Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{390 is het kleinste gemene veelvoud van 6, 13 en 5} \\ \begin{align} & \frac{x}{6}-\frac{2}{13}& = & \frac{1}{5}x-7 \\\Leftrightarrow & \color{blue}{390.} (\frac{65x}{ \color{blue}{390} }- \frac{ 60 }{ \color{blue}{390} })& = & (\frac{78}{ \color{blue}{390} }x-\frac{2730}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 65x-60& = & 78x-2730 \\\Leftrightarrow & 65x \color{red}{-60} \color{blue}{+60} \color{blue}{-78x} & = & \color{red}{78x} -2730 \color{blue}{-78x} \color{blue}{+60} \\\Leftrightarrow & -13x& = & -2670 \\\Leftrightarrow & \frac{-13x}{ \color{red}{-13} }& = & \frac{-2670}{-13} \\\Leftrightarrow & x = \frac{2670}{13} & & \\ & V = \left\{ \frac{2670}{13} \right\} & \\\end{align}\)
2. \(\text{18 is het kleinste gemene veelvoud van 2, 9 en 3} \\ \begin{align} & \frac{x}{2}-\frac{2}{9}& = & \frac{-2}{3}x-8 \\\Leftrightarrow & \color{blue}{18.} (\frac{9x}{ \color{blue}{18} }- \frac{ 4 }{ \color{blue}{18} })& = & (\frac{-12}{ \color{blue}{18} }x-\frac{144}{ \color{blue}{18} }) \color{blue}{.18} \\\Leftrightarrow & 9x-4& = & -12x-144 \\\Leftrightarrow & 9x \color{red}{-4} \color{blue}{+4} \color{blue}{+12x} & = & \color{red}{-12x} -144 \color{blue}{+12x} \color{blue}{+4} \\\Leftrightarrow & 21x& = & -140 \\\Leftrightarrow & \frac{21x}{ \color{red}{21} }& = & \frac{-140}{21} \\\Leftrightarrow & x = \frac{-20}{3} & & \\ & V = \left\{ \frac{-20}{3} \right\} & \\\end{align}\)
3. \(\text{45 is het kleinste gemene veelvoud van 3, 9 en 5} \\ \begin{align} & \frac{x}{3}-\frac{2}{9}& = & \frac{6}{5}x+1 \\\Leftrightarrow & \color{blue}{45.} (\frac{15x}{ \color{blue}{45} }- \frac{ 10 }{ \color{blue}{45} })& = & (\frac{54}{ \color{blue}{45} }x+\frac{45}{ \color{blue}{45} }) \color{blue}{.45} \\\Leftrightarrow & 15x-10& = & 54x+45 \\\Leftrightarrow & 15x \color{red}{-10} \color{blue}{+10} \color{blue}{-54x} & = & \color{red}{54x} +45 \color{blue}{-54x} \color{blue}{+10} \\\Leftrightarrow & -39x& = & 55 \\\Leftrightarrow & \frac{-39x}{ \color{red}{-39} }& = & \frac{55}{-39} \\\Leftrightarrow & x = \frac{-55}{39} & & \\ & V = \left\{ \frac{-55}{39} \right\} & \\\end{align}\)
4. \(\text{35 is het kleinste gemene veelvoud van 7, 7 en 5} \\ \begin{align} & \frac{x}{7}+\frac{2}{7}& = & \frac{7}{5}x-8 \\\Leftrightarrow & \color{blue}{35.} (\frac{5x}{ \color{blue}{35} }+ \frac{ 10 }{ \color{blue}{35} })& = & (\frac{49}{ \color{blue}{35} }x-\frac{280}{ \color{blue}{35} }) \color{blue}{.35} \\\Leftrightarrow & 5x+10& = & 49x-280 \\\Leftrightarrow & 5x \color{red}{+10} \color{blue}{-10} \color{blue}{-49x} & = & \color{red}{49x} -280 \color{blue}{-49x} \color{blue}{-10} \\\Leftrightarrow & -44x& = & -290 \\\Leftrightarrow & \frac{-44x}{ \color{red}{-44} }& = & \frac{-290}{-44} \\\Leftrightarrow & x = \frac{145}{22} & & \\ & V = \left\{ \frac{145}{22} \right\} & \\\end{align}\)
