Vgln. eerste graad (reeks 4)

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1. \(\frac{x}{6}+\frac{3}{11}=\frac{-7}{5}x-2\)
2. \(\frac{x}{4}+\frac{3}{16}=\frac{-2}{3}x-8\)
3. \(\frac{x}{4}-\frac{4}{9}=\frac{7}{3}x-2\)
4. \(\frac{x}{2}-\frac{5}{12}=\frac{4}{3}x-3\)
5. \(\frac{x}{2}-\frac{3}{14}=\frac{-2}{3}x-5\)
6. \(\frac{x}{5}+\frac{3}{13}=\frac{1}{6}x+6\)
7. \(\frac{x}{4}-\frac{3}{11}=\frac{-2}{3}x-2\)
8. \(\frac{x}{6}-\frac{4}{9}=\frac{-4}{5}x+5\)
9. \(\frac{x}{6}-\frac{5}{6}=\frac{2}{5}x-2\)
10. \(\frac{x}{4}+\frac{3}{10}=\frac{-5}{3}x+3\)
11. \(\frac{x}{5}+\frac{4}{15}=\frac{1}{3}x-4\)
12. \(\frac{x}{5}-\frac{5}{6}=\frac{1}{2}x+8\)

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

Verbetersleutel

1. \(\text{330 is het kleinste gemene veelvoud van 6, 11 en 5} \\ \begin{align} & \frac{x}{6}+\frac{3}{11}& = & \frac{-7}{5}x-2 \\\Leftrightarrow & \color{blue}{330.} (\frac{55x}{ \color{blue}{330} }+ \frac{ 90 }{ \color{blue}{330} })& = & (\frac{-462}{ \color{blue}{330} }x-\frac{660}{ \color{blue}{330} }) \color{blue}{.330} \\\Leftrightarrow & 55x+90& = & -462x-660 \\\Leftrightarrow & 55x \color{red}{+90} \color{blue}{-90} \color{blue}{+462x} & = & \color{red}{-462x} -660 \color{blue}{+462x} \color{blue}{-90} \\\Leftrightarrow & 517x& = & -750 \\\Leftrightarrow & \frac{517x}{ \color{red}{517} }& = & \frac{-750}{517} \\\Leftrightarrow & x = \frac{-750}{517} & & \\ & V = \left\{ \frac{-750}{517} \right\} & \\\end{align}\)
2. \(\text{48 is het kleinste gemene veelvoud van 4, 16 en 3} \\ \begin{align} & \frac{x}{4}+\frac{3}{16}& = & \frac{-2}{3}x-8 \\\Leftrightarrow & \color{blue}{48.} (\frac{12x}{ \color{blue}{48} }+ \frac{ 9 }{ \color{blue}{48} })& = & (\frac{-32}{ \color{blue}{48} }x-\frac{384}{ \color{blue}{48} }) \color{blue}{.48} \\\Leftrightarrow & 12x+9& = & -32x-384 \\\Leftrightarrow & 12x \color{red}{+9} \color{blue}{-9} \color{blue}{+32x} & = & \color{red}{-32x} -384 \color{blue}{+32x} \color{blue}{-9} \\\Leftrightarrow & 44x& = & -393 \\\Leftrightarrow & \frac{44x}{ \color{red}{44} }& = & \frac{-393}{44} \\\Leftrightarrow & x = \frac{-393}{44} & & \\ & V = \left\{ \frac{-393}{44} \right\} & \\\end{align}\)
3. \(\text{36 is het kleinste gemene veelvoud van 4, 9 en 3} \\ \begin{align} & \frac{x}{4}-\frac{4}{9}& = & \frac{7}{3}x-2 \\\Leftrightarrow & \color{blue}{36.} (\frac{9x}{ \color{blue}{36} }- \frac{ 16 }{ \color{blue}{36} })& = & (\frac{84}{ \color{blue}{36} }x-\frac{72}{ \color{blue}{36} }) \color{blue}{.36} \\\Leftrightarrow & 9x-16& = & 84x-72 \\\Leftrightarrow & 9x \color{red}{-16} \color{blue}{+16} \color{blue}{-84x} & = & \color{red}{84x} -72 \color{blue}{-84x} \color{blue}{+16} \\\Leftrightarrow & -75x& = & -56 \\\Leftrightarrow & \frac{-75x}{ \color{red}{-75} }& = & \frac{-56}{-75} \\\Leftrightarrow & x = \frac{56}{75} & & \\ & V = \left\{ \frac{56}{75} \right\} & \\\end{align}\)
