Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{35 is het kleinste gemene veelvoud van 7, 7 en 5} \\ \begin{align} & \frac{x}{7}-\frac{2}{7}& = & \frac{2}{5}x+8 \\\Leftrightarrow & \color{blue}{35.} (\frac{5x}{ \color{blue}{35} }- \frac{ 10 }{ \color{blue}{35} })& = & (\frac{14}{ \color{blue}{35} }x+\frac{280}{ \color{blue}{35} }) \color{blue}{.35} \\\Leftrightarrow & 5x-10& = & 14x+280 \\\Leftrightarrow & 5x \color{red}{-10} \color{blue}{+10} \color{blue}{-14x} & = & \color{red}{14x} +280 \color{blue}{-14x} \color{blue}{+10} \\\Leftrightarrow & -9x& = & 290 \\\Leftrightarrow & \frac{-9x}{ \color{red}{-9} }& = & \frac{290}{-9} \\\Leftrightarrow & x = \frac{-290}{9} & & \\ & V = \left\{ \frac{-290}{9} \right\} & \\\end{align}\)
2. \(\text{42 is het kleinste gemene veelvoud van 3, 7 en 2} \\ \begin{align} & \frac{x}{3}+\frac{4}{7}& = & \frac{1}{2}x-6 \\\Leftrightarrow & \color{blue}{42.} (\frac{14x}{ \color{blue}{42} }+ \frac{ 24 }{ \color{blue}{42} })& = & (\frac{21}{ \color{blue}{42} }x-\frac{252}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 14x+24& = & 21x-252 \\\Leftrightarrow & 14x \color{red}{+24} \color{blue}{-24} \color{blue}{-21x} & = & \color{red}{21x} -252 \color{blue}{-21x} \color{blue}{-24} \\\Leftrightarrow & -7x& = & -276 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{-276}{-7} \\\Leftrightarrow & x = \frac{276}{7} & & \\ & V = \left\{ \frac{276}{7} \right\} & \\\end{align}\)
3. \(\text{60 is het kleinste gemene veelvoud van 3, 15 en 4} \\ \begin{align} & \frac{x}{3}+\frac{2}{15}& = & \frac{-7}{4}x+4 \\\Leftrightarrow & \color{blue}{60.} (\frac{20x}{ \color{blue}{60} }+ \frac{ 8 }{ \color{blue}{60} })& = & (\frac{-105}{ \color{blue}{60} }x+\frac{240}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 20x+8& = & -105x+240 \\\Leftrightarrow & 20x \color{red}{+8} \color{blue}{-8} \color{blue}{+105x} & = & \color{red}{-105x} +240 \color{blue}{+105x} \color{blue}{-8} \\\Leftrightarrow & 125x& = & 232 \\\Leftrightarrow & \frac{125x}{ \color{red}{125} }& = & \frac{232}{125} \\\Leftrightarrow & x = \frac{232}{125} & & \\ & V = \left\{ \frac{232}{125} \right\} & \\\end{align}\)
4. \(\text{156 is het kleinste gemene veelvoud van 3, 13 en 4} \\ \begin{align} & \frac{x}{3}+\frac{5}{13}& = & \frac{-3}{4}x+1 \\\Leftrightarrow & \color{blue}{156.} (\frac{52x}{ \color{blue}{156} }+ \frac{ 60 }{ \color{blue}{156} })& = & (\frac{-117}{ \color{blue}{156} }x+\frac{156}{ \color{blue}{156} }) \color{blue}{.156} \\\Leftrightarrow & 52x+60& = & -117x+156 \\\Leftrightarrow & 52x \color{red}{+60} \color{blue}{-60} \color{blue}{+117x} & = & \color{red}{-117x} +156 \color{blue}{+117x} \color{blue}{-60} \\\Leftrightarrow & 169x& = & 96 \\\Leftrightarrow & \frac{169x}{ \color{red}{169} }& = & \frac{96}{169} \\\Leftrightarrow & x = \frac{96}{169} & & \\ & V = \left\{ \frac{96}{169} \right\} & \\\end{align}\)
