Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}+\frac{2}{7}& = & \frac{5}{3}x-5 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }+ \frac{ 24 }{ \color{blue}{84} })& = & (\frac{140}{ \color{blue}{84} }x-\frac{420}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x+24& = & 140x-420 \\\Leftrightarrow & 21x \color{red}{+24} \color{blue}{-24} \color{blue}{-140x} & = & \color{red}{140x} -420 \color{blue}{-140x} \color{blue}{-24} \\\Leftrightarrow & -119x& = & -444 \\\Leftrightarrow & \frac{-119x}{ \color{red}{-119} }& = & \frac{-444}{-119} \\\Leftrightarrow & x = \frac{444}{119} & & \\ & V = \left\{ \frac{444}{119} \right\} & \\\end{align}\)
2. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}-\frac{5}{7}& = & \frac{-7}{5}x-4 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }- \frac{ 150 }{ \color{blue}{210} })& = & (\frac{-294}{ \color{blue}{210} }x-\frac{840}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x-150& = & -294x-840 \\\Leftrightarrow & 35x \color{red}{-150} \color{blue}{+150} \color{blue}{+294x} & = & \color{red}{-294x} -840 \color{blue}{+294x} \color{blue}{+150} \\\Leftrightarrow & 329x& = & -690 \\\Leftrightarrow & \frac{329x}{ \color{red}{329} }& = & \frac{-690}{329} \\\Leftrightarrow & x = \frac{-690}{329} & & \\ & V = \left\{ \frac{-690}{329} \right\} & \\\end{align}\)
3. \(\text{18 is het kleinste gemene veelvoud van 2, 9 en 3} \\ \begin{align} & \frac{x}{2}-\frac{4}{9}& = & \frac{-8}{3}x-1 \\\Leftrightarrow & \color{blue}{18.} (\frac{9x}{ \color{blue}{18} }- \frac{ 8 }{ \color{blue}{18} })& = & (\frac{-48}{ \color{blue}{18} }x-\frac{18}{ \color{blue}{18} }) \color{blue}{.18} \\\Leftrightarrow & 9x-8& = & -48x-18 \\\Leftrightarrow & 9x \color{red}{-8} \color{blue}{+8} \color{blue}{+48x} & = & \color{red}{-48x} -18 \color{blue}{+48x} \color{blue}{+8} \\\Leftrightarrow & 57x& = & -10 \\\Leftrightarrow & \frac{57x}{ \color{red}{57} }& = & \frac{-10}{57} \\\Leftrightarrow & x = \frac{-10}{57} & & \\ & V = \left\{ \frac{-10}{57} \right\} & \\\end{align}\)
4. \(\text{390 is het kleinste gemene veelvoud van 6, 13 en 5} \\ \begin{align} & \frac{x}{6}+\frac{3}{13}& = & \frac{3}{5}x-7 \\\Leftrightarrow & \color{blue}{390.} (\frac{65x}{ \color{blue}{390} }+ \frac{ 90 }{ \color{blue}{390} })& = & (\frac{234}{ \color{blue}{390} }x-\frac{2730}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 65x+90& = & 234x-2730 \\\Leftrightarrow & 65x \color{red}{+90} \color{blue}{-90} \color{blue}{-234x} & = & \color{red}{234x} -2730 \color{blue}{-234x} \color{blue}{-90} \\\Leftrightarrow & -169x& = & -2820 \\\Leftrightarrow & \frac{-169x}{ \color{red}{-169} }& = & \frac{-2820}{-169} \\\Leftrightarrow & x = \frac{2820}{169} & & \\ & V = \left\{ \frac{2820}{169} \right\} & \\\end{align}\)
