Vgln. eerste graad (reeks 4)

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1. \(\frac{x}{5}+\frac{5}{16}=\frac{-5}{6}x+3\)
2. \(\frac{x}{5}-\frac{3}{7}=\frac{5}{6}x-1\)
3. \(\frac{x}{7}+\frac{5}{11}=\frac{7}{6}x+8\)
4. \(\frac{x}{5}+\frac{4}{13}=\frac{1}{2}x-4\)
5. \(\frac{x}{7}+\frac{5}{8}=\frac{-5}{3}x-6\)
6. \(\frac{x}{4}-\frac{4}{15}=\frac{-5}{3}x-1\)
7. \(\frac{x}{6}+\frac{2}{7}=\frac{-7}{5}x-6\)
8. \(\frac{x}{2}+\frac{4}{11}=\frac{7}{3}x+7\)
9. \(\frac{x}{2}-\frac{3}{16}=\frac{7}{3}x+7\)
10. \(\frac{x}{2}+\frac{3}{7}=\frac{-2}{5}x-5\)
11. \(\frac{x}{3}-\frac{2}{13}=\frac{1}{2}x+3\)
12. \(\frac{x}{7}-\frac{2}{7}=\frac{1}{2}x+5\)

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

Verbetersleutel

1. \(\text{240 is het kleinste gemene veelvoud van 5, 16 en 6} \\ \begin{align} & \frac{x}{5}+\frac{5}{16}& = & \frac{-5}{6}x+3 \\\Leftrightarrow & \color{blue}{240.} (\frac{48x}{ \color{blue}{240} }+ \frac{ 75 }{ \color{blue}{240} })& = & (\frac{-200}{ \color{blue}{240} }x+\frac{720}{ \color{blue}{240} }) \color{blue}{.240} \\\Leftrightarrow & 48x+75& = & -200x+720 \\\Leftrightarrow & 48x \color{red}{+75} \color{blue}{-75} \color{blue}{+200x} & = & \color{red}{-200x} +720 \color{blue}{+200x} \color{blue}{-75} \\\Leftrightarrow & 248x& = & 645 \\\Leftrightarrow & \frac{248x}{ \color{red}{248} }& = & \frac{645}{248} \\\Leftrightarrow & x = \frac{645}{248} & & \\ & V = \left\{ \frac{645}{248} \right\} & \\\end{align}\)
2. \(\text{210 is het kleinste gemene veelvoud van 5, 7 en 6} \\ \begin{align} & \frac{x}{5}-\frac{3}{7}& = & \frac{5}{6}x-1 \\\Leftrightarrow & \color{blue}{210.} (\frac{42x}{ \color{blue}{210} }- \frac{ 90 }{ \color{blue}{210} })& = & (\frac{175}{ \color{blue}{210} }x-\frac{210}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 42x-90& = & 175x-210 \\\Leftrightarrow & 42x \color{red}{-90} \color{blue}{+90} \color{blue}{-175x} & = & \color{red}{175x} -210 \color{blue}{-175x} \color{blue}{+90} \\\Leftrightarrow & -133x& = & -120 \\\Leftrightarrow & \frac{-133x}{ \color{red}{-133} }& = & \frac{-120}{-133} \\\Leftrightarrow & x = \frac{120}{133} & & \\ & V = \left\{ \frac{120}{133} \right\} & \\\end{align}\)
3. \(\text{462 is het kleinste gemene veelvoud van 7, 11 en 6} \\ \begin{align} & \frac{x}{7}+\frac{5}{11}& = & \frac{7}{6}x+8 \\\Leftrightarrow & \color{blue}{462.} (\frac{66x}{ \color{blue}{462} }+ \frac{ 210 }{ \color{blue}{462} })& = & (\frac{539}{ \color{blue}{462} }x+\frac{3696}{ \color{blue}{462} }) \color{blue}{.462} \\\Leftrightarrow & 66x+210& = & 539x+3696 \\\Leftrightarrow & 66x \color{red}{+210} \color{blue}{-210} \color{blue}{-539x} & = & \color{red}{539x} +3696 \color{blue}{-539x} \color{blue}{-210} \\\Leftrightarrow & -473x& = & 3486 \\\Leftrightarrow & \frac{-473x}{ \color{red}{-473} }& = & \frac{3486}{-473} \\\Leftrightarrow & x = \frac{-3486}{473} & & \\ & V = \left\{ \frac{-3486}{473} \right\} & \\\end{align}\)