4. \(\text{12 is het kleinste gemene veelvoud van 2, 12 en 3} \\ \begin{align} & \frac{x}{2}-\frac{5}{12}& = & \frac{4}{3}x-3 \\\Leftrightarrow & \color{blue}{12.} (\frac{6x}{ \color{blue}{12} }- \frac{ 5 }{ \color{blue}{12} })& = & (\frac{16}{ \color{blue}{12} }x-\frac{36}{ \color{blue}{12} }) \color{blue}{.12} \\\Leftrightarrow & 6x-5& = & 16x-36 \\\Leftrightarrow & 6x \color{red}{-5} \color{blue}{+5} \color{blue}{-16x} & = & \color{red}{16x} -36 \color{blue}{-16x} \color{blue}{+5} \\\Leftrightarrow & -10x& = & -31 \\\Leftrightarrow & \frac{-10x}{ \color{red}{-10} }& = & \frac{-31}{-10} \\\Leftrightarrow & x = \frac{31}{10} & & \\ & V = \left\{ \frac{31}{10} \right\} & \\\end{align}\)
5. \(\text{42 is het kleinste gemene veelvoud van 2, 14 en 3} \\ \begin{align} & \frac{x}{2}-\frac{3}{14}& = & \frac{-2}{3}x-5 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }- \frac{ 9 }{ \color{blue}{42} })& = & (\frac{-28}{ \color{blue}{42} }x-\frac{210}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x-9& = & -28x-210 \\\Leftrightarrow & 21x \color{red}{-9} \color{blue}{+9} \color{blue}{+28x} & = & \color{red}{-28x} -210 \color{blue}{+28x} \color{blue}{+9} \\\Leftrightarrow & 49x& = & -201 \\\Leftrightarrow & \frac{49x}{ \color{red}{49} }& = & \frac{-201}{49} \\\Leftrightarrow & x = \frac{-201}{49} & & \\ & V = \left\{ \frac{-201}{49} \right\} & \\\end{align}\)
6. \(\text{390 is het kleinste gemene veelvoud van 5, 13 en 6} \\ \begin{align} & \frac{x}{5}+\frac{3}{13}& = & \frac{1}{6}x+6 \\\Leftrightarrow & \color{blue}{390.} (\frac{78x}{ \color{blue}{390} }+ \frac{ 90 }{ \color{blue}{390} })& = & (\frac{65}{ \color{blue}{390} }x+\frac{2340}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 78x+90& = & 65x+2340 \\\Leftrightarrow & 78x \color{red}{+90} \color{blue}{-90} \color{blue}{-65x} & = & \color{red}{65x} +2340 \color{blue}{-65x} \color{blue}{-90} \\\Leftrightarrow & 13x& = & 2250 \\\Leftrightarrow & \frac{13x}{ \color{red}{13} }& = & \frac{2250}{13} \\\Leftrightarrow & x = \frac{2250}{13} & & \\ & V = \left\{ \frac{2250}{13} \right\} & \\\end{align}\)
7. \(\text{132 is het kleinste gemene veelvoud van 4, 11 en 3} \\ \begin{align} & \frac{x}{4}-\frac{3}{11}& = & \frac{-2}{3}x-2 \\\Leftrightarrow & \color{blue}{132.} (\frac{33x}{ \color{blue}{132} }- \frac{ 36 }{ \color{blue}{132} })& = & (\frac{-88}{ \color{blue}{132} }x-\frac{264}{ \color{blue}{132} }) \color{blue}{.132} \\\Leftrightarrow & 33x-36& = & -88x-264 \\\Leftrightarrow & 33x \color{red}{-36} \color{blue}{+36} \color{blue}{+88x} & = & \color{red}{-88x} -264 \color{blue}{+88x} \color{blue}{+36} \\\Leftrightarrow & 121x& = & -228 \\\Leftrightarrow & \frac{121x}{ \color{red}{121} }& = & \frac{-228}{121} \\\Leftrightarrow & x = \frac{-228}{121} & & \\ & V = \left\{ \frac{-228}{121} \right\} & \\\end{align}\)
8. \(\text{90 is het kleinste gemene veelvoud van 6, 9 en 5} \\ \begin{align} & \frac{x}{6}-\frac{4}{9}& = & \frac{-4}{5}x+5 \\\Leftrightarrow & \color{blue}{90.} (\frac{15x}{ \color{blue}{90} }- \frac{ 40 }{ \color{blue}{90} })& = & (\frac{-72}{ \color{blue}{90} }x+\frac{450}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 15x-40& = & -72x+450 \\\Leftrightarrow & 15x \color{red}{-40} \color{blue}{+40} \color{blue}{+72x} & = & \color{red}{-72x} +450 \color{blue}{+72x} \color{blue}{+40} \\\Leftrightarrow & 87x& = & 490 \\\Leftrightarrow & \frac{87x}{ \color{red}{87} }& = & \frac{490}{87} \\\Leftrightarrow & x = \frac{490}{87} & & \\ & V = \left\{ \frac{490}{87} \right\} & \\\end{align}\)