5. \(\text{80 is het kleinste gemene veelvoud van 2, 16 en 5} \\ \begin{align} & \frac{x}{2}+\frac{5}{16}& = & \frac{-7}{5}x-4 \\\Leftrightarrow & \color{blue}{80.} (\frac{40x}{ \color{blue}{80} }+ \frac{ 25 }{ \color{blue}{80} })& = & (\frac{-112}{ \color{blue}{80} }x-\frac{320}{ \color{blue}{80} }) \color{blue}{.80} \\\Leftrightarrow & 40x+25& = & -112x-320 \\\Leftrightarrow & 40x \color{red}{+25} \color{blue}{-25} \color{blue}{+112x} & = & \color{red}{-112x} -320 \color{blue}{+112x} \color{blue}{-25} \\\Leftrightarrow & 152x& = & -345 \\\Leftrightarrow & \frac{152x}{ \color{red}{152} }& = & \frac{-345}{152} \\\Leftrightarrow & x = \frac{-345}{152} & & \\ & V = \left\{ \frac{-345}{152} \right\} & \\\end{align}\)
6. \(\text{220 is het kleinste gemene veelvoud van 5, 11 en 4} \\ \begin{align} & \frac{x}{5}-\frac{2}{11}& = & \frac{1}{4}x+1 \\\Leftrightarrow & \color{blue}{220.} (\frac{44x}{ \color{blue}{220} }- \frac{ 40 }{ \color{blue}{220} })& = & (\frac{55}{ \color{blue}{220} }x+\frac{220}{ \color{blue}{220} }) \color{blue}{.220} \\\Leftrightarrow & 44x-40& = & 55x+220 \\\Leftrightarrow & 44x \color{red}{-40} \color{blue}{+40} \color{blue}{-55x} & = & \color{red}{55x} +220 \color{blue}{-55x} \color{blue}{+40} \\\Leftrightarrow & -11x& = & 260 \\\Leftrightarrow & \frac{-11x}{ \color{red}{-11} }& = & \frac{260}{-11} \\\Leftrightarrow & x = \frac{-260}{11} & & \\ & V = \left\{ \frac{-260}{11} \right\} & \\\end{align}\)
7. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}+\frac{5}{7}& = & \frac{7}{3}x+7 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }+ \frac{ 60 }{ \color{blue}{84} })& = & (\frac{196}{ \color{blue}{84} }x+\frac{588}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x+60& = & 196x+588 \\\Leftrightarrow & 21x \color{red}{+60} \color{blue}{-60} \color{blue}{-196x} & = & \color{red}{196x} +588 \color{blue}{-196x} \color{blue}{-60} \\\Leftrightarrow & -175x& = & 528 \\\Leftrightarrow & \frac{-175x}{ \color{red}{-175} }& = & \frac{528}{-175} \\\Leftrightarrow & x = \frac{-528}{175} & & \\ & V = \left\{ \frac{-528}{175} \right\} & \\\end{align}\)
8. \(\text{210 is het kleinste gemene veelvoud van 5, 7 en 6} \\ \begin{align} & \frac{x}{5}+\frac{4}{7}& = & \frac{-5}{6}x+3 \\\Leftrightarrow & \color{blue}{210.} (\frac{42x}{ \color{blue}{210} }+ \frac{ 120 }{ \color{blue}{210} })& = & (\frac{-175}{ \color{blue}{210} }x+\frac{630}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 42x+120& = & -175x+630 \\\Leftrightarrow & 42x \color{red}{+120} \color{blue}{-120} \color{blue}{+175x} & = & \color{red}{-175x} +630 \color{blue}{+175x} \color{blue}{-120} \\\Leftrightarrow & 217x& = & 510 \\\Leftrightarrow & \frac{217x}{ \color{red}{217} }& = & \frac{510}{217} \\\Leftrightarrow & x = \frac{510}{217} & & \\ & V = \left\{ \frac{510}{217} \right\} & \\\end{align}\)
