Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{42 is het kleinste gemene veelvoud van 2, 7 en 3} \\ \begin{align} & \frac{x}{2}-\frac{3}{7}& = & \frac{-5}{3}x-4 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }- \frac{ 18 }{ \color{blue}{42} })& = & (\frac{-70}{ \color{blue}{42} }x-\frac{168}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x-18& = & -70x-168 \\\Leftrightarrow & 21x \color{red}{-18} \color{blue}{+18} \color{blue}{+70x} & = & \color{red}{-70x} -168 \color{blue}{+70x} \color{blue}{+18} \\\Leftrightarrow & 91x& = & -150 \\\Leftrightarrow & \frac{91x}{ \color{red}{91} }& = & \frac{-150}{91} \\\Leftrightarrow & x = \frac{-150}{91} & & \\ & V = \left\{ \frac{-150}{91} \right\} & \\\end{align}\)
2. \(\text{18 is het kleinste gemene veelvoud van 2, 9 en 3} \\ \begin{align} & \frac{x}{2}+\frac{4}{9}& = & \frac{1}{3}x-7 \\\Leftrightarrow & \color{blue}{18.} (\frac{9x}{ \color{blue}{18} }+ \frac{ 8 }{ \color{blue}{18} })& = & (\frac{6}{ \color{blue}{18} }x-\frac{126}{ \color{blue}{18} }) \color{blue}{.18} \\\Leftrightarrow & 9x+8& = & 6x-126 \\\Leftrightarrow & 9x \color{red}{+8} \color{blue}{-8} \color{blue}{-6x} & = & \color{red}{6x} -126 \color{blue}{-6x} \color{blue}{-8} \\\Leftrightarrow & 3x& = & -134 \\\Leftrightarrow & \frac{3x}{ \color{red}{3} }& = & \frac{-134}{3} \\\Leftrightarrow & x = \frac{-134}{3} & & \\ & V = \left\{ \frac{-134}{3} \right\} & \\\end{align}\)
3. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}+\frac{2}{7}& = & \frac{6}{5}x+5 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }+ \frac{ 60 }{ \color{blue}{210} })& = & (\frac{252}{ \color{blue}{210} }x+\frac{1050}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x+60& = & 252x+1050 \\\Leftrightarrow & 35x \color{red}{+60} \color{blue}{-60} \color{blue}{-252x} & = & \color{red}{252x} +1050 \color{blue}{-252x} \color{blue}{-60} \\\Leftrightarrow & -217x& = & 990 \\\Leftrightarrow & \frac{-217x}{ \color{red}{-217} }& = & \frac{990}{-217} \\\Leftrightarrow & x = \frac{-990}{217} & & \\ & V = \left\{ \frac{-990}{217} \right\} & \\\end{align}\)
4. \(\text{330 is het kleinste gemene veelvoud van 6, 11 en 5} \\ \begin{align} & \frac{x}{6}-\frac{5}{11}& = & \frac{-7}{5}x-5 \\\Leftrightarrow & \color{blue}{330.} (\frac{55x}{ \color{blue}{330} }- \frac{ 150 }{ \color{blue}{330} })& = & (\frac{-462}{ \color{blue}{330} }x-\frac{1650}{ \color{blue}{330} }) \color{blue}{.330} \\\Leftrightarrow & 55x-150& = & -462x-1650 \\\Leftrightarrow & 55x \color{red}{-150} \color{blue}{+150} \color{blue}{+462x} & = & \color{red}{-462x} -1650 \color{blue}{+462x} \color{blue}{+150} \\\Leftrightarrow & 517x& = & -1500 \\\Leftrightarrow & \frac{517x}{ \color{red}{517} }& = & \frac{-1500}{517} \\\Leftrightarrow & x = \frac{-1500}{517} & & \\ & V = \left\{ \frac{-1500}{517} \right\} & \\\end{align}\)
