x+30& = & 39x-975 \\\Leftrightarrow & 65x \color{red}{+30} \color{blue}{-30} \color{blue}{-39x} & = & \color{red}{39x} -975 \color{blue}{-39x} \color{blue}{-30} \\\Leftrightarrow & 26x& = & -1005 \\\Leftrightarrow & \frac{26x}{ \color{red}{26} }& = & \frac{-1005}{26} \\\Leftrightarrow & x = \frac{-1005}{26} & & \\ & V = \left\{ \frac{-1005}{26} \right\} & \\\end{align}\)
5. \(\text{70 is het kleinste gemene veelvoud van 7, 10 en 5} \\ \begin{align} & \frac{x}{7}-\frac{3}{10}& = & \frac{6}{5}x+6 \\\Leftrightarrow & \color{blue}{70.} (\frac{10x}{ \color{blue}{70} }- \frac{ 21 }{ \color{blue}{70} })& = & (\frac{84}{ \color{blue}{70} }x+\frac{420}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 10x-21& = & 84x+420 \\\Leftrightarrow & 10x \color{red}{-21} \color{blue}{+21} \color{blue}{-84x} & = & \color{red}{84x} +420 \color{blue}{-84x} \color{blue}{+21} \\\Leftrightarrow & -74x& = & 441 \\\Leftrightarrow & \frac{-74x}{ \color{red}{-74} }& = & \frac{441}{-74} \\\Leftrightarrow & x = \frac{-441}{74} & & \\ & V = \left\{ \frac{-441}{74} \right\} & \\\end{align}\)
6. \(\text{132 is het kleinste gemene veelvoud van 4, 11 en 3} \\ \begin{align} & \frac{x}{4}-\frac{4}{11}& = & \frac{-8}{3}x-2 \\\Leftrightarrow & \color{blue}{132.} (\frac{33x}{ \color{blue}{132} }- \frac{ 48 }{ \color{blue}{132} })& = & (\frac{-352}{ \color{blue}{132} }x-\frac{264}{ \color{blue}{132} }) \color{blue}{.132} \\\Leftrightarrow & 33x-48& = & -352x-264 \\\Leftrightarrow & 33x \color{red}{-48} \color{blue}{+48} \color{blue}{+352x} & = & \color{red}{-352x} -264 \color{blue}{+352x} \color{blue}{+48} \\\Leftrightarrow & 385x& = & -216 \\\Leftrightarrow & \frac{385x}{ \color{red}{385} }& = & \frac{-216}{385} \\\Leftrightarrow & x = \frac{-216}{385} & & \\ & V = \left\{ \frac{-216}{385} \right\} & \\\end{align}\)
7. \(\text{140 is het kleinste gemene veelvoud van 4, 14 en 5} \\ \begin{align} & \frac{x}{4}+\frac{5}{14}& = & \frac{-7}{5}x-3 \\\Leftrightarrow & \color{blue}{140.} (\frac{35x}{ \color{blue}{140} }+ \frac{ 50 }{ \color{blue}{140} })& = & (\frac{-196}{ \color{blue}{140} }x-\frac{420}{ \color{blue}{140} }) \color{blue}{.140} \\\Leftrightarrow & 35x+50& = & -196x-420 \\\Leftrightarrow & 35x \color{red}{+50} \color{blue}{-50} \color{blue}{+196x} & = & \color{red}{-196x} -420 \color{blue}{+196x} \color{blue}{-50} \\\Leftrightarrow & 231x& = & -470 \\\Leftrightarrow & \frac{231x}{ \color{red}{231} }& = & \frac{-470}{231} \\\Leftrightarrow & x = \frac{-470}{231} & & \\ & V = \left\{ \frac{-470}{231} \right\} & \\\end{align}\)
8. \(\text{330 is het kleinste gemene veelvoud van 6, 11 en 5} \\ \begin{align} & \frac{x}{6}-\frac{5}{11}& = & \frac{-4}{5}x+3 \\\Leftrightarrow & \color{blue}{330.} (\frac{55x}{ \color{blue}{330} }- \frac{ 150 }{ \color{blue}{330} })& = & (\frac{-264}{ \color{blue}{330} }x+\frac{990}{ \color{blue}{330} }) \color{blue}{.330} \\\Leftrightarrow & 55x-150& = & -264x+990 \\\Leftrightarrow & 55x \color{red}{-150} \color{blue}{+150} \color{blue}{+264x} & = & \color{red}{-264x} +990 \color{blue}{+264x} \color{blue}{+150} \\\Leftrightarrow & 319x& = & 1140 \\\Leftrightarrow & \frac{319x}{ \color{red}{319} }& = & \frac{1140}{319} \\\Leftrightarrow & x = \frac{1140}{319} & & \\ & V = \left\{ \frac{1140}{319} \right\} & \\\end{align}\)
