Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{48 is het kleinste gemene veelvoud van 4, 16 en 3} \\ \begin{align} & \frac{x}{4}+\frac{3}{16}& = & \frac{7}{3}x-5 \\\Leftrightarrow & \color{blue}{48.} (\frac{12x}{ \color{blue}{48} }+ \frac{ 9 }{ \color{blue}{48} })& = & (\frac{112}{ \color{blue}{48} }x-\frac{240}{ \color{blue}{48} }) \color{blue}{.48} \\\Leftrightarrow & 12x+9& = & 112x-240 \\\Leftrightarrow & 12x \color{red}{+9} \color{blue}{-9} \color{blue}{-112x} & = & \color{red}{112x} -240 \color{blue}{-112x} \color{blue}{-9} \\\Leftrightarrow & -100x& = & -249 \\\Leftrightarrow & \frac{-100x}{ \color{red}{-100} }& = & \frac{-249}{-100} \\\Leftrightarrow & x = \frac{249}{100} & & \\ & V = \left\{ \frac{249}{100} \right\} & \\\end{align}\)
2. \(\text{390 is het kleinste gemene veelvoud van 6, 13 en 5} \\ \begin{align} & \frac{x}{6}-\frac{4}{13}& = & \frac{6}{5}x-5 \\\Leftrightarrow & \color{blue}{390.} (\frac{65x}{ \color{blue}{390} }- \frac{ 120 }{ \color{blue}{390} })& = & (\frac{468}{ \color{blue}{390} }x-\frac{1950}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 65x-120& = & 468x-1950 \\\Leftrightarrow & 65x \color{red}{-120} \color{blue}{+120} \color{blue}{-468x} & = & \color{red}{468x} -1950 \color{blue}{-468x} \color{blue}{+120} \\\Leftrightarrow & -403x& = & -1830 \\\Leftrightarrow & \frac{-403x}{ \color{red}{-403} }& = & \frac{-1830}{-403} \\\Leftrightarrow & x = \frac{1830}{403} & & \\ & V = \left\{ \frac{1830}{403} \right\} & \\\end{align}\)
3. \(\text{140 is het kleinste gemene veelvoud van 4, 7 en 5} \\ \begin{align} & \frac{x}{4}+\frac{4}{7}& = & \frac{2}{5}x-3 \\\Leftrightarrow & \color{blue}{140.} (\frac{35x}{ \color{blue}{140} }+ \frac{ 80 }{ \color{blue}{140} })& = & (\frac{56}{ \color{blue}{140} }x-\frac{420}{ \color{blue}{140} }) \color{blue}{.140} \\\Leftrightarrow & 35x+80& = & 56x-420 \\\Leftrightarrow & 35x \color{red}{+80} \color{blue}{-80} \color{blue}{-56x} & = & \color{red}{56x} -420 \color{blue}{-56x} \color{blue}{-80} \\\Leftrightarrow & -21x& = & -500 \\\Leftrightarrow & \frac{-21x}{ \color{red}{-21} }& = & \frac{-500}{-21} \\\Leftrightarrow & x = \frac{500}{21} & & \\ & V = \left\{ \frac{500}{21} \right\} & \\\end{align}\)
4. \(\text{30 is het kleinste gemene veelvoud van 2, 15 en 3} \\ \begin{align} & \frac{x}{2}-\frac{2}{15}& = & \frac{7}{3}x+6 \\\Leftrightarrow & \color{blue}{30.} (\frac{15x}{ \color{blue}{30} }- \frac{ 4 }{ \color{blue}{30} })& = & (\frac{70}{ \color{blue}{30} }x+\frac{180}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 15x-4& = & 70x+180 \\\Leftrightarrow & 15x \color{red}{-4} \color{blue}{+4} \color{blue}{-70x} & = & \color{red}{70x} +180 \color{blue}{-70x} \color{blue}{+4} \\\Leftrightarrow & -55x& = & 184 \\\Leftrightarrow & \frac{-55x}{ \color{red}{-55} }& = & \frac{184}{-55} \\\Leftrightarrow & x = \frac{-184}{55} & & \\ & V = \left\{ \frac{-184}{55} \right\} & \\\end{align}\)
5. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}-\frac{2}{7}& = & \frac{-7}{5}x-8 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }- \frac{ 60 }{ \color{blue}{210} })& = & (\frac{-294}{ \color{blue}{210} }x-\frac{1680}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x-60& = & -294x-1680 \\\Leftrightarrow & 35x \color{red}{-60} \color{blue}{+60} \color{blue}{+294x} & = & \color{red}{-294x} -1680 \color{blue}{+294x} \color{blue}{+60} \\\Leftrightarrow & 329x& = & -1620 \\\Leftrightarrow & \frac{329x}{ \color{red}{329} }& = & \frac{-1620}{329} \\\Leftrightarrow & x = \frac{-1620}{329} & & \\ & V = \left\{ \frac{-1620}{329} \right\} & \\\end{align}\)
6. \(\text{30 is het kleinste gemene veelvoud van 6, 6 en 5} \\ \begin{align} & \frac{x}{6}-\frac{5}{6}& = & \frac{-4}{5}x+5 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }- \frac{ 25 }{ \color{blue}{30} })& = & (\frac{-24}{ \color{blue}{30} }x+\frac{150}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x-25& = & -24x+150 \\\Leftrightarrow & 5x \color{red}{-25} \color{blue}{+25} \color{blue}{+24x} & = & \color{red}{-24x} +150 \color{blue}{+24x} \color{blue}{+25} \\\Leftrightarrow & 29x& = & 175 \\\Leftrightarrow & \frac{29x}{ \color{red}{29} }& = & \frac{175}{29} \\\Leftrightarrow & x = \frac{175}{29} & & \\ & V = \left\{ \frac{175}{29} \right\} & \\\end{align}\)
7. \(\text{48 is het kleinste gemene veelvoud van 2, 16 en 3} \\ \begin{align} & \frac{x}{2}-\frac{5}{16}& = & \frac{-5}{3}x+7 \\\Leftrightarrow & \color{blue}{48.} (\frac{24x}{ \color{blue}{48} }- \frac{ 15 }{ \color{blue}{48} })& = & (\frac{-80}{ \color{blue}{48} }x+\frac{336}{ \color{blue}{48} }) \color{blue}{.48} \\\Leftrightarrow & 24x-15& = & -80x+336 \\\Leftrightarrow & 24x \color{red}{-15} \color{blue}{+15} \color{blue}{+80x} & = & \color{red}{-80x} +336 \color{blue}{+80x} \color{blue}{+15} \\\Leftrightarrow & 104x& = & 351 \\\Leftrightarrow & \frac{104x}{ \color{red}{104} }& = & \frac{351}{104} \\\Leftrightarrow & x = \frac{27}{8} & & \\ & V = \left\{ \frac{27}{8} \right\} & \\\end{align}\)
8. \(\text{154 is het kleinste gemene veelvoud van 7, 11 en 2} \\ \begin{align} & \frac{x}{7}+\frac{2}{11}& = & \frac{7}{2}x+3 \\\Leftrightarrow & \color{blue}{154.} (\frac{22x}{ \color{blue}{154} }+ \frac{ 28 }{ \color{blue}{154} })& = & (\frac{539}{ \color{blue}{154} }x+\frac{462}{ \color{blue}{154} }) \color{blue}{.154} \\\Leftrightarrow & 22x+28& = & 539x+462 \\\Leftrightarrow & 22x \color{red}{+28} \color{blue}{-28} \color{blue}{-539x} & = & \color{red}{539x} +462 \color{blue}{-539x} \color{blue}{-28} \\\Leftrightarrow & -517x& = & 434 \\\Leftrightarrow & \frac{-517x}{ \color{red}{-517} }& = & \frac{434}{-517} \\\Leftrightarrow & x = \frac{-434}{517} & & \\ & V = \left\{ \frac{-434}{517} \right\} & \\\end{align}\)
