Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{130 is het kleinste gemene veelvoud van 2, 13 en 5} \\ \begin{align} & \frac{x}{2}-\frac{4}{13}& = & \frac{-4}{5}x-7 \\\Leftrightarrow & \color{blue}{130.} (\frac{65x}{ \color{blue}{130} }- \frac{ 40 }{ \color{blue}{130} })& = & (\frac{-104}{ \color{blue}{130} }x-\frac{910}{ \color{blue}{130} }) \color{blue}{.130} \\\Leftrightarrow & 65x-40& = & -104x-910 \\\Leftrightarrow & 65x \color{red}{-40} \color{blue}{+40} \color{blue}{+104x} & = & \color{red}{-104x} -910 \color{blue}{+104x} \color{blue}{+40} \\\Leftrightarrow & 169x& = & -870 \\\Leftrightarrow & \frac{169x}{ \color{red}{169} }& = & \frac{-870}{169} \\\Leftrightarrow & x = \frac{-870}{169} & & \\ & V = \left\{ \frac{-870}{169} \right\} & \\\end{align}\)
2. \(\text{560 is het kleinste gemene veelvoud van 7, 16 en 5} \\ \begin{align} & \frac{x}{7}+\frac{3}{16}& = & \frac{1}{5}x+8 \\\Leftrightarrow & \color{blue}{560.} (\frac{80x}{ \color{blue}{560} }+ \frac{ 105 }{ \color{blue}{560} })& = & (\frac{112}{ \color{blue}{560} }x+\frac{4480}{ \color{blue}{560} }) \color{blue}{.560} \\\Leftrightarrow & 80x+105& = & 112x+4480 \\\Leftrightarrow & 80x \color{red}{+105} \color{blue}{-105} \color{blue}{-112x} & = & \color{red}{112x} +4480 \color{blue}{-112x} \color{blue}{-105} \\\Leftrightarrow & -32x& = & 4375 \\\Leftrightarrow & \frac{-32x}{ \color{red}{-32} }& = & \frac{4375}{-32} \\\Leftrightarrow & x = \frac{-4375}{32} & & \\ & V = \left\{ \frac{-4375}{32} \right\} & \\\end{align}\)
3. \(\text{18 is het kleinste gemene veelvoud van 2, 9 en 3} \\ \begin{align} & \frac{x}{2}-\frac{4}{9}& = & \frac{2}{3}x+1 \\\Leftrightarrow & \color{blue}{18.} (\frac{9x}{ \color{blue}{18} }- \frac{ 8 }{ \color{blue}{18} })& = & (\frac{12}{ \color{blue}{18} }x+\frac{18}{ \color{blue}{18} }) \color{blue}{.18} \\\Leftrightarrow & 9x-8& = & 12x+18 \\\Leftrightarrow & 9x \color{red}{-8} \color{blue}{+8} \color{blue}{-12x} & = & \color{red}{12x} +18 \color{blue}{-12x} \color{blue}{+8} \\\Leftrightarrow & -3x& = & 26 \\\Leftrightarrow & \frac{-3x}{ \color{red}{-3} }& = & \frac{26}{-3} \\\Leftrightarrow & x = \frac{-26}{3} & & \\ & V = \left\{ \frac{-26}{3} \right\} & \\\end{align}\)
4. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}+\frac{5}{7}& = & \frac{-7}{5}x+8 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }+ \frac{ 150 }{ \color{blue}{210} })& = & (\frac{-294}{ \color{blue}{210} }x+\frac{1680}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x+150& = & -294x+1680 \\\Leftrightarrow & 35x \color{red}{+150} \color{blue}{-150} \color{blue}{+294x} & = & \color{red}{-294x} +1680 \color{blue}{+294x} \color{blue}{-150} \\\Leftrightarrow & 329x& = & 1530 \\\Leftrightarrow & \frac{329x}{ \color{red}{329} }& = & \frac{1530}{329} \\\Leftrightarrow & x = \frac{1530}{329} & & \\ & V = \left\{ \frac{1530}{329} \right\} & \\\end{align}\)
