Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{78 is het kleinste gemene veelvoud van 3, 13 en 2} \\ \begin{align} & \frac{x}{3}+\frac{2}{13}& = & \frac{1}{2}x-3 \\\Leftrightarrow & \color{blue}{78.} (\frac{26x}{ \color{blue}{78} }+ \frac{ 12 }{ \color{blue}{78} })& = & (\frac{39}{ \color{blue}{78} }x-\frac{234}{ \color{blue}{78} }) \color{blue}{.78} \\\Leftrightarrow & 26x+12& = & 39x-234 \\\Leftrightarrow & 26x \color{red}{+12} \color{blue}{-12} \color{blue}{-39x} & = & \color{red}{39x} -234 \color{blue}{-39x} \color{blue}{-12} \\\Leftrightarrow & -13x& = & -246 \\\Leftrightarrow & \frac{-13x}{ \color{red}{-13} }& = & \frac{-246}{-13} \\\Leftrightarrow & x = \frac{246}{13} & & \\ & V = \left\{ \frac{246}{13} \right\} & \\\end{align}\)
2. \(\text{165 is het kleinste gemene veelvoud van 3, 11 en 5} \\ \begin{align} & \frac{x}{3}-\frac{2}{11}& = & \frac{-4}{5}x-6 \\\Leftrightarrow & \color{blue}{165.} (\frac{55x}{ \color{blue}{165} }- \frac{ 30 }{ \color{blue}{165} })& = & (\frac{-132}{ \color{blue}{165} }x-\frac{990}{ \color{blue}{165} }) \color{blue}{.165} \\\Leftrightarrow & 55x-30& = & -132x-990 \\\Leftrightarrow & 55x \color{red}{-30} \color{blue}{+30} \color{blue}{+132x} & = & \color{red}{-132x} -990 \color{blue}{+132x} \color{blue}{+30} \\\Leftrightarrow & 187x& = & -960 \\\Leftrightarrow & \frac{187x}{ \color{red}{187} }& = & \frac{-960}{187} \\\Leftrightarrow & x = \frac{-960}{187} & & \\ & V = \left\{ \frac{-960}{187} \right\} & \\\end{align}\)
3. \(\text{210 is het kleinste gemene veelvoud van 5, 7 en 6} \\ \begin{align} & \frac{x}{5}-\frac{2}{7}& = & \frac{5}{6}x-5 \\\Leftrightarrow & \color{blue}{210.} (\frac{42x}{ \color{blue}{210} }- \frac{ 60 }{ \color{blue}{210} })& = & (\frac{175}{ \color{blue}{210} }x-\frac{1050}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 42x-60& = & 175x-1050 \\\Leftrightarrow & 42x \color{red}{-60} \color{blue}{+60} \color{blue}{-175x} & = & \color{red}{175x} -1050 \color{blue}{-175x} \color{blue}{+60} \\\Leftrightarrow & -133x& = & -990 \\\Leftrightarrow & \frac{-133x}{ \color{red}{-133} }& = & \frac{-990}{-133} \\\Leftrightarrow & x = \frac{990}{133} & & \\ & V = \left\{ \frac{990}{133} \right\} & \\\end{align}\)
4. \(\text{156 is het kleinste gemene veelvoud van 4, 13 en 3} \\ \begin{align} & \frac{x}{4}-\frac{5}{13}& = & \frac{1}{3}x-6 \\\Leftrightarrow & \color{blue}{156.} (\frac{39x}{ \color{blue}{156} }- \frac{ 60 }{ \color{blue}{156} })& = & (\frac{52}{ \color{blue}{156} }x-\frac{936}{ \color{blue}{156} }) \color{blue}{.156} \\\Leftrightarrow & 39x-60& = & 52x-936 \\\Leftrightarrow & 39x \color{red}{-60} \color{blue}{+60} \color{blue}{-52x} & = & \color{red}{52x} -936 \color{blue}{-52x} \color{blue}{+60} \\\Leftrightarrow & -13x& = & -876 \\\Leftrightarrow & \frac{-13x}{ \color{red}{-13} }& = & \frac{-876}{-13} \\\Leftrightarrow & x = \frac{876}{13} & & \\ & V = \left\{ \frac{876}{13} \right\} & \\\end{align}\)
