Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}-\frac{4}{7}& = & \frac{-2}{3}x+3 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }- \frac{ 48 }{ \color{blue}{84} })& = & (\frac{-56}{ \color{blue}{84} }x+\frac{252}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x-48& = & -56x+252 \\\Leftrightarrow & 21x \color{red}{-48} \color{blue}{+48} \color{blue}{+56x} & = & \color{red}{-56x} +252 \color{blue}{+56x} \color{blue}{+48} \\\Leftrightarrow & 77x& = & 300 \\\Leftrightarrow & \frac{77x}{ \color{red}{77} }& = & \frac{300}{77} \\\Leftrightarrow & x = \frac{300}{77} & & \\ & V = \left\{ \frac{300}{77} \right\} & \\\end{align}\)
2. \(\text{48 is het kleinste gemene veelvoud van 3, 16 en 4} \\ \begin{align} & \frac{x}{3}-\frac{5}{16}& = & \frac{-7}{4}x+8 \\\Leftrightarrow & \color{blue}{48.} (\frac{16x}{ \color{blue}{48} }- \frac{ 15 }{ \color{blue}{48} })& = & (\frac{-84}{ \color{blue}{48} }x+\frac{384}{ \color{blue}{48} }) \color{blue}{.48} \\\Leftrightarrow & 16x-15& = & -84x+384 \\\Leftrightarrow & 16x \color{red}{-15} \color{blue}{+15} \color{blue}{+84x} & = & \color{red}{-84x} +384 \color{blue}{+84x} \color{blue}{+15} \\\Leftrightarrow & 100x& = & 399 \\\Leftrightarrow & \frac{100x}{ \color{red}{100} }& = & \frac{399}{100} \\\Leftrightarrow & x = \frac{399}{100} & & \\ & V = \left\{ \frac{399}{100} \right\} & \\\end{align}\)
3. \(\text{240 is het kleinste gemene veelvoud van 5, 16 en 3} \\ \begin{align} & \frac{x}{5}+\frac{3}{16}& = & \frac{-2}{3}x-6 \\\Leftrightarrow & \color{blue}{240.} (\frac{48x}{ \color{blue}{240} }+ \frac{ 45 }{ \color{blue}{240} })& = & (\frac{-160}{ \color{blue}{240} }x-\frac{1440}{ \color{blue}{240} }) \color{blue}{.240} \\\Leftrightarrow & 48x+45& = & -160x-1440 \\\Leftrightarrow & 48x \color{red}{+45} \color{blue}{-45} \color{blue}{+160x} & = & \color{red}{-160x} -1440 \color{blue}{+160x} \color{blue}{-45} \\\Leftrightarrow & 208x& = & -1485 \\\Leftrightarrow & \frac{208x}{ \color{red}{208} }& = & \frac{-1485}{208} \\\Leftrightarrow & x = \frac{-1485}{208} & & \\ & V = \left\{ \frac{-1485}{208} \right\} & \\\end{align}\)
4. \(\text{42 is het kleinste gemene veelvoud van 2, 7 en 3} \\ \begin{align} & \frac{x}{2}-\frac{3}{7}& = & \frac{-5}{3}x+3 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }- \frac{ 18 }{ \color{blue}{42} })& = & (\frac{-70}{ \color{blue}{42} }x+\frac{126}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x-18& = & -70x+126 \\\Leftrightarrow & 21x \color{red}{-18} \color{blue}{+18} \color{blue}{+70x} & = & \color{red}{-70x} +126 \color{blue}{+70x} \color{blue}{+18} \\\Leftrightarrow & 91x& = & 144 \\\Leftrightarrow & \frac{91x}{ \color{red}{91} }& = & \frac{144}{91} \\\Leftrightarrow & x = \frac{144}{91} & & \\ & V = \left\{ \frac{144}{91} \right\} & \\\end{align}\)
5. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}+\frac{4}{7}& = & \frac{1}{5}x+5 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }+ \frac{ 120 }{ \color{blue}{210} })& = & (\frac{42}{ \color{blue}{210} }x+\frac{1050}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x+120& = & 42x+1050 \\\Leftrightarrow & 35x \color{red}{+120} \color{blue}{-120} \color{blue}{-42x} & = & \color{red}{42x} +1050 \color{blue}{-42x} \color{blue}{-120} \\\Leftrightarrow & -7x& = & 930 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{930}{-7} \\\Leftrightarrow & x = \frac{-930}{7} & & \\ & V = \left\{ \frac{-930}{7} \right\} & \\\end{align}\)