4. \(\text{130 is het kleinste gemene veelvoud van 5, 13 en 2} \\ \begin{align} & \frac{x}{5}+\frac{4}{13}& = & \frac{1}{2}x-4 \\\Leftrightarrow & \color{blue}{130.} (\frac{26x}{ \color{blue}{130} }+ \frac{ 40 }{ \color{blue}{130} })& = & (\frac{65}{ \color{blue}{130} }x-\frac{520}{ \color{blue}{130} }) \color{blue}{.130} \\\Leftrightarrow & 26x+40& = & 65x-520 \\\Leftrightarrow & 26x \color{red}{+40} \color{blue}{-40} \color{blue}{-65x} & = & \color{red}{65x} -520 \color{blue}{-65x} \color{blue}{-40} \\\Leftrightarrow & -39x& = & -560 \\\Leftrightarrow & \frac{-39x}{ \color{red}{-39} }& = & \frac{-560}{-39} \\\Leftrightarrow & x = \frac{560}{39} & & \\ & V = \left\{ \frac{560}{39} \right\} & \\\end{align}\)
5. \(\text{168 is het kleinste gemene veelvoud van 7, 8 en 3} \\ \begin{align} & \frac{x}{7}+\frac{5}{8}& = & \frac{-5}{3}x-6 \\\Leftrightarrow & \color{blue}{168.} (\frac{24x}{ \color{blue}{168} }+ \frac{ 105 }{ \color{blue}{168} })& = & (\frac{-280}{ \color{blue}{168} }x-\frac{1008}{ \color{blue}{168} }) \color{blue}{.168} \\\Leftrightarrow & 24x+105& = & -280x-1008 \\\Leftrightarrow & 24x \color{red}{+105} \color{blue}{-105} \color{blue}{+280x} & = & \color{red}{-280x} -1008 \color{blue}{+280x} \color{blue}{-105} \\\Leftrightarrow & 304x& = & -1113 \\\Leftrightarrow & \frac{304x}{ \color{red}{304} }& = & \frac{-1113}{304} \\\Leftrightarrow & x = \frac{-1113}{304} & & \\ & V = \left\{ \frac{-1113}{304} \right\} & \\\end{align}\)
6. \(\text{60 is het kleinste gemene veelvoud van 4, 15 en 3} \\ \begin{align} & \frac{x}{4}-\frac{4}{15}& = & \frac{-5}{3}x-1 \\\Leftrightarrow & \color{blue}{60.} (\frac{15x}{ \color{blue}{60} }- \frac{ 16 }{ \color{blue}{60} })& = & (\frac{-100}{ \color{blue}{60} }x-\frac{60}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 15x-16& = & -100x-60 \\\Leftrightarrow & 15x \color{red}{-16} \color{blue}{+16} \color{blue}{+100x} & = & \color{red}{-100x} -60 \color{blue}{+100x} \color{blue}{+16} \\\Leftrightarrow & 115x& = & -44 \\\Leftrightarrow & \frac{115x}{ \color{red}{115} }& = & \frac{-44}{115} \\\Leftrightarrow & x = \frac{-44}{115} & & \\ & V = \left\{ \frac{-44}{115} \right\} & \\\end{align}\)
7. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}+\frac{2}{7}& = & \frac{-7}{5}x-6 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }+ \frac{ 60 }{ \color{blue}{210} })& = & (\frac{-294}{ \color{blue}{210} }x-\frac{1260}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x+60& = & -294x-1260 \\\Leftrightarrow & 35x \color{red}{+60} \color{blue}{-60} \color{blue}{+294x} & = & \color{red}{-294x} -1260 \color{blue}{+294x} \color{blue}{-60} \\\Leftrightarrow & 329x& = & -1320 \\\Leftrightarrow & \frac{329x}{ \color{red}{329} }& = & \frac{-1320}{329} \\\Leftrightarrow & x = \frac{-1320}{329} & & \\ & V = \left\{ \frac{-1320}{329} \right\} & \\\end{align}\)
8. \(\text{66 is het kleinste gemene veelvoud van 2, 11 en 3} \\ \begin{align} & \frac{x}{2}+\frac{4}{11}& = & \frac{7}{3}x+7 \\\Leftrightarrow & \color{blue}{66.} (\frac{33x}{ \color{blue}{66} }+ \frac{ 24 }{ \color{blue}{66} })& = & (\frac{154}{ \color{blue}{66} }x+\frac{462}{ \color{blue}{66} }) \color{blue}{.66} \\\Leftrightarrow & 33x+24& = & 154x+462 \\\Leftrightarrow & 33x \color{red}{+24} \color{blue}{-24} \color{blue}{-154x} & = & \color{red}{154x} +462 \color{blue}{-154x} \color{blue}{-24} \\\Leftrightarrow & -121x& = & 438 \\\Leftrightarrow & \frac{-121x}{ \color{red}{-121} }& = & \frac{438}{-121} \\\Leftrightarrow & x = \frac{-438}{121} & & \\ & V = \left\{ \frac{-438}{121} \right\} & \\\end{align}\)
