Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{24 is het kleinste gemene veelvoud van 3, 8 en 2} \\ \begin{align} & \frac{x}{3}-\frac{3}{8}& = & \frac{1}{2}x-8 \\\Leftrightarrow & \color{blue}{24.} (\frac{8x}{ \color{blue}{24} }- \frac{ 9 }{ \color{blue}{24} })& = & (\frac{12}{ \color{blue}{24} }x-\frac{192}{ \color{blue}{24} }) \color{blue}{.24} \\\Leftrightarrow & 8x-9& = & 12x-192 \\\Leftrightarrow & 8x \color{red}{-9} \color{blue}{+9} \color{blue}{-12x} & = & \color{red}{12x} -192 \color{blue}{-12x} \color{blue}{+9} \\\Leftrightarrow & -4x& = & -183 \\\Leftrightarrow & \frac{-4x}{ \color{red}{-4} }& = & \frac{-183}{-4} \\\Leftrightarrow & x = \frac{183}{4} & & \\ & V = \left\{ \frac{183}{4} \right\} & \\\end{align}\)
2. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}+\frac{4}{7}& = & \frac{-2}{3}x-2 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }+ \frac{ 48 }{ \color{blue}{84} })& = & (\frac{-56}{ \color{blue}{84} }x-\frac{168}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x+48& = & -56x-168 \\\Leftrightarrow & 21x \color{red}{+48} \color{blue}{-48} \color{blue}{+56x} & = & \color{red}{-56x} -168 \color{blue}{+56x} \color{blue}{-48} \\\Leftrightarrow & 77x& = & -216 \\\Leftrightarrow & \frac{77x}{ \color{red}{77} }& = & \frac{-216}{77} \\\Leftrightarrow & x = \frac{-216}{77} & & \\ & V = \left\{ \frac{-216}{77} \right\} & \\\end{align}\)
3. \(\text{110 is het kleinste gemene veelvoud van 5, 11 en 2} \\ \begin{align} & \frac{x}{5}-\frac{4}{11}& = & \frac{1}{2}x-7 \\\Leftrightarrow & \color{blue}{110.} (\frac{22x}{ \color{blue}{110} }- \frac{ 40 }{ \color{blue}{110} })& = & (\frac{55}{ \color{blue}{110} }x-\frac{770}{ \color{blue}{110} }) \color{blue}{.110} \\\Leftrightarrow & 22x-40& = & 55x-770 \\\Leftrightarrow & 22x \color{red}{-40} \color{blue}{+40} \color{blue}{-55x} & = & \color{red}{55x} -770 \color{blue}{-55x} \color{blue}{+40} \\\Leftrightarrow & -33x& = & -730 \\\Leftrightarrow & \frac{-33x}{ \color{red}{-33} }& = & \frac{-730}{-33} \\\Leftrightarrow & x = \frac{730}{33} & & \\ & V = \left\{ \frac{730}{33} \right\} & \\\end{align}\)
4. \(\text{30 is het kleinste gemene veelvoud van 6, 6 en 5} \\ \begin{align} & \frac{x}{6}+\frac{5}{6}& = & \frac{-7}{5}x-3 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }+ \frac{ 25 }{ \color{blue}{30} })& = & (\frac{-42}{ \color{blue}{30} }x-\frac{90}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x+25& = & -42x-90 \\\Leftrightarrow & 5x \color{red}{+25} \color{blue}{-25} \color{blue}{+42x} & = & \color{red}{-42x} -90 \color{blue}{+42x} \color{blue}{-25} \\\Leftrightarrow & 47x& = & -115 \\\Leftrightarrow & \frac{47x}{ \color{red}{47} }& = & \frac{-115}{47} \\\Leftrightarrow & x = \frac{-115}{47} & & \\ & V = \left\{ \frac{-115}{47} \right\} & \\\end{align}\)
