Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{66 is het kleinste gemene veelvoud van 3, 11 en 2} \\ \begin{align} & \frac{x}{3}+\frac{4}{11}& = & \frac{1}{2}x+2 \\\Leftrightarrow & \color{blue}{66.} (\frac{22x}{ \color{blue}{66} }+ \frac{ 24 }{ \color{blue}{66} })& = & (\frac{33}{ \color{blue}{66} }x+\frac{132}{ \color{blue}{66} }) \color{blue}{.66} \\\Leftrightarrow & 22x+24& = & 33x+132 \\\Leftrightarrow & 22x \color{red}{+24} \color{blue}{-24} \color{blue}{-33x} & = & \color{red}{33x} +132 \color{blue}{-33x} \color{blue}{-24} \\\Leftrightarrow & -11x& = & 108 \\\Leftrightarrow & \frac{-11x}{ \color{red}{-11} }& = & \frac{108}{-11} \\\Leftrightarrow & x = \frac{-108}{11} & & \\ & V = \left\{ \frac{-108}{11} \right\} & \\\end{align}\)
2. \(\text{10 is het kleinste gemene veelvoud van 2, 10 en 5} \\ \begin{align} & \frac{x}{2}-\frac{3}{10}& = & \frac{-7}{5}x+2 \\\Leftrightarrow & \color{blue}{10.} (\frac{5x}{ \color{blue}{10} }- \frac{ 3 }{ \color{blue}{10} })& = & (\frac{-14}{ \color{blue}{10} }x+\frac{20}{ \color{blue}{10} }) \color{blue}{.10} \\\Leftrightarrow & 5x-3& = & -14x+20 \\\Leftrightarrow & 5x \color{red}{-3} \color{blue}{+3} \color{blue}{+14x} & = & \color{red}{-14x} +20 \color{blue}{+14x} \color{blue}{+3} \\\Leftrightarrow & 19x& = & 23 \\\Leftrightarrow & \frac{19x}{ \color{red}{19} }& = & \frac{23}{19} \\\Leftrightarrow & x = \frac{23}{19} & & \\ & V = \left\{ \frac{23}{19} \right\} & \\\end{align}\)
3. \(\text{6 is het kleinste gemene veelvoud van 2, 6 en 3} \\ \begin{align} & \frac{x}{2}-\frac{5}{6}& = & \frac{-8}{3}x-1 \\\Leftrightarrow & \color{blue}{6.} (\frac{3x}{ \color{blue}{6} }- \frac{ 5 }{ \color{blue}{6} })& = & (\frac{-16}{ \color{blue}{6} }x-\frac{6}{ \color{blue}{6} }) \color{blue}{.6} \\\Leftrightarrow & 3x-5& = & -16x-6 \\\Leftrightarrow & 3x \color{red}{-5} \color{blue}{+5} \color{blue}{+16x} & = & \color{red}{-16x} -6 \color{blue}{+16x} \color{blue}{+5} \\\Leftrightarrow & 19x& = & -1 \\\Leftrightarrow & \frac{19x}{ \color{red}{19} }& = & \frac{-1}{19} \\\Leftrightarrow & x = \frac{-1}{19} & & \\ & V = \left\{ \frac{-1}{19} \right\} & \\\end{align}\)
4. \(\text{330 is het kleinste gemene veelvoud van 6, 11 en 5} \\ \begin{align} & \frac{x}{6}-\frac{2}{11}& = & \frac{7}{5}x-3 \\\Leftrightarrow & \color{blue}{330.} (\frac{55x}{ \color{blue}{330} }- \frac{ 60 }{ \color{blue}{330} })& = & (\frac{462}{ \color{blue}{330} }x-\frac{990}{ \color{blue}{330} }) \color{blue}{.330} \\\Leftrightarrow & 55x-60& = & 462x-990 \\\Leftrightarrow & 55x \color{red}{-60} \color{blue}{+60} \color{blue}{-462x} & = & \color{red}{462x} -990 \color{blue}{-462x} \color{blue}{+60} \\\Leftrightarrow & -407x& = & -930 \\\Leftrightarrow & \frac{-407x}{ \color{red}{-407} }& = & \frac{-930}{-407} \\\Leftrightarrow & x = \frac{930}{407} & & \\ & V = \left\{ \frac{930}{407} \right\} & \\\end{align}\)