5. \(\text{90 is het kleinste gemene veelvoud van 6, 9 en 5} \\ \begin{align} & \frac{x}{6}+\frac{5}{9}& = & \frac{-4}{5}x+4 \\\Leftrightarrow & \color{blue}{90.} (\frac{15x}{ \color{blue}{90} }+ \frac{ 50 }{ \color{blue}{90} })& = & (\frac{-72}{ \color{blue}{90} }x+\frac{360}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 15x+50& = & -72x+360 \\\Leftrightarrow & 15x \color{red}{+50} \color{blue}{-50} \color{blue}{+72x} & = & \color{red}{-72x} +360 \color{blue}{+72x} \color{blue}{-50} \\\Leftrightarrow & 87x& = & 310 \\\Leftrightarrow & \frac{87x}{ \color{red}{87} }& = & \frac{310}{87} \\\Leftrightarrow & x = \frac{310}{87} & & \\ & V = \left\{ \frac{310}{87} \right\} & \\\end{align}\)
6. \(\text{165 is het kleinste gemene veelvoud van 3, 11 en 5} \\ \begin{align} & \frac{x}{3}-\frac{5}{11}& = & \frac{1}{5}x-2 \\\Leftrightarrow & \color{blue}{165.} (\frac{55x}{ \color{blue}{165} }- \frac{ 75 }{ \color{blue}{165} })& = & (\frac{33}{ \color{blue}{165} }x-\frac{330}{ \color{blue}{165} }) \color{blue}{.165} \\\Leftrightarrow & 55x-75& = & 33x-330 \\\Leftrightarrow & 55x \color{red}{-75} \color{blue}{+75} \color{blue}{-33x} & = & \color{red}{33x} -330 \color{blue}{-33x} \color{blue}{+75} \\\Leftrightarrow & 22x& = & -255 \\\Leftrightarrow & \frac{22x}{ \color{red}{22} }& = & \frac{-255}{22} \\\Leftrightarrow & x = \frac{-255}{22} & & \\ & V = \left\{ \frac{-255}{22} \right\} & \\\end{align}\)
7. \(\text{330 is het kleinste gemene veelvoud van 6, 11 en 5} \\ \begin{align} & \frac{x}{6}-\frac{5}{11}& = & \frac{-2}{5}x-8 \\\Leftrightarrow & \color{blue}{330.} (\frac{55x}{ \color{blue}{330} }- \frac{ 150 }{ \color{blue}{330} })& = & (\frac{-132}{ \color{blue}{330} }x-\frac{2640}{ \color{blue}{330} }) \color{blue}{.330} \\\Leftrightarrow & 55x-150& = & -132x-2640 \\\Leftrightarrow & 55x \color{red}{-150} \color{blue}{+150} \color{blue}{+132x} & = & \color{red}{-132x} -2640 \color{blue}{+132x} \color{blue}{+150} \\\Leftrightarrow & 187x& = & -2490 \\\Leftrightarrow & \frac{187x}{ \color{red}{187} }& = & \frac{-2490}{187} \\\Leftrightarrow & x = \frac{-2490}{187} & & \\ & V = \left\{ \frac{-2490}{187} \right\} & \\\end{align}\)
8. \(\text{30 is het kleinste gemene veelvoud van 3, 15 en 2} \\ \begin{align} & \frac{x}{3}-\frac{2}{15}& = & \frac{1}{2}x-5 \\\Leftrightarrow & \color{blue}{30.} (\frac{10x}{ \color{blue}{30} }- \frac{ 4 }{ \color{blue}{30} })& = & (\frac{15}{ \color{blue}{30} }x-\frac{150}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 10x-4& = & 15x-150 \\\Leftrightarrow & 10x \color{red}{-4} \color{blue}{+4} \color{blue}{-15x} & = & \color{red}{15x} -150 \color{blue}{-15x} \color{blue}{+4} \\\Leftrightarrow & -5x& = & -146 \\\Leftrightarrow & \frac{-5x}{ \color{red}{-5} }& = & \frac{-146}{-5} \\\Leftrightarrow & x = \frac{146}{5} & & \\ & V = \left\{ \frac{146}{5} \right\} & \\\end{align}\)
