Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{48 is het kleinste gemene veelvoud van 3, 16 en 4} \\ \begin{align} & \frac{x}{3}-\frac{3}{16}& = & \frac{1}{4}x-5 \\\Leftrightarrow & \color{blue}{48.} (\frac{16x}{ \color{blue}{48} }- \frac{ 9 }{ \color{blue}{48} })& = & (\frac{12}{ \color{blue}{48} }x-\frac{240}{ \color{blue}{48} }) \color{blue}{.48} \\\Leftrightarrow & 16x-9& = & 12x-240 \\\Leftrightarrow & 16x \color{red}{-9} \color{blue}{+9} \color{blue}{-12x} & = & \color{red}{12x} -240 \color{blue}{-12x} \color{blue}{+9} \\\Leftrightarrow & 4x& = & -231 \\\Leftrightarrow & \frac{4x}{ \color{red}{4} }& = & \frac{-231}{4} \\\Leftrightarrow & x = \frac{-231}{4} & & \\ & V = \left\{ \frac{-231}{4} \right\} & \\\end{align}\)
2. \(\text{30 is het kleinste gemene veelvoud van 5, 15 en 6} \\ \begin{align} & \frac{x}{5}+\frac{2}{15}& = & \frac{-5}{6}x+6 \\\Leftrightarrow & \color{blue}{30.} (\frac{6x}{ \color{blue}{30} }+ \frac{ 4 }{ \color{blue}{30} })& = & (\frac{-25}{ \color{blue}{30} }x+\frac{180}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 6x+4& = & -25x+180 \\\Leftrightarrow & 6x \color{red}{+4} \color{blue}{-4} \color{blue}{+25x} & = & \color{red}{-25x} +180 \color{blue}{+25x} \color{blue}{-4} \\\Leftrightarrow & 31x& = & 176 \\\Leftrightarrow & \frac{31x}{ \color{red}{31} }& = & \frac{176}{31} \\\Leftrightarrow & x = \frac{176}{31} & & \\ & V = \left\{ \frac{176}{31} \right\} & \\\end{align}\)
3. \(\text{30 is het kleinste gemene veelvoud van 5, 15 en 6} \\ \begin{align} & \frac{x}{5}-\frac{4}{15}& = & \frac{7}{6}x-7 \\\Leftrightarrow & \color{blue}{30.} (\frac{6x}{ \color{blue}{30} }- \frac{ 8 }{ \color{blue}{30} })& = & (\frac{35}{ \color{blue}{30} }x-\frac{210}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 6x-8& = & 35x-210 \\\Leftrightarrow & 6x \color{red}{-8} \color{blue}{+8} \color{blue}{-35x} & = & \color{red}{35x} -210 \color{blue}{-35x} \color{blue}{+8} \\\Leftrightarrow & -29x& = & -202 \\\Leftrightarrow & \frac{-29x}{ \color{red}{-29} }& = & \frac{-202}{-29} \\\Leftrightarrow & x = \frac{202}{29} & & \\ & V = \left\{ \frac{202}{29} \right\} & \\\end{align}\)
4. \(\text{14 is het kleinste gemene veelvoud van 7, 7 en 2} \\ \begin{align} & \frac{x}{7}-\frac{3}{7}& = & \frac{1}{2}x+4 \\\Leftrightarrow & \color{blue}{14.} (\frac{2x}{ \color{blue}{14} }- \frac{ 6 }{ \color{blue}{14} })& = & (\frac{7}{ \color{blue}{14} }x+\frac{56}{ \color{blue}{14} }) \color{blue}{.14} \\\Leftrightarrow & 2x-6& = & 7x+56 \\\Leftrightarrow & 2x \color{red}{-6} \color{blue}{+6} \color{blue}{-7x} & = & \color{red}{7x} +56 \color{blue}{-7x} \color{blue}{+6} \\\Leftrightarrow & -5x& = & 62 \\\Leftrightarrow & \frac{-5x}{ \color{red}{-5} }& = & \frac{62}{-5} \\\Leftrightarrow & x = \frac{-62}{5} & & \\ & V = \left\{ \frac{-62}{5} \right\} & \\\end{align}\)