5. \(\text{132 is het kleinste gemene veelvoud van 3, 11 en 4} \\ \begin{align} & \frac{x}{3}+\frac{4}{11}& = & \frac{-7}{4}x-8 \\\Leftrightarrow & \color{blue}{132.} (\frac{44x}{ \color{blue}{132} }+ \frac{ 48 }{ \color{blue}{132} })& = & (\frac{-231}{ \color{blue}{132} }x-\frac{1056}{ \color{blue}{132} }) \color{blue}{.132} \\\Leftrightarrow & 44x+48& = & -231x-1056 \\\Leftrightarrow & 44x \color{red}{+48} \color{blue}{-48} \color{blue}{+231x} & = & \color{red}{-231x} -1056 \color{blue}{+231x} \color{blue}{-48} \\\Leftrightarrow & 275x& = & -1104 \\\Leftrightarrow & \frac{275x}{ \color{red}{275} }& = & \frac{-1104}{275} \\\Leftrightarrow & x = \frac{-1104}{275} & & \\ & V = \left\{ \frac{-1104}{275} \right\} & \\\end{align}\)
6. \(\text{70 is het kleinste gemene veelvoud van 5, 7 en 2} \\ \begin{align} & \frac{x}{5}+\frac{4}{7}& = & \frac{1}{2}x-3 \\\Leftrightarrow & \color{blue}{70.} (\frac{14x}{ \color{blue}{70} }+ \frac{ 40 }{ \color{blue}{70} })& = & (\frac{35}{ \color{blue}{70} }x-\frac{210}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 14x+40& = & 35x-210 \\\Leftrightarrow & 14x \color{red}{+40} \color{blue}{-40} \color{blue}{-35x} & = & \color{red}{35x} -210 \color{blue}{-35x} \color{blue}{-40} \\\Leftrightarrow & -21x& = & -250 \\\Leftrightarrow & \frac{-21x}{ \color{red}{-21} }& = & \frac{-250}{-21} \\\Leftrightarrow & x = \frac{250}{21} & & \\ & V = \left\{ \frac{250}{21} \right\} & \\\end{align}\)
7. \(\text{30 is het kleinste gemene veelvoud van 6, 15 en 5} \\ \begin{align} & \frac{x}{6}-\frac{4}{15}& = & \frac{1}{5}x-8 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }- \frac{ 8 }{ \color{blue}{30} })& = & (\frac{6}{ \color{blue}{30} }x-\frac{240}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x-8& = & 6x-240 \\\Leftrightarrow & 5x \color{red}{-8} \color{blue}{+8} \color{blue}{-6x} & = & \color{red}{6x} -240 \color{blue}{-6x} \color{blue}{+8} \\\Leftrightarrow & -x& = & -232 \\\Leftrightarrow & \frac{-x}{ \color{red}{-1} }& = & \frac{-232}{-1} \\\Leftrightarrow & x = 232 & & \\ & V = \left\{ 232 \right\} & \\\end{align}\)
8. \(\text{60 is het kleinste gemene veelvoud van 3, 15 en 4} \\ \begin{align} & \frac{x}{3}-\frac{4}{15}& = & \frac{3}{4}x-7 \\\Leftrightarrow & \color{blue}{60.} (\frac{20x}{ \color{blue}{60} }- \frac{ 16 }{ \color{blue}{60} })& = & (\frac{45}{ \color{blue}{60} }x-\frac{420}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 20x-16& = & 45x-420 \\\Leftrightarrow & 20x \color{red}{-16} \color{blue}{+16} \color{blue}{-45x} & = & \color{red}{45x} -420 \color{blue}{-45x} \color{blue}{+16} \\\Leftrightarrow & -25x& = & -404 \\\Leftrightarrow & \frac{-25x}{ \color{red}{-25} }& = & \frac{-404}{-25} \\\Leftrightarrow & x = \frac{404}{25} & & \\ & V = \left\{ \frac{404}{25} \right\} & \\\end{align}\)
