Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{30 is het kleinste gemene veelvoud van 3, 10 en 5} \\ \begin{align} & \frac{x}{3}+\frac{3}{10}& = & \frac{3}{5}x-7 \\\Leftrightarrow & \color{blue}{30.} (\frac{10x}{ \color{blue}{30} }+ \frac{ 9 }{ \color{blue}{30} })& = & (\frac{18}{ \color{blue}{30} }x-\frac{210}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 10x+9& = & 18x-210 \\\Leftrightarrow & 10x \color{red}{+9} \color{blue}{-9} \color{blue}{-18x} & = & \color{red}{18x} -210 \color{blue}{-18x} \color{blue}{-9} \\\Leftrightarrow & -8x& = & -219 \\\Leftrightarrow & \frac{-8x}{ \color{red}{-8} }& = & \frac{-219}{-8} \\\Leftrightarrow & x = \frac{219}{8} & & \\ & V = \left\{ \frac{219}{8} \right\} & \\\end{align}\)
2. \(\text{156 is het kleinste gemene veelvoud van 4, 13 en 3} \\ \begin{align} & \frac{x}{4}+\frac{3}{13}& = & \frac{4}{3}x-5 \\\Leftrightarrow & \color{blue}{156.} (\frac{39x}{ \color{blue}{156} }+ \frac{ 36 }{ \color{blue}{156} })& = & (\frac{208}{ \color{blue}{156} }x-\frac{780}{ \color{blue}{156} }) \color{blue}{.156} \\\Leftrightarrow & 39x+36& = & 208x-780 \\\Leftrightarrow & 39x \color{red}{+36} \color{blue}{-36} \color{blue}{-208x} & = & \color{red}{208x} -780 \color{blue}{-208x} \color{blue}{-36} \\\Leftrightarrow & -169x& = & -816 \\\Leftrightarrow & \frac{-169x}{ \color{red}{-169} }& = & \frac{-816}{-169} \\\Leftrightarrow & x = \frac{816}{169} & & \\ & V = \left\{ \frac{816}{169} \right\} & \\\end{align}\)
3. \(\text{462 is het kleinste gemene veelvoud van 7, 11 en 6} \\ \begin{align} & \frac{x}{7}-\frac{4}{11}& = & \frac{1}{6}x+6 \\\Leftrightarrow & \color{blue}{462.} (\frac{66x}{ \color{blue}{462} }- \frac{ 168 }{ \color{blue}{462} })& = & (\frac{77}{ \color{blue}{462} }x+\frac{2772}{ \color{blue}{462} }) \color{blue}{.462} \\\Leftrightarrow & 66x-168& = & 77x+2772 \\\Leftrightarrow & 66x \color{red}{-168} \color{blue}{+168} \color{blue}{-77x} & = & \color{red}{77x} +2772 \color{blue}{-77x} \color{blue}{+168} \\\Leftrightarrow & -11x& = & 2940 \\\Leftrightarrow & \frac{-11x}{ \color{red}{-11} }& = & \frac{2940}{-11} \\\Leftrightarrow & x = \frac{-2940}{11} & & \\ & V = \left\{ \frac{-2940}{11} \right\} & \\\end{align}\)
4. \(\text{210 is het kleinste gemene veelvoud van 5, 7 en 6} \\ \begin{align} & \frac{x}{5}+\frac{2}{7}& = & \frac{1}{6}x+2 \\\Leftrightarrow & \color{blue}{210.} (\frac{42x}{ \color{blue}{210} }+ \frac{ 60 }{ \color{blue}{210} })& = & (\frac{35}{ \color{blue}{210} }x+\frac{420}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 42x+60& = & 35x+420 \\\Leftrightarrow & 42x \color{red}{+60} \color{blue}{-60} \color{blue}{-35x} & = & \color{red}{35x} +420 \color{blue}{-35x} \color{blue}{-60} \\\Leftrightarrow & 7x& = & 360 \\\Leftrightarrow & \frac{7x}{ \color{red}{7} }& = & \frac{360}{7} \\\Leftrightarrow & x = \frac{360}{7} & & \\ & V = \left\{ \frac{360}{7} \right\} & \\\end{align}\)
