Vgln. eerste graad (reeks 4)

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

1. \(\frac{x}{6}-\frac{4}{7}=\frac{7}{5}x+1\)
2. \(\frac{x}{7}+\frac{3}{7}=\frac{1}{2}x-5\)
3. \(\frac{x}{3}-\frac{3}{14}=\frac{1}{2}x-2\)
4. \(\frac{x}{7}-\frac{2}{15}=\frac{1}{3}x+4\)
5. \(\frac{x}{6}+\frac{4}{9}=\frac{1}{5}x-7\)
6. \(\frac{x}{7}-\frac{3}{10}=\frac{1}{2}x-7\)
7. \(\frac{x}{2}-\frac{5}{16}=\frac{-2}{5}x-5\)
8. \(\frac{x}{2}-\frac{4}{7}=\frac{4}{3}x-2\)
9. \(\frac{x}{2}-\frac{5}{6}=\frac{7}{3}x+2\)
10. \(\frac{x}{4}+\frac{4}{11}=\frac{1}{3}x-1\)
11. \(\frac{x}{5}-\frac{2}{11}=\frac{-5}{6}x+7\)
12. \(\frac{x}{2}+\frac{2}{13}=\frac{-5}{3}x+4\)

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

Verbetersleutel

1. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}-\frac{4}{7}& = & \frac{7}{5}x+1 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }- \frac{ 120 }{ \color{blue}{210} })& = & (\frac{294}{ \color{blue}{210} }x+\frac{210}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x-120& = & 294x+210 \\\Leftrightarrow & 35x \color{red}{-120} \color{blue}{+120} \color{blue}{-294x} & = & \color{red}{294x} +210 \color{blue}{-294x} \color{blue}{+120} \\\Leftrightarrow & -259x& = & 330 \\\Leftrightarrow & \frac{-259x}{ \color{red}{-259} }& = & \frac{330}{-259} \\\Leftrightarrow & x = \frac{-330}{259} & & \\ & V = \left\{ \frac{-330}{259} \right\} & \\\end{align}\)
2. \(\text{14 is het kleinste gemene veelvoud van 7, 7 en 2} \\ \begin{align} & \frac{x}{7}+\frac{3}{7}& = & \frac{1}{2}x-5 \\\Leftrightarrow & \color{blue}{14.} (\frac{2x}{ \color{blue}{14} }+ \frac{ 6 }{ \color{blue}{14} })& = & (\frac{7}{ \color{blue}{14} }x-\frac{70}{ \color{blue}{14} }) \color{blue}{.14} \\\Leftrightarrow & 2x+6& = & 7x-70 \\\Leftrightarrow & 2x \color{red}{+6} \color{blue}{-6} \color{blue}{-7x} & = & \color{red}{7x} -70 \color{blue}{-7x} \color{blue}{-6} \\\Leftrightarrow & -5x& = & -76 \\\Leftrightarrow & \frac{-5x}{ \color{red}{-5} }& = & \frac{-76}{-5} \\\Leftrightarrow & x = \frac{76}{5} & & \\ & V = \left\{ \frac{76}{5} \right\} & \\\end{align}\)
3. \(\text{42 is het kleinste gemene veelvoud van 3, 14 en 2} \\ \begin{align} & \frac{x}{3}-\frac{3}{14}& = & \frac{1}{2}x-2 \\\Leftrightarrow & \color{blue}{42.} (\frac{14x}{ \color{blue}{42} }- \frac{ 9 }{ \color{blue}{42} })& = & (\frac{21}{ \color{blue}{42} }x-\frac{84}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 14x-9& = & 21x-84 \\\Leftrightarrow & 14x \color{red}{-9} \color{blue}{+9} \color{blue}{-21x} & = & \color{red}{21x} -84 \color{blue}{-21x} \color{blue}{+9} \\\Leftrightarrow & -7x& = & -75 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{-75}{-7} \\\Leftrightarrow & x = \frac{75}{7} & & \\ & V = \left\{ \frac{75}{7} \right\} & \\\end{align}\)