9. \(\text{84 is het kleinste gemene veelvoud van 3, 7 en 4} \\ \begin{align} & \frac{x}{3}+\frac{4}{7}& = & \frac{5}{4}x-7 \\\Leftrightarrow & \color{blue}{84.} (\frac{28x}{ \color{blue}{84} }+ \frac{ 48 }{ \color{blue}{84} })& = & (\frac{105}{ \color{blue}{84} }x-\frac{588}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 28x+48& = & 105x-588 \\\Leftrightarrow & 28x \color{red}{+48} \color{blue}{-48} \color{blue}{-105x} & = & \color{red}{105x} -588 \color{blue}{-105x} \color{blue}{-48} \\\Leftrightarrow & -77x& = & -636 \\\Leftrightarrow & \frac{-77x}{ \color{red}{-77} }& = & \frac{-636}{-77} \\\Leftrightarrow & x = \frac{636}{77} & & \\ & V = \left\{ \frac{636}{77} \right\} & \\\end{align}\)
10. \(\text{18 is het kleinste gemene veelvoud van 2, 9 en 3} \\ \begin{align} & \frac{x}{2}-\frac{4}{9}& = & \frac{-5}{3}x+2 \\\Leftrightarrow & \color{blue}{18.} (\frac{9x}{ \color{blue}{18} }- \frac{ 8 }{ \color{blue}{18} })& = & (\frac{-30}{ \color{blue}{18} }x+\frac{36}{ \color{blue}{18} }) \color{blue}{.18} \\\Leftrightarrow & 9x-8& = & -30x+36 \\\Leftrightarrow & 9x \color{red}{-8} \color{blue}{+8} \color{blue}{+30x} & = & \color{red}{-30x} +36 \color{blue}{+30x} \color{blue}{+8} \\\Leftrightarrow & 39x& = & 44 \\\Leftrightarrow & \frac{39x}{ \color{red}{39} }& = & \frac{44}{39} \\\Leftrightarrow & x = \frac{44}{39} & & \\ & V = \left\{ \frac{44}{39} \right\} & \\\end{align}\)
11. \(\text{60 is het kleinste gemene veelvoud van 4, 10 en 3} \\ \begin{align} & \frac{x}{4}-\frac{3}{10}& = & \frac{4}{3}x+7 \\\Leftrightarrow & \color{blue}{60.} (\frac{15x}{ \color{blue}{60} }- \frac{ 18 }{ \color{blue}{60} })& = & (\frac{80}{ \color{blue}{60} }x+\frac{420}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 15x-18& = & 80x+420 \\\Leftrightarrow & 15x \color{red}{-18} \color{blue}{+18} \color{blue}{-80x} & = & \color{red}{80x} +420 \color{blue}{-80x} \color{blue}{+18} \\\Leftrightarrow & -65x& = & 438 \\\Leftrightarrow & \frac{-65x}{ \color{red}{-65} }& = & \frac{438}{-65} \\\Leftrightarrow & x = \frac{-438}{65} & & \\ & V = \left\{ \frac{-438}{65} \right\} & \\\end{align}\)
12. \(\text{140 is het kleinste gemene veelvoud van 4, 7 en 5} \\ \begin{align} & \frac{x}{4}-\frac{2}{7}& = & \frac{6}{5}x+6 \\\Leftrightarrow & \color{blue}{140.} (\frac{35x}{ \color{blue}{140} }- \frac{ 40 }{ \color{blue}{140} })& = & (\frac{168}{ \color{blue}{140} }x+\frac{840}{ \color{blue}{140} }) \color{blue}{.140} \\\Leftrightarrow & 35x-40& = & 168x+840 \\\Leftrightarrow & 35x \color{red}{-40} \color{blue}{+40} \color{blue}{-168x} & = & \color{red}{168x} +840 \color{blue}{-168x} \color{blue}{+40} \\\Leftrightarrow & -133x& = & 880 \\\Leftrightarrow & \frac{-133x}{ \color{red}{-133} }& = & \frac{880}{-133} \\\Leftrightarrow & x = \frac{-880}{133} & & \\ & V = \left\{ \frac{-880}{133} \right\} & \\\end{align}\)