Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{66 is het kleinste gemene veelvoud van 3, 11 en 2} \\ \begin{align} & \frac{x}{3}+\frac{4}{11}& = & \frac{1}{2}x+3 \\\Leftrightarrow & \color{blue}{66.} (\frac{22x}{ \color{blue}{66} }+ \frac{ 24 }{ \color{blue}{66} })& = & (\frac{33}{ \color{blue}{66} }x+\frac{198}{ \color{blue}{66} }) \color{blue}{.66} \\\Leftrightarrow & 22x+24& = & 33x+198 \\\Leftrightarrow & 22x \color{red}{+24} \color{blue}{-24} \color{blue}{-33x} & = & \color{red}{33x} +198 \color{blue}{-33x} \color{blue}{-24} \\\Leftrightarrow & -11x& = & 174 \\\Leftrightarrow & \frac{-11x}{ \color{red}{-11} }& = & \frac{174}{-11} \\\Leftrightarrow & x = \frac{-174}{11} & & \\ & V = \left\{ \frac{-174}{11} \right\} & \\\end{align}\)
2. \(\text{70 is het kleinste gemene veelvoud van 5, 7 en 2} \\ \begin{align} & \frac{x}{5}+\frac{2}{7}& = & \frac{1}{2}x+7 \\\Leftrightarrow & \color{blue}{70.} (\frac{14x}{ \color{blue}{70} }+ \frac{ 20 }{ \color{blue}{70} })& = & (\frac{35}{ \color{blue}{70} }x+\frac{490}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 14x+20& = & 35x+490 \\\Leftrightarrow & 14x \color{red}{+20} \color{blue}{-20} \color{blue}{-35x} & = & \color{red}{35x} +490 \color{blue}{-35x} \color{blue}{-20} \\\Leftrightarrow & -21x& = & 470 \\\Leftrightarrow & \frac{-21x}{ \color{red}{-21} }& = & \frac{470}{-21} \\\Leftrightarrow & x = \frac{-470}{21} & & \\ & V = \left\{ \frac{-470}{21} \right\} & \\\end{align}\)
3. \(\text{12 is het kleinste gemene veelvoud van 4, 6 en 3} \\ \begin{align} & \frac{x}{4}-\frac{5}{6}& = & \frac{-8}{3}x+6 \\\Leftrightarrow & \color{blue}{12.} (\frac{3x}{ \color{blue}{12} }- \frac{ 10 }{ \color{blue}{12} })& = & (\frac{-32}{ \color{blue}{12} }x+\frac{72}{ \color{blue}{12} }) \color{blue}{.12} \\\Leftrightarrow & 3x-10& = & -32x+72 \\\Leftrightarrow & 3x \color{red}{-10} \color{blue}{+10} \color{blue}{+32x} & = & \color{red}{-32x} +72 \color{blue}{+32x} \color{blue}{+10} \\\Leftrightarrow & 35x& = & 82 \\\Leftrightarrow & \frac{35x}{ \color{red}{35} }& = & \frac{82}{35} \\\Leftrightarrow & x = \frac{82}{35} & & \\ & V = \left\{ \frac{82}{35} \right\} & \\\end{align}\)
4. \(\text{24 is het kleinste gemene veelvoud van 3, 8 en 4} \\ \begin{align} & \frac{x}{3}+\frac{3}{8}& = & \frac{3}{4}x+6 \\\Leftrightarrow & \color{blue}{24.} (\frac{8x}{ \color{blue}{24} }+ \frac{ 9 }{ \color{blue}{24} })& = & (\frac{18}{ \color{blue}{24} }x+\frac{144}{ \color{blue}{24} }) \color{blue}{.24} \\\Leftrightarrow & 8x+9& = & 18x+144 \\\Leftrightarrow & 8x \color{red}{+9} \color{blue}{-9} \color{blue}{-18x} & = & \color{red}{18x} +144 \color{blue}{-18x} \color{blue}{-9} \\\Leftrightarrow & -10x& = & 135 \\\Leftrightarrow & \frac{-10x}{ \color{red}{-10} }& = & \frac{135}{-10} \\\Leftrightarrow & x = \frac{-27}{2} & & \\ & V = \left\{ \frac{-27}{2} \right\} & \\\end{align}\)
