Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{165 is het kleinste gemene veelvoud van 3, 11 en 5} \\ \begin{align} & \frac{x}{3}+\frac{2}{11}& = & \frac{1}{5}x+5 \\\Leftrightarrow & \color{blue}{165.} (\frac{55x}{ \color{blue}{165} }+ \frac{ 30 }{ \color{blue}{165} })& = & (\frac{33}{ \color{blue}{165} }x+\frac{825}{ \color{blue}{165} }) \color{blue}{.165} \\\Leftrightarrow & 55x+30& = & 33x+825 \\\Leftrightarrow & 55x \color{red}{+30} \color{blue}{-30} \color{blue}{-33x} & = & \color{red}{33x} +825 \color{blue}{-33x} \color{blue}{-30} \\\Leftrightarrow & 22x& = & 795 \\\Leftrightarrow & \frac{22x}{ \color{red}{22} }& = & \frac{795}{22} \\\Leftrightarrow & x = \frac{795}{22} & & \\ & V = \left\{ \frac{795}{22} \right\} & \\\end{align}\)
2. \(\text{48 is het kleinste gemene veelvoud van 2, 16 en 3} \\ \begin{align} & \frac{x}{2}+\frac{3}{16}& = & \frac{4}{3}x-5 \\\Leftrightarrow & \color{blue}{48.} (\frac{24x}{ \color{blue}{48} }+ \frac{ 9 }{ \color{blue}{48} })& = & (\frac{64}{ \color{blue}{48} }x-\frac{240}{ \color{blue}{48} }) \color{blue}{.48} \\\Leftrightarrow & 24x+9& = & 64x-240 \\\Leftrightarrow & 24x \color{red}{+9} \color{blue}{-9} \color{blue}{-64x} & = & \color{red}{64x} -240 \color{blue}{-64x} \color{blue}{-9} \\\Leftrightarrow & -40x& = & -249 \\\Leftrightarrow & \frac{-40x}{ \color{red}{-40} }& = & \frac{-249}{-40} \\\Leftrightarrow & x = \frac{249}{40} & & \\ & V = \left\{ \frac{249}{40} \right\} & \\\end{align}\)
3. \(\text{455 is het kleinste gemene veelvoud van 7, 13 en 5} \\ \begin{align} & \frac{x}{7}-\frac{2}{13}& = & \frac{-7}{5}x-2 \\\Leftrightarrow & \color{blue}{455.} (\frac{65x}{ \color{blue}{455} }- \frac{ 70 }{ \color{blue}{455} })& = & (\frac{-637}{ \color{blue}{455} }x-\frac{910}{ \color{blue}{455} }) \color{blue}{.455} \\\Leftrightarrow & 65x-70& = & -637x-910 \\\Leftrightarrow & 65x \color{red}{-70} \color{blue}{+70} \color{blue}{+637x} & = & \color{red}{-637x} -910 \color{blue}{+637x} \color{blue}{+70} \\\Leftrightarrow & 702x& = & -840 \\\Leftrightarrow & \frac{702x}{ \color{red}{702} }& = & \frac{-840}{702} \\\Leftrightarrow & x = \frac{-140}{117} & & \\ & V = \left\{ \frac{-140}{117} \right\} & \\\end{align}\)
4. \(\text{45 is het kleinste gemene veelvoud van 3, 9 en 5} \\ \begin{align} & \frac{x}{3}-\frac{4}{9}& = & \frac{1}{5}x-7 \\\Leftrightarrow & \color{blue}{45.} (\frac{15x}{ \color{blue}{45} }- \frac{ 20 }{ \color{blue}{45} })& = & (\frac{9}{ \color{blue}{45} }x-\frac{315}{ \color{blue}{45} }) \color{blue}{.45} \\\Leftrightarrow & 15x-20& = & 9x-315 \\\Leftrightarrow & 15x \color{red}{-20} \color{blue}{+20} \color{blue}{-9x} & = & \color{red}{9x} -315 \color{blue}{-9x} \color{blue}{+20} \\\Leftrightarrow & 6x& = & -295 \\\Leftrightarrow & \frac{6x}{ \color{red}{6} }& = & \frac{-295}{6} \\\Leftrightarrow & x = \frac{-295}{6} & & \\ & V = \left\{ \frac{-295}{6} \right\} & \\\end{align}\)
5. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}+\frac{2}{7}& = & \frac{6}{5}x+3 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }+ \frac{ 60 }{ \color{blue}{210} })& = & (\frac{252}{ \color{blue}{210} }x+\frac{630}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x+60& = & 252x+630 \\\Leftrightarrow & 35x \color{red}{+60} \color{blue}{-60} \color{blue}{-252x} & = & \color{red}{252x} +630 \color{blue}{-252x} \color{blue}{-60} \\\Leftrightarrow & -217x& = & 570 \\\Leftrightarrow & \frac{-217x}{ \color{red}{-217} }& = & \frac{570}{-217} \\\Leftrightarrow & x = \frac{-570}{217} & & \\ & V = \left\{ \frac{-570}{217} \right\} & \\\end{align}\)
6. \(\text{156 is het kleinste gemene veelvoud van 4, 13 en 3} \\ \begin{align} & \frac{x}{4}-\frac{4}{13}& = & \frac{-8}{3}x+3 \\\Leftrightarrow & \color{blue}{156.} (\frac{39x}{ \color{blue}{156} }- \frac{ 48 }{ \color{blue}{156} })& = & (\frac{-416}{ \color{blue}{156} }x+\frac{468}{ \color{blue}{156} }) \color{blue}{.156} \\\Leftrightarrow & 39x-48& = & -416x+468 \\\Leftrightarrow & 39x \color{red}{-48} \color{blue}{+48} \color{blue}{+416x} & = & \color{red}{-416x} +468 \color{blue}{+416x} \color{blue}{+48} \\\Leftrightarrow & 455x& = & 516 \\\Leftrightarrow & \frac{455x}{ \color{red}{455} }& = & \frac{516}{455} \\\Leftrightarrow & x = \frac{516}{455} & & \\ & V = \left\{ \frac{516}{455} \right\} & \\\end{align}\)
7. \(\text{60 is het kleinste gemene veelvoud van 4, 15 en 3} \\ \begin{align} & \frac{x}{4}+\frac{4}{15}& = & \frac{4}{3}x-4 \\\Leftrightarrow & \color{blue}{60.} (\frac{15x}{ \color{blue}{60} }+ \frac{ 16 }{ \color{blue}{60} })& = & (\frac{80}{ \color{blue}{60} }x-\frac{240}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 15x+16& = & 80x-240 \\\Leftrightarrow & 15x \color{red}{+16} \color{blue}{-16} \color{blue}{-80x} & = & \color{red}{80x} -240 \color{blue}{-80x} \color{blue}{-16} \\\Leftrightarrow & -65x& = & -256 \\\Leftrightarrow & \frac{-65x}{ \color{red}{-65} }& = & \frac{-256}{-65} \\\Leftrightarrow & x = \frac{256}{65} & & \\ & V = \left\{ \frac{256}{65} \right\} & \\\end{align}\)
8. \(\text{21 is het kleinste gemene veelvoud van 7, 7 en 3} \\ \begin{align} & \frac{x}{7}-\frac{5}{7}& = & \frac{4}{3}x+8 \\\Leftrightarrow & \color{blue}{21.} (\frac{3x}{ \color{blue}{21} }- \frac{ 15 }{ \color{blue}{21} })& = & (\frac{28}{ \color{blue}{21} }x+\frac{168}{ \color{blue}{21} }) \color{blue}{.21} \\\Leftrightarrow & 3x-15& = & 28x+168 \\\Leftrightarrow & 3x \color{red}{-15} \color{blue}{+15} \color{blue}{-28x} & = & \color{red}{28x} +168 \color{blue}{-28x} \color{blue}{+15} \\\Leftrightarrow & -25x& = & 183 \\\Leftrightarrow & \frac{-25x}{ \color{red}{-25} }& = & \frac{183}{-25} \\\Leftrightarrow & x = \frac{-183}{25} & & \\ & V = \left\{ \frac{-183}{25} \right\} & \\\end{align}\)