5. \(\text{18 is het kleinste gemene veelvoud van 3, 9 en 2} \\ \begin{align} & \frac{x}{3}-\frac{5}{9}& = & \frac{1}{2}x+5 \\\Leftrightarrow & \color{blue}{18.} (\frac{6x}{ \color{blue}{18} }- \frac{ 10 }{ \color{blue}{18} })& = & (\frac{9}{ \color{blue}{18} }x+\frac{90}{ \color{blue}{18} }) \color{blue}{.18} \\\Leftrightarrow & 6x-10& = & 9x+90 \\\Leftrightarrow & 6x \color{red}{-10} \color{blue}{+10} \color{blue}{-9x} & = & \color{red}{9x} +90 \color{blue}{-9x} \color{blue}{+10} \\\Leftrightarrow & -3x& = & 100 \\\Leftrightarrow & \frac{-3x}{ \color{red}{-3} }& = & \frac{100}{-3} \\\Leftrightarrow & x = \frac{-100}{3} & & \\ & V = \left\{ \frac{-100}{3} \right\} & \\\end{align}\)
6. \(\text{336 is het kleinste gemene veelvoud van 7, 16 en 6} \\ \begin{align} & \frac{x}{7}-\frac{3}{16}& = & \frac{7}{6}x+4 \\\Leftrightarrow & \color{blue}{336.} (\frac{48x}{ \color{blue}{336} }- \frac{ 63 }{ \color{blue}{336} })& = & (\frac{392}{ \color{blue}{336} }x+\frac{1344}{ \color{blue}{336} }) \color{blue}{.336} \\\Leftrightarrow & 48x-63& = & 392x+1344 \\\Leftrightarrow & 48x \color{red}{-63} \color{blue}{+63} \color{blue}{-392x} & = & \color{red}{392x} +1344 \color{blue}{-392x} \color{blue}{+63} \\\Leftrightarrow & -344x& = & 1407 \\\Leftrightarrow & \frac{-344x}{ \color{red}{-344} }& = & \frac{1407}{-344} \\\Leftrightarrow & x = \frac{-1407}{344} & & \\ & V = \left\{ \frac{-1407}{344} \right\} & \\\end{align}\)
7. \(\text{12 is het kleinste gemene veelvoud van 3, 6 en 4} \\ \begin{align} & \frac{x}{3}-\frac{5}{6}& = & \frac{7}{4}x+5 \\\Leftrightarrow & \color{blue}{12.} (\frac{4x}{ \color{blue}{12} }- \frac{ 10 }{ \color{blue}{12} })& = & (\frac{21}{ \color{blue}{12} }x+\frac{60}{ \color{blue}{12} }) \color{blue}{.12} \\\Leftrightarrow & 4x-10& = & 21x+60 \\\Leftrightarrow & 4x \color{red}{-10} \color{blue}{+10} \color{blue}{-21x} & = & \color{red}{21x} +60 \color{blue}{-21x} \color{blue}{+10} \\\Leftrightarrow & -17x& = & 70 \\\Leftrightarrow & \frac{-17x}{ \color{red}{-17} }& = & \frac{70}{-17} \\\Leftrightarrow & x = \frac{-70}{17} & & \\ & V = \left\{ \frac{-70}{17} \right\} & \\\end{align}\)
8. \(\text{280 is het kleinste gemene veelvoud van 7, 8 en 5} \\ \begin{align} & \frac{x}{7}+\frac{3}{8}& = & \frac{1}{5}x-6 \\\Leftrightarrow & \color{blue}{280.} (\frac{40x}{ \color{blue}{280} }+ \frac{ 105 }{ \color{blue}{280} })& = & (\frac{56}{ \color{blue}{280} }x-\frac{1680}{ \color{blue}{280} }) \color{blue}{.280} \\\Leftrightarrow & 40x+105& = & 56x-1680 \\\Leftrightarrow & 40x \color{red}{+105} \color{blue}{-105} \color{blue}{-56x} & = & \color{red}{56x} -1680 \color{blue}{-56x} \color{blue}{-105} \\\Leftrightarrow & -16x& = & -1785 \\\Leftrightarrow & \frac{-16x}{ \color{red}{-16} }& = & \frac{-1785}{-16} \\\Leftrightarrow & x = \frac{1785}{16} & & \\ & V = \left\{ \frac{1785}{16} \right\} & \\\end{align}\)
