Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{18 is het kleinste gemene veelvoud van 2, 9 en 3} \\ \begin{align} & \frac{x}{2}+\frac{5}{9}& = & \frac{-5}{3}x-8 \\\Leftrightarrow & \color{blue}{18.} (\frac{9x}{ \color{blue}{18} }+ \frac{ 10 }{ \color{blue}{18} })& = & (\frac{-30}{ \color{blue}{18} }x-\frac{144}{ \color{blue}{18} }) \color{blue}{.18} \\\Leftrightarrow & 9x+10& = & -30x-144 \\\Leftrightarrow & 9x \color{red}{+10} \color{blue}{-10} \color{blue}{+30x} & = & \color{red}{-30x} -144 \color{blue}{+30x} \color{blue}{-10} \\\Leftrightarrow & 39x& = & -154 \\\Leftrightarrow & \frac{39x}{ \color{red}{39} }& = & \frac{-154}{39} \\\Leftrightarrow & x = \frac{-154}{39} & & \\ & V = \left\{ \frac{-154}{39} \right\} & \\\end{align}\)
2. \(\text{84 is het kleinste gemene veelvoud van 3, 7 en 4} \\ \begin{align} & \frac{x}{3}-\frac{3}{7}& = & \frac{7}{4}x-6 \\\Leftrightarrow & \color{blue}{84.} (\frac{28x}{ \color{blue}{84} }- \frac{ 36 }{ \color{blue}{84} })& = & (\frac{147}{ \color{blue}{84} }x-\frac{504}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 28x-36& = & 147x-504 \\\Leftrightarrow & 28x \color{red}{-36} \color{blue}{+36} \color{blue}{-147x} & = & \color{red}{147x} -504 \color{blue}{-147x} \color{blue}{+36} \\\Leftrightarrow & -119x& = & -468 \\\Leftrightarrow & \frac{-119x}{ \color{red}{-119} }& = & \frac{-468}{-119} \\\Leftrightarrow & x = \frac{468}{119} & & \\ & V = \left\{ \frac{468}{119} \right\} & \\\end{align}\)
3. \(\text{90 is het kleinste gemene veelvoud van 6, 9 en 5} \\ \begin{align} & \frac{x}{6}-\frac{2}{9}& = & \frac{2}{5}x+6 \\\Leftrightarrow & \color{blue}{90.} (\frac{15x}{ \color{blue}{90} }- \frac{ 20 }{ \color{blue}{90} })& = & (\frac{36}{ \color{blue}{90} }x+\frac{540}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 15x-20& = & 36x+540 \\\Leftrightarrow & 15x \color{red}{-20} \color{blue}{+20} \color{blue}{-36x} & = & \color{red}{36x} +540 \color{blue}{-36x} \color{blue}{+20} \\\Leftrightarrow & -21x& = & 560 \\\Leftrightarrow & \frac{-21x}{ \color{red}{-21} }& = & \frac{560}{-21} \\\Leftrightarrow & x = \frac{-80}{3} & & \\ & V = \left\{ \frac{-80}{3} \right\} & \\\end{align}\)
4. \(\text{210 is het kleinste gemene veelvoud van 5, 7 en 6} \\ \begin{align} & \frac{x}{5}-\frac{2}{7}& = & \frac{1}{6}x+8 \\\Leftrightarrow & \color{blue}{210.} (\frac{42x}{ \color{blue}{210} }- \frac{ 60 }{ \color{blue}{210} })& = & (\frac{35}{ \color{blue}{210} }x+\frac{1680}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 42x-60& = & 35x+1680 \\\Leftrightarrow & 42x \color{red}{-60} \color{blue}{+60} \color{blue}{-35x} & = & \color{red}{35x} +1680 \color{blue}{-35x} \color{blue}{+60} \\\Leftrightarrow & 7x& = & 1740 \\\Leftrightarrow & \frac{7x}{ \color{red}{7} }& = & \frac{1740}{7} \\\Leftrightarrow & x = \frac{1740}{7} & & \\ & V = \left\{ \frac{1740}{7} \right\} & \\\end{align}\)