5. \(\text{330 is het kleinste gemene veelvoud van 6, 11 en 5} \\ \begin{align} & \frac{x}{6}+\frac{2}{11}& = & \frac{-7}{5}x+8 \\\Leftrightarrow & \color{blue}{330.} (\frac{55x}{ \color{blue}{330} }+ \frac{ 60 }{ \color{blue}{330} })& = & (\frac{-462}{ \color{blue}{330} }x+\frac{2640}{ \color{blue}{330} }) \color{blue}{.330} \\\Leftrightarrow & 55x+60& = & -462x+2640 \\\Leftrightarrow & 55x \color{red}{+60} \color{blue}{-60} \color{blue}{+462x} & = & \color{red}{-462x} +2640 \color{blue}{+462x} \color{blue}{-60} \\\Leftrightarrow & 517x& = & 2580 \\\Leftrightarrow & \frac{517x}{ \color{red}{517} }& = & \frac{2580}{517} \\\Leftrightarrow & x = \frac{2580}{517} & & \\ & V = \left\{ \frac{2580}{517} \right\} & \\\end{align}\)
6. \(\text{30 is het kleinste gemene veelvoud van 6, 10 en 5} \\ \begin{align} & \frac{x}{6}+\frac{3}{10}& = & \frac{6}{5}x-5 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }+ \frac{ 9 }{ \color{blue}{30} })& = & (\frac{36}{ \color{blue}{30} }x-\frac{150}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x+9& = & 36x-150 \\\Leftrightarrow & 5x \color{red}{+9} \color{blue}{-9} \color{blue}{-36x} & = & \color{red}{36x} -150 \color{blue}{-36x} \color{blue}{-9} \\\Leftrightarrow & -31x& = & -159 \\\Leftrightarrow & \frac{-31x}{ \color{red}{-31} }& = & \frac{-159}{-31} \\\Leftrightarrow & x = \frac{159}{31} & & \\ & V = \left\{ \frac{159}{31} \right\} & \\\end{align}\)
7. \(\text{30 is het kleinste gemene veelvoud van 3, 10 en 2} \\ \begin{align} & \frac{x}{3}-\frac{3}{10}& = & \frac{1}{2}x-3 \\\Leftrightarrow & \color{blue}{30.} (\frac{10x}{ \color{blue}{30} }- \frac{ 9 }{ \color{blue}{30} })& = & (\frac{15}{ \color{blue}{30} }x-\frac{90}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 10x-9& = & 15x-90 \\\Leftrightarrow & 10x \color{red}{-9} \color{blue}{+9} \color{blue}{-15x} & = & \color{red}{15x} -90 \color{blue}{-15x} \color{blue}{+9} \\\Leftrightarrow & -5x& = & -81 \\\Leftrightarrow & \frac{-5x}{ \color{red}{-5} }& = & \frac{-81}{-5} \\\Leftrightarrow & x = \frac{81}{5} & & \\ & V = \left\{ \frac{81}{5} \right\} & \\\end{align}\)
8. \(\text{132 is het kleinste gemene veelvoud van 3, 11 en 4} \\ \begin{align} & \frac{x}{3}-\frac{4}{11}& = & \frac{-3}{4}x-3 \\\Leftrightarrow & \color{blue}{132.} (\frac{44x}{ \color{blue}{132} }- \frac{ 48 }{ \color{blue}{132} })& = & (\frac{-99}{ \color{blue}{132} }x-\frac{396}{ \color{blue}{132} }) \color{blue}{.132} \\\Leftrightarrow & 44x-48& = & -99x-396 \\\Leftrightarrow & 44x \color{red}{-48} \color{blue}{+48} \color{blue}{+99x} & = & \color{red}{-99x} -396 \color{blue}{+99x} \color{blue}{+48} \\\Leftrightarrow & 143x& = & -348 \\\Leftrightarrow & \frac{143x}{ \color{red}{143} }& = & \frac{-348}{143} \\\Leftrightarrow & x = \frac{-348}{143} & & \\ & V = \left\{ \frac{-348}{143} \right\} & \\\end{align}\)