9. \(\text{48 is het kleinste gemene veelvoud van 2, 16 en 3} \\ \begin{align} & \frac{x}{2}-\frac{3}{16}& = & \frac{7}{3}x+7 \\\Leftrightarrow & \color{blue}{48.} (\frac{24x}{ \color{blue}{48} }- \frac{ 9 }{ \color{blue}{48} })& = & (\frac{112}{ \color{blue}{48} }x+\frac{336}{ \color{blue}{48} }) \color{blue}{.48} \\\Leftrightarrow & 24x-9& = & 112x+336 \\\Leftrightarrow & 24x \color{red}{-9} \color{blue}{+9} \color{blue}{-112x} & = & \color{red}{112x} +336 \color{blue}{-112x} \color{blue}{+9} \\\Leftrightarrow & -88x& = & 345 \\\Leftrightarrow & \frac{-88x}{ \color{red}{-88} }& = & \frac{345}{-88} \\\Leftrightarrow & x = \frac{-345}{88} & & \\ & V = \left\{ \frac{-345}{88} \right\} & \\\end{align}\)
10. \(\text{70 is het kleinste gemene veelvoud van 2, 7 en 5} \\ \begin{align} & \frac{x}{2}+\frac{3}{7}& = & \frac{-2}{5}x-5 \\\Leftrightarrow & \color{blue}{70.} (\frac{35x}{ \color{blue}{70} }+ \frac{ 30 }{ \color{blue}{70} })& = & (\frac{-28}{ \color{blue}{70} }x-\frac{350}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 35x+30& = & -28x-350 \\\Leftrightarrow & 35x \color{red}{+30} \color{blue}{-30} \color{blue}{+28x} & = & \color{red}{-28x} -350 \color{blue}{+28x} \color{blue}{-30} \\\Leftrightarrow & 63x& = & -380 \\\Leftrightarrow & \frac{63x}{ \color{red}{63} }& = & \frac{-380}{63} \\\Leftrightarrow & x = \frac{-380}{63} & & \\ & V = \left\{ \frac{-380}{63} \right\} & \\\end{align}\)
11. \(\text{78 is het kleinste gemene veelvoud van 3, 13 en 2} \\ \begin{align} & \frac{x}{3}-\frac{2}{13}& = & \frac{1}{2}x+3 \\\Leftrightarrow & \color{blue}{78.} (\frac{26x}{ \color{blue}{78} }- \frac{ 12 }{ \color{blue}{78} })& = & (\frac{39}{ \color{blue}{78} }x+\frac{234}{ \color{blue}{78} }) \color{blue}{.78} \\\Leftrightarrow & 26x-12& = & 39x+234 \\\Leftrightarrow & 26x \color{red}{-12} \color{blue}{+12} \color{blue}{-39x} & = & \color{red}{39x} +234 \color{blue}{-39x} \color{blue}{+12} \\\Leftrightarrow & -13x& = & 246 \\\Leftrightarrow & \frac{-13x}{ \color{red}{-13} }& = & \frac{246}{-13} \\\Leftrightarrow & x = \frac{-246}{13} & & \\ & V = \left\{ \frac{-246}{13} \right\} & \\\end{align}\)
12. \(\text{14 is het kleinste gemene veelvoud van 7, 7 en 2} \\ \begin{align} & \frac{x}{7}-\frac{2}{7}& = & \frac{1}{2}x+5 \\\Leftrightarrow & \color{blue}{14.} (\frac{2x}{ \color{blue}{14} }- \frac{ 4 }{ \color{blue}{14} })& = & (\frac{7}{ \color{blue}{14} }x+\frac{70}{ \color{blue}{14} }) \color{blue}{.14} \\\Leftrightarrow & 2x-4& = & 7x+70 \\\Leftrightarrow & 2x \color{red}{-4} \color{blue}{+4} \color{blue}{-7x} & = & \color{red}{7x} +70 \color{blue}{-7x} \color{blue}{+4} \\\Leftrightarrow & -5x& = & 74 \\\Leftrightarrow & \frac{-5x}{ \color{red}{-5} }& = & \frac{74}{-5} \\\Leftrightarrow & x = \frac{-74}{5} & & \\ & V = \left\{ \frac{-74}{5} \right\} & \\\end{align}\)