Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{330 is het kleinste gemene veelvoud van 6, 11 en 5} \\ \begin{align} & \frac{x}{6}-\frac{3}{11}& = & \frac{1}{5}x+4 \\\Leftrightarrow & \color{blue}{330.} (\frac{55x}{ \color{blue}{330} }- \frac{ 90 }{ \color{blue}{330} })& = & (\frac{66}{ \color{blue}{330} }x+\frac{1320}{ \color{blue}{330} }) \color{blue}{.330} \\\Leftrightarrow & 55x-90& = & 66x+1320 \\\Leftrightarrow & 55x \color{red}{-90} \color{blue}{+90} \color{blue}{-66x} & = & \color{red}{66x} +1320 \color{blue}{-66x} \color{blue}{+90} \\\Leftrightarrow & -11x& = & 1410 \\\Leftrightarrow & \frac{-11x}{ \color{red}{-11} }& = & \frac{1410}{-11} \\\Leftrightarrow & x = \frac{-1410}{11} & & \\ & V = \left\{ \frac{-1410}{11} \right\} & \\\end{align}\)
2. \(\text{462 is het kleinste gemene veelvoud van 7, 11 en 6} \\ \begin{align} & \frac{x}{7}+\frac{4}{11}& = & \frac{-5}{6}x-2 \\\Leftrightarrow & \color{blue}{462.} (\frac{66x}{ \color{blue}{462} }+ \frac{ 168 }{ \color{blue}{462} })& = & (\frac{-385}{ \color{blue}{462} }x-\frac{924}{ \color{blue}{462} }) \color{blue}{.462} \\\Leftrightarrow & 66x+168& = & -385x-924 \\\Leftrightarrow & 66x \color{red}{+168} \color{blue}{-168} \color{blue}{+385x} & = & \color{red}{-385x} -924 \color{blue}{+385x} \color{blue}{-168} \\\Leftrightarrow & 451x& = & -1092 \\\Leftrightarrow & \frac{451x}{ \color{red}{451} }& = & \frac{-1092}{451} \\\Leftrightarrow & x = \frac{-1092}{451} & & \\ & V = \left\{ \frac{-1092}{451} \right\} & \\\end{align}\)
3. \(\text{66 is het kleinste gemene veelvoud van 2, 11 en 3} \\ \begin{align} & \frac{x}{2}-\frac{4}{11}& = & \frac{-8}{3}x+5 \\\Leftrightarrow & \color{blue}{66.} (\frac{33x}{ \color{blue}{66} }- \frac{ 24 }{ \color{blue}{66} })& = & (\frac{-176}{ \color{blue}{66} }x+\frac{330}{ \color{blue}{66} }) \color{blue}{.66} \\\Leftrightarrow & 33x-24& = & -176x+330 \\\Leftrightarrow & 33x \color{red}{-24} \color{blue}{+24} \color{blue}{+176x} & = & \color{red}{-176x} +330 \color{blue}{+176x} \color{blue}{+24} \\\Leftrightarrow & 209x& = & 354 \\\Leftrightarrow & \frac{209x}{ \color{red}{209} }& = & \frac{354}{209} \\\Leftrightarrow & x = \frac{354}{209} & & \\ & V = \left\{ \frac{354}{209} \right\} & \\\end{align}\)
4. \(\text{156 is het kleinste gemene veelvoud van 4, 13 en 3} \\ \begin{align} & \frac{x}{4}-\frac{5}{13}& = & \frac{-2}{3}x-7 \\\Leftrightarrow & \color{blue}{156.} (\frac{39x}{ \color{blue}{156} }- \frac{ 60 }{ \color{blue}{156} })& = & (\frac{-104}{ \color{blue}{156} }x-\frac{1092}{ \color{blue}{156} }) \color{blue}{.156} \\\Leftrightarrow & 39x-60& = & -104x-1092 \\\Leftrightarrow & 39x \color{red}{-60} \color{blue}{+60} \color{blue}{+104x} & = & \color{red}{-104x} -1092 \color{blue}{+104x} \color{blue}{+60} \\\Leftrightarrow & 143x& = & -1032 \\\Leftrightarrow & \frac{143x}{ \color{red}{143} }& = & \frac{-1032}{143} \\\Leftrightarrow & x = \frac{-1032}{143} & & \\ & V = \left\{ \frac{-1032}{143} \right\} & \\\end{align}\)
