Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}+\frac{4}{7}& = & \frac{-8}{3}x-1 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }+ \frac{ 48 }{ \color{blue}{84} })& = & (\frac{-224}{ \color{blue}{84} }x-\frac{84}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x+48& = & -224x-84 \\\Leftrightarrow & 21x \color{red}{+48} \color{blue}{-48} \color{blue}{+224x} & = & \color{red}{-224x} -84 \color{blue}{+224x} \color{blue}{-48} \\\Leftrightarrow & 245x& = & -132 \\\Leftrightarrow & \frac{245x}{ \color{red}{245} }& = & \frac{-132}{245} \\\Leftrightarrow & x = \frac{-132}{245} & & \\ & V = \left\{ \frac{-132}{245} \right\} & \\\end{align}\)
2. \(\text{70 is het kleinste gemene veelvoud van 2, 7 en 5} \\ \begin{align} & \frac{x}{2}+\frac{2}{7}& = & \frac{-2}{5}x+2 \\\Leftrightarrow & \color{blue}{70.} (\frac{35x}{ \color{blue}{70} }+ \frac{ 20 }{ \color{blue}{70} })& = & (\frac{-28}{ \color{blue}{70} }x+\frac{140}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 35x+20& = & -28x+140 \\\Leftrightarrow & 35x \color{red}{+20} \color{blue}{-20} \color{blue}{+28x} & = & \color{red}{-28x} +140 \color{blue}{+28x} \color{blue}{-20} \\\Leftrightarrow & 63x& = & 120 \\\Leftrightarrow & \frac{63x}{ \color{red}{63} }& = & \frac{120}{63} \\\Leftrightarrow & x = \frac{40}{21} & & \\ & V = \left\{ \frac{40}{21} \right\} & \\\end{align}\)
3. \(\text{18 is het kleinste gemene veelvoud van 3, 9 en 2} \\ \begin{align} & \frac{x}{3}+\frac{2}{9}& = & \frac{1}{2}x+3 \\\Leftrightarrow & \color{blue}{18.} (\frac{6x}{ \color{blue}{18} }+ \frac{ 4 }{ \color{blue}{18} })& = & (\frac{9}{ \color{blue}{18} }x+\frac{54}{ \color{blue}{18} }) \color{blue}{.18} \\\Leftrightarrow & 6x+4& = & 9x+54 \\\Leftrightarrow & 6x \color{red}{+4} \color{blue}{-4} \color{blue}{-9x} & = & \color{red}{9x} +54 \color{blue}{-9x} \color{blue}{-4} \\\Leftrightarrow & -3x& = & 50 \\\Leftrightarrow & \frac{-3x}{ \color{red}{-3} }& = & \frac{50}{-3} \\\Leftrightarrow & x = \frac{-50}{3} & & \\ & V = \left\{ \frac{-50}{3} \right\} & \\\end{align}\)
4. \(\text{66 is het kleinste gemene veelvoud van 3, 11 en 2} \\ \begin{align} & \frac{x}{3}-\frac{4}{11}& = & \frac{1}{2}x+5 \\\Leftrightarrow & \color{blue}{66.} (\frac{22x}{ \color{blue}{66} }- \frac{ 24 }{ \color{blue}{66} })& = & (\frac{33}{ \color{blue}{66} }x+\frac{330}{ \color{blue}{66} }) \color{blue}{.66} \\\Leftrightarrow & 22x-24& = & 33x+330 \\\Leftrightarrow & 22x \color{red}{-24} \color{blue}{+24} \color{blue}{-33x} & = & \color{red}{33x} +330 \color{blue}{-33x} \color{blue}{+24} \\\Leftrightarrow & -11x& = & 354 \\\Leftrightarrow & \frac{-11x}{ \color{red}{-11} }& = & \frac{354}{-11} \\\Leftrightarrow & x = \frac{-354}{11} & & \\ & V = \left\{ \frac{-354}{11} \right\} & \\\end{align}\)