5. \(\text{156 is het kleinste gemene veelvoud van 3, 13 en 4} \\ \begin{align} & \frac{x}{3}-\frac{2}{13}& = & \frac{1}{4}x-6 \\\Leftrightarrow & \color{blue}{156.} (\frac{52x}{ \color{blue}{156} }- \frac{ 24 }{ \color{blue}{156} })& = & (\frac{39}{ \color{blue}{156} }x-\frac{936}{ \color{blue}{156} }) \color{blue}{.156} \\\Leftrightarrow & 52x-24& = & 39x-936 \\\Leftrightarrow & 52x \color{red}{-24} \color{blue}{+24} \color{blue}{-39x} & = & \color{red}{39x} -936 \color{blue}{-39x} \color{blue}{+24} \\\Leftrightarrow & 13x& = & -912 \\\Leftrightarrow & \frac{13x}{ \color{red}{13} }& = & \frac{-912}{13} \\\Leftrightarrow & x = \frac{-912}{13} & & \\ & V = \left\{ \frac{-912}{13} \right\} & \\\end{align}\)
6. \(\text{18 is het kleinste gemene veelvoud van 3, 9 en 2} \\ \begin{align} & \frac{x}{3}-\frac{4}{9}& = & \frac{3}{2}x-6 \\\Leftrightarrow & \color{blue}{18.} (\frac{6x}{ \color{blue}{18} }- \frac{ 8 }{ \color{blue}{18} })& = & (\frac{27}{ \color{blue}{18} }x-\frac{108}{ \color{blue}{18} }) \color{blue}{.18} \\\Leftrightarrow & 6x-8& = & 27x-108 \\\Leftrightarrow & 6x \color{red}{-8} \color{blue}{+8} \color{blue}{-27x} & = & \color{red}{27x} -108 \color{blue}{-27x} \color{blue}{+8} \\\Leftrightarrow & -21x& = & -100 \\\Leftrightarrow & \frac{-21x}{ \color{red}{-21} }& = & \frac{-100}{-21} \\\Leftrightarrow & x = \frac{100}{21} & & \\ & V = \left\{ \frac{100}{21} \right\} & \\\end{align}\)
7. \(\text{60 is het kleinste gemene veelvoud van 4, 15 en 3} \\ \begin{align} & \frac{x}{4}+\frac{2}{15}& = & \frac{-5}{3}x+6 \\\Leftrightarrow & \color{blue}{60.} (\frac{15x}{ \color{blue}{60} }+ \frac{ 8 }{ \color{blue}{60} })& = & (\frac{-100}{ \color{blue}{60} }x+\frac{360}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 15x+8& = & -100x+360 \\\Leftrightarrow & 15x \color{red}{+8} \color{blue}{-8} \color{blue}{+100x} & = & \color{red}{-100x} +360 \color{blue}{+100x} \color{blue}{-8} \\\Leftrightarrow & 115x& = & 352 \\\Leftrightarrow & \frac{115x}{ \color{red}{115} }& = & \frac{352}{115} \\\Leftrightarrow & x = \frac{352}{115} & & \\ & V = \left\{ \frac{352}{115} \right\} & \\\end{align}\)
8. \(\text{126 is het kleinste gemene veelvoud van 7, 9 en 6} \\ \begin{align} & \frac{x}{7}+\frac{2}{9}& = & \frac{-5}{6}x-3 \\\Leftrightarrow & \color{blue}{126.} (\frac{18x}{ \color{blue}{126} }+ \frac{ 28 }{ \color{blue}{126} })& = & (\frac{-105}{ \color{blue}{126} }x-\frac{378}{ \color{blue}{126} }) \color{blue}{.126} \\\Leftrightarrow & 18x+28& = & -105x-378 \\\Leftrightarrow & 18x \color{red}{+28} \color{blue}{-28} \color{blue}{+105x} & = & \color{red}{-105x} -378 \color{blue}{+105x} \color{blue}{-28} \\\Leftrightarrow & 123x& = & -406 \\\Leftrightarrow & \frac{123x}{ \color{red}{123} }& = & \frac{-406}{123} \\\Leftrightarrow & x = \frac{-406}{123} & & \\ & V = \left\{ \frac{-406}{123} \right\} & \\\end{align}\)