9. \(\text{70 is het kleinste gemene veelvoud van 5, 7 en 2} \\ \begin{align} & \frac{x}{5}+\frac{4}{7}& = & \frac{1}{2}x+4 \\\Leftrightarrow & \color{blue}{70.} (\frac{14x}{ \color{blue}{70} }+ \frac{ 40 }{ \color{blue}{70} })& = & (\frac{35}{ \color{blue}{70} }x+\frac{280}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 14x+40& = & 35x+280 \\\Leftrightarrow & 14x \color{red}{+40} \color{blue}{-40} \color{blue}{-35x} & = & \color{red}{35x} +280 \color{blue}{-35x} \color{blue}{-40} \\\Leftrightarrow & -21x& = & 240 \\\Leftrightarrow & \frac{-21x}{ \color{red}{-21} }& = & \frac{240}{-21} \\\Leftrightarrow & x = \frac{-80}{7} & & \\ & V = \left\{ \frac{-80}{7} \right\} & \\\end{align}\)
10. \(\text{546 is het kleinste gemene veelvoud van 7, 13 en 6} \\ \begin{align} & \frac{x}{7}-\frac{4}{13}& = & \frac{7}{6}x+1 \\\Leftrightarrow & \color{blue}{546.} (\frac{78x}{ \color{blue}{546} }- \frac{ 168 }{ \color{blue}{546} })& = & (\frac{637}{ \color{blue}{546} }x+\frac{546}{ \color{blue}{546} }) \color{blue}{.546} \\\Leftrightarrow & 78x-168& = & 637x+546 \\\Leftrightarrow & 78x \color{red}{-168} \color{blue}{+168} \color{blue}{-637x} & = & \color{red}{637x} +546 \color{blue}{-637x} \color{blue}{+168} \\\Leftrightarrow & -559x& = & 714 \\\Leftrightarrow & \frac{-559x}{ \color{red}{-559} }& = & \frac{714}{-559} \\\Leftrightarrow & x = \frac{-714}{559} & & \\ & V = \left\{ \frac{-714}{559} \right\} & \\\end{align}\)
11. \(\text{260 is het kleinste gemene veelvoud van 4, 13 en 5} \\ \begin{align} & \frac{x}{4}+\frac{2}{13}& = & \frac{-7}{5}x-2 \\\Leftrightarrow & \color{blue}{260.} (\frac{65x}{ \color{blue}{260} }+ \frac{ 40 }{ \color{blue}{260} })& = & (\frac{-364}{ \color{blue}{260} }x-\frac{520}{ \color{blue}{260} }) \color{blue}{.260} \\\Leftrightarrow & 65x+40& = & -364x-520 \\\Leftrightarrow & 65x \color{red}{+40} \color{blue}{-40} \color{blue}{+364x} & = & \color{red}{-364x} -520 \color{blue}{+364x} \color{blue}{-40} \\\Leftrightarrow & 429x& = & -560 \\\Leftrightarrow & \frac{429x}{ \color{red}{429} }& = & \frac{-560}{429} \\\Leftrightarrow & x = \frac{-560}{429} & & \\ & V = \left\{ \frac{-560}{429} \right\} & \\\end{align}\)
12. \(\text{30 is het kleinste gemene veelvoud van 6, 15 en 5} \\ \begin{align} & \frac{x}{6}-\frac{4}{15}& = & \frac{2}{5}x+5 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }- \frac{ 8 }{ \color{blue}{30} })& = & (\frac{12}{ \color{blue}{30} }x+\frac{150}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x-8& = & 12x+150 \\\Leftrightarrow & 5x \color{red}{-8} \color{blue}{+8} \color{blue}{-12x} & = & \color{red}{12x} +150 \color{blue}{-12x} \color{blue}{+8} \\\Leftrightarrow & -7x& = & 158 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{158}{-7} \\\Leftrightarrow & x = \frac{-158}{7} & & \\ & V = \left\{ \frac{-158}{7} \right\} & \\\end{align}\)