5. \(\text{156 is het kleinste gemene veelvoud van 3, 13 en 4} \\ \begin{align} & \frac{x}{3}+\frac{4}{13}& = & \frac{-7}{4}x-1 \\\Leftrightarrow & \color{blue}{156.} (\frac{52x}{ \color{blue}{156} }+ \frac{ 48 }{ \color{blue}{156} })& = & (\frac{-273}{ \color{blue}{156} }x-\frac{156}{ \color{blue}{156} }) \color{blue}{.156} \\\Leftrightarrow & 52x+48& = & -273x-156 \\\Leftrightarrow & 52x \color{red}{+48} \color{blue}{-48} \color{blue}{+273x} & = & \color{red}{-273x} -156 \color{blue}{+273x} \color{blue}{-48} \\\Leftrightarrow & 325x& = & -204 \\\Leftrightarrow & \frac{325x}{ \color{red}{325} }& = & \frac{-204}{325} \\\Leftrightarrow & x = \frac{-204}{325} & & \\ & V = \left\{ \frac{-204}{325} \right\} & \\\end{align}\)
6. \(\text{154 is het kleinste gemene veelvoud van 7, 11 en 2} \\ \begin{align} & \frac{x}{7}+\frac{4}{11}& = & \frac{3}{2}x+7 \\\Leftrightarrow & \color{blue}{154.} (\frac{22x}{ \color{blue}{154} }+ \frac{ 56 }{ \color{blue}{154} })& = & (\frac{231}{ \color{blue}{154} }x+\frac{1078}{ \color{blue}{154} }) \color{blue}{.154} \\\Leftrightarrow & 22x+56& = & 231x+1078 \\\Leftrightarrow & 22x \color{red}{+56} \color{blue}{-56} \color{blue}{-231x} & = & \color{red}{231x} +1078 \color{blue}{-231x} \color{blue}{-56} \\\Leftrightarrow & -209x& = & 1022 \\\Leftrightarrow & \frac{-209x}{ \color{red}{-209} }& = & \frac{1022}{-209} \\\Leftrightarrow & x = \frac{-1022}{209} & & \\ & V = \left\{ \frac{-1022}{209} \right\} & \\\end{align}\)
7. \(\text{21 is het kleinste gemene veelvoud van 7, 7 en 3} \\ \begin{align} & \frac{x}{7}-\frac{5}{7}& = & \frac{7}{3}x+4 \\\Leftrightarrow & \color{blue}{21.} (\frac{3x}{ \color{blue}{21} }- \frac{ 15 }{ \color{blue}{21} })& = & (\frac{49}{ \color{blue}{21} }x+\frac{84}{ \color{blue}{21} }) \color{blue}{.21} \\\Leftrightarrow & 3x-15& = & 49x+84 \\\Leftrightarrow & 3x \color{red}{-15} \color{blue}{+15} \color{blue}{-49x} & = & \color{red}{49x} +84 \color{blue}{-49x} \color{blue}{+15} \\\Leftrightarrow & -46x& = & 99 \\\Leftrightarrow & \frac{-46x}{ \color{red}{-46} }& = & \frac{99}{-46} \\\Leftrightarrow & x = \frac{-99}{46} & & \\ & V = \left\{ \frac{-99}{46} \right\} & \\\end{align}\)
8. \(\text{240 is het kleinste gemene veelvoud van 6, 16 en 5} \\ \begin{align} & \frac{x}{6}-\frac{5}{16}& = & \frac{-4}{5}x+8 \\\Leftrightarrow & \color{blue}{240.} (\frac{40x}{ \color{blue}{240} }- \frac{ 75 }{ \color{blue}{240} })& = & (\frac{-192}{ \color{blue}{240} }x+\frac{1920}{ \color{blue}{240} }) \color{blue}{.240} \\\Leftrightarrow & 40x-75& = & -192x+1920 \\\Leftrightarrow & 40x \color{red}{-75} \color{blue}{+75} \color{blue}{+192x} & = & \color{red}{-192x} +1920 \color{blue}{+192x} \color{blue}{+75} \\\Leftrightarrow & 232x& = & 1995 \\\Leftrightarrow & \frac{232x}{ \color{red}{232} }& = & \frac{1995}{232} \\\Leftrightarrow & x = \frac{1995}{232} & & \\ & V = \left\{ \frac{1995}{232} \right\} & \\\end{align}\)