9. \(\text{240 is het kleinste gemene veelvoud van 5, 16 en 6} \\ \begin{align} & \frac{x}{5}-\frac{5}{16}& = & \frac{-5}{6}x+3 \\\Leftrightarrow & \color{blue}{240.} (\frac{48x}{ \color{blue}{240} }- \frac{ 75 }{ \color{blue}{240} })& = & (\frac{-200}{ \color{blue}{240} }x+\frac{720}{ \color{blue}{240} }) \color{blue}{.240} \\\Leftrightarrow & 48x-75& = & -200x+720 \\\Leftrightarrow & 48x \color{red}{-75} \color{blue}{+75} \color{blue}{+200x} & = & \color{red}{-200x} +720 \color{blue}{+200x} \color{blue}{+75} \\\Leftrightarrow & 248x& = & 795 \\\Leftrightarrow & \frac{248x}{ \color{red}{248} }& = & \frac{795}{248} \\\Leftrightarrow & x = \frac{795}{248} & & \\ & V = \left\{ \frac{795}{248} \right\} & \\\end{align}\)
10. \(\text{210 is het kleinste gemene veelvoud van 6, 14 en 5} \\ \begin{align} & \frac{x}{6}+\frac{3}{14}& = & \frac{-7}{5}x-6 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }+ \frac{ 45 }{ \color{blue}{210} })& = & (\frac{-294}{ \color{blue}{210} }x-\frac{1260}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x+45& = & -294x-1260 \\\Leftrightarrow & 35x \color{red}{+45} \color{blue}{-45} \color{blue}{+294x} & = & \color{red}{-294x} -1260 \color{blue}{+294x} \color{blue}{-45} \\\Leftrightarrow & 329x& = & -1305 \\\Leftrightarrow & \frac{329x}{ \color{red}{329} }& = & \frac{-1305}{329} \\\Leftrightarrow & x = \frac{-1305}{329} & & \\ & V = \left\{ \frac{-1305}{329} \right\} & \\\end{align}\)
11. \(\text{42 is het kleinste gemene veelvoud van 2, 7 en 3} \\ \begin{align} & \frac{x}{2}+\frac{2}{7}& = & \frac{-2}{3}x+8 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }+ \frac{ 12 }{ \color{blue}{42} })& = & (\frac{-28}{ \color{blue}{42} }x+\frac{336}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x+12& = & -28x+336 \\\Leftrightarrow & 21x \color{red}{+12} \color{blue}{-12} \color{blue}{+28x} & = & \color{red}{-28x} +336 \color{blue}{+28x} \color{blue}{-12} \\\Leftrightarrow & 49x& = & 324 \\\Leftrightarrow & \frac{49x}{ \color{red}{49} }& = & \frac{324}{49} \\\Leftrightarrow & x = \frac{324}{49} & & \\ & V = \left\{ \frac{324}{49} \right\} & \\\end{align}\)
12. \(\text{330 is het kleinste gemene veelvoud van 6, 11 en 5} \\ \begin{align} & \frac{x}{6}+\frac{4}{11}& = & \frac{1}{5}x+5 \\\Leftrightarrow & \color{blue}{330.} (\frac{55x}{ \color{blue}{330} }+ \frac{ 120 }{ \color{blue}{330} })& = & (\frac{66}{ \color{blue}{330} }x+\frac{1650}{ \color{blue}{330} }) \color{blue}{.330} \\\Leftrightarrow & 55x+120& = & 66x+1650 \\\Leftrightarrow & 55x \color{red}{+120} \color{blue}{-120} \color{blue}{-66x} & = & \color{red}{66x} +1650 \color{blue}{-66x} \color{blue}{-120} \\\Leftrightarrow & -11x& = & 1530 \\\Leftrightarrow & \frac{-11x}{ \color{red}{-11} }& = & \frac{1530}{-11} \\\Leftrightarrow & x = \frac{-1530}{11} & & \\ & V = \left\{ \frac{-1530}{11} \right\} & \\\end{align}\)