5. \(\text{130 is het kleinste gemene veelvoud van 5, 13 en 2} \\ \begin{align} & \frac{x}{5}+\frac{3}{13}& = & \frac{1}{2}x-6 \\\Leftrightarrow & \color{blue}{130.} (\frac{26x}{ \color{blue}{130} }+ \frac{ 30 }{ \color{blue}{130} })& = & (\frac{65}{ \color{blue}{130} }x-\frac{780}{ \color{blue}{130} }) \color{blue}{.130} \\\Leftrightarrow & 26x+30& = & 65x-780 \\\Leftrightarrow & 26x \color{red}{+30} \color{blue}{-30} \color{blue}{-65x} & = & \color{red}{65x} -780 \color{blue}{-65x} \color{blue}{-30} \\\Leftrightarrow & -39x& = & -810 \\\Leftrightarrow & \frac{-39x}{ \color{red}{-39} }& = & \frac{-810}{-39} \\\Leftrightarrow & x = \frac{270}{13} & & \\ & V = \left\{ \frac{270}{13} \right\} & \\\end{align}\)
6. \(\text{12 is het kleinste gemene veelvoud van 3, 12 en 4} \\ \begin{align} & \frac{x}{3}+\frac{5}{12}& = & \frac{3}{4}x-6 \\\Leftrightarrow & \color{blue}{12.} (\frac{4x}{ \color{blue}{12} }+ \frac{ 5 }{ \color{blue}{12} })& = & (\frac{9}{ \color{blue}{12} }x-\frac{72}{ \color{blue}{12} }) \color{blue}{.12} \\\Leftrightarrow & 4x+5& = & 9x-72 \\\Leftrightarrow & 4x \color{red}{+5} \color{blue}{-5} \color{blue}{-9x} & = & \color{red}{9x} -72 \color{blue}{-9x} \color{blue}{-5} \\\Leftrightarrow & -5x& = & -77 \\\Leftrightarrow & \frac{-5x}{ \color{red}{-5} }& = & \frac{-77}{-5} \\\Leftrightarrow & x = \frac{77}{5} & & \\ & V = \left\{ \frac{77}{5} \right\} & \\\end{align}\)
7. \(\text{260 is het kleinste gemene veelvoud van 4, 13 en 5} \\ \begin{align} & \frac{x}{4}-\frac{4}{13}& = & \frac{-7}{5}x-8 \\\Leftrightarrow & \color{blue}{260.} (\frac{65x}{ \color{blue}{260} }- \frac{ 80 }{ \color{blue}{260} })& = & (\frac{-364}{ \color{blue}{260} }x-\frac{2080}{ \color{blue}{260} }) \color{blue}{.260} \\\Leftrightarrow & 65x-80& = & -364x-2080 \\\Leftrightarrow & 65x \color{red}{-80} \color{blue}{+80} \color{blue}{+364x} & = & \color{red}{-364x} -2080 \color{blue}{+364x} \color{blue}{+80} \\\Leftrightarrow & 429x& = & -2000 \\\Leftrightarrow & \frac{429x}{ \color{red}{429} }& = & \frac{-2000}{429} \\\Leftrightarrow & x = \frac{-2000}{429} & & \\ & V = \left\{ \frac{-2000}{429} \right\} & \\\end{align}\)
8. \(\text{42 is het kleinste gemene veelvoud van 7, 6 en 2} \\ \begin{align} & \frac{x}{7}-\frac{5}{6}& = & \frac{1}{2}x+8 \\\Leftrightarrow & \color{blue}{42.} (\frac{6x}{ \color{blue}{42} }- \frac{ 35 }{ \color{blue}{42} })& = & (\frac{21}{ \color{blue}{42} }x+\frac{336}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 6x-35& = & 21x+336 \\\Leftrightarrow & 6x \color{red}{-35} \color{blue}{+35} \color{blue}{-21x} & = & \color{red}{21x} +336 \color{blue}{-21x} \color{blue}{+35} \\\Leftrightarrow & -15x& = & 371 \\\Leftrightarrow & \frac{-15x}{ \color{red}{-15} }& = & \frac{371}{-15} \\\Leftrightarrow & x = \frac{-371}{15} & & \\ & V = \left\{ \frac{-371}{15} \right\} & \\\end{align}\)