9. \(\text{36 is het kleinste gemene veelvoud van 4, 9 en 3} \\ \begin{align} & \frac{x}{4}-\frac{4}{9}& = & \frac{-5}{3}x+5 \\\Leftrightarrow & \color{blue}{36.} (\frac{9x}{ \color{blue}{36} }- \frac{ 16 }{ \color{blue}{36} })& = & (\frac{-60}{ \color{blue}{36} }x+\frac{180}{ \color{blue}{36} }) \color{blue}{.36} \\\Leftrightarrow & 9x-16& = & -60x+180 \\\Leftrightarrow & 9x \color{red}{-16} \color{blue}{+16} \color{blue}{+60x} & = & \color{red}{-60x} +180 \color{blue}{+60x} \color{blue}{+16} \\\Leftrightarrow & 69x& = & 196 \\\Leftrightarrow & \frac{69x}{ \color{red}{69} }& = & \frac{196}{69} \\\Leftrightarrow & x = \frac{196}{69} & & \\ & V = \left\{ \frac{196}{69} \right\} & \\\end{align}\)
10. \(\text{60 is het kleinste gemene veelvoud van 3, 10 en 4} \\ \begin{align} & \frac{x}{3}-\frac{3}{10}& = & \frac{-3}{4}x+2 \\\Leftrightarrow & \color{blue}{60.} (\frac{20x}{ \color{blue}{60} }- \frac{ 18 }{ \color{blue}{60} })& = & (\frac{-45}{ \color{blue}{60} }x+\frac{120}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 20x-18& = & -45x+120 \\\Leftrightarrow & 20x \color{red}{-18} \color{blue}{+18} \color{blue}{+45x} & = & \color{red}{-45x} +120 \color{blue}{+45x} \color{blue}{+18} \\\Leftrightarrow & 65x& = & 138 \\\Leftrightarrow & \frac{65x}{ \color{red}{65} }& = & \frac{138}{65} \\\Leftrightarrow & x = \frac{138}{65} & & \\ & V = \left\{ \frac{138}{65} \right\} & \\\end{align}\)
11. \(\text{30 is het kleinste gemene veelvoud van 2, 15 en 3} \\ \begin{align} & \frac{x}{2}-\frac{4}{15}& = & \frac{-2}{3}x+7 \\\Leftrightarrow & \color{blue}{30.} (\frac{15x}{ \color{blue}{30} }- \frac{ 8 }{ \color{blue}{30} })& = & (\frac{-20}{ \color{blue}{30} }x+\frac{210}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 15x-8& = & -20x+210 \\\Leftrightarrow & 15x \color{red}{-8} \color{blue}{+8} \color{blue}{+20x} & = & \color{red}{-20x} +210 \color{blue}{+20x} \color{blue}{+8} \\\Leftrightarrow & 35x& = & 218 \\\Leftrightarrow & \frac{35x}{ \color{red}{35} }& = & \frac{218}{35} \\\Leftrightarrow & x = \frac{218}{35} & & \\ & V = \left\{ \frac{218}{35} \right\} & \\\end{align}\)
12. \(\text{42 is het kleinste gemene veelvoud van 3, 7 en 2} \\ \begin{align} & \frac{x}{3}-\frac{2}{7}& = & \frac{1}{2}x-5 \\\Leftrightarrow & \color{blue}{42.} (\frac{14x}{ \color{blue}{42} }- \frac{ 12 }{ \color{blue}{42} })& = & (\frac{21}{ \color{blue}{42} }x-\frac{210}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 14x-12& = & 21x-210 \\\Leftrightarrow & 14x \color{red}{-12} \color{blue}{+12} \color{blue}{-21x} & = & \color{red}{21x} -210 \color{blue}{-21x} \color{blue}{+12} \\\Leftrightarrow & -7x& = & -198 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{-198}{-7} \\\Leftrightarrow & x = \frac{198}{7} & & \\ & V = \left\{ \frac{198}{7} \right\} & \\\end{align}\)