5. \(\text{240 is het kleinste gemene veelvoud van 6, 16 en 5} \\ \begin{align} & \frac{x}{6}+\frac{5}{16}& = & \frac{6}{5}x+4 \\\Leftrightarrow & \color{blue}{240.} (\frac{40x}{ \color{blue}{240} }+ \frac{ 75 }{ \color{blue}{240} })& = & (\frac{288}{ \color{blue}{240} }x+\frac{960}{ \color{blue}{240} }) \color{blue}{.240} \\\Leftrightarrow & 40x+75& = & 288x+960 \\\Leftrightarrow & 40x \color{red}{+75} \color{blue}{-75} \color{blue}{-288x} & = & \color{red}{288x} +960 \color{blue}{-288x} \color{blue}{-75} \\\Leftrightarrow & -248x& = & 885 \\\Leftrightarrow & \frac{-248x}{ \color{red}{-248} }& = & \frac{885}{-248} \\\Leftrightarrow & x = \frac{-885}{248} & & \\ & V = \left\{ \frac{-885}{248} \right\} & \\\end{align}\)
6. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}+\frac{4}{7}& = & \frac{5}{3}x-3 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }+ \frac{ 48 }{ \color{blue}{84} })& = & (\frac{140}{ \color{blue}{84} }x-\frac{252}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x+48& = & 140x-252 \\\Leftrightarrow & 21x \color{red}{+48} \color{blue}{-48} \color{blue}{-140x} & = & \color{red}{140x} -252 \color{blue}{-140x} \color{blue}{-48} \\\Leftrightarrow & -119x& = & -300 \\\Leftrightarrow & \frac{-119x}{ \color{red}{-119} }& = & \frac{-300}{-119} \\\Leftrightarrow & x = \frac{300}{119} & & \\ & V = \left\{ \frac{300}{119} \right\} & \\\end{align}\)
7. \(\text{42 is het kleinste gemene veelvoud van 3, 7 en 2} \\ \begin{align} & \frac{x}{3}-\frac{4}{7}& = & \frac{1}{2}x-6 \\\Leftrightarrow & \color{blue}{42.} (\frac{14x}{ \color{blue}{42} }- \frac{ 24 }{ \color{blue}{42} })& = & (\frac{21}{ \color{blue}{42} }x-\frac{252}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 14x-24& = & 21x-252 \\\Leftrightarrow & 14x \color{red}{-24} \color{blue}{+24} \color{blue}{-21x} & = & \color{red}{21x} -252 \color{blue}{-21x} \color{blue}{+24} \\\Leftrightarrow & -7x& = & -228 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{-228}{-7} \\\Leftrightarrow & x = \frac{228}{7} & & \\ & V = \left\{ \frac{228}{7} \right\} & \\\end{align}\)
8. \(\text{40 is het kleinste gemene veelvoud van 5, 8 en 2} \\ \begin{align} & \frac{x}{5}+\frac{5}{8}& = & \frac{3}{2}x-6 \\\Leftrightarrow & \color{blue}{40.} (\frac{8x}{ \color{blue}{40} }+ \frac{ 25 }{ \color{blue}{40} })& = & (\frac{60}{ \color{blue}{40} }x-\frac{240}{ \color{blue}{40} }) \color{blue}{.40} \\\Leftrightarrow & 8x+25& = & 60x-240 \\\Leftrightarrow & 8x \color{red}{+25} \color{blue}{-25} \color{blue}{-60x} & = & \color{red}{60x} -240 \color{blue}{-60x} \color{blue}{-25} \\\Leftrightarrow & -52x& = & -265 \\\Leftrightarrow & \frac{-52x}{ \color{red}{-52} }& = & \frac{-265}{-52} \\\Leftrightarrow & x = \frac{265}{52} & & \\ & V = \left\{ \frac{265}{52} \right\} & \\\end{align}\)
