Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{240 is het kleinste gemene veelvoud van 6, 16 en 5} \\ \begin{align} & \frac{x}{6}+\frac{5}{16}& = & \frac{-7}{5}x+2 \\\Leftrightarrow & \color{blue}{240.} (\frac{40x}{ \color{blue}{240} }+ \frac{ 75 }{ \color{blue}{240} })& = & (\frac{-336}{ \color{blue}{240} }x+\frac{480}{ \color{blue}{240} }) \color{blue}{.240} \\\Leftrightarrow & 40x+75& = & -336x+480 \\\Leftrightarrow & 40x \color{red}{+75} \color{blue}{-75} \color{blue}{+336x} & = & \color{red}{-336x} +480 \color{blue}{+336x} \color{blue}{-75} \\\Leftrightarrow & 376x& = & 405 \\\Leftrightarrow & \frac{376x}{ \color{red}{376} }& = & \frac{405}{376} \\\Leftrightarrow & x = \frac{405}{376} & & \\ & V = \left\{ \frac{405}{376} \right\} & \\\end{align}\)
2. \(\text{546 is het kleinste gemene veelvoud van 7, 13 en 6} \\ \begin{align} & \frac{x}{7}+\frac{2}{13}& = & \frac{1}{6}x-4 \\\Leftrightarrow & \color{blue}{546.} (\frac{78x}{ \color{blue}{546} }+ \frac{ 84 }{ \color{blue}{546} })& = & (\frac{91}{ \color{blue}{546} }x-\frac{2184}{ \color{blue}{546} }) \color{blue}{.546} \\\Leftrightarrow & 78x+84& = & 91x-2184 \\\Leftrightarrow & 78x \color{red}{+84} \color{blue}{-84} \color{blue}{-91x} & = & \color{red}{91x} -2184 \color{blue}{-91x} \color{blue}{-84} \\\Leftrightarrow & -13x& = & -2268 \\\Leftrightarrow & \frac{-13x}{ \color{red}{-13} }& = & \frac{-2268}{-13} \\\Leftrightarrow & x = \frac{2268}{13} & & \\ & V = \left\{ \frac{2268}{13} \right\} & \\\end{align}\)
3. \(\text{90 is het kleinste gemene veelvoud van 6, 9 en 5} \\ \begin{align} & \frac{x}{6}+\frac{2}{9}& = & \frac{-7}{5}x-8 \\\Leftrightarrow & \color{blue}{90.} (\frac{15x}{ \color{blue}{90} }+ \frac{ 20 }{ \color{blue}{90} })& = & (\frac{-126}{ \color{blue}{90} }x-\frac{720}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 15x+20& = & -126x-720 \\\Leftrightarrow & 15x \color{red}{+20} \color{blue}{-20} \color{blue}{+126x} & = & \color{red}{-126x} -720 \color{blue}{+126x} \color{blue}{-20} \\\Leftrightarrow & 141x& = & -740 \\\Leftrightarrow & \frac{141x}{ \color{red}{141} }& = & \frac{-740}{141} \\\Leftrightarrow & x = \frac{-740}{141} & & \\ & V = \left\{ \frac{-740}{141} \right\} & \\\end{align}\)
4. \(\text{240 is het kleinste gemene veelvoud van 6, 16 en 5} \\ \begin{align} & \frac{x}{6}+\frac{3}{16}& = & \frac{-4}{5}x-7 \\\Leftrightarrow & \color{blue}{240.} (\frac{40x}{ \color{blue}{240} }+ \frac{ 45 }{ \color{blue}{240} })& = & (\frac{-192}{ \color{blue}{240} }x-\frac{1680}{ \color{blue}{240} }) \color{blue}{.240} \\\Leftrightarrow & 40x+45& = & -192x-1680 \\\Leftrightarrow & 40x \color{red}{+45} \color{blue}{-45} \color{blue}{+192x} & = & \color{red}{-192x} -1680 \color{blue}{+192x} \color{blue}{-45} \\\Leftrightarrow & 232x& = & -1725 \\\Leftrightarrow & \frac{232x}{ \color{red}{232} }& = & \frac{-1725}{232} \\\Leftrightarrow & x = \frac{-1725}{232} & & \\ & V = \left\{ \frac{-1725}{232} \right\} & \\\end{align}\)