4. \(\text{105 is het kleinste gemene veelvoud van 7, 15 en 3} \\ \begin{align} & \frac{x}{7}-\frac{2}{15}& = & \frac{1}{3}x+4 \\\Leftrightarrow & \color{blue}{105.} (\frac{15x}{ \color{blue}{105} }- \frac{ 14 }{ \color{blue}{105} })& = & (\frac{35}{ \color{blue}{105} }x+\frac{420}{ \color{blue}{105} }) \color{blue}{.105} \\\Leftrightarrow & 15x-14& = & 35x+420 \\\Leftrightarrow & 15x \color{red}{-14} \color{blue}{+14} \color{blue}{-35x} & = & \color{red}{35x} +420 \color{blue}{-35x} \color{blue}{+14} \\\Leftrightarrow & -20x& = & 434 \\\Leftrightarrow & \frac{-20x}{ \color{red}{-20} }& = & \frac{434}{-20} \\\Leftrightarrow & x = \frac{-217}{10} & & \\ & V = \left\{ \frac{-217}{10} \right\} & \\\end{align}\)
5. \(\text{90 is het kleinste gemene veelvoud van 6, 9 en 5} \\ \begin{align} & \frac{x}{6}+\frac{4}{9}& = & \frac{1}{5}x-7 \\\Leftrightarrow & \color{blue}{90.} (\frac{15x}{ \color{blue}{90} }+ \frac{ 40 }{ \color{blue}{90} })& = & (\frac{18}{ \color{blue}{90} }x-\frac{630}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 15x+40& = & 18x-630 \\\Leftrightarrow & 15x \color{red}{+40} \color{blue}{-40} \color{blue}{-18x} & = & \color{red}{18x} -630 \color{blue}{-18x} \color{blue}{-40} \\\Leftrightarrow & -3x& = & -670 \\\Leftrightarrow & \frac{-3x}{ \color{red}{-3} }& = & \frac{-670}{-3} \\\Leftrightarrow & x = \frac{670}{3} & & \\ & V = \left\{ \frac{670}{3} \right\} & \\\end{align}\)
6. \(\text{70 is het kleinste gemene veelvoud van 7, 10 en 2} \\ \begin{align} & \frac{x}{7}-\frac{3}{10}& = & \frac{1}{2}x-7 \\\Leftrightarrow & \color{blue}{70.} (\frac{10x}{ \color{blue}{70} }- \frac{ 21 }{ \color{blue}{70} })& = & (\frac{35}{ \color{blue}{70} }x-\frac{490}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 10x-21& = & 35x-490 \\\Leftrightarrow & 10x \color{red}{-21} \color{blue}{+21} \color{blue}{-35x} & = & \color{red}{35x} -490 \color{blue}{-35x} \color{blue}{+21} \\\Leftrightarrow & -25x& = & -469 \\\Leftrightarrow & \frac{-25x}{ \color{red}{-25} }& = & \frac{-469}{-25} \\\Leftrightarrow & x = \frac{469}{25} & & \\ & V = \left\{ \frac{469}{25} \right\} & \\\end{align}\)
7. \(\text{80 is het kleinste gemene veelvoud van 2, 16 en 5} \\ \begin{align} & \frac{x}{2}-\frac{5}{16}& = & \frac{-2}{5}x-5 \\\Leftrightarrow & \color{blue}{80.} (\frac{40x}{ \color{blue}{80} }- \frac{ 25 }{ \color{blue}{80} })& = & (\frac{-32}{ \color{blue}{80} }x-\frac{400}{ \color{blue}{80} }) \color{blue}{.80} \\\Leftrightarrow & 40x-25& = & -32x-400 \\\Leftrightarrow & 40x \color{red}{-25} \color{blue}{+25} \color{blue}{+32x} & = & \color{red}{-32x} -400 \color{blue}{+32x} \color{blue}{+25} \\\Leftrightarrow & 72x& = & -375 \\\Leftrightarrow & \frac{72x}{ \color{red}{72} }& = & \frac{-375}{72} \\\Leftrightarrow & x = \frac{-125}{24} & & \\ & V = \left\{ \frac{-125}{24} \right\} & \\\end{align}\)
8. \(\text{42 is het kleinste gemene veelvoud van 2, 7 en 3} \\ \begin{align} & \frac{x}{2}-\frac{4}{7}& = & \frac{4}{3}x-2 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }- \frac{ 24 }{ \color{blue}{42} })& = & (\frac{56}{ \color{blue}{42} }x-\frac{84}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x-24& = & 56x-84 \\\Leftrightarrow & 21x \color{red}{-24} \color{blue}{+24} \color{blue}{-56x} & = & \color{red}{56x} -84 \color{blue}{-56x} \color{blue}{+24} \\\Leftrightarrow & -35x& = & -60 \\\Leftrightarrow & \frac{-35x}{ \color{red}{-35} }& = & \frac{-60}{-35} \\\Leftrightarrow & x = \frac{12}{7} & & \\ & V = \left\{ \frac{12}{7} \right\} & \\\end{align}\)