9. \(\text{273 is het kleinste gemene veelvoud van 7, 13 en 3} \\ \begin{align} & \frac{x}{7}-\frac{2}{13}& = & \frac{1}{3}x+4 \\\Leftrightarrow & \color{blue}{273.} (\frac{39x}{ \color{blue}{273} }- \frac{ 42 }{ \color{blue}{273} })& = & (\frac{91}{ \color{blue}{273} }x+\frac{1092}{ \color{blue}{273} }) \color{blue}{.273} \\\Leftrightarrow & 39x-42& = & 91x+1092 \\\Leftrightarrow & 39x \color{red}{-42} \color{blue}{+42} \color{blue}{-91x} & = & \color{red}{91x} +1092 \color{blue}{-91x} \color{blue}{+42} \\\Leftrightarrow & -52x& = & 1134 \\\Leftrightarrow & \frac{-52x}{ \color{red}{-52} }& = & \frac{1134}{-52} \\\Leftrightarrow & x = \frac{-567}{26} & & \\ & V = \left\{ \frac{-567}{26} \right\} & \\\end{align}\)
10. \(\text{78 is het kleinste gemene veelvoud van 2, 13 en 3} \\ \begin{align} & \frac{x}{2}+\frac{2}{13}& = & \frac{1}{3}x-6 \\\Leftrightarrow & \color{blue}{78.} (\frac{39x}{ \color{blue}{78} }+ \frac{ 12 }{ \color{blue}{78} })& = & (\frac{26}{ \color{blue}{78} }x-\frac{468}{ \color{blue}{78} }) \color{blue}{.78} \\\Leftrightarrow & 39x+12& = & 26x-468 \\\Leftrightarrow & 39x \color{red}{+12} \color{blue}{-12} \color{blue}{-26x} & = & \color{red}{26x} -468 \color{blue}{-26x} \color{blue}{-12} \\\Leftrightarrow & 13x& = & -480 \\\Leftrightarrow & \frac{13x}{ \color{red}{13} }& = & \frac{-480}{13} \\\Leftrightarrow & x = \frac{-480}{13} & & \\ & V = \left\{ \frac{-480}{13} \right\} & \\\end{align}\)
11. \(\text{66 is het kleinste gemene veelvoud van 2, 11 en 3} \\ \begin{align} & \frac{x}{2}+\frac{3}{11}& = & \frac{7}{3}x-5 \\\Leftrightarrow & \color{blue}{66.} (\frac{33x}{ \color{blue}{66} }+ \frac{ 18 }{ \color{blue}{66} })& = & (\frac{154}{ \color{blue}{66} }x-\frac{330}{ \color{blue}{66} }) \color{blue}{.66} \\\Leftrightarrow & 33x+18& = & 154x-330 \\\Leftrightarrow & 33x \color{red}{+18} \color{blue}{-18} \color{blue}{-154x} & = & \color{red}{154x} -330 \color{blue}{-154x} \color{blue}{-18} \\\Leftrightarrow & -121x& = & -348 \\\Leftrightarrow & \frac{-121x}{ \color{red}{-121} }& = & \frac{-348}{-121} \\\Leftrightarrow & x = \frac{348}{121} & & \\ & V = \left\{ \frac{348}{121} \right\} & \\\end{align}\)
12. \(\text{336 is het kleinste gemene veelvoud van 7, 16 en 3} \\ \begin{align} & \frac{x}{7}-\frac{5}{16}& = & \frac{-5}{3}x-7 \\\Leftrightarrow & \color{blue}{336.} (\frac{48x}{ \color{blue}{336} }- \frac{ 105 }{ \color{blue}{336} })& = & (\frac{-560}{ \color{blue}{336} }x-\frac{2352}{ \color{blue}{336} }) \color{blue}{.336} \\\Leftrightarrow & 48x-105& = & -560x-2352 \\\Leftrightarrow & 48x \color{red}{-105} \color{blue}{+105} \color{blue}{+560x} & = & \color{red}{-560x} -2352 \color{blue}{+560x} \color{blue}{+105} \\\Leftrightarrow & 608x& = & -2247 \\\Leftrightarrow & \frac{608x}{ \color{red}{608} }& = & \frac{-2247}{608} \\\Leftrightarrow & x = \frac{-2247}{608} & & \\ & V = \left\{ \frac{-2247}{608} \right\} & \\\end{align}\)