Vgln. eerste graad (reeks 4)

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

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

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

Verbetersleutel

1. \(\text{6 is het kleinste gemene veelvoud van 3, 6 en 2} \\ \begin{align} & \frac{x}{3}+\frac{5}{6}& = & \frac{3}{2}x-2 \\\Leftrightarrow & \color{blue}{6.} (\frac{2x}{ \color{blue}{6} }+ \frac{ 5 }{ \color{blue}{6} })& = & (\frac{9}{ \color{blue}{6} }x-\frac{12}{ \color{blue}{6} }) \color{blue}{.6} \\\Leftrightarrow & 2x+5& = & 9x-12 \\\Leftrightarrow & 2x \color{red}{+5} \color{blue}{-5} \color{blue}{-9x} & = & \color{red}{9x} -12 \color{blue}{-9x} \color{blue}{-5} \\\Leftrightarrow & -7x& = & -17 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{-17}{-7} \\\Leftrightarrow & x = \frac{17}{7} & & \\ & V = \left\{ \frac{17}{7} \right\} & \\\end{align}\)
2. \(\text{126 is het kleinste gemene veelvoud van 7, 9 en 2} \\ \begin{align} & \frac{x}{7}+\frac{4}{9}& = & \frac{1}{2}x+2 \\\Leftrightarrow & \color{blue}{126.} (\frac{18x}{ \color{blue}{126} }+ \frac{ 56 }{ \color{blue}{126} })& = & (\frac{63}{ \color{blue}{126} }x+\frac{252}{ \color{blue}{126} }) \color{blue}{.126} \\\Leftrightarrow & 18x+56& = & 63x+252 \\\Leftrightarrow & 18x \color{red}{+56} \color{blue}{-56} \color{blue}{-63x} & = & \color{red}{63x} +252 \color{blue}{-63x} \color{blue}{-56} \\\Leftrightarrow & -45x& = & 196 \\\Leftrightarrow & \frac{-45x}{ \color{red}{-45} }& = & \frac{196}{-45} \\\Leftrightarrow & x = \frac{-196}{45} & & \\ & V = \left\{ \frac{-196}{45} \right\} & \\\end{align}\)
3. \(\text{84 is het kleinste gemene veelvoud van 3, 7 en 4} \\ \begin{align} & \frac{x}{3}+\frac{4}{7}& = & \frac{1}{4}x-6 \\\Leftrightarrow & \color{blue}{84.} (\frac{28x}{ \color{blue}{84} }+ \frac{ 48 }{ \color{blue}{84} })& = & (\frac{21}{ \color{blue}{84} }x-\frac{504}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 28x+48& = & 21x-504 \\\Leftrightarrow & 28x \color{red}{+48} \color{blue}{-48} \color{blue}{-21x} & = & \color{red}{21x} -504 \color{blue}{-21x} \color{blue}{-48} \\\Leftrightarrow & 7x& = & -552 \\\Leftrightarrow & \frac{7x}{ \color{red}{7} }& = & \frac{-552}{7} \\\Leftrightarrow & x = \frac{-552}{7} & & \\ & V = \left\{ \frac{-552}{7} \right\} & \\\end{align}\)
4. \(\text{66 is het kleinste gemene veelvoud van 2, 11 en 3} \\ \begin{align} & \frac{x}{2}-\frac{3}{11}& = & \frac{4}{3}x+1 \\\Leftrightarrow & \color{blue}{66.} (\frac{33x}{ \color{blue}{66} }- \frac{ 18 }{ \color{blue}{66} })& = & (\frac{88}{ \color{blue}{66} }x+\frac{66}{ \color{blue}{66} }) \color{blue}{.66} \\\Leftrightarrow & 33x-18& = & 88x+66 \\\Leftrightarrow & 33x \color{red}{-18} \color{blue}{+18} \color{blue}{-88x} & = & \color{red}{88x} +66 \color{blue}{-88x} \color{blue}{+18} \\\Leftrightarrow & -55x& = & 84 \\\Leftrightarrow & \frac{-55x}{ \color{red}{-55} }& = & \frac{84}{-55} \\\Leftrightarrow & x = \frac{-84}{55} & & \\ & V = \left\{ \frac{-84}{55} \right\} & \\\end{align}\)