9. \(\text{30 is het kleinste gemene veelvoud van 6, 6 en 5} \\ \begin{align} & \frac{x}{6}-\frac{5}{6}& = & \frac{2}{5}x-2 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }- \frac{ 25 }{ \color{blue}{30} })& = & (\frac{12}{ \color{blue}{30} }x-\frac{60}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x-25& = & 12x-60 \\\Leftrightarrow & 5x \color{red}{-25} \color{blue}{+25} \color{blue}{-12x} & = & \color{red}{12x} -60 \color{blue}{-12x} \color{blue}{+25} \\\Leftrightarrow & -7x& = & -35 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{-35}{-7} \\\Leftrightarrow & x = 5 & & \\ & V = \left\{ 5 \right\} & \\\end{align}\)
10. \(\text{60 is het kleinste gemene veelvoud van 4, 10 en 3} \\ \begin{align} & \frac{x}{4}+\frac{3}{10}& = & \frac{-5}{3}x+3 \\\Leftrightarrow & \color{blue}{60.} (\frac{15x}{ \color{blue}{60} }+ \frac{ 18 }{ \color{blue}{60} })& = & (\frac{-100}{ \color{blue}{60} }x+\frac{180}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 15x+18& = & -100x+180 \\\Leftrightarrow & 15x \color{red}{+18} \color{blue}{-18} \color{blue}{+100x} & = & \color{red}{-100x} +180 \color{blue}{+100x} \color{blue}{-18} \\\Leftrightarrow & 115x& = & 162 \\\Leftrightarrow & \frac{115x}{ \color{red}{115} }& = & \frac{162}{115} \\\Leftrightarrow & x = \frac{162}{115} & & \\ & V = \left\{ \frac{162}{115} \right\} & \\\end{align}\)
11. \(\text{15 is het kleinste gemene veelvoud van 5, 15 en 3} \\ \begin{align} & \frac{x}{5}+\frac{4}{15}& = & \frac{1}{3}x-4 \\\Leftrightarrow & \color{blue}{15.} (\frac{3x}{ \color{blue}{15} }+ \frac{ 4 }{ \color{blue}{15} })& = & (\frac{5}{ \color{blue}{15} }x-\frac{60}{ \color{blue}{15} }) \color{blue}{.15} \\\Leftrightarrow & 3x+4& = & 5x-60 \\\Leftrightarrow & 3x \color{red}{+4} \color{blue}{-4} \color{blue}{-5x} & = & \color{red}{5x} -60 \color{blue}{-5x} \color{blue}{-4} \\\Leftrightarrow & -2x& = & -64 \\\Leftrightarrow & \frac{-2x}{ \color{red}{-2} }& = & \frac{-64}{-2} \\\Leftrightarrow & x = 32 & & \\ & V = \left\{ 32 \right\} & \\\end{align}\)
12. \(\text{30 is het kleinste gemene veelvoud van 5, 6 en 2} \\ \begin{align} & \frac{x}{5}-\frac{5}{6}& = & \frac{1}{2}x+8 \\\Leftrightarrow & \color{blue}{30.} (\frac{6x}{ \color{blue}{30} }- \frac{ 25 }{ \color{blue}{30} })& = & (\frac{15}{ \color{blue}{30} }x+\frac{240}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 6x-25& = & 15x+240 \\\Leftrightarrow & 6x \color{red}{-25} \color{blue}{+25} \color{blue}{-15x} & = & \color{red}{15x} +240 \color{blue}{-15x} \color{blue}{+25} \\\Leftrightarrow & -9x& = & 265 \\\Leftrightarrow & \frac{-9x}{ \color{red}{-9} }& = & \frac{265}{-9} \\\Leftrightarrow & x = \frac{-265}{9} & & \\ & V = \left\{ \frac{-265}{9} \right\} & \\\end{align}\)