9. \(\text{165 is het kleinste gemene veelvoud van 3, 11 en 5} \\ \begin{align} & \frac{x}{3}+\frac{5}{11}& = & \frac{2}{5}x-4 \\\Leftrightarrow & \color{blue}{165.} (\frac{55x}{ \color{blue}{165} }+ \frac{ 75 }{ \color{blue}{165} })& = & (\frac{66}{ \color{blue}{165} }x-\frac{660}{ \color{blue}{165} }) \color{blue}{.165} \\\Leftrightarrow & 55x+75& = & 66x-660 \\\Leftrightarrow & 55x \color{red}{+75} \color{blue}{-75} \color{blue}{-66x} & = & \color{red}{66x} -660 \color{blue}{-66x} \color{blue}{-75} \\\Leftrightarrow & -11x& = & -735 \\\Leftrightarrow & \frac{-11x}{ \color{red}{-11} }& = & \frac{-735}{-11} \\\Leftrightarrow & x = \frac{735}{11} & & \\ & V = \left\{ \frac{735}{11} \right\} & \\\end{align}\)
10. \(\text{70 is het kleinste gemene veelvoud van 2, 7 en 5} \\ \begin{align} & \frac{x}{2}+\frac{2}{7}& = & \frac{6}{5}x-5 \\\Leftrightarrow & \color{blue}{70.} (\frac{35x}{ \color{blue}{70} }+ \frac{ 20 }{ \color{blue}{70} })& = & (\frac{84}{ \color{blue}{70} }x-\frac{350}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 35x+20& = & 84x-350 \\\Leftrightarrow & 35x \color{red}{+20} \color{blue}{-20} \color{blue}{-84x} & = & \color{red}{84x} -350 \color{blue}{-84x} \color{blue}{-20} \\\Leftrightarrow & -49x& = & -370 \\\Leftrightarrow & \frac{-49x}{ \color{red}{-49} }& = & \frac{-370}{-49} \\\Leftrightarrow & x = \frac{370}{49} & & \\ & V = \left\{ \frac{370}{49} \right\} & \\\end{align}\)
11. \(\text{180 is het kleinste gemene veelvoud van 4, 9 en 5} \\ \begin{align} & \frac{x}{4}+\frac{5}{9}& = & \frac{6}{5}x+5 \\\Leftrightarrow & \color{blue}{180.} (\frac{45x}{ \color{blue}{180} }+ \frac{ 100 }{ \color{blue}{180} })& = & (\frac{216}{ \color{blue}{180} }x+\frac{900}{ \color{blue}{180} }) \color{blue}{.180} \\\Leftrightarrow & 45x+100& = & 216x+900 \\\Leftrightarrow & 45x \color{red}{+100} \color{blue}{-100} \color{blue}{-216x} & = & \color{red}{216x} +900 \color{blue}{-216x} \color{blue}{-100} \\\Leftrightarrow & -171x& = & 800 \\\Leftrightarrow & \frac{-171x}{ \color{red}{-171} }& = & \frac{800}{-171} \\\Leftrightarrow & x = \frac{-800}{171} & & \\ & V = \left\{ \frac{-800}{171} \right\} & \\\end{align}\)
12. \(\text{42 is het kleinste gemene veelvoud van 2, 7 en 3} \\ \begin{align} & \frac{x}{2}+\frac{4}{7}& = & \frac{-5}{3}x+8 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }+ \frac{ 24 }{ \color{blue}{42} })& = & (\frac{-70}{ \color{blue}{42} }x+\frac{336}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x+24& = & -70x+336 \\\Leftrightarrow & 21x \color{red}{+24} \color{blue}{-24} \color{blue}{+70x} & = & \color{red}{-70x} +336 \color{blue}{+70x} \color{blue}{-24} \\\Leftrightarrow & 91x& = & 312 \\\Leftrightarrow & \frac{91x}{ \color{red}{91} }& = & \frac{312}{91} \\\Leftrightarrow & x = \frac{24}{7} & & \\ & V = \left\{ \frac{24}{7} \right\} & \\\end{align}\)