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

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

Verbetersleutel

1. \(\text{130 is het kleinste gemene veelvoud van 5, 13 en 2} \\ \begin{align} & \frac{x}{5}+\frac{5}{13}& = & \frac{1}{2}x+5 \\\Leftrightarrow & \color{blue}{130.} (\frac{26x}{ \color{blue}{130} }+ \frac{ 50 }{ \color{blue}{130} })& = & (\frac{65}{ \color{blue}{130} }x+\frac{650}{ \color{blue}{130} }) \color{blue}{.130} \\\Leftrightarrow & 26x+50& = & 65x+650 \\\Leftrightarrow & 26x \color{red}{+50} \color{blue}{-50} \color{blue}{-65x} & = & \color{red}{65x} +650 \color{blue}{-65x} \color{blue}{-50} \\\Leftrightarrow & -39x& = & 600 \\\Leftrightarrow & \frac{-39x}{ \color{red}{-39} }& = & \frac{600}{-39} \\\Leftrightarrow & x = \frac{-200}{13} & & \\ & V = \left\{ \frac{-200}{13} \right\} & \\\end{align}\)
2. \(\text{42 is het kleinste gemene veelvoud van 3, 7 en 2} \\ \begin{align} & \frac{x}{3}-\frac{4}{7}& = & \frac{3}{2}x+2 \\\Leftrightarrow & \color{blue}{42.} (\frac{14x}{ \color{blue}{42} }- \frac{ 24 }{ \color{blue}{42} })& = & (\frac{63}{ \color{blue}{42} }x+\frac{84}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 14x-24& = & 63x+84 \\\Leftrightarrow & 14x \color{red}{-24} \color{blue}{+24} \color{blue}{-63x} & = & \color{red}{63x} +84 \color{blue}{-63x} \color{blue}{+24} \\\Leftrightarrow & -49x& = & 108 \\\Leftrightarrow & \frac{-49x}{ \color{red}{-49} }& = & \frac{108}{-49} \\\Leftrightarrow & x = \frac{-108}{49} & & \\ & V = \left\{ \frac{-108}{49} \right\} & \\\end{align}\)
3. \(\text{240 is het kleinste gemene veelvoud van 5, 16 en 6} \\ \begin{align} & \frac{x}{5}+\frac{5}{16}& = & \frac{5}{6}x-6 \\\Leftrightarrow & \color{blue}{240.} (\frac{48x}{ \color{blue}{240} }+ \frac{ 75 }{ \color{blue}{240} })& = & (\frac{200}{ \color{blue}{240} }x-\frac{1440}{ \color{blue}{240} }) \color{blue}{.240} \\\Leftrightarrow & 48x+75& = & 200x-1440 \\\Leftrightarrow & 48x \color{red}{+75} \color{blue}{-75} \color{blue}{-200x} & = & \color{red}{200x} -1440 \color{blue}{-200x} \color{blue}{-75} \\\Leftrightarrow & -152x& = & -1515 \\\Leftrightarrow & \frac{-152x}{ \color{red}{-152} }& = & \frac{-1515}{-152} \\\Leftrightarrow & x = \frac{1515}{152} & & \\ & V = \left\{ \frac{1515}{152} \right\} & \\\end{align}\)
4. \(\text{140 is het kleinste gemene veelvoud van 7, 10 en 4} \\ \begin{align} & \frac{x}{7}-\frac{3}{10}& = & \frac{1}{4}x+4 \\\Leftrightarrow & \color{blue}{140.} (\frac{20x}{ \color{blue}{140} }- \frac{ 42 }{ \color{blue}{140} })& = & (\frac{35}{ \color{blue}{140} }x+\frac{560}{ \color{blue}{140} }) \color{blue}{.140} \\\Leftrightarrow & 20x-42& = & 35x+560 \\\Leftrightarrow & 20x \color{red}{-42} \color{blue}{+42} \color{blue}{-35x} & = & \color{red}{35x} +560 \color{blue}{-35x} \color{blue}{+42} \\\Leftrightarrow & -15x& = & 602 \\\Leftrightarrow & \frac{-15x}{ \color{red}{-15} }& = & \frac{602}{-15} \\\Leftrightarrow & x = \frac{-602}{15} & & \\ & V = \left\{ \frac{-602}{15} \right\} & \\\end{align}\)