9. \(\text{24 is het kleinste gemene veelvoud van 3, 8 en 2} \\ \begin{align} & \frac{x}{3}+\frac{3}{8}& = & \frac{1}{2}x-8 \\\Leftrightarrow & \color{blue}{24.} (\frac{8x}{ \color{blue}{24} }+ \frac{ 9 }{ \color{blue}{24} })& = & (\frac{12}{ \color{blue}{24} }x-\frac{192}{ \color{blue}{24} }) \color{blue}{.24} \\\Leftrightarrow & 8x+9& = & 12x-192 \\\Leftrightarrow & 8x \color{red}{+9} \color{blue}{-9} \color{blue}{-12x} & = & \color{red}{12x} -192 \color{blue}{-12x} \color{blue}{-9} \\\Leftrightarrow & -4x& = & -201 \\\Leftrightarrow & \frac{-4x}{ \color{red}{-4} }& = & \frac{-201}{-4} \\\Leftrightarrow & x = \frac{201}{4} & & \\ & V = \left\{ \frac{201}{4} \right\} & \\\end{align}\)
10. \(\text{546 is het kleinste gemene veelvoud van 7, 13 en 6} \\ \begin{align} & \frac{x}{7}+\frac{2}{13}& = & \frac{7}{6}x+5 \\\Leftrightarrow & \color{blue}{546.} (\frac{78x}{ \color{blue}{546} }+ \frac{ 84 }{ \color{blue}{546} })& = & (\frac{637}{ \color{blue}{546} }x+\frac{2730}{ \color{blue}{546} }) \color{blue}{.546} \\\Leftrightarrow & 78x+84& = & 637x+2730 \\\Leftrightarrow & 78x \color{red}{+84} \color{blue}{-84} \color{blue}{-637x} & = & \color{red}{637x} +2730 \color{blue}{-637x} \color{blue}{-84} \\\Leftrightarrow & -559x& = & 2646 \\\Leftrightarrow & \frac{-559x}{ \color{red}{-559} }& = & \frac{2646}{-559} \\\Leftrightarrow & x = \frac{-2646}{559} & & \\ & V = \left\{ \frac{-2646}{559} \right\} & \\\end{align}\)
11. \(\text{330 is het kleinste gemene veelvoud van 6, 11 en 5} \\ \begin{align} & \frac{x}{6}+\frac{5}{11}& = & \frac{6}{5}x-3 \\\Leftrightarrow & \color{blue}{330.} (\frac{55x}{ \color{blue}{330} }+ \frac{ 150 }{ \color{blue}{330} })& = & (\frac{396}{ \color{blue}{330} }x-\frac{990}{ \color{blue}{330} }) \color{blue}{.330} \\\Leftrightarrow & 55x+150& = & 396x-990 \\\Leftrightarrow & 55x \color{red}{+150} \color{blue}{-150} \color{blue}{-396x} & = & \color{red}{396x} -990 \color{blue}{-396x} \color{blue}{-150} \\\Leftrightarrow & -341x& = & -1140 \\\Leftrightarrow & \frac{-341x}{ \color{red}{-341} }& = & \frac{-1140}{-341} \\\Leftrightarrow & x = \frac{1140}{341} & & \\ & V = \left\{ \frac{1140}{341} \right\} & \\\end{align}\)
12. \(\text{12 is het kleinste gemene veelvoud van 4, 12 en 3} \\ \begin{align} & \frac{x}{4}+\frac{5}{12}& = & \frac{-5}{3}x-5 \\\Leftrightarrow & \color{blue}{12.} (\frac{3x}{ \color{blue}{12} }+ \frac{ 5 }{ \color{blue}{12} })& = & (\frac{-20}{ \color{blue}{12} }x-\frac{60}{ \color{blue}{12} }) \color{blue}{.12} \\\Leftrightarrow & 3x+5& = & -20x-60 \\\Leftrightarrow & 3x \color{red}{+5} \color{blue}{-5} \color{blue}{+20x} & = & \color{red}{-20x} -60 \color{blue}{+20x} \color{blue}{-5} \\\Leftrightarrow & 23x& = & -65 \\\Leftrightarrow & \frac{23x}{ \color{red}{23} }& = & \frac{-65}{23} \\\Leftrightarrow & x = \frac{-65}{23} & & \\ & V = \left\{ \frac{-65}{23} \right\} & \\\end{align}\)