9. \(\text{6 is het kleinste gemene veelvoud van 2, 6 en 3} \\ \begin{align} & \frac{x}{2}-\frac{5}{6}& = & \frac{7}{3}x+2 \\\Leftrightarrow & \color{blue}{6.} (\frac{3x}{ \color{blue}{6} }- \frac{ 5 }{ \color{blue}{6} })& = & (\frac{14}{ \color{blue}{6} }x+\frac{12}{ \color{blue}{6} }) \color{blue}{.6} \\\Leftrightarrow & 3x-5& = & 14x+12 \\\Leftrightarrow & 3x \color{red}{-5} \color{blue}{+5} \color{blue}{-14x} & = & \color{red}{14x} +12 \color{blue}{-14x} \color{blue}{+5} \\\Leftrightarrow & -11x& = & 17 \\\Leftrightarrow & \frac{-11x}{ \color{red}{-11} }& = & \frac{17}{-11} \\\Leftrightarrow & x = \frac{-17}{11} & & \\ & V = \left\{ \frac{-17}{11} \right\} & \\\end{align}\)
10. \(\text{132 is het kleinste gemene veelvoud van 4, 11 en 3} \\ \begin{align} & \frac{x}{4}+\frac{4}{11}& = & \frac{1}{3}x-1 \\\Leftrightarrow & \color{blue}{132.} (\frac{33x}{ \color{blue}{132} }+ \frac{ 48 }{ \color{blue}{132} })& = & (\frac{44}{ \color{blue}{132} }x-\frac{132}{ \color{blue}{132} }) \color{blue}{.132} \\\Leftrightarrow & 33x+48& = & 44x-132 \\\Leftrightarrow & 33x \color{red}{+48} \color{blue}{-48} \color{blue}{-44x} & = & \color{red}{44x} -132 \color{blue}{-44x} \color{blue}{-48} \\\Leftrightarrow & -11x& = & -180 \\\Leftrightarrow & \frac{-11x}{ \color{red}{-11} }& = & \frac{-180}{-11} \\\Leftrightarrow & x = \frac{180}{11} & & \\ & V = \left\{ \frac{180}{11} \right\} & \\\end{align}\)
11. \(\text{330 is het kleinste gemene veelvoud van 5, 11 en 6} \\ \begin{align} & \frac{x}{5}-\frac{2}{11}& = & \frac{-5}{6}x+7 \\\Leftrightarrow & \color{blue}{330.} (\frac{66x}{ \color{blue}{330} }- \frac{ 60 }{ \color{blue}{330} })& = & (\frac{-275}{ \color{blue}{330} }x+\frac{2310}{ \color{blue}{330} }) \color{blue}{.330} \\\Leftrightarrow & 66x-60& = & -275x+2310 \\\Leftrightarrow & 66x \color{red}{-60} \color{blue}{+60} \color{blue}{+275x} & = & \color{red}{-275x} +2310 \color{blue}{+275x} \color{blue}{+60} \\\Leftrightarrow & 341x& = & 2370 \\\Leftrightarrow & \frac{341x}{ \color{red}{341} }& = & \frac{2370}{341} \\\Leftrightarrow & x = \frac{2370}{341} & & \\ & V = \left\{ \frac{2370}{341} \right\} & \\\end{align}\)
12. \(\text{78 is het kleinste gemene veelvoud van 2, 13 en 3} \\ \begin{align} & \frac{x}{2}+\frac{2}{13}& = & \frac{-5}{3}x+4 \\\Leftrightarrow & \color{blue}{78.} (\frac{39x}{ \color{blue}{78} }+ \frac{ 12 }{ \color{blue}{78} })& = & (\frac{-130}{ \color{blue}{78} }x+\frac{312}{ \color{blue}{78} }) \color{blue}{.78} \\\Leftrightarrow & 39x+12& = & -130x+312 \\\Leftrightarrow & 39x \color{red}{+12} \color{blue}{-12} \color{blue}{+130x} & = & \color{red}{-130x} +312 \color{blue}{+130x} \color{blue}{-12} \\\Leftrightarrow & 169x& = & 300 \\\Leftrightarrow & \frac{169x}{ \color{red}{169} }& = & \frac{300}{169} \\\Leftrightarrow & x = \frac{300}{169} & & \\ & V = \left\{ \frac{300}{169} \right\} & \\\end{align}\)