5. \(\text{154 is het kleinste gemene veelvoud van 7, 11 en 2} \\ \begin{align} & \frac{x}{7}-\frac{3}{11}& = & \frac{1}{2}x-4 \\\Leftrightarrow & \color{blue}{154.} (\frac{22x}{ \color{blue}{154} }- \frac{ 42 }{ \color{blue}{154} })& = & (\frac{77}{ \color{blue}{154} }x-\frac{616}{ \color{blue}{154} }) \color{blue}{.154} \\\Leftrightarrow & 22x-42& = & 77x-616 \\\Leftrightarrow & 22x \color{red}{-42} \color{blue}{+42} \color{blue}{-77x} & = & \color{red}{77x} -616 \color{blue}{-77x} \color{blue}{+42} \\\Leftrightarrow & -55x& = & -574 \\\Leftrightarrow & \frac{-55x}{ \color{red}{-55} }& = & \frac{-574}{-55} \\\Leftrightarrow & x = \frac{574}{55} & & \\ & V = \left\{ \frac{574}{55} \right\} & \\\end{align}\)
6. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}+\frac{4}{7}& = & \frac{4}{3}x+1 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }+ \frac{ 48 }{ \color{blue}{84} })& = & (\frac{112}{ \color{blue}{84} }x+\frac{84}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x+48& = & 112x+84 \\\Leftrightarrow & 21x \color{red}{+48} \color{blue}{-48} \color{blue}{-112x} & = & \color{red}{112x} +84 \color{blue}{-112x} \color{blue}{-48} \\\Leftrightarrow & -91x& = & 36 \\\Leftrightarrow & \frac{-91x}{ \color{red}{-91} }& = & \frac{36}{-91} \\\Leftrightarrow & x = \frac{-36}{91} & & \\ & V = \left\{ \frac{-36}{91} \right\} & \\\end{align}\)
7. \(\text{156 is het kleinste gemene veelvoud van 4, 13 en 3} \\ \begin{align} & \frac{x}{4}+\frac{2}{13}& = & \frac{4}{3}x-2 \\\Leftrightarrow & \color{blue}{156.} (\frac{39x}{ \color{blue}{156} }+ \frac{ 24 }{ \color{blue}{156} })& = & (\frac{208}{ \color{blue}{156} }x-\frac{312}{ \color{blue}{156} }) \color{blue}{.156} \\\Leftrightarrow & 39x+24& = & 208x-312 \\\Leftrightarrow & 39x \color{red}{+24} \color{blue}{-24} \color{blue}{-208x} & = & \color{red}{208x} -312 \color{blue}{-208x} \color{blue}{-24} \\\Leftrightarrow & -169x& = & -336 \\\Leftrightarrow & \frac{-169x}{ \color{red}{-169} }& = & \frac{-336}{-169} \\\Leftrightarrow & x = \frac{336}{169} & & \\ & V = \left\{ \frac{336}{169} \right\} & \\\end{align}\)
8. \(\text{130 is het kleinste gemene veelvoud van 2, 13 en 5} \\ \begin{align} & \frac{x}{2}-\frac{4}{13}& = & \frac{7}{5}x-5 \\\Leftrightarrow & \color{blue}{130.} (\frac{65x}{ \color{blue}{130} }- \frac{ 40 }{ \color{blue}{130} })& = & (\frac{182}{ \color{blue}{130} }x-\frac{650}{ \color{blue}{130} }) \color{blue}{.130} \\\Leftrightarrow & 65x-40& = & 182x-650 \\\Leftrightarrow & 65x \color{red}{-40} \color{blue}{+40} \color{blue}{-182x} & = & \color{red}{182x} -650 \color{blue}{-182x} \color{blue}{+40} \\\Leftrightarrow & -117x& = & -610 \\\Leftrightarrow & \frac{-117x}{ \color{red}{-117} }& = & \frac{-610}{-117} \\\Leftrightarrow & x = \frac{610}{117} & & \\ & V = \left\{ \frac{610}{117} \right\} & \\\end{align}\)
