Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{42 is het kleinste gemene veelvoud van 2, 7 en 3} \\ \begin{align} & \frac{x}{2}-\frac{4}{7}& = & \frac{-8}{3}x+1 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }- \frac{ 24 }{ \color{blue}{42} })& = & (\frac{-112}{ \color{blue}{42} }x+\frac{42}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x-24& = & -112x+42 \\\Leftrightarrow & 21x \color{red}{-24} \color{blue}{+24} \color{blue}{+112x} & = & \color{red}{-112x} +42 \color{blue}{+112x} \color{blue}{+24} \\\Leftrightarrow & 133x& = & 66 \\\Leftrightarrow & \frac{133x}{ \color{red}{133} }& = & \frac{66}{133} \\\Leftrightarrow & x = \frac{66}{133} & & \\ & V = \left\{ \frac{66}{133} \right\} & \\\end{align}\)
2. \(\text{84 is het kleinste gemene veelvoud van 3, 7 en 4} \\ \begin{align} & \frac{x}{3}-\frac{2}{7}& = & \frac{-3}{4}x+8 \\\Leftrightarrow & \color{blue}{84.} (\frac{28x}{ \color{blue}{84} }- \frac{ 24 }{ \color{blue}{84} })& = & (\frac{-63}{ \color{blue}{84} }x+\frac{672}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 28x-24& = & -63x+672 \\\Leftrightarrow & 28x \color{red}{-24} \color{blue}{+24} \color{blue}{+63x} & = & \color{red}{-63x} +672 \color{blue}{+63x} \color{blue}{+24} \\\Leftrightarrow & 91x& = & 696 \\\Leftrightarrow & \frac{91x}{ \color{red}{91} }& = & \frac{696}{91} \\\Leftrightarrow & x = \frac{696}{91} & & \\ & V = \left\{ \frac{696}{91} \right\} & \\\end{align}\)
3. \(\text{546 is het kleinste gemene veelvoud van 7, 13 en 6} \\ \begin{align} & \frac{x}{7}+\frac{4}{13}& = & \frac{-5}{6}x-5 \\\Leftrightarrow & \color{blue}{546.} (\frac{78x}{ \color{blue}{546} }+ \frac{ 168 }{ \color{blue}{546} })& = & (\frac{-455}{ \color{blue}{546} }x-\frac{2730}{ \color{blue}{546} }) \color{blue}{.546} \\\Leftrightarrow & 78x+168& = & -455x-2730 \\\Leftrightarrow & 78x \color{red}{+168} \color{blue}{-168} \color{blue}{+455x} & = & \color{red}{-455x} -2730 \color{blue}{+455x} \color{blue}{-168} \\\Leftrightarrow & 533x& = & -2898 \\\Leftrightarrow & \frac{533x}{ \color{red}{533} }& = & \frac{-2898}{533} \\\Leftrightarrow & x = \frac{-2898}{533} & & \\ & V = \left\{ \frac{-2898}{533} \right\} & \\\end{align}\)
4. \(\text{30 is het kleinste gemene veelvoud van 6, 10 en 5} \\ \begin{align} & \frac{x}{6}+\frac{3}{10}& = & \frac{1}{5}x+5 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }+ \frac{ 9 }{ \color{blue}{30} })& = & (\frac{6}{ \color{blue}{30} }x+\frac{150}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x+9& = & 6x+150 \\\Leftrightarrow & 5x \color{red}{+9} \color{blue}{-9} \color{blue}{-6x} & = & \color{red}{6x} +150 \color{blue}{-6x} \color{blue}{-9} \\\Leftrightarrow & -x& = & 141 \\\Leftrightarrow & \frac{-x}{ \color{red}{-1} }& = & \frac{141}{-1} \\\Leftrightarrow & x = -141 & & \\ & V = \left\{ -141 \right\} & \\\end{align}\)