9. \(\text{30 is het kleinste gemene veelvoud van 6, 10 en 5} \\ \begin{align} & \frac{x}{6}+\frac{3}{10}& = & \frac{1}{5}x+2 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }+ \frac{ 9 }{ \color{blue}{30} })& = & (\frac{6}{ \color{blue}{30} }x+\frac{60}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x+9& = & 6x+60 \\\Leftrightarrow & 5x \color{red}{+9} \color{blue}{-9} \color{blue}{-6x} & = & \color{red}{6x} +60 \color{blue}{-6x} \color{blue}{-9} \\\Leftrightarrow & -x& = & 51 \\\Leftrightarrow & \frac{-x}{ \color{red}{-1} }& = & \frac{51}{-1} \\\Leftrightarrow & x = -51 & & \\ & V = \left\{ -51 \right\} & \\\end{align}\)
10. \(\text{35 is het kleinste gemene veelvoud van 7, 7 en 5} \\ \begin{align} & \frac{x}{7}-\frac{4}{7}& = & \frac{-4}{5}x+6 \\\Leftrightarrow & \color{blue}{35.} (\frac{5x}{ \color{blue}{35} }- \frac{ 20 }{ \color{blue}{35} })& = & (\frac{-28}{ \color{blue}{35} }x+\frac{210}{ \color{blue}{35} }) \color{blue}{.35} \\\Leftrightarrow & 5x-20& = & -28x+210 \\\Leftrightarrow & 5x \color{red}{-20} \color{blue}{+20} \color{blue}{+28x} & = & \color{red}{-28x} +210 \color{blue}{+28x} \color{blue}{+20} \\\Leftrightarrow & 33x& = & 230 \\\Leftrightarrow & \frac{33x}{ \color{red}{33} }& = & \frac{230}{33} \\\Leftrightarrow & x = \frac{230}{33} & & \\ & V = \left\{ \frac{230}{33} \right\} & \\\end{align}\)
11. \(\text{66 is het kleinste gemene veelvoud van 2, 11 en 3} \\ \begin{align} & \frac{x}{2}-\frac{4}{11}& = & \frac{1}{3}x-2 \\\Leftrightarrow & \color{blue}{66.} (\frac{33x}{ \color{blue}{66} }- \frac{ 24 }{ \color{blue}{66} })& = & (\frac{22}{ \color{blue}{66} }x-\frac{132}{ \color{blue}{66} }) \color{blue}{.66} \\\Leftrightarrow & 33x-24& = & 22x-132 \\\Leftrightarrow & 33x \color{red}{-24} \color{blue}{+24} \color{blue}{-22x} & = & \color{red}{22x} -132 \color{blue}{-22x} \color{blue}{+24} \\\Leftrightarrow & 11x& = & -108 \\\Leftrightarrow & \frac{11x}{ \color{red}{11} }& = & \frac{-108}{11} \\\Leftrightarrow & x = \frac{-108}{11} & & \\ & V = \left\{ \frac{-108}{11} \right\} & \\\end{align}\)
12. \(\text{105 is het kleinste gemene veelvoud van 7, 15 en 5} \\ \begin{align} & \frac{x}{7}-\frac{2}{15}& = & \frac{7}{5}x+7 \\\Leftrightarrow & \color{blue}{105.} (\frac{15x}{ \color{blue}{105} }- \frac{ 14 }{ \color{blue}{105} })& = & (\frac{147}{ \color{blue}{105} }x+\frac{735}{ \color{blue}{105} }) \color{blue}{.105} \\\Leftrightarrow & 15x-14& = & 147x+735 \\\Leftrightarrow & 15x \color{red}{-14} \color{blue}{+14} \color{blue}{-147x} & = & \color{red}{147x} +735 \color{blue}{-147x} \color{blue}{+14} \\\Leftrightarrow & -132x& = & 749 \\\Leftrightarrow & \frac{-132x}{ \color{red}{-132} }& = & \frac{749}{-132} \\\Leftrightarrow & x = \frac{-749}{132} & & \\ & V = \left\{ \frac{-749}{132} \right\} & \\\end{align}\)