5. \(\text{30 is het kleinste gemene veelvoud van 5, 15 en 6} \\ \begin{align} & \frac{x}{5}-\frac{4}{15}& = & \frac{5}{6}x+7 \\\Leftrightarrow & \color{blue}{30.} (\frac{6x}{ \color{blue}{30} }- \frac{ 8 }{ \color{blue}{30} })& = & (\frac{25}{ \color{blue}{30} }x+\frac{210}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 6x-8& = & 25x+210 \\\Leftrightarrow & 6x \color{red}{-8} \color{blue}{+8} \color{blue}{-25x} & = & \color{red}{25x} +210 \color{blue}{-25x} \color{blue}{+8} \\\Leftrightarrow & -19x& = & 218 \\\Leftrightarrow & \frac{-19x}{ \color{red}{-19} }& = & \frac{218}{-19} \\\Leftrightarrow & x = \frac{-218}{19} & & \\ & V = \left\{ \frac{-218}{19} \right\} & \\\end{align}\)
6. \(\text{240 is het kleinste gemene veelvoud van 6, 16 en 5} \\ \begin{align} & \frac{x}{6}-\frac{5}{16}& = & \frac{1}{5}x+3 \\\Leftrightarrow & \color{blue}{240.} (\frac{40x}{ \color{blue}{240} }- \frac{ 75 }{ \color{blue}{240} })& = & (\frac{48}{ \color{blue}{240} }x+\frac{720}{ \color{blue}{240} }) \color{blue}{.240} \\\Leftrightarrow & 40x-75& = & 48x+720 \\\Leftrightarrow & 40x \color{red}{-75} \color{blue}{+75} \color{blue}{-48x} & = & \color{red}{48x} +720 \color{blue}{-48x} \color{blue}{+75} \\\Leftrightarrow & -8x& = & 795 \\\Leftrightarrow & \frac{-8x}{ \color{red}{-8} }& = & \frac{795}{-8} \\\Leftrightarrow & x = \frac{-795}{8} & & \\ & V = \left\{ \frac{-795}{8} \right\} & \\\end{align}\)
7. \(\text{78 is het kleinste gemene veelvoud van 2, 13 en 3} \\ \begin{align} & \frac{x}{2}+\frac{3}{13}& = & \frac{-2}{3}x-6 \\\Leftrightarrow & \color{blue}{78.} (\frac{39x}{ \color{blue}{78} }+ \frac{ 18 }{ \color{blue}{78} })& = & (\frac{-52}{ \color{blue}{78} }x-\frac{468}{ \color{blue}{78} }) \color{blue}{.78} \\\Leftrightarrow & 39x+18& = & -52x-468 \\\Leftrightarrow & 39x \color{red}{+18} \color{blue}{-18} \color{blue}{+52x} & = & \color{red}{-52x} -468 \color{blue}{+52x} \color{blue}{-18} \\\Leftrightarrow & 91x& = & -486 \\\Leftrightarrow & \frac{91x}{ \color{red}{91} }& = & \frac{-486}{91} \\\Leftrightarrow & x = \frac{-486}{91} & & \\ & V = \left\{ \frac{-486}{91} \right\} & \\\end{align}\)
8. \(\text{70 is het kleinste gemene veelvoud van 7, 10 en 5} \\ \begin{align} & \frac{x}{7}-\frac{3}{10}& = & \frac{3}{5}x+2 \\\Leftrightarrow & \color{blue}{70.} (\frac{10x}{ \color{blue}{70} }- \frac{ 21 }{ \color{blue}{70} })& = & (\frac{42}{ \color{blue}{70} }x+\frac{140}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 10x-21& = & 42x+140 \\\Leftrightarrow & 10x \color{red}{-21} \color{blue}{+21} \color{blue}{-42x} & = & \color{red}{42x} +140 \color{blue}{-42x} \color{blue}{+21} \\\Leftrightarrow & -32x& = & 161 \\\Leftrightarrow & \frac{-32x}{ \color{red}{-32} }& = & \frac{161}{-32} \\\Leftrightarrow & x = \frac{-161}{32} & & \\ & V = \left\{ \frac{-161}{32} \right\} & \\\end{align}\)