5. \(\text{60 is het kleinste gemene veelvoud van 6, 12 en 5} \\ \begin{align} & \frac{x}{6}-\frac{5}{12}& = & \frac{7}{5}x-3 \\\Leftrightarrow & \color{blue}{60.} (\frac{10x}{ \color{blue}{60} }- \frac{ 25 }{ \color{blue}{60} })& = & (\frac{84}{ \color{blue}{60} }x-\frac{180}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 10x-25& = & 84x-180 \\\Leftrightarrow & 10x \color{red}{-25} \color{blue}{+25} \color{blue}{-84x} & = & \color{red}{84x} -180 \color{blue}{-84x} \color{blue}{+25} \\\Leftrightarrow & -74x& = & -155 \\\Leftrightarrow & \frac{-74x}{ \color{red}{-74} }& = & \frac{-155}{-74} \\\Leftrightarrow & x = \frac{155}{74} & & \\ & V = \left\{ \frac{155}{74} \right\} & \\\end{align}\)
6. \(\text{105 is het kleinste gemene veelvoud van 7, 15 en 3} \\ \begin{align} & \frac{x}{7}+\frac{4}{15}& = & \frac{7}{3}x-3 \\\Leftrightarrow & \color{blue}{105.} (\frac{15x}{ \color{blue}{105} }+ \frac{ 28 }{ \color{blue}{105} })& = & (\frac{245}{ \color{blue}{105} }x-\frac{315}{ \color{blue}{105} }) \color{blue}{.105} \\\Leftrightarrow & 15x+28& = & 245x-315 \\\Leftrightarrow & 15x \color{red}{+28} \color{blue}{-28} \color{blue}{-245x} & = & \color{red}{245x} -315 \color{blue}{-245x} \color{blue}{-28} \\\Leftrightarrow & -230x& = & -343 \\\Leftrightarrow & \frac{-230x}{ \color{red}{-230} }& = & \frac{-343}{-230} \\\Leftrightarrow & x = \frac{343}{230} & & \\ & V = \left\{ \frac{343}{230} \right\} & \\\end{align}\)
7. \(\text{105 is het kleinste gemene veelvoud van 5, 7 en 3} \\ \begin{align} & \frac{x}{5}-\frac{2}{7}& = & \frac{-2}{3}x-4 \\\Leftrightarrow & \color{blue}{105.} (\frac{21x}{ \color{blue}{105} }- \frac{ 30 }{ \color{blue}{105} })& = & (\frac{-70}{ \color{blue}{105} }x-\frac{420}{ \color{blue}{105} }) \color{blue}{.105} \\\Leftrightarrow & 21x-30& = & -70x-420 \\\Leftrightarrow & 21x \color{red}{-30} \color{blue}{+30} \color{blue}{+70x} & = & \color{red}{-70x} -420 \color{blue}{+70x} \color{blue}{+30} \\\Leftrightarrow & 91x& = & -390 \\\Leftrightarrow & \frac{91x}{ \color{red}{91} }& = & \frac{-390}{91} \\\Leftrightarrow & x = \frac{-30}{7} & & \\ & V = \left\{ \frac{-30}{7} \right\} & \\\end{align}\)
8. \(\text{546 is het kleinste gemene veelvoud van 7, 13 en 6} \\ \begin{align} & \frac{x}{7}+\frac{2}{13}& = & \frac{7}{6}x+3 \\\Leftrightarrow & \color{blue}{546.} (\frac{78x}{ \color{blue}{546} }+ \frac{ 84 }{ \color{blue}{546} })& = & (\frac{637}{ \color{blue}{546} }x+\frac{1638}{ \color{blue}{546} }) \color{blue}{.546} \\\Leftrightarrow & 78x+84& = & 637x+1638 \\\Leftrightarrow & 78x \color{red}{+84} \color{blue}{-84} \color{blue}{-637x} & = & \color{red}{637x} +1638 \color{blue}{-637x} \color{blue}{-84} \\\Leftrightarrow & -559x& = & 1554 \\\Leftrightarrow & \frac{-559x}{ \color{red}{-559} }& = & \frac{1554}{-559} \\\Leftrightarrow & x = \frac{-1554}{559} & & \\ & V = \left\{ \frac{-1554}{559} \right\} & \\\end{align}\)