5. \(\text{182 is het kleinste gemene veelvoud van 7, 13 en 2} \\ \begin{align} & \frac{x}{7}+\frac{2}{13}& = & \frac{1}{2}x+5 \\\Leftrightarrow & \color{blue}{182.} (\frac{26x}{ \color{blue}{182} }+ \frac{ 28 }{ \color{blue}{182} })& = & (\frac{91}{ \color{blue}{182} }x+\frac{910}{ \color{blue}{182} }) \color{blue}{.182} \\\Leftrightarrow & 26x+28& = & 91x+910 \\\Leftrightarrow & 26x \color{red}{+28} \color{blue}{-28} \color{blue}{-91x} & = & \color{red}{91x} +910 \color{blue}{-91x} \color{blue}{-28} \\\Leftrightarrow & -65x& = & 882 \\\Leftrightarrow & \frac{-65x}{ \color{red}{-65} }& = & \frac{882}{-65} \\\Leftrightarrow & x = \frac{-882}{65} & & \\ & V = \left\{ \frac{-882}{65} \right\} & \\\end{align}\)
6. \(\text{240 is het kleinste gemene veelvoud van 5, 16 en 6} \\ \begin{align} & \frac{x}{5}+\frac{5}{16}& = & \frac{5}{6}x+2 \\\Leftrightarrow & \color{blue}{240.} (\frac{48x}{ \color{blue}{240} }+ \frac{ 75 }{ \color{blue}{240} })& = & (\frac{200}{ \color{blue}{240} }x+\frac{480}{ \color{blue}{240} }) \color{blue}{.240} \\\Leftrightarrow & 48x+75& = & 200x+480 \\\Leftrightarrow & 48x \color{red}{+75} \color{blue}{-75} \color{blue}{-200x} & = & \color{red}{200x} +480 \color{blue}{-200x} \color{blue}{-75} \\\Leftrightarrow & -152x& = & 405 \\\Leftrightarrow & \frac{-152x}{ \color{red}{-152} }& = & \frac{405}{-152} \\\Leftrightarrow & x = \frac{-405}{152} & & \\ & V = \left\{ \frac{-405}{152} \right\} & \\\end{align}\)
7. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}+\frac{5}{7}& = & \frac{-4}{5}x-5 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }+ \frac{ 150 }{ \color{blue}{210} })& = & (\frac{-168}{ \color{blue}{210} }x-\frac{1050}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x+150& = & -168x-1050 \\\Leftrightarrow & 35x \color{red}{+150} \color{blue}{-150} \color{blue}{+168x} & = & \color{red}{-168x} -1050 \color{blue}{+168x} \color{blue}{-150} \\\Leftrightarrow & 203x& = & -1200 \\\Leftrightarrow & \frac{203x}{ \color{red}{203} }& = & \frac{-1200}{203} \\\Leftrightarrow & x = \frac{-1200}{203} & & \\ & V = \left\{ \frac{-1200}{203} \right\} & \\\end{align}\)
8. \(\text{210 is het kleinste gemene veelvoud van 5, 7 en 6} \\ \begin{align} & \frac{x}{5}+\frac{2}{7}& = & \frac{7}{6}x+1 \\\Leftrightarrow & \color{blue}{210.} (\frac{42x}{ \color{blue}{210} }+ \frac{ 60 }{ \color{blue}{210} })& = & (\frac{245}{ \color{blue}{210} }x+\frac{210}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 42x+60& = & 245x+210 \\\Leftrightarrow & 42x \color{red}{+60} \color{blue}{-60} \color{blue}{-245x} & = & \color{red}{245x} +210 \color{blue}{-245x} \color{blue}{-60} \\\Leftrightarrow & -203x& = & 150 \\\Leftrightarrow & \frac{-203x}{ \color{red}{-203} }& = & \frac{150}{-203} \\\Leftrightarrow & x = \frac{-150}{203} & & \\ & V = \left\{ \frac{-150}{203} \right\} & \\\end{align}\)