5. \(\text{385 is het kleinste gemene veelvoud van 7, 11 en 5} \\ \begin{align} & \frac{x}{7}-\frac{5}{11}& = & \frac{-7}{5}x+1 \\\Leftrightarrow & \color{blue}{385.} (\frac{55x}{ \color{blue}{385} }- \frac{ 175 }{ \color{blue}{385} })& = & (\frac{-539}{ \color{blue}{385} }x+\frac{385}{ \color{blue}{385} }) \color{blue}{.385} \\\Leftrightarrow & 55x-175& = & -539x+385 \\\Leftrightarrow & 55x \color{red}{-175} \color{blue}{+175} \color{blue}{+539x} & = & \color{red}{-539x} +385 \color{blue}{+539x} \color{blue}{+175} \\\Leftrightarrow & 594x& = & 560 \\\Leftrightarrow & \frac{594x}{ \color{red}{594} }& = & \frac{560}{594} \\\Leftrightarrow & x = \frac{280}{297} & & \\ & V = \left\{ \frac{280}{297} \right\} & \\\end{align}\)
6. \(\text{14 is het kleinste gemene veelvoud van 7, 7 en 2} \\ \begin{align} & \frac{x}{7}-\frac{2}{7}& = & \frac{1}{2}x+7 \\\Leftrightarrow & \color{blue}{14.} (\frac{2x}{ \color{blue}{14} }- \frac{ 4 }{ \color{blue}{14} })& = & (\frac{7}{ \color{blue}{14} }x+\frac{98}{ \color{blue}{14} }) \color{blue}{.14} \\\Leftrightarrow & 2x-4& = & 7x+98 \\\Leftrightarrow & 2x \color{red}{-4} \color{blue}{+4} \color{blue}{-7x} & = & \color{red}{7x} +98 \color{blue}{-7x} \color{blue}{+4} \\\Leftrightarrow & -5x& = & 102 \\\Leftrightarrow & \frac{-5x}{ \color{red}{-5} }& = & \frac{102}{-5} \\\Leftrightarrow & x = \frac{-102}{5} & & \\ & V = \left\{ \frac{-102}{5} \right\} & \\\end{align}\)
7. \(\text{84 is het kleinste gemene veelvoud van 3, 14 en 4} \\ \begin{align} & \frac{x}{3}+\frac{5}{14}& = & \frac{-3}{4}x-5 \\\Leftrightarrow & \color{blue}{84.} (\frac{28x}{ \color{blue}{84} }+ \frac{ 30 }{ \color{blue}{84} })& = & (\frac{-63}{ \color{blue}{84} }x-\frac{420}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 28x+30& = & -63x-420 \\\Leftrightarrow & 28x \color{red}{+30} \color{blue}{-30} \color{blue}{+63x} & = & \color{red}{-63x} -420 \color{blue}{+63x} \color{blue}{-30} \\\Leftrightarrow & 91x& = & -450 \\\Leftrightarrow & \frac{91x}{ \color{red}{91} }& = & \frac{-450}{91} \\\Leftrightarrow & x = \frac{-450}{91} & & \\ & V = \left\{ \frac{-450}{91} \right\} & \\\end{align}\)
8. \(\text{42 is het kleinste gemene veelvoud van 3, 7 en 2} \\ \begin{align} & \frac{x}{3}+\frac{2}{7}& = & \frac{1}{2}x+6 \\\Leftrightarrow & \color{blue}{42.} (\frac{14x}{ \color{blue}{42} }+ \frac{ 12 }{ \color{blue}{42} })& = & (\frac{21}{ \color{blue}{42} }x+\frac{252}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 14x+12& = & 21x+252 \\\Leftrightarrow & 14x \color{red}{+12} \color{blue}{-12} \color{blue}{-21x} & = & \color{red}{21x} +252 \color{blue}{-21x} \color{blue}{-12} \\\Leftrightarrow & -7x& = & 240 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{240}{-7} \\\Leftrightarrow & x = \frac{-240}{7} & & \\ & V = \left\{ \frac{-240}{7} \right\} & \\\end{align}\)
9. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}-\frac{2}{7}& = & \frac{1}{5}x-3 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }- \frac{ 60 }{ \color{blue}{210} })& = & (\frac{42}{ \color{blue}{210} }x-\frac{630}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x-60& = & 42x-630 \\\Leftrightarrow & 35x \color{red}{-60} \color{blue}{+60} \color{blue}{-42x} & = & \color{red}{42x} -630 \color{blue}{-42x} \color{blue}{+60} \\\Leftrightarrow & -7x& = & -570 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{-570}{-7} \\\Leftrightarrow & x = \frac{570}{7} & & \\ & V = \left\{ \frac{570}{7} \right\} & \\\end{align}\)
10. \(\text{180 is het kleinste gemene veelvoud van 4, 9 en 5} \\ \begin{align} & \frac{x}{4}-\frac{5}{9}& = & \frac{1}{5}x+8 \\\Leftrightarrow & \color{blue}{180.} (\frac{45x}{ \color{blue}{180} }- \frac{ 100 }{ \color{blue}{180} })& = & (\frac{36}{ \color{blue}{180} }x+\frac{1440}{ \color{blue}{180} }) \color{blue}{.180} \\\Leftrightarrow & 45x-100& = & 36x+1440 \\\Leftrightarrow & 45x \color{red}{-100} \color{blue}{+100} \color{blue}{-36x} & = & \color{red}{36x} +1440 \color{blue}{-36x} \color{blue}{+100} \\\Leftrightarrow & 9x& = & 1540 \\\Leftrightarrow & \frac{9x}{ \color{red}{9} }& = & \frac{1540}{9} \\\Leftrightarrow & x = \frac{1540}{9} & & \\ & V = \left\{ \frac{1540}{9} \right\} & \\\end{align}\)
11. \(\text{14 is het kleinste gemene veelvoud van 7, 7 en 2} \\ \begin{align} & \frac{x}{7}+\frac{3}{7}& = & \frac{5}{2}x-7 \\\Leftrightarrow & \color{blue}{14.} (\frac{2x}{ \color{blue}{14} }+ \frac{ 6 }{ \color{blue}{14} })& = & (\frac{35}{ \color{blue}{14} }x-\frac{98}{ \color{blue}{14} }) \color{blue}{.14} \\\Leftrightarrow & 2x+6& = & 35x-98 \\\Leftrightarrow & 2x \color{red}{+6} \color{blue}{-6} \color{blue}{-35x} & = & \color{red}{35x} -98 \color{blue}{-35x} \color{blue}{-6} \\\Leftrightarrow & -33x& = & -104 \\\Leftrightarrow & \frac{-33x}{ \color{red}{-33} }& = & \frac{-104}{-33} \\\Leftrightarrow & x = \frac{104}{33} & & \\ & V = \left\{ \frac{104}{33} \right\} & \\\end{align}\)
12. \(\text{18 is het kleinste gemene veelvoud van 3, 9 en 2} \\ \begin{align} & \frac{x}{3}+\frac{4}{9}& = & \frac{1}{2}x-2 \\\Leftrightarrow & \color{blue}{18.} (\frac{6x}{ \color{blue}{18} }+ \frac{ 8 }{ \color{blue}{18} })& = & (\frac{9}{ \color{blue}{18} }x-\frac{36}{ \color{blue}{18} }) \color{blue}{.18} \\\Leftrightarrow & 6x+8& = & 9x-36 \\\Leftrightarrow & 6x \color{red}{+8} \color{blue}{-8} \color{blue}{-9x} & = & \color{red}{9x} -36 \color{blue}{-9x} \color{blue}{-8} \\\Leftrightarrow & -3x& = & -44 \\\Leftrightarrow & \frac{-3x}{ \color{red}{-3} }& = & \frac{-44}{-3} \\\Leftrightarrow & x = \frac{44}{3} & & \\ & V = \left\{ \frac{44}{3} \right\} & \\\end{align}\)