5. \(\text{364 is het kleinste gemene veelvoud van 7, 13 en 4} \\ \begin{align} & \frac{x}{7}-\frac{3}{13}& = & \frac{-3}{4}x-1 \\\Leftrightarrow & \color{blue}{364.} (\frac{52x}{ \color{blue}{364} }- \frac{ 84 }{ \color{blue}{364} })& = & (\frac{-273}{ \color{blue}{364} }x-\frac{364}{ \color{blue}{364} }) \color{blue}{.364} \\\Leftrightarrow & 52x-84& = & -273x-364 \\\Leftrightarrow & 52x \color{red}{-84} \color{blue}{+84} \color{blue}{+273x} & = & \color{red}{-273x} -364 \color{blue}{+273x} \color{blue}{+84} \\\Leftrightarrow & 325x& = & -280 \\\Leftrightarrow & \frac{325x}{ \color{red}{325} }& = & \frac{-280}{325} \\\Leftrightarrow & x = \frac{-56}{65} & & \\ & V = \left\{ \frac{-56}{65} \right\} & \\\end{align}\)
6. \(\text{240 is het kleinste gemene veelvoud van 5, 16 en 6} \\ \begin{align} & \frac{x}{5}-\frac{5}{16}& = & \frac{7}{6}x+2 \\\Leftrightarrow & \color{blue}{240.} (\frac{48x}{ \color{blue}{240} }- \frac{ 75 }{ \color{blue}{240} })& = & (\frac{280}{ \color{blue}{240} }x+\frac{480}{ \color{blue}{240} }) \color{blue}{.240} \\\Leftrightarrow & 48x-75& = & 280x+480 \\\Leftrightarrow & 48x \color{red}{-75} \color{blue}{+75} \color{blue}{-280x} & = & \color{red}{280x} +480 \color{blue}{-280x} \color{blue}{+75} \\\Leftrightarrow & -232x& = & 555 \\\Leftrightarrow & \frac{-232x}{ \color{red}{-232} }& = & \frac{555}{-232} \\\Leftrightarrow & x = \frac{-555}{232} & & \\ & V = \left\{ \frac{-555}{232} \right\} & \\\end{align}\)
7. \(\text{70 is het kleinste gemene veelvoud van 5, 7 en 2} \\ \begin{align} & \frac{x}{5}+\frac{2}{7}& = & \frac{1}{2}x-8 \\\Leftrightarrow & \color{blue}{70.} (\frac{14x}{ \color{blue}{70} }+ \frac{ 20 }{ \color{blue}{70} })& = & (\frac{35}{ \color{blue}{70} }x-\frac{560}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 14x+20& = & 35x-560 \\\Leftrightarrow & 14x \color{red}{+20} \color{blue}{-20} \color{blue}{-35x} & = & \color{red}{35x} -560 \color{blue}{-35x} \color{blue}{-20} \\\Leftrightarrow & -21x& = & -580 \\\Leftrightarrow & \frac{-21x}{ \color{red}{-21} }& = & \frac{-580}{-21} \\\Leftrightarrow & x = \frac{580}{21} & & \\ & V = \left\{ \frac{580}{21} \right\} & \\\end{align}\)
8. \(\text{30 is het kleinste gemene veelvoud van 3, 10 en 5} \\ \begin{align} & \frac{x}{3}+\frac{3}{10}& = & \frac{1}{5}x+1 \\\Leftrightarrow & \color{blue}{30.} (\frac{10x}{ \color{blue}{30} }+ \frac{ 9 }{ \color{blue}{30} })& = & (\frac{6}{ \color{blue}{30} }x+\frac{30}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 10x+9& = & 6x+30 \\\Leftrightarrow & 10x \color{red}{+9} \color{blue}{-9} \color{blue}{-6x} & = & \color{red}{6x} +30 \color{blue}{-6x} \color{blue}{-9} \\\Leftrightarrow & 4x& = & 21 \\\Leftrightarrow & \frac{4x}{ \color{red}{4} }& = & \frac{21}{4} \\\Leftrightarrow & x = \frac{21}{4} & & \\ & V = \left\{ \frac{21}{4} \right\} & \\\end{align}\)
