Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{240 is het kleinste gemene veelvoud van 6, 16 en 5} \\ \begin{align} & \frac{x}{6}-\frac{5}{16}& = & \frac{7}{5}x+2 \\\Leftrightarrow & \color{blue}{240.} (\frac{40x}{ \color{blue}{240} }- \frac{ 75 }{ \color{blue}{240} })& = & (\frac{336}{ \color{blue}{240} }x+\frac{480}{ \color{blue}{240} }) \color{blue}{.240} \\\Leftrightarrow & 40x-75& = & 336x+480 \\\Leftrightarrow & 40x \color{red}{-75} \color{blue}{+75} \color{blue}{-336x} & = & \color{red}{336x} +480 \color{blue}{-336x} \color{blue}{+75} \\\Leftrightarrow & -296x& = & 555 \\\Leftrightarrow & \frac{-296x}{ \color{red}{-296} }& = & \frac{555}{-296} \\\Leftrightarrow & x = \frac{-15}{8} & & \\ & V = \left\{ \frac{-15}{8} \right\} & \\\end{align}\)
2. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}+\frac{4}{7}& = & \frac{4}{5}x-2 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }+ \frac{ 120 }{ \color{blue}{210} })& = & (\frac{168}{ \color{blue}{210} }x-\frac{420}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x+120& = & 168x-420 \\\Leftrightarrow & 35x \color{red}{+120} \color{blue}{-120} \color{blue}{-168x} & = & \color{red}{168x} -420 \color{blue}{-168x} \color{blue}{-120} \\\Leftrightarrow & -133x& = & -540 \\\Leftrightarrow & \frac{-133x}{ \color{red}{-133} }& = & \frac{-540}{-133} \\\Leftrightarrow & x = \frac{540}{133} & & \\ & V = \left\{ \frac{540}{133} \right\} & \\\end{align}\)
3. \(\text{28 is het kleinste gemene veelvoud van 7, 7 en 4} \\ \begin{align} & \frac{x}{7}-\frac{4}{7}& = & \frac{-3}{4}x+8 \\\Leftrightarrow & \color{blue}{28.} (\frac{4x}{ \color{blue}{28} }- \frac{ 16 }{ \color{blue}{28} })& = & (\frac{-21}{ \color{blue}{28} }x+\frac{224}{ \color{blue}{28} }) \color{blue}{.28} \\\Leftrightarrow & 4x-16& = & -21x+224 \\\Leftrightarrow & 4x \color{red}{-16} \color{blue}{+16} \color{blue}{+21x} & = & \color{red}{-21x} +224 \color{blue}{+21x} \color{blue}{+16} \\\Leftrightarrow & 25x& = & 240 \\\Leftrightarrow & \frac{25x}{ \color{red}{25} }& = & \frac{240}{25} \\\Leftrightarrow & x = \frac{48}{5} & & \\ & V = \left\{ \frac{48}{5} \right\} & \\\end{align}\)
4. \(\text{240 is het kleinste gemene veelvoud van 6, 16 en 5} \\ \begin{align} & \frac{x}{6}-\frac{3}{16}& = & \frac{-2}{5}x-3 \\\Leftrightarrow & \color{blue}{240.} (\frac{40x}{ \color{blue}{240} }- \frac{ 45 }{ \color{blue}{240} })& = & (\frac{-96}{ \color{blue}{240} }x-\frac{720}{ \color{blue}{240} }) \color{blue}{.240} \\\Leftrightarrow & 40x-45& = & -96x-720 \\\Leftrightarrow & 40x \color{red}{-45} \color{blue}{+45} \color{blue}{+96x} & = & \color{red}{-96x} -720 \color{blue}{+96x} \color{blue}{+45} \\\Leftrightarrow & 136x& = & -675 \\\Leftrightarrow & \frac{136x}{ \color{red}{136} }& = & \frac{-675}{136} \\\Leftrightarrow & x = \frac{-675}{136} & & \\ & V = \left\{ \frac{-675}{136} \right\} & \\\end{align}\)
5. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}+\frac{2}{7}& = & \frac{-7}{5}x+3 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }+ \frac{ 60 }{ \color{blue}{210} })& = & (\frac{-294}{ \color{blue}{210} }x+\frac{630}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x+60& = & -294x+630 \\\Leftrightarrow & 35x \color{red}{+60} \color{blue}{-60} \color{blue}{+294x} & = & \color{red}{-294x} +630 \color{blue}{+294x} \color{blue}{-60} \\\Leftrightarrow & 329x& = & 570 \\\Leftrightarrow & \frac{329x}{ \color{red}{329} }& = & \frac{570}{329} \\\Leftrightarrow & x = \frac{570}{329} & & \\ & V = \left\{ \frac{570}{329} \right\} & \\\end{align}\)
6. \(\text{364 is het kleinste gemene veelvoud van 7, 13 en 4} \\ \begin{align} & \frac{x}{7}-\frac{4}{13}& = & \frac{1}{4}x-3 \\\Leftrightarrow & \color{blue}{364.} (\frac{52x}{ \color{blue}{364} }- \frac{ 112 }{ \color{blue}{364} })& = & (\frac{91}{ \color{blue}{364} }x-\frac{1092}{ \color{blue}{364} }) \color{blue}{.364} \\\Leftrightarrow & 52x-112& = & 91x-1092 \\\Leftrightarrow & 52x \color{red}{-112} \color{blue}{+112} \color{blue}{-91x} & = & \color{red}{91x} -1092 \color{blue}{-91x} \color{blue}{+112} \\\Leftrightarrow & -39x& = & -980 \\\Leftrightarrow & \frac{-39x}{ \color{red}{-39} }& = & \frac{-980}{-39} \\\Leftrightarrow & x = \frac{980}{39} & & \\ & V = \left\{ \frac{980}{39} \right\} & \\\end{align}\)
7. \(\text{120 is het kleinste gemene veelvoud van 6, 8 en 5} \\ \begin{align} & \frac{x}{6}+\frac{5}{8}& = & \frac{-2}{5}x-3 \\\Leftrightarrow & \color{blue}{120.} (\frac{20x}{ \color{blue}{120} }+ \frac{ 75 }{ \color{blue}{120} })& = & (\frac{-48}{ \color{blue}{120} }x-\frac{360}{ \color{blue}{120} }) \color{blue}{.120} \\\Leftrightarrow & 20x+75& = & -48x-360 \\\Leftrightarrow & 20x \color{red}{+75} \color{blue}{-75} \color{blue}{+48x} & = & \color{red}{-48x} -360 \color{blue}{+48x} \color{blue}{-75} \\\Leftrightarrow & 68x& = & -435 \\\Leftrightarrow & \frac{68x}{ \color{red}{68} }& = & \frac{-435}{68} \\\Leftrightarrow & x = \frac{-435}{68} & & \\ & V = \left\{ \frac{-435}{68} \right\} & \\\end{align}\)
8. \(\text{30 is het kleinste gemene veelvoud van 5, 6 en 3} \\ \begin{align} & \frac{x}{5}+\frac{5}{6}& = & \frac{1}{3}x+4 \\\Leftrightarrow & \color{blue}{30.} (\frac{6x}{ \color{blue}{30} }+ \frac{ 25 }{ \color{blue}{30} })& = & (\frac{10}{ \color{blue}{30} }x+\frac{120}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 6x+25& = & 10x+120 \\\Leftrightarrow & 6x \color{red}{+25} \color{blue}{-25} \color{blue}{-10x} & = & \color{red}{10x} +120 \color{blue}{-10x} \color{blue}{-25} \\\Leftrightarrow & -4x& = & 95 \\\Leftrightarrow & \frac{-4x}{ \color{red}{-4} }& = & \frac{95}{-4} \\\Leftrightarrow & x = \frac{-95}{4} & & \\ & V = \left\{ \frac{-95}{4} \right\} & \\\end{align}\)