5. \(\text{390 is het kleinste gemene veelvoud van 6, 13 en 5} \\ \begin{align} & \frac{x}{6}+\frac{2}{13}& = & \frac{-7}{5}x+6 \\\Leftrightarrow & \color{blue}{390.} (\frac{65x}{ \color{blue}{390} }+ \frac{ 60 }{ \color{blue}{390} })& = & (\frac{-546}{ \color{blue}{390} }x+\frac{2340}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 65x+60& = & -546x+2340 \\\Leftrightarrow & 65x \color{red}{+60} \color{blue}{-60} \color{blue}{+546x} & = & \color{red}{-546x} +2340 \color{blue}{+546x} \color{blue}{-60} \\\Leftrightarrow & 611x& = & 2280 \\\Leftrightarrow & \frac{611x}{ \color{red}{611} }& = & \frac{2280}{611} \\\Leftrightarrow & x = \frac{2280}{611} & & \\ & V = \left\{ \frac{2280}{611} \right\} & \\\end{align}\)
6. \(\text{364 is het kleinste gemene veelvoud van 7, 13 en 4} \\ \begin{align} & \frac{x}{7}+\frac{3}{13}& = & \frac{1}{4}x-7 \\\Leftrightarrow & \color{blue}{364.} (\frac{52x}{ \color{blue}{364} }+ \frac{ 84 }{ \color{blue}{364} })& = & (\frac{91}{ \color{blue}{364} }x-\frac{2548}{ \color{blue}{364} }) \color{blue}{.364} \\\Leftrightarrow & 52x+84& = & 91x-2548 \\\Leftrightarrow & 52x \color{red}{+84} \color{blue}{-84} \color{blue}{-91x} & = & \color{red}{91x} -2548 \color{blue}{-91x} \color{blue}{-84} \\\Leftrightarrow & -39x& = & -2632 \\\Leftrightarrow & \frac{-39x}{ \color{red}{-39} }& = & \frac{-2632}{-39} \\\Leftrightarrow & x = \frac{2632}{39} & & \\ & V = \left\{ \frac{2632}{39} \right\} & \\\end{align}\)
7. \(\text{336 is het kleinste gemene veelvoud van 7, 16 en 6} \\ \begin{align} & \frac{x}{7}+\frac{3}{16}& = & \frac{1}{6}x-7 \\\Leftrightarrow & \color{blue}{336.} (\frac{48x}{ \color{blue}{336} }+ \frac{ 63 }{ \color{blue}{336} })& = & (\frac{56}{ \color{blue}{336} }x-\frac{2352}{ \color{blue}{336} }) \color{blue}{.336} \\\Leftrightarrow & 48x+63& = & 56x-2352 \\\Leftrightarrow & 48x \color{red}{+63} \color{blue}{-63} \color{blue}{-56x} & = & \color{red}{56x} -2352 \color{blue}{-56x} \color{blue}{-63} \\\Leftrightarrow & -8x& = & -2415 \\\Leftrightarrow & \frac{-8x}{ \color{red}{-8} }& = & \frac{-2415}{-8} \\\Leftrightarrow & x = \frac{2415}{8} & & \\ & V = \left\{ \frac{2415}{8} \right\} & \\\end{align}\)
8. \(\text{42 is het kleinste gemene veelvoud van 7, 6 en 2} \\ \begin{align} & \frac{x}{7}-\frac{5}{6}& = & \frac{5}{2}x+6 \\\Leftrightarrow & \color{blue}{42.} (\frac{6x}{ \color{blue}{42} }- \frac{ 35 }{ \color{blue}{42} })& = & (\frac{105}{ \color{blue}{42} }x+\frac{252}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 6x-35& = & 105x+252 \\\Leftrightarrow & 6x \color{red}{-35} \color{blue}{+35} \color{blue}{-105x} & = & \color{red}{105x} +252 \color{blue}{-105x} \color{blue}{+35} \\\Leftrightarrow & -99x& = & 287 \\\Leftrightarrow & \frac{-99x}{ \color{red}{-99} }& = & \frac{287}{-99} \\\Leftrightarrow & x = \frac{-287}{99} & & \\ & V = \left\{ \frac{-287}{99} \right\} & \\\end{align}\)
9. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}+\frac{5}{7}& = & \frac{-8}{3}x-3 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }+ \frac{ 60 }{ \color{blue}{84} })& = & (\frac{-224}{ \color{blue}{84} }x-\frac{252}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x+60& = & -224x-252 \\\Leftrightarrow & 21x \color{red}{+60} \color{blue}{-60} \color{blue}{+224x} & = & \color{red}{-224x} -252 \color{blue}{+224x} \color{blue}{-60} \\\Leftrightarrow & 245x& = & -312 \\\Leftrightarrow & \frac{245x}{ \color{red}{245} }& = & \frac{-312}{245} \\\Leftrightarrow & x = \frac{-312}{245} & & \\ & V = \left\{ \frac{-312}{245} \right\} & \\\end{align}\)
10. \(\text{12 is het kleinste gemene veelvoud van 3, 12 en 2} \\ \begin{align} & \frac{x}{3}-\frac{5}{12}& = & \frac{5}{2}x-6 \\\Leftrightarrow & \color{blue}{12.} (\frac{4x}{ \color{blue}{12} }- \frac{ 5 }{ \color{blue}{12} })& = & (\frac{30}{ \color{blue}{12} }x-\frac{72}{ \color{blue}{12} }) \color{blue}{.12} \\\Leftrightarrow & 4x-5& = & 30x-72 \\\Leftrightarrow & 4x \color{red}{-5} \color{blue}{+5} \color{blue}{-30x} & = & \color{red}{30x} -72 \color{blue}{-30x} \color{blue}{+5} \\\Leftrightarrow & -26x& = & -67 \\\Leftrightarrow & \frac{-26x}{ \color{red}{-26} }& = & \frac{-67}{-26} \\\Leftrightarrow & x = \frac{67}{26} & & \\ & V = \left\{ \frac{67}{26} \right\} & \\\end{align}\)
11. \(\text{48 is het kleinste gemene veelvoud van 3, 16 en 4} \\ \begin{align} & \frac{x}{3}-\frac{3}{16}& = & \frac{-3}{4}x-2 \\\Leftrightarrow & \color{blue}{48.} (\frac{16x}{ \color{blue}{48} }- \frac{ 9 }{ \color{blue}{48} })& = & (\frac{-36}{ \color{blue}{48} }x-\frac{96}{ \color{blue}{48} }) \color{blue}{.48} \\\Leftrightarrow & 16x-9& = & -36x-96 \\\Leftrightarrow & 16x \color{red}{-9} \color{blue}{+9} \color{blue}{+36x} & = & \color{red}{-36x} -96 \color{blue}{+36x} \color{blue}{+9} \\\Leftrightarrow & 52x& = & -87 \\\Leftrightarrow & \frac{52x}{ \color{red}{52} }& = & \frac{-87}{52} \\\Leftrightarrow & x = \frac{-87}{52} & & \\ & V = \left\{ \frac{-87}{52} \right\} & \\\end{align}\)
12. \(\text{105 is het kleinste gemene veelvoud van 5, 7 en 3} \\ \begin{align} & \frac{x}{5}-\frac{5}{7}& = & \frac{7}{3}x+4 \\\Leftrightarrow & \color{blue}{105.} (\frac{21x}{ \color{blue}{105} }- \frac{ 75 }{ \color{blue}{105} })& = & (\frac{245}{ \color{blue}{105} }x+\frac{420}{ \color{blue}{105} }) \color{blue}{.105} \\\Leftrightarrow & 21x-75& = & 245x+420 \\\Leftrightarrow & 21x \color{red}{-75} \color{blue}{+75} \color{blue}{-245x} & = & \color{red}{245x} +420 \color{blue}{-245x} \color{blue}{+75} \\\Leftrightarrow & -224x& = & 495 \\\Leftrightarrow & \frac{-224x}{ \color{red}{-224} }& = & \frac{495}{-224} \\\Leftrightarrow & x = \frac{-495}{224} & & \\ & V = \left\{ \frac{-495}{224} \right\} & \\\end{align}\)