Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{240 is het kleinste gemene veelvoud van 6, 16 en 5} \\ \begin{align} & \frac{x}{6}-\frac{5}{16}& = & \frac{1}{5}x-5 \\\Leftrightarrow & \color{blue}{240.} (\frac{40x}{ \color{blue}{240} }- \frac{ 75 }{ \color{blue}{240} })& = & (\frac{48}{ \color{blue}{240} }x-\frac{1200}{ \color{blue}{240} }) \color{blue}{.240} \\\Leftrightarrow & 40x-75& = & 48x-1200 \\\Leftrightarrow & 40x \color{red}{-75} \color{blue}{+75} \color{blue}{-48x} & = & \color{red}{48x} -1200 \color{blue}{-48x} \color{blue}{+75} \\\Leftrightarrow & -8x& = & -1125 \\\Leftrightarrow & \frac{-8x}{ \color{red}{-8} }& = & \frac{-1125}{-8} \\\Leftrightarrow & x = \frac{1125}{8} & & \\ & V = \left\{ \frac{1125}{8} \right\} & \\\end{align}\)
2. \(\text{120 is het kleinste gemene veelvoud van 6, 8 en 5} \\ \begin{align} & \frac{x}{6}-\frac{5}{8}& = & \frac{-4}{5}x+3 \\\Leftrightarrow & \color{blue}{120.} (\frac{20x}{ \color{blue}{120} }- \frac{ 75 }{ \color{blue}{120} })& = & (\frac{-96}{ \color{blue}{120} }x+\frac{360}{ \color{blue}{120} }) \color{blue}{.120} \\\Leftrightarrow & 20x-75& = & -96x+360 \\\Leftrightarrow & 20x \color{red}{-75} \color{blue}{+75} \color{blue}{+96x} & = & \color{red}{-96x} +360 \color{blue}{+96x} \color{blue}{+75} \\\Leftrightarrow & 116x& = & 435 \\\Leftrightarrow & \frac{116x}{ \color{red}{116} }& = & \frac{435}{116} \\\Leftrightarrow & x = \frac{15}{4} & & \\ & V = \left\{ \frac{15}{4} \right\} & \\\end{align}\)
3. \(\text{240 is het kleinste gemene veelvoud van 5, 16 en 6} \\ \begin{align} & \frac{x}{5}+\frac{5}{16}& = & \frac{-5}{6}x-2 \\\Leftrightarrow & \color{blue}{240.} (\frac{48x}{ \color{blue}{240} }+ \frac{ 75 }{ \color{blue}{240} })& = & (\frac{-200}{ \color{blue}{240} }x-\frac{480}{ \color{blue}{240} }) \color{blue}{.240} \\\Leftrightarrow & 48x+75& = & -200x-480 \\\Leftrightarrow & 48x \color{red}{+75} \color{blue}{-75} \color{blue}{+200x} & = & \color{red}{-200x} -480 \color{blue}{+200x} \color{blue}{-75} \\\Leftrightarrow & 248x& = & -555 \\\Leftrightarrow & \frac{248x}{ \color{red}{248} }& = & \frac{-555}{248} \\\Leftrightarrow & x = \frac{-555}{248} & & \\ & V = \left\{ \frac{-555}{248} \right\} & \\\end{align}\)
4. \(\text{42 is het kleinste gemene veelvoud van 7, 6 en 2} \\ \begin{align} & \frac{x}{7}-\frac{5}{6}& = & \frac{1}{2}x-5 \\\Leftrightarrow & \color{blue}{42.} (\frac{6x}{ \color{blue}{42} }- \frac{ 35 }{ \color{blue}{42} })& = & (\frac{21}{ \color{blue}{42} }x-\frac{210}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 6x-35& = & 21x-210 \\\Leftrightarrow & 6x \color{red}{-35} \color{blue}{+35} \color{blue}{-21x} & = & \color{red}{21x} -210 \color{blue}{-21x} \color{blue}{+35} \\\Leftrightarrow & -15x& = & -175 \\\Leftrightarrow & \frac{-15x}{ \color{red}{-15} }& = & \frac{-175}{-15} \\\Leftrightarrow & x = \frac{35}{3} & & \\ & V = \left\{ \frac{35}{3} \right\} & \\\end{align}\)