9. \(\text{18 is het kleinste gemene veelvoud van 2, 9 en 3} \\ \begin{align} & \frac{x}{2}-\frac{2}{9}& = & \frac{-8}{3}x+7 \\\Leftrightarrow & \color{blue}{18.} (\frac{9x}{ \color{blue}{18} }- \frac{ 4 }{ \color{blue}{18} })& = & (\frac{-48}{ \color{blue}{18} }x+\frac{126}{ \color{blue}{18} }) \color{blue}{.18} \\\Leftrightarrow & 9x-4& = & -48x+126 \\\Leftrightarrow & 9x \color{red}{-4} \color{blue}{+4} \color{blue}{+48x} & = & \color{red}{-48x} +126 \color{blue}{+48x} \color{blue}{+4} \\\Leftrightarrow & 57x& = & 130 \\\Leftrightarrow & \frac{57x}{ \color{red}{57} }& = & \frac{130}{57} \\\Leftrightarrow & x = \frac{130}{57} & & \\ & V = \left\{ \frac{130}{57} \right\} & \\\end{align}\)
10. \(\text{130 is het kleinste gemene veelvoud van 2, 13 en 5} \\ \begin{align} & \frac{x}{2}+\frac{4}{13}& = & \frac{-2}{5}x-5 \\\Leftrightarrow & \color{blue}{130.} (\frac{65x}{ \color{blue}{130} }+ \frac{ 40 }{ \color{blue}{130} })& = & (\frac{-52}{ \color{blue}{130} }x-\frac{650}{ \color{blue}{130} }) \color{blue}{.130} \\\Leftrightarrow & 65x+40& = & -52x-650 \\\Leftrightarrow & 65x \color{red}{+40} \color{blue}{-40} \color{blue}{+52x} & = & \color{red}{-52x} -650 \color{blue}{+52x} \color{blue}{-40} \\\Leftrightarrow & 117x& = & -690 \\\Leftrightarrow & \frac{117x}{ \color{red}{117} }& = & \frac{-690}{117} \\\Leftrightarrow & x = \frac{-230}{39} & & \\ & V = \left\{ \frac{-230}{39} \right\} & \\\end{align}\)
11. \(\text{220 is het kleinste gemene veelvoud van 5, 11 en 4} \\ \begin{align} & \frac{x}{5}+\frac{4}{11}& = & \frac{1}{4}x-4 \\\Leftrightarrow & \color{blue}{220.} (\frac{44x}{ \color{blue}{220} }+ \frac{ 80 }{ \color{blue}{220} })& = & (\frac{55}{ \color{blue}{220} }x-\frac{880}{ \color{blue}{220} }) \color{blue}{.220} \\\Leftrightarrow & 44x+80& = & 55x-880 \\\Leftrightarrow & 44x \color{red}{+80} \color{blue}{-80} \color{blue}{-55x} & = & \color{red}{55x} -880 \color{blue}{-55x} \color{blue}{-80} \\\Leftrightarrow & -11x& = & -960 \\\Leftrightarrow & \frac{-11x}{ \color{red}{-11} }& = & \frac{-960}{-11} \\\Leftrightarrow & x = \frac{960}{11} & & \\ & V = \left\{ \frac{960}{11} \right\} & \\\end{align}\)
12. \(\text{130 is het kleinste gemene veelvoud van 2, 13 en 5} \\ \begin{align} & \frac{x}{2}-\frac{3}{13}& = & \frac{-4}{5}x-1 \\\Leftrightarrow & \color{blue}{130.} (\frac{65x}{ \color{blue}{130} }- \frac{ 30 }{ \color{blue}{130} })& = & (\frac{-104}{ \color{blue}{130} }x-\frac{130}{ \color{blue}{130} }) \color{blue}{.130} \\\Leftrightarrow & 65x-30& = & -104x-130 \\\Leftrightarrow & 65x \color{red}{-30} \color{blue}{+30} \color{blue}{+104x} & = & \color{red}{-104x} -130 \color{blue}{+104x} \color{blue}{+30} \\\Leftrightarrow & 169x& = & -100 \\\Leftrightarrow & \frac{169x}{ \color{red}{169} }& = & \frac{-100}{169} \\\Leftrightarrow & x = \frac{-100}{169} & & \\ & V = \left\{ \frac{-100}{169} \right\} & \\\end{align}\)