5. \(\text{18 is het kleinste gemene veelvoud van 2, 9 en 3} \\ \begin{align} & \frac{x}{2}+\frac{4}{9}& = & \frac{-8}{3}x-2 \\\Leftrightarrow & \color{blue}{18.} (\frac{9x}{ \color{blue}{18} }+ \frac{ 8 }{ \color{blue}{18} })& = & (\frac{-48}{ \color{blue}{18} }x-\frac{36}{ \color{blue}{18} }) \color{blue}{.18} \\\Leftrightarrow & 9x+8& = & -48x-36 \\\Leftrightarrow & 9x \color{red}{+8} \color{blue}{-8} \color{blue}{+48x} & = & \color{red}{-48x} -36 \color{blue}{+48x} \color{blue}{-8} \\\Leftrightarrow & 57x& = & -44 \\\Leftrightarrow & \frac{57x}{ \color{red}{57} }& = & \frac{-44}{57} \\\Leftrightarrow & x = \frac{-44}{57} & & \\ & V = \left\{ \frac{-44}{57} \right\} & \\\end{align}\)
6. \(\text{42 is het kleinste gemene veelvoud van 3, 7 en 2} \\ \begin{align} & \frac{x}{3}+\frac{2}{7}& = & \frac{1}{2}x-1 \\\Leftrightarrow & \color{blue}{42.} (\frac{14x}{ \color{blue}{42} }+ \frac{ 12 }{ \color{blue}{42} })& = & (\frac{21}{ \color{blue}{42} }x-\frac{42}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 14x+12& = & 21x-42 \\\Leftrightarrow & 14x \color{red}{+12} \color{blue}{-12} \color{blue}{-21x} & = & \color{red}{21x} -42 \color{blue}{-21x} \color{blue}{-12} \\\Leftrightarrow & -7x& = & -54 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{-54}{-7} \\\Leftrightarrow & x = \frac{54}{7} & & \\ & V = \left\{ \frac{54}{7} \right\} & \\\end{align}\)
7. \(\text{30 is het kleinste gemene veelvoud van 6, 6 en 5} \\ \begin{align} & \frac{x}{6}-\frac{5}{6}& = & \frac{-2}{5}x-5 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }- \frac{ 25 }{ \color{blue}{30} })& = & (\frac{-12}{ \color{blue}{30} }x-\frac{150}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x-25& = & -12x-150 \\\Leftrightarrow & 5x \color{red}{-25} \color{blue}{+25} \color{blue}{+12x} & = & \color{red}{-12x} -150 \color{blue}{+12x} \color{blue}{+25} \\\Leftrightarrow & 17x& = & -125 \\\Leftrightarrow & \frac{17x}{ \color{red}{17} }& = & \frac{-125}{17} \\\Leftrightarrow & x = \frac{-125}{17} & & \\ & V = \left\{ \frac{-125}{17} \right\} & \\\end{align}\)
8. \(\text{420 is het kleinste gemene veelvoud van 7, 15 en 4} \\ \begin{align} & \frac{x}{7}-\frac{2}{15}& = & \frac{7}{4}x-3 \\\Leftrightarrow & \color{blue}{420.} (\frac{60x}{ \color{blue}{420} }- \frac{ 56 }{ \color{blue}{420} })& = & (\frac{735}{ \color{blue}{420} }x-\frac{1260}{ \color{blue}{420} }) \color{blue}{.420} \\\Leftrightarrow & 60x-56& = & 735x-1260 \\\Leftrightarrow & 60x \color{red}{-56} \color{blue}{+56} \color{blue}{-735x} & = & \color{red}{735x} -1260 \color{blue}{-735x} \color{blue}{+56} \\\Leftrightarrow & -675x& = & -1204 \\\Leftrightarrow & \frac{-675x}{ \color{red}{-675} }& = & \frac{-1204}{-675} \\\Leftrightarrow & x = \frac{1204}{675} & & \\ & V = \left\{ \frac{1204}{675} \right\} & \\\end{align}\)