9. \(\text{120 is het kleinste gemene veelvoud van 6, 8 en 5} \\ \begin{align} & \frac{x}{6}+\frac{5}{8}& = & \frac{2}{5}x+2 \\\Leftrightarrow & \color{blue}{120.} (\frac{20x}{ \color{blue}{120} }+ \frac{ 75 }{ \color{blue}{120} })& = & (\frac{48}{ \color{blue}{120} }x+\frac{240}{ \color{blue}{120} }) \color{blue}{.120} \\\Leftrightarrow & 20x+75& = & 48x+240 \\\Leftrightarrow & 20x \color{red}{+75} \color{blue}{-75} \color{blue}{-48x} & = & \color{red}{48x} +240 \color{blue}{-48x} \color{blue}{-75} \\\Leftrightarrow & -28x& = & 165 \\\Leftrightarrow & \frac{-28x}{ \color{red}{-28} }& = & \frac{165}{-28} \\\Leftrightarrow & x = \frac{-165}{28} & & \\ & V = \left\{ \frac{-165}{28} \right\} & \\\end{align}\)
10. \(\text{30 is het kleinste gemene veelvoud van 6, 15 en 5} \\ \begin{align} & \frac{x}{6}-\frac{2}{15}& = & \frac{1}{5}x+4 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }- \frac{ 4 }{ \color{blue}{30} })& = & (\frac{6}{ \color{blue}{30} }x+\frac{120}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x-4& = & 6x+120 \\\Leftrightarrow & 5x \color{red}{-4} \color{blue}{+4} \color{blue}{-6x} & = & \color{red}{6x} +120 \color{blue}{-6x} \color{blue}{+4} \\\Leftrightarrow & -x& = & 124 \\\Leftrightarrow & \frac{-x}{ \color{red}{-1} }& = & \frac{124}{-1} \\\Leftrightarrow & x = -124 & & \\ & V = \left\{ -124 \right\} & \\\end{align}\)
11. \(\text{273 is het kleinste gemene veelvoud van 7, 13 en 3} \\ \begin{align} & \frac{x}{7}+\frac{5}{13}& = & \frac{1}{3}x-4 \\\Leftrightarrow & \color{blue}{273.} (\frac{39x}{ \color{blue}{273} }+ \frac{ 105 }{ \color{blue}{273} })& = & (\frac{91}{ \color{blue}{273} }x-\frac{1092}{ \color{blue}{273} }) \color{blue}{.273} \\\Leftrightarrow & 39x+105& = & 91x-1092 \\\Leftrightarrow & 39x \color{red}{+105} \color{blue}{-105} \color{blue}{-91x} & = & \color{red}{91x} -1092 \color{blue}{-91x} \color{blue}{-105} \\\Leftrightarrow & -52x& = & -1197 \\\Leftrightarrow & \frac{-52x}{ \color{red}{-52} }& = & \frac{-1197}{-52} \\\Leftrightarrow & x = \frac{1197}{52} & & \\ & V = \left\{ \frac{1197}{52} \right\} & \\\end{align}\)
12. \(\text{14 is het kleinste gemene veelvoud van 7, 7 en 2} \\ \begin{align} & \frac{x}{7}-\frac{4}{7}& = & \frac{1}{2}x+7 \\\Leftrightarrow & \color{blue}{14.} (\frac{2x}{ \color{blue}{14} }- \frac{ 8 }{ \color{blue}{14} })& = & (\frac{7}{ \color{blue}{14} }x+\frac{98}{ \color{blue}{14} }) \color{blue}{.14} \\\Leftrightarrow & 2x-8& = & 7x+98 \\\Leftrightarrow & 2x \color{red}{-8} \color{blue}{+8} \color{blue}{-7x} & = & \color{red}{7x} +98 \color{blue}{-7x} \color{blue}{+8} \\\Leftrightarrow & -5x& = & 106 \\\Leftrightarrow & \frac{-5x}{ \color{red}{-5} }& = & \frac{106}{-5} \\\Leftrightarrow & x = \frac{-106}{5} & & \\ & V = \left\{ \frac{-106}{5} \right\} & \\\end{align}\)