9. \(\text{70 is het kleinste gemene veelvoud van 5, 7 en 2} \\ \begin{align} & \frac{x}{5}-\frac{3}{7}& = & \frac{5}{2}x-3 \\\Leftrightarrow & \color{blue}{70.} (\frac{14x}{ \color{blue}{70} }- \frac{ 30 }{ \color{blue}{70} })& = & (\frac{175}{ \color{blue}{70} }x-\frac{210}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 14x-30& = & 175x-210 \\\Leftrightarrow & 14x \color{red}{-30} \color{blue}{+30} \color{blue}{-175x} & = & \color{red}{175x} -210 \color{blue}{-175x} \color{blue}{+30} \\\Leftrightarrow & -161x& = & -180 \\\Leftrightarrow & \frac{-161x}{ \color{red}{-161} }& = & \frac{-180}{-161} \\\Leftrightarrow & x = \frac{180}{161} & & \\ & V = \left\{ \frac{180}{161} \right\} & \\\end{align}\)
10. \(\text{14 is het kleinste gemene veelvoud van 7, 7 en 2} \\ \begin{align} & \frac{x}{7}+\frac{4}{7}& = & \frac{3}{2}x+1 \\\Leftrightarrow & \color{blue}{14.} (\frac{2x}{ \color{blue}{14} }+ \frac{ 8 }{ \color{blue}{14} })& = & (\frac{21}{ \color{blue}{14} }x+\frac{14}{ \color{blue}{14} }) \color{blue}{.14} \\\Leftrightarrow & 2x+8& = & 21x+14 \\\Leftrightarrow & 2x \color{red}{+8} \color{blue}{-8} \color{blue}{-21x} & = & \color{red}{21x} +14 \color{blue}{-21x} \color{blue}{-8} \\\Leftrightarrow & -19x& = & 6 \\\Leftrightarrow & \frac{-19x}{ \color{red}{-19} }& = & \frac{6}{-19} \\\Leftrightarrow & x = \frac{-6}{19} & & \\ & V = \left\{ \frac{-6}{19} \right\} & \\\end{align}\)
11. \(\text{30 is het kleinste gemene veelvoud van 2, 15 en 3} \\ \begin{align} & \frac{x}{2}-\frac{2}{15}& = & \frac{7}{3}x-7 \\\Leftrightarrow & \color{blue}{30.} (\frac{15x}{ \color{blue}{30} }- \frac{ 4 }{ \color{blue}{30} })& = & (\frac{70}{ \color{blue}{30} }x-\frac{210}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 15x-4& = & 70x-210 \\\Leftrightarrow & 15x \color{red}{-4} \color{blue}{+4} \color{blue}{-70x} & = & \color{red}{70x} -210 \color{blue}{-70x} \color{blue}{+4} \\\Leftrightarrow & -55x& = & -206 \\\Leftrightarrow & \frac{-55x}{ \color{red}{-55} }& = & \frac{-206}{-55} \\\Leftrightarrow & x = \frac{206}{55} & & \\ & V = \left\{ \frac{206}{55} \right\} & \\\end{align}\)
12. \(\text{240 is het kleinste gemene veelvoud van 5, 16 en 6} \\ \begin{align} & \frac{x}{5}+\frac{5}{16}& = & \frac{7}{6}x-1 \\\Leftrightarrow & \color{blue}{240.} (\frac{48x}{ \color{blue}{240} }+ \frac{ 75 }{ \color{blue}{240} })& = & (\frac{280}{ \color{blue}{240} }x-\frac{240}{ \color{blue}{240} }) \color{blue}{.240} \\\Leftrightarrow & 48x+75& = & 280x-240 \\\Leftrightarrow & 48x \color{red}{+75} \color{blue}{-75} \color{blue}{-280x} & = & \color{red}{280x} -240 \color{blue}{-280x} \color{blue}{-75} \\\Leftrightarrow & -232x& = & -315 \\\Leftrightarrow & \frac{-232x}{ \color{red}{-232} }& = & \frac{-315}{-232} \\\Leftrightarrow & x = \frac{315}{232} & & \\ & V = \left\{ \frac{315}{232} \right\} & \\\end{align}\)