9. \(\text{60 is het kleinste gemene veelvoud van 6, 12 en 5} \\ \begin{align} & \frac{x}{6}+\frac{5}{12}& = & \frac{3}{5}x-2 \\\Leftrightarrow & \color{blue}{60.} (\frac{10x}{ \color{blue}{60} }+ \frac{ 25 }{ \color{blue}{60} })& = & (\frac{36}{ \color{blue}{60} }x-\frac{120}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 10x+25& = & 36x-120 \\\Leftrightarrow & 10x \color{red}{+25} \color{blue}{-25} \color{blue}{-36x} & = & \color{red}{36x} -120 \color{blue}{-36x} \color{blue}{-25} \\\Leftrightarrow & -26x& = & -145 \\\Leftrightarrow & \frac{-26x}{ \color{red}{-26} }& = & \frac{-145}{-26} \\\Leftrightarrow & x = \frac{145}{26} & & \\ & V = \left\{ \frac{145}{26} \right\} & \\\end{align}\)
10. \(\text{140 is het kleinste gemene veelvoud van 5, 14 en 4} \\ \begin{align} & \frac{x}{5}-\frac{3}{14}& = & \frac{5}{4}x-6 \\\Leftrightarrow & \color{blue}{140.} (\frac{28x}{ \color{blue}{140} }- \frac{ 30 }{ \color{blue}{140} })& = & (\frac{175}{ \color{blue}{140} }x-\frac{840}{ \color{blue}{140} }) \color{blue}{.140} \\\Leftrightarrow & 28x-30& = & 175x-840 \\\Leftrightarrow & 28x \color{red}{-30} \color{blue}{+30} \color{blue}{-175x} & = & \color{red}{175x} -840 \color{blue}{-175x} \color{blue}{+30} \\\Leftrightarrow & -147x& = & -810 \\\Leftrightarrow & \frac{-147x}{ \color{red}{-147} }& = & \frac{-810}{-147} \\\Leftrightarrow & x = \frac{270}{49} & & \\ & V = \left\{ \frac{270}{49} \right\} & \\\end{align}\)
11. \(\text{21 is het kleinste gemene veelvoud van 7, 7 en 3} \\ \begin{align} & \frac{x}{7}-\frac{2}{7}& = & \frac{5}{3}x+6 \\\Leftrightarrow & \color{blue}{21.} (\frac{3x}{ \color{blue}{21} }- \frac{ 6 }{ \color{blue}{21} })& = & (\frac{35}{ \color{blue}{21} }x+\frac{126}{ \color{blue}{21} }) \color{blue}{.21} \\\Leftrightarrow & 3x-6& = & 35x+126 \\\Leftrightarrow & 3x \color{red}{-6} \color{blue}{+6} \color{blue}{-35x} & = & \color{red}{35x} +126 \color{blue}{-35x} \color{blue}{+6} \\\Leftrightarrow & -32x& = & 132 \\\Leftrightarrow & \frac{-32x}{ \color{red}{-32} }& = & \frac{132}{-32} \\\Leftrightarrow & x = \frac{-33}{8} & & \\ & V = \left\{ \frac{-33}{8} \right\} & \\\end{align}\)
12. \(\text{48 is het kleinste gemene veelvoud van 3, 16 en 2} \\ \begin{align} & \frac{x}{3}-\frac{3}{16}& = & \frac{3}{2}x+6 \\\Leftrightarrow & \color{blue}{48.} (\frac{16x}{ \color{blue}{48} }- \frac{ 9 }{ \color{blue}{48} })& = & (\frac{72}{ \color{blue}{48} }x+\frac{288}{ \color{blue}{48} }) \color{blue}{.48} \\\Leftrightarrow & 16x-9& = & 72x+288 \\\Leftrightarrow & 16x \color{red}{-9} \color{blue}{+9} \color{blue}{-72x} & = & \color{red}{72x} +288 \color{blue}{-72x} \color{blue}{+9} \\\Leftrightarrow & -56x& = & 297 \\\Leftrightarrow & \frac{-56x}{ \color{red}{-56} }& = & \frac{297}{-56} \\\Leftrightarrow & x = \frac{-297}{56} & & \\ & V = \left\{ \frac{-297}{56} \right\} & \\\end{align}\)