9. \(\text{14 is het kleinste gemene veelvoud van 7, 7 en 2} \\ \begin{align} & \frac{x}{7}-\frac{3}{7}& = & \frac{1}{2}x-5 \\\Leftrightarrow & \color{blue}{14.} (\frac{2x}{ \color{blue}{14} }- \frac{ 6 }{ \color{blue}{14} })& = & (\frac{7}{ \color{blue}{14} }x-\frac{70}{ \color{blue}{14} }) \color{blue}{.14} \\\Leftrightarrow & 2x-6& = & 7x-70 \\\Leftrightarrow & 2x \color{red}{-6} \color{blue}{+6} \color{blue}{-7x} & = & \color{red}{7x} -70 \color{blue}{-7x} \color{blue}{+6} \\\Leftrightarrow & -5x& = & -64 \\\Leftrightarrow & \frac{-5x}{ \color{red}{-5} }& = & \frac{-64}{-5} \\\Leftrightarrow & x = \frac{64}{5} & & \\ & V = \left\{ \frac{64}{5} \right\} & \\\end{align}\)
10. \(\text{210 is het kleinste gemene veelvoud van 7, 15 en 2} \\ \begin{align} & \frac{x}{7}-\frac{4}{15}& = & \frac{1}{2}x+2 \\\Leftrightarrow & \color{blue}{210.} (\frac{30x}{ \color{blue}{210} }- \frac{ 56 }{ \color{blue}{210} })& = & (\frac{105}{ \color{blue}{210} }x+\frac{420}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 30x-56& = & 105x+420 \\\Leftrightarrow & 30x \color{red}{-56} \color{blue}{+56} \color{blue}{-105x} & = & \color{red}{105x} +420 \color{blue}{-105x} \color{blue}{+56} \\\Leftrightarrow & -75x& = & 476 \\\Leftrightarrow & \frac{-75x}{ \color{red}{-75} }& = & \frac{476}{-75} \\\Leftrightarrow & x = \frac{-476}{75} & & \\ & V = \left\{ \frac{-476}{75} \right\} & \\\end{align}\)
11. \(\text{48 is het kleinste gemene veelvoud van 3, 16 en 2} \\ \begin{align} & \frac{x}{3}+\frac{5}{16}& = & \frac{1}{2}x-5 \\\Leftrightarrow & \color{blue}{48.} (\frac{16x}{ \color{blue}{48} }+ \frac{ 15 }{ \color{blue}{48} })& = & (\frac{24}{ \color{blue}{48} }x-\frac{240}{ \color{blue}{48} }) \color{blue}{.48} \\\Leftrightarrow & 16x+15& = & 24x-240 \\\Leftrightarrow & 16x \color{red}{+15} \color{blue}{-15} \color{blue}{-24x} & = & \color{red}{24x} -240 \color{blue}{-24x} \color{blue}{-15} \\\Leftrightarrow & -8x& = & -255 \\\Leftrightarrow & \frac{-8x}{ \color{red}{-8} }& = & \frac{-255}{-8} \\\Leftrightarrow & x = \frac{255}{8} & & \\ & V = \left\{ \frac{255}{8} \right\} & \\\end{align}\)
12. \(\text{70 is het kleinste gemene veelvoud van 2, 7 en 5} \\ \begin{align} & \frac{x}{2}+\frac{3}{7}& = & \frac{6}{5}x-5 \\\Leftrightarrow & \color{blue}{70.} (\frac{35x}{ \color{blue}{70} }+ \frac{ 30 }{ \color{blue}{70} })& = & (\frac{84}{ \color{blue}{70} }x-\frac{350}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 35x+30& = & 84x-350 \\\Leftrightarrow & 35x \color{red}{+30} \color{blue}{-30} \color{blue}{-84x} & = & \color{red}{84x} -350 \color{blue}{-84x} \color{blue}{-30} \\\Leftrightarrow & -49x& = & -380 \\\Leftrightarrow & \frac{-49x}{ \color{red}{-49} }& = & \frac{-380}{-49} \\\Leftrightarrow & x = \frac{380}{49} & & \\ & V = \left\{ \frac{380}{49} \right\} & \\\end{align}\)