9. \(\text{210 is het kleinste gemene veelvoud van 5, 7 en 6} \\ \begin{align} & \frac{x}{5}+\frac{3}{7}& = & \frac{5}{6}x-7 \\\Leftrightarrow & \color{blue}{210.} (\frac{42x}{ \color{blue}{210} }+ \frac{ 90 }{ \color{blue}{210} })& = & (\frac{175}{ \color{blue}{210} }x-\frac{1470}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 42x+90& = & 175x-1470 \\\Leftrightarrow & 42x \color{red}{+90} \color{blue}{-90} \color{blue}{-175x} & = & \color{red}{175x} -1470 \color{blue}{-175x} \color{blue}{-90} \\\Leftrightarrow & -133x& = & -1560 \\\Leftrightarrow & \frac{-133x}{ \color{red}{-133} }& = & \frac{-1560}{-133} \\\Leftrightarrow & x = \frac{1560}{133} & & \\ & V = \left\{ \frac{1560}{133} \right\} & \\\end{align}\)
10. \(\text{120 is het kleinste gemene veelvoud van 3, 8 en 5} \\ \begin{align} & \frac{x}{3}+\frac{3}{8}& = & \frac{-2}{5}x-3 \\\Leftrightarrow & \color{blue}{120.} (\frac{40x}{ \color{blue}{120} }+ \frac{ 45 }{ \color{blue}{120} })& = & (\frac{-48}{ \color{blue}{120} }x-\frac{360}{ \color{blue}{120} }) \color{blue}{.120} \\\Leftrightarrow & 40x+45& = & -48x-360 \\\Leftrightarrow & 40x \color{red}{+45} \color{blue}{-45} \color{blue}{+48x} & = & \color{red}{-48x} -360 \color{blue}{+48x} \color{blue}{-45} \\\Leftrightarrow & 88x& = & -405 \\\Leftrightarrow & \frac{88x}{ \color{red}{88} }& = & \frac{-405}{88} \\\Leftrightarrow & x = \frac{-405}{88} & & \\ & V = \left\{ \frac{-405}{88} \right\} & \\\end{align}\)
11. \(\text{220 is het kleinste gemene veelvoud van 4, 11 en 5} \\ \begin{align} & \frac{x}{4}-\frac{2}{11}& = & \frac{-2}{5}x+5 \\\Leftrightarrow & \color{blue}{220.} (\frac{55x}{ \color{blue}{220} }- \frac{ 40 }{ \color{blue}{220} })& = & (\frac{-88}{ \color{blue}{220} }x+\frac{1100}{ \color{blue}{220} }) \color{blue}{.220} \\\Leftrightarrow & 55x-40& = & -88x+1100 \\\Leftrightarrow & 55x \color{red}{-40} \color{blue}{+40} \color{blue}{+88x} & = & \color{red}{-88x} +1100 \color{blue}{+88x} \color{blue}{+40} \\\Leftrightarrow & 143x& = & 1140 \\\Leftrightarrow & \frac{143x}{ \color{red}{143} }& = & \frac{1140}{143} \\\Leftrightarrow & x = \frac{1140}{143} & & \\ & V = \left\{ \frac{1140}{143} \right\} & \\\end{align}\)
12. \(\text{42 is het kleinste gemene veelvoud van 2, 7 en 3} \\ \begin{align} & \frac{x}{2}-\frac{4}{7}& = & \frac{-5}{3}x+2 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }- \frac{ 24 }{ \color{blue}{42} })& = & (\frac{-70}{ \color{blue}{42} }x+\frac{84}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x-24& = & -70x+84 \\\Leftrightarrow & 21x \color{red}{-24} \color{blue}{+24} \color{blue}{+70x} & = & \color{red}{-70x} +84 \color{blue}{+70x} \color{blue}{+24} \\\Leftrightarrow & 91x& = & 108 \\\Leftrightarrow & \frac{91x}{ \color{red}{91} }& = & \frac{108}{91} \\\Leftrightarrow & x = \frac{108}{91} & & \\ & V = \left\{ \frac{108}{91} \right\} & \\\end{align}\)