9. \(\text{105 is het kleinste gemene veelvoud van 5, 7 en 3} \\ \begin{align} & \frac{x}{5}+\frac{3}{7}& = & \frac{5}{3}x+7 \\\Leftrightarrow & \color{blue}{105.} (\frac{21x}{ \color{blue}{105} }+ \frac{ 45 }{ \color{blue}{105} })& = & (\frac{175}{ \color{blue}{105} }x+\frac{735}{ \color{blue}{105} }) \color{blue}{.105} \\\Leftrightarrow & 21x+45& = & 175x+735 \\\Leftrightarrow & 21x \color{red}{+45} \color{blue}{-45} \color{blue}{-175x} & = & \color{red}{175x} +735 \color{blue}{-175x} \color{blue}{-45} \\\Leftrightarrow & -154x& = & 690 \\\Leftrightarrow & \frac{-154x}{ \color{red}{-154} }& = & \frac{690}{-154} \\\Leftrightarrow & x = \frac{-345}{77} & & \\ & V = \left\{ \frac{-345}{77} \right\} & \\\end{align}\)
10. \(\text{130 is het kleinste gemene veelvoud van 2, 13 en 5} \\ \begin{align} & \frac{x}{2}+\frac{3}{13}& = & \frac{6}{5}x+7 \\\Leftrightarrow & \color{blue}{130.} (\frac{65x}{ \color{blue}{130} }+ \frac{ 30 }{ \color{blue}{130} })& = & (\frac{156}{ \color{blue}{130} }x+\frac{910}{ \color{blue}{130} }) \color{blue}{.130} \\\Leftrightarrow & 65x+30& = & 156x+910 \\\Leftrightarrow & 65x \color{red}{+30} \color{blue}{-30} \color{blue}{-156x} & = & \color{red}{156x} +910 \color{blue}{-156x} \color{blue}{-30} \\\Leftrightarrow & -91x& = & 880 \\\Leftrightarrow & \frac{-91x}{ \color{red}{-91} }& = & \frac{880}{-91} \\\Leftrightarrow & x = \frac{-880}{91} & & \\ & V = \left\{ \frac{-880}{91} \right\} & \\\end{align}\)
11. \(\text{120 is het kleinste gemene veelvoud van 3, 8 en 5} \\ \begin{align} & \frac{x}{3}-\frac{5}{8}& = & \frac{4}{5}x+1 \\\Leftrightarrow & \color{blue}{120.} (\frac{40x}{ \color{blue}{120} }- \frac{ 75 }{ \color{blue}{120} })& = & (\frac{96}{ \color{blue}{120} }x+\frac{120}{ \color{blue}{120} }) \color{blue}{.120} \\\Leftrightarrow & 40x-75& = & 96x+120 \\\Leftrightarrow & 40x \color{red}{-75} \color{blue}{+75} \color{blue}{-96x} & = & \color{red}{96x} +120 \color{blue}{-96x} \color{blue}{+75} \\\Leftrightarrow & -56x& = & 195 \\\Leftrightarrow & \frac{-56x}{ \color{red}{-56} }& = & \frac{195}{-56} \\\Leftrightarrow & x = \frac{-195}{56} & & \\ & V = \left\{ \frac{-195}{56} \right\} & \\\end{align}\)
12. \(\text{330 is het kleinste gemene veelvoud van 6, 11 en 5} \\ \begin{align} & \frac{x}{6}+\frac{4}{11}& = & \frac{1}{5}x+5 \\\Leftrightarrow & \color{blue}{330.} (\frac{55x}{ \color{blue}{330} }+ \frac{ 120 }{ \color{blue}{330} })& = & (\frac{66}{ \color{blue}{330} }x+\frac{1650}{ \color{blue}{330} }) \color{blue}{.330} \\\Leftrightarrow & 55x+120& = & 66x+1650 \\\Leftrightarrow & 55x \color{red}{+120} \color{blue}{-120} \color{blue}{-66x} & = & \color{red}{66x} +1650 \color{blue}{-66x} \color{blue}{-120} \\\Leftrightarrow & -11x& = & 1530 \\\Leftrightarrow & \frac{-11x}{ \color{red}{-11} }& = & \frac{1530}{-11} \\\Leftrightarrow & x = \frac{-1530}{11} & & \\ & V = \left\{ \frac{-1530}{11} \right\} & \\\end{align}\)