5. \(\text{60 is het kleinste gemene veelvoud van 4, 6 en 5} \\ \begin{align} & \frac{x}{4}+\frac{5}{6}& = & \frac{1}{5}x+7 \\\Leftrightarrow & \color{blue}{60.} (\frac{15x}{ \color{blue}{60} }+ \frac{ 50 }{ \color{blue}{60} })& = & (\frac{12}{ \color{blue}{60} }x+\frac{420}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 15x+50& = & 12x+420 \\\Leftrightarrow & 15x \color{red}{+50} \color{blue}{-50} \color{blue}{-12x} & = & \color{red}{12x} +420 \color{blue}{-12x} \color{blue}{-50} \\\Leftrightarrow & 3x& = & 370 \\\Leftrightarrow & \frac{3x}{ \color{red}{3} }& = & \frac{370}{3} \\\Leftrightarrow & x = \frac{370}{3} & & \\ & V = \left\{ \frac{370}{3} \right\} & \\\end{align}\)
6. \(\text{18 is het kleinste gemene veelvoud van 2, 9 en 3} \\ \begin{align} & \frac{x}{2}-\frac{4}{9}& = & \frac{-2}{3}x+5 \\\Leftrightarrow & \color{blue}{18.} (\frac{9x}{ \color{blue}{18} }- \frac{ 8 }{ \color{blue}{18} })& = & (\frac{-12}{ \color{blue}{18} }x+\frac{90}{ \color{blue}{18} }) \color{blue}{.18} \\\Leftrightarrow & 9x-8& = & -12x+90 \\\Leftrightarrow & 9x \color{red}{-8} \color{blue}{+8} \color{blue}{+12x} & = & \color{red}{-12x} +90 \color{blue}{+12x} \color{blue}{+8} \\\Leftrightarrow & 21x& = & 98 \\\Leftrightarrow & \frac{21x}{ \color{red}{21} }& = & \frac{98}{21} \\\Leftrightarrow & x = \frac{14}{3} & & \\ & V = \left\{ \frac{14}{3} \right\} & \\\end{align}\)
7. \(\text{220 is het kleinste gemene veelvoud van 5, 11 en 4} \\ \begin{align} & \frac{x}{5}-\frac{4}{11}& = & \frac{3}{4}x-1 \\\Leftrightarrow & \color{blue}{220.} (\frac{44x}{ \color{blue}{220} }- \frac{ 80 }{ \color{blue}{220} })& = & (\frac{165}{ \color{blue}{220} }x-\frac{220}{ \color{blue}{220} }) \color{blue}{.220} \\\Leftrightarrow & 44x-80& = & 165x-220 \\\Leftrightarrow & 44x \color{red}{-80} \color{blue}{+80} \color{blue}{-165x} & = & \color{red}{165x} -220 \color{blue}{-165x} \color{blue}{+80} \\\Leftrightarrow & -121x& = & -140 \\\Leftrightarrow & \frac{-121x}{ \color{red}{-121} }& = & \frac{-140}{-121} \\\Leftrightarrow & x = \frac{140}{121} & & \\ & V = \left\{ \frac{140}{121} \right\} & \\\end{align}\)
8. \(\text{30 is het kleinste gemene veelvoud van 5, 15 en 2} \\ \begin{align} & \frac{x}{5}+\frac{2}{15}& = & \frac{1}{2}x-7 \\\Leftrightarrow & \color{blue}{30.} (\frac{6x}{ \color{blue}{30} }+ \frac{ 4 }{ \color{blue}{30} })& = & (\frac{15}{ \color{blue}{30} }x-\frac{210}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 6x+4& = & 15x-210 \\\Leftrightarrow & 6x \color{red}{+4} \color{blue}{-4} \color{blue}{-15x} & = & \color{red}{15x} -210 \color{blue}{-15x} \color{blue}{-4} \\\Leftrightarrow & -9x& = & -214 \\\Leftrightarrow & \frac{-9x}{ \color{red}{-9} }& = & \frac{-214}{-9} \\\Leftrightarrow & x = \frac{214}{9} & & \\ & V = \left\{ \frac{214}{9} \right\} & \\\end{align}\)