5. \(\text{330 is het kleinste gemene veelvoud van 6, 11 en 5} \\ \begin{align} & \frac{x}{6}-\frac{4}{11}& = & \frac{1}{5}x-8 \\\Leftrightarrow & \color{blue}{330.} (\frac{55x}{ \color{blue}{330} }- \frac{ 120 }{ \color{blue}{330} })& = & (\frac{66}{ \color{blue}{330} }x-\frac{2640}{ \color{blue}{330} }) \color{blue}{.330} \\\Leftrightarrow & 55x-120& = & 66x-2640 \\\Leftrightarrow & 55x \color{red}{-120} \color{blue}{+120} \color{blue}{-66x} & = & \color{red}{66x} -2640 \color{blue}{-66x} \color{blue}{+120} \\\Leftrightarrow & -11x& = & -2520 \\\Leftrightarrow & \frac{-11x}{ \color{red}{-11} }& = & \frac{-2520}{-11} \\\Leftrightarrow & x = \frac{2520}{11} & & \\ & V = \left\{ \frac{2520}{11} \right\} & \\\end{align}\)
6. \(\text{42 is het kleinste gemene veelvoud van 2, 7 en 3} \\ \begin{align} & \frac{x}{2}+\frac{4}{7}& = & \frac{-2}{3}x-6 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }+ \frac{ 24 }{ \color{blue}{42} })& = & (\frac{-28}{ \color{blue}{42} }x-\frac{252}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x+24& = & -28x-252 \\\Leftrightarrow & 21x \color{red}{+24} \color{blue}{-24} \color{blue}{+28x} & = & \color{red}{-28x} -252 \color{blue}{+28x} \color{blue}{-24} \\\Leftrightarrow & 49x& = & -276 \\\Leftrightarrow & \frac{49x}{ \color{red}{49} }& = & \frac{-276}{49} \\\Leftrightarrow & x = \frac{-276}{49} & & \\ & V = \left\{ \frac{-276}{49} \right\} & \\\end{align}\)
7. \(\text{40 is het kleinste gemene veelvoud van 5, 8 en 2} \\ \begin{align} & \frac{x}{5}-\frac{5}{8}& = & \frac{1}{2}x-3 \\\Leftrightarrow & \color{blue}{40.} (\frac{8x}{ \color{blue}{40} }- \frac{ 25 }{ \color{blue}{40} })& = & (\frac{20}{ \color{blue}{40} }x-\frac{120}{ \color{blue}{40} }) \color{blue}{.40} \\\Leftrightarrow & 8x-25& = & 20x-120 \\\Leftrightarrow & 8x \color{red}{-25} \color{blue}{+25} \color{blue}{-20x} & = & \color{red}{20x} -120 \color{blue}{-20x} \color{blue}{+25} \\\Leftrightarrow & -12x& = & -95 \\\Leftrightarrow & \frac{-12x}{ \color{red}{-12} }& = & \frac{-95}{-12} \\\Leftrightarrow & x = \frac{95}{12} & & \\ & V = \left\{ \frac{95}{12} \right\} & \\\end{align}\)
8. \(\text{30 is het kleinste gemene veelvoud van 6, 15 en 5} \\ \begin{align} & \frac{x}{6}-\frac{4}{15}& = & \frac{1}{5}x+1 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }- \frac{ 8 }{ \color{blue}{30} })& = & (\frac{6}{ \color{blue}{30} }x+\frac{30}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x-8& = & 6x+30 \\\Leftrightarrow & 5x \color{red}{-8} \color{blue}{+8} \color{blue}{-6x} & = & \color{red}{6x} +30 \color{blue}{-6x} \color{blue}{+8} \\\Leftrightarrow & -x& = & 38 \\\Leftrightarrow & \frac{-x}{ \color{red}{-1} }& = & \frac{38}{-1} \\\Leftrightarrow & x = -38 & & \\ & V = \left\{ -38 \right\} & \\\end{align}\)