9. \(\text{330 is het kleinste gemene veelvoud van 6, 11 en 5} \\ \begin{align} & \frac{x}{6}-\frac{5}{11}& = & \frac{-7}{5}x+6 \\\Leftrightarrow & \color{blue}{330.} (\frac{55x}{ \color{blue}{330} }- \frac{ 150 }{ \color{blue}{330} })& = & (\frac{-462}{ \color{blue}{330} }x+\frac{1980}{ \color{blue}{330} }) \color{blue}{.330} \\\Leftrightarrow & 55x-150& = & -462x+1980 \\\Leftrightarrow & 55x \color{red}{-150} \color{blue}{+150} \color{blue}{+462x} & = & \color{red}{-462x} +1980 \color{blue}{+462x} \color{blue}{+150} \\\Leftrightarrow & 517x& = & 2130 \\\Leftrightarrow & \frac{517x}{ \color{red}{517} }& = & \frac{2130}{517} \\\Leftrightarrow & x = \frac{2130}{517} & & \\ & V = \left\{ \frac{2130}{517} \right\} & \\\end{align}\)
10. \(\text{42 is het kleinste gemene veelvoud van 2, 7 en 3} \\ \begin{align} & \frac{x}{2}+\frac{4}{7}& = & \frac{1}{3}x+3 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }+ \frac{ 24 }{ \color{blue}{42} })& = & (\frac{14}{ \color{blue}{42} }x+\frac{126}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x+24& = & 14x+126 \\\Leftrightarrow & 21x \color{red}{+24} \color{blue}{-24} \color{blue}{-14x} & = & \color{red}{14x} +126 \color{blue}{-14x} \color{blue}{-24} \\\Leftrightarrow & 7x& = & 102 \\\Leftrightarrow & \frac{7x}{ \color{red}{7} }& = & \frac{102}{7} \\\Leftrightarrow & x = \frac{102}{7} & & \\ & V = \left\{ \frac{102}{7} \right\} & \\\end{align}\)
11. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}+\frac{4}{7}& = & \frac{-2}{5}x+1 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }+ \frac{ 120 }{ \color{blue}{210} })& = & (\frac{-84}{ \color{blue}{210} }x+\frac{210}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x+120& = & -84x+210 \\\Leftrightarrow & 35x \color{red}{+120} \color{blue}{-120} \color{blue}{+84x} & = & \color{red}{-84x} +210 \color{blue}{+84x} \color{blue}{-120} \\\Leftrightarrow & 119x& = & 90 \\\Leftrightarrow & \frac{119x}{ \color{red}{119} }& = & \frac{90}{119} \\\Leftrightarrow & x = \frac{90}{119} & & \\ & V = \left\{ \frac{90}{119} \right\} & \\\end{align}\)
12. \(\text{42 is het kleinste gemene veelvoud van 2, 7 en 3} \\ \begin{align} & \frac{x}{2}+\frac{3}{7}& = & \frac{-5}{3}x-7 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }+ \frac{ 18 }{ \color{blue}{42} })& = & (\frac{-70}{ \color{blue}{42} }x-\frac{294}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x+18& = & -70x-294 \\\Leftrightarrow & 21x \color{red}{+18} \color{blue}{-18} \color{blue}{+70x} & = & \color{red}{-70x} -294 \color{blue}{+70x} \color{blue}{-18} \\\Leftrightarrow & 91x& = & -312 \\\Leftrightarrow & \frac{91x}{ \color{red}{91} }& = & \frac{-312}{91} \\\Leftrightarrow & x = \frac{-24}{7} & & \\ & V = \left\{ \frac{-24}{7} \right\} & \\\end{align}\)