9. \(\text{78 is het kleinste gemene veelvoud van 2, 13 en 3} \\ \begin{align} & \frac{x}{2}+\frac{3}{13}& = & \frac{-5}{3}x+1 \\\Leftrightarrow & \color{blue}{78.} (\frac{39x}{ \color{blue}{78} }+ \frac{ 18 }{ \color{blue}{78} })& = & (\frac{-130}{ \color{blue}{78} }x+\frac{78}{ \color{blue}{78} }) \color{blue}{.78} \\\Leftrightarrow & 39x+18& = & -130x+78 \\\Leftrightarrow & 39x \color{red}{+18} \color{blue}{-18} \color{blue}{+130x} & = & \color{red}{-130x} +78 \color{blue}{+130x} \color{blue}{-18} \\\Leftrightarrow & 169x& = & 60 \\\Leftrightarrow & \frac{169x}{ \color{red}{169} }& = & \frac{60}{169} \\\Leftrightarrow & x = \frac{60}{169} & & \\ & V = \left\{ \frac{60}{169} \right\} & \\\end{align}\)
10. \(\text{48 is het kleinste gemene veelvoud van 2, 16 en 3} \\ \begin{align} & \frac{x}{2}-\frac{3}{16}& = & \frac{-5}{3}x-8 \\\Leftrightarrow & \color{blue}{48.} (\frac{24x}{ \color{blue}{48} }- \frac{ 9 }{ \color{blue}{48} })& = & (\frac{-80}{ \color{blue}{48} }x-\frac{384}{ \color{blue}{48} }) \color{blue}{.48} \\\Leftrightarrow & 24x-9& = & -80x-384 \\\Leftrightarrow & 24x \color{red}{-9} \color{blue}{+9} \color{blue}{+80x} & = & \color{red}{-80x} -384 \color{blue}{+80x} \color{blue}{+9} \\\Leftrightarrow & 104x& = & -375 \\\Leftrightarrow & \frac{104x}{ \color{red}{104} }& = & \frac{-375}{104} \\\Leftrightarrow & x = \frac{-375}{104} & & \\ & V = \left\{ \frac{-375}{104} \right\} & \\\end{align}\)
11. \(\text{112 is het kleinste gemene veelvoud van 7, 16 en 4} \\ \begin{align} & \frac{x}{7}+\frac{5}{16}& = & \frac{-3}{4}x+7 \\\Leftrightarrow & \color{blue}{112.} (\frac{16x}{ \color{blue}{112} }+ \frac{ 35 }{ \color{blue}{112} })& = & (\frac{-84}{ \color{blue}{112} }x+\frac{784}{ \color{blue}{112} }) \color{blue}{.112} \\\Leftrightarrow & 16x+35& = & -84x+784 \\\Leftrightarrow & 16x \color{red}{+35} \color{blue}{-35} \color{blue}{+84x} & = & \color{red}{-84x} +784 \color{blue}{+84x} \color{blue}{-35} \\\Leftrightarrow & 100x& = & 749 \\\Leftrightarrow & \frac{100x}{ \color{red}{100} }& = & \frac{749}{100} \\\Leftrightarrow & x = \frac{749}{100} & & \\ & V = \left\{ \frac{749}{100} \right\} & \\\end{align}\)
12. \(\text{42 is het kleinste gemene veelvoud van 3, 7 en 2} \\ \begin{align} & \frac{x}{3}-\frac{3}{7}& = & \frac{1}{2}x-2 \\\Leftrightarrow & \color{blue}{42.} (\frac{14x}{ \color{blue}{42} }- \frac{ 18 }{ \color{blue}{42} })& = & (\frac{21}{ \color{blue}{42} }x-\frac{84}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 14x-18& = & 21x-84 \\\Leftrightarrow & 14x \color{red}{-18} \color{blue}{+18} \color{blue}{-21x} & = & \color{red}{21x} -84 \color{blue}{-21x} \color{blue}{+18} \\\Leftrightarrow & -7x& = & -66 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{-66}{-7} \\\Leftrightarrow & x = \frac{66}{7} & & \\ & V = \left\{ \frac{66}{7} \right\} & \\\end{align}\)