9. \(\text{195 is het kleinste gemene veelvoud van 5, 13 en 3} \\ \begin{align} & \frac{x}{5}+\frac{4}{13}& = & \frac{5}{3}x+3 \\\Leftrightarrow & \color{blue}{195.} (\frac{39x}{ \color{blue}{195} }+ \frac{ 60 }{ \color{blue}{195} })& = & (\frac{325}{ \color{blue}{195} }x+\frac{585}{ \color{blue}{195} }) \color{blue}{.195} \\\Leftrightarrow & 39x+60& = & 325x+585 \\\Leftrightarrow & 39x \color{red}{+60} \color{blue}{-60} \color{blue}{-325x} & = & \color{red}{325x} +585 \color{blue}{-325x} \color{blue}{-60} \\\Leftrightarrow & -286x& = & 525 \\\Leftrightarrow & \frac{-286x}{ \color{red}{-286} }& = & \frac{525}{-286} \\\Leftrightarrow & x = \frac{-525}{286} & & \\ & V = \left\{ \frac{-525}{286} \right\} & \\\end{align}\)
10. \(\text{24 is het kleinste gemene veelvoud van 3, 8 en 2} \\ \begin{align} & \frac{x}{3}-\frac{5}{8}& = & \frac{1}{2}x-8 \\\Leftrightarrow & \color{blue}{24.} (\frac{8x}{ \color{blue}{24} }- \frac{ 15 }{ \color{blue}{24} })& = & (\frac{12}{ \color{blue}{24} }x-\frac{192}{ \color{blue}{24} }) \color{blue}{.24} \\\Leftrightarrow & 8x-15& = & 12x-192 \\\Leftrightarrow & 8x \color{red}{-15} \color{blue}{+15} \color{blue}{-12x} & = & \color{red}{12x} -192 \color{blue}{-12x} \color{blue}{+15} \\\Leftrightarrow & -4x& = & -177 \\\Leftrightarrow & \frac{-4x}{ \color{red}{-4} }& = & \frac{-177}{-4} \\\Leftrightarrow & x = \frac{177}{4} & & \\ & V = \left\{ \frac{177}{4} \right\} & \\\end{align}\)
11. \(\text{42 is het kleinste gemene veelvoud van 7, 7 en 6} \\ \begin{align} & \frac{x}{7}-\frac{3}{7}& = & \frac{1}{6}x-3 \\\Leftrightarrow & \color{blue}{42.} (\frac{6x}{ \color{blue}{42} }- \frac{ 18 }{ \color{blue}{42} })& = & (\frac{7}{ \color{blue}{42} }x-\frac{126}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 6x-18& = & 7x-126 \\\Leftrightarrow & 6x \color{red}{-18} \color{blue}{+18} \color{blue}{-7x} & = & \color{red}{7x} -126 \color{blue}{-7x} \color{blue}{+18} \\\Leftrightarrow & -x& = & -108 \\\Leftrightarrow & \frac{-x}{ \color{red}{-1} }& = & \frac{-108}{-1} \\\Leftrightarrow & x = 108 & & \\ & V = \left\{ 108 \right\} & \\\end{align}\)
12. \(\text{78 is het kleinste gemene veelvoud van 2, 13 en 3} \\ \begin{align} & \frac{x}{2}-\frac{3}{13}& = & \frac{-5}{3}x+2 \\\Leftrightarrow & \color{blue}{78.} (\frac{39x}{ \color{blue}{78} }- \frac{ 18 }{ \color{blue}{78} })& = & (\frac{-130}{ \color{blue}{78} }x+\frac{156}{ \color{blue}{78} }) \color{blue}{.78} \\\Leftrightarrow & 39x-18& = & -130x+156 \\\Leftrightarrow & 39x \color{red}{-18} \color{blue}{+18} \color{blue}{+130x} & = & \color{red}{-130x} +156 \color{blue}{+130x} \color{blue}{+18} \\\Leftrightarrow & 169x& = & 174 \\\Leftrightarrow & \frac{169x}{ \color{red}{169} }& = & \frac{174}{169} \\\Leftrightarrow & x = \frac{174}{169} & & \\ & V = \left\{ \frac{174}{169} \right\} & \\\end{align}\)