9. \(\text{48 is het kleinste gemene veelvoud van 2, 16 en 3} \\ \begin{align} & \frac{x}{2}+\frac{3}{16}& = & \frac{1}{3}x+5 \\\Leftrightarrow & \color{blue}{48.} (\frac{24x}{ \color{blue}{48} }+ \frac{ 9 }{ \color{blue}{48} })& = & (\frac{16}{ \color{blue}{48} }x+\frac{240}{ \color{blue}{48} }) \color{blue}{.48} \\\Leftrightarrow & 24x+9& = & 16x+240 \\\Leftrightarrow & 24x \color{red}{+9} \color{blue}{-9} \color{blue}{-16x} & = & \color{red}{16x} +240 \color{blue}{-16x} \color{blue}{-9} \\\Leftrightarrow & 8x& = & 231 \\\Leftrightarrow & \frac{8x}{ \color{red}{8} }& = & \frac{231}{8} \\\Leftrightarrow & x = \frac{231}{8} & & \\ & V = \left\{ \frac{231}{8} \right\} & \\\end{align}\)
10. \(\text{42 is het kleinste gemene veelvoud van 2, 7 en 3} \\ \begin{align} & \frac{x}{2}-\frac{3}{7}& = & \frac{-5}{3}x+1 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }- \frac{ 18 }{ \color{blue}{42} })& = & (\frac{-70}{ \color{blue}{42} }x+\frac{42}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x-18& = & -70x+42 \\\Leftrightarrow & 21x \color{red}{-18} \color{blue}{+18} \color{blue}{+70x} & = & \color{red}{-70x} +42 \color{blue}{+70x} \color{blue}{+18} \\\Leftrightarrow & 91x& = & 60 \\\Leftrightarrow & \frac{91x}{ \color{red}{91} }& = & \frac{60}{91} \\\Leftrightarrow & x = \frac{60}{91} & & \\ & V = \left\{ \frac{60}{91} \right\} & \\\end{align}\)
11. \(\text{132 is het kleinste gemene veelvoud van 3, 11 en 4} \\ \begin{align} & \frac{x}{3}+\frac{2}{11}& = & \frac{-7}{4}x+5 \\\Leftrightarrow & \color{blue}{132.} (\frac{44x}{ \color{blue}{132} }+ \frac{ 24 }{ \color{blue}{132} })& = & (\frac{-231}{ \color{blue}{132} }x+\frac{660}{ \color{blue}{132} }) \color{blue}{.132} \\\Leftrightarrow & 44x+24& = & -231x+660 \\\Leftrightarrow & 44x \color{red}{+24} \color{blue}{-24} \color{blue}{+231x} & = & \color{red}{-231x} +660 \color{blue}{+231x} \color{blue}{-24} \\\Leftrightarrow & 275x& = & 636 \\\Leftrightarrow & \frac{275x}{ \color{red}{275} }& = & \frac{636}{275} \\\Leftrightarrow & x = \frac{636}{275} & & \\ & V = \left\{ \frac{636}{275} \right\} & \\\end{align}\)
12. \(\text{90 is het kleinste gemene veelvoud van 6, 9 en 5} \\ \begin{align} & \frac{x}{6}+\frac{2}{9}& = & \frac{2}{5}x-8 \\\Leftrightarrow & \color{blue}{90.} (\frac{15x}{ \color{blue}{90} }+ \frac{ 20 }{ \color{blue}{90} })& = & (\frac{36}{ \color{blue}{90} }x-\frac{720}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 15x+20& = & 36x-720 \\\Leftrightarrow & 15x \color{red}{+20} \color{blue}{-20} \color{blue}{-36x} & = & \color{red}{36x} -720 \color{blue}{-36x} \color{blue}{-20} \\\Leftrightarrow & -21x& = & -740 \\\Leftrightarrow & \frac{-21x}{ \color{red}{-21} }& = & \frac{-740}{-21} \\\Leftrightarrow & x = \frac{740}{21} & & \\ & V = \left\{ \frac{740}{21} \right\} & \\\end{align}\)