9. \(\text{240 is het kleinste gemene veelvoud van 5, 16 en 6} \\ \begin{align} & \frac{x}{5}+\frac{3}{16}& = & \frac{1}{6}x-6 \\\Leftrightarrow & \color{blue}{240.} (\frac{48x}{ \color{blue}{240} }+ \frac{ 45 }{ \color{blue}{240} })& = & (\frac{40}{ \color{blue}{240} }x-\frac{1440}{ \color{blue}{240} }) \color{blue}{.240} \\\Leftrightarrow & 48x+45& = & 40x-1440 \\\Leftrightarrow & 48x \color{red}{+45} \color{blue}{-45} \color{blue}{-40x} & = & \color{red}{40x} -1440 \color{blue}{-40x} \color{blue}{-45} \\\Leftrightarrow & 8x& = & -1485 \\\Leftrightarrow & \frac{8x}{ \color{red}{8} }& = & \frac{-1485}{8} \\\Leftrightarrow & x = \frac{-1485}{8} & & \\ & V = \left\{ \frac{-1485}{8} \right\} & \\\end{align}\)
10. \(\text{280 is het kleinste gemene veelvoud van 7, 8 en 5} \\ \begin{align} & \frac{x}{7}+\frac{3}{8}& = & \frac{1}{5}x+7 \\\Leftrightarrow & \color{blue}{280.} (\frac{40x}{ \color{blue}{280} }+ \frac{ 105 }{ \color{blue}{280} })& = & (\frac{56}{ \color{blue}{280} }x+\frac{1960}{ \color{blue}{280} }) \color{blue}{.280} \\\Leftrightarrow & 40x+105& = & 56x+1960 \\\Leftrightarrow & 40x \color{red}{+105} \color{blue}{-105} \color{blue}{-56x} & = & \color{red}{56x} +1960 \color{blue}{-56x} \color{blue}{-105} \\\Leftrightarrow & -16x& = & 1855 \\\Leftrightarrow & \frac{-16x}{ \color{red}{-16} }& = & \frac{1855}{-16} \\\Leftrightarrow & x = \frac{-1855}{16} & & \\ & V = \left\{ \frac{-1855}{16} \right\} & \\\end{align}\)
11. \(\text{60 is het kleinste gemene veelvoud van 4, 15 en 3} \\ \begin{align} & \frac{x}{4}-\frac{2}{15}& = & \frac{2}{3}x+4 \\\Leftrightarrow & \color{blue}{60.} (\frac{15x}{ \color{blue}{60} }- \frac{ 8 }{ \color{blue}{60} })& = & (\frac{40}{ \color{blue}{60} }x+\frac{240}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 15x-8& = & 40x+240 \\\Leftrightarrow & 15x \color{red}{-8} \color{blue}{+8} \color{blue}{-40x} & = & \color{red}{40x} +240 \color{blue}{-40x} \color{blue}{+8} \\\Leftrightarrow & -25x& = & 248 \\\Leftrightarrow & \frac{-25x}{ \color{red}{-25} }& = & \frac{248}{-25} \\\Leftrightarrow & x = \frac{-248}{25} & & \\ & V = \left\{ \frac{-248}{25} \right\} & \\\end{align}\)
12. \(\text{30 is het kleinste gemene veelvoud van 2, 15 en 3} \\ \begin{align} & \frac{x}{2}-\frac{4}{15}& = & \frac{-5}{3}x+1 \\\Leftrightarrow & \color{blue}{30.} (\frac{15x}{ \color{blue}{30} }- \frac{ 8 }{ \color{blue}{30} })& = & (\frac{-50}{ \color{blue}{30} }x+\frac{30}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 15x-8& = & -50x+30 \\\Leftrightarrow & 15x \color{red}{-8} \color{blue}{+8} \color{blue}{+50x} & = & \color{red}{-50x} +30 \color{blue}{+50x} \color{blue}{+8} \\\Leftrightarrow & 65x& = & 38 \\\Leftrightarrow & \frac{65x}{ \color{red}{65} }& = & \frac{38}{65} \\\Leftrightarrow & x = \frac{38}{65} & & \\ & V = \left\{ \frac{38}{65} \right\} & \\\end{align}\)