9. \(\text{70 is het kleinste gemene veelvoud van 2, 7 en 5} \\ \begin{align} & \frac{x}{2}+\frac{2}{7}& = & \frac{1}{5}x-5 \\\Leftrightarrow & \color{blue}{70.} (\frac{35x}{ \color{blue}{70} }+ \frac{ 20 }{ \color{blue}{70} })& = & (\frac{14}{ \color{blue}{70} }x-\frac{350}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 35x+20& = & 14x-350 \\\Leftrightarrow & 35x \color{red}{+20} \color{blue}{-20} \color{blue}{-14x} & = & \color{red}{14x} -350 \color{blue}{-14x} \color{blue}{-20} \\\Leftrightarrow & 21x& = & -370 \\\Leftrightarrow & \frac{21x}{ \color{red}{21} }& = & \frac{-370}{21} \\\Leftrightarrow & x = \frac{-370}{21} & & \\ & V = \left\{ \frac{-370}{21} \right\} & \\\end{align}\)
10. \(\text{195 is het kleinste gemene veelvoud van 3, 13 en 5} \\ \begin{align} & \frac{x}{3}+\frac{4}{13}& = & \frac{1}{5}x+3 \\\Leftrightarrow & \color{blue}{195.} (\frac{65x}{ \color{blue}{195} }+ \frac{ 60 }{ \color{blue}{195} })& = & (\frac{39}{ \color{blue}{195} }x+\frac{585}{ \color{blue}{195} }) \color{blue}{.195} \\\Leftrightarrow & 65x+60& = & 39x+585 \\\Leftrightarrow & 65x \color{red}{+60} \color{blue}{-60} \color{blue}{-39x} & = & \color{red}{39x} +585 \color{blue}{-39x} \color{blue}{-60} \\\Leftrightarrow & 26x& = & 525 \\\Leftrightarrow & \frac{26x}{ \color{red}{26} }& = & \frac{525}{26} \\\Leftrightarrow & x = \frac{525}{26} & & \\ & V = \left\{ \frac{525}{26} \right\} & \\\end{align}\)
11. \(\text{252 is het kleinste gemene veelvoud van 7, 9 en 4} \\ \begin{align} & \frac{x}{7}+\frac{5}{9}& = & \frac{-3}{4}x-7 \\\Leftrightarrow & \color{blue}{252.} (\frac{36x}{ \color{blue}{252} }+ \frac{ 140 }{ \color{blue}{252} })& = & (\frac{-189}{ \color{blue}{252} }x-\frac{1764}{ \color{blue}{252} }) \color{blue}{.252} \\\Leftrightarrow & 36x+140& = & -189x-1764 \\\Leftrightarrow & 36x \color{red}{+140} \color{blue}{-140} \color{blue}{+189x} & = & \color{red}{-189x} -1764 \color{blue}{+189x} \color{blue}{-140} \\\Leftrightarrow & 225x& = & -1904 \\\Leftrightarrow & \frac{225x}{ \color{red}{225} }& = & \frac{-1904}{225} \\\Leftrightarrow & x = \frac{-1904}{225} & & \\ & V = \left\{ \frac{-1904}{225} \right\} & \\\end{align}\)
12. \(\text{30 is het kleinste gemene veelvoud van 5, 15 en 6} \\ \begin{align} & \frac{x}{5}+\frac{4}{15}& = & \frac{1}{6}x-7 \\\Leftrightarrow & \color{blue}{30.} (\frac{6x}{ \color{blue}{30} }+ \frac{ 8 }{ \color{blue}{30} })& = & (\frac{5}{ \color{blue}{30} }x-\frac{210}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 6x+8& = & 5x-210 \\\Leftrightarrow & 6x \color{red}{+8} \color{blue}{-8} \color{blue}{-5x} & = & \color{red}{5x} -210 \color{blue}{-5x} \color{blue}{-8} \\\Leftrightarrow & x& = & -218 \\ & V = \left\{ -218 \right\} & \\\end{align}\)