6. \(\text{80 is het kleinste gemene veelvoud van 4, 16 en 5} \\ \begin{align} & \frac{x}{4}+\frac{5}{16}& = & \frac{-7}{5}x-3 \\\Leftrightarrow & \color{blue}{80.} (\frac{20x}{ \color{blue}{80} }+ \frac{ 25 }{ \color{blue}{80} })& = & (\frac{-112}{ \color{blue}{80} }x-\frac{240}{ \color{blue}{80} }) \color{blue}{.80} \\\Leftrightarrow & 20x+25& = & -112x-240 \\\Leftrightarrow & 20x \color{red}{+25} \color{blue}{-25} \color{blue}{+112x} & = & \color{red}{-112x} -240 \color{blue}{+112x} \color{blue}{-25} \\\Leftrightarrow & 132x& = & -265 \\\Leftrightarrow & \frac{132x}{ \color{red}{132} }& = & \frac{-265}{132} \\\Leftrightarrow & x = \frac{-265}{132} & & \\ & V = \left\{ \frac{-265}{132} \right\} & \\\end{align}\)
7. \(\text{28 is het kleinste gemene veelvoud van 7, 14 en 4} \\ \begin{align} & \frac{x}{7}+\frac{3}{14}& = & \frac{-3}{4}x+1 \\\Leftrightarrow & \color{blue}{28.} (\frac{4x}{ \color{blue}{28} }+ \frac{ 6 }{ \color{blue}{28} })& = & (\frac{-21}{ \color{blue}{28} }x+\frac{28}{ \color{blue}{28} }) \color{blue}{.28} \\\Leftrightarrow & 4x+6& = & -21x+28 \\\Leftrightarrow & 4x \color{red}{+6} \color{blue}{-6} \color{blue}{+21x} & = & \color{red}{-21x} +28 \color{blue}{+21x} \color{blue}{-6} \\\Leftrightarrow & 25x& = & 22 \\\Leftrightarrow & \frac{25x}{ \color{red}{25} }& = & \frac{22}{25} \\\Leftrightarrow & x = \frac{22}{25} & & \\ & V = \left\{ \frac{22}{25} \right\} & \\\end{align}\)
8. \(\text{42 is het kleinste gemene veelvoud van 3, 7 en 2} \\ \begin{align} & \frac{x}{3}-\frac{2}{7}& = & \frac{1}{2}x+7 \\\Leftrightarrow & \color{blue}{42.} (\frac{14x}{ \color{blue}{42} }- \frac{ 12 }{ \color{blue}{42} })& = & (\frac{21}{ \color{blue}{42} }x+\frac{294}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 14x-12& = & 21x+294 \\\Leftrightarrow & 14x \color{red}{-12} \color{blue}{+12} \color{blue}{-21x} & = & \color{red}{21x} +294 \color{blue}{-21x} \color{blue}{+12} \\\Leftrightarrow & -7x& = & 306 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{306}{-7} \\\Leftrightarrow & x = \frac{-306}{7} & & \\ & V = \left\{ \frac{-306}{7} \right\} & \\\end{align}\)
9. \(\text{30 is het kleinste gemene veelvoud van 3, 6 en 5} \\ \begin{align} & \frac{x}{3}+\frac{5}{6}& = & \frac{-2}{5}x+7 \\\Leftrightarrow & \color{blue}{30.} (\frac{10x}{ \color{blue}{30} }+ \frac{ 25 }{ \color{blue}{30} })& = & (\frac{-12}{ \color{blue}{30} }x+\frac{210}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 10x+25& = & -12x+210 \\\Leftrightarrow & 10x \color{red}{+25} \color{blue}{-25} \color{blue}{+12x} & = & \color{red}{-12x} +210 \color{blue}{+12x} \color{blue}{-25} \\\Leftrightarrow & 22x& = & 185 \\\Leftrightarrow & \frac{22x}{ \color{red}{22} }& = & \frac{185}{22} \\\Leftrightarrow & x = \frac{185}{22} & & \\ & V = \left\{ \frac{185}{22} \right\} & \\\end{align}\)
10. \(\text{42 is het kleinste gemene veelvoud van 3, 7 en 2} \\ \begin{align} & \frac{x}{3}+\frac{4}{7}& = & \frac{1}{2}x-8 \\\Leftrightarrow & \color{blue}{42.} (\frac{14x}{ \color{blue}{42} }+ \frac{ 24 }{ \color{blue}{42} })& = & (\frac{21}{ \color{blue}{42} }x-\frac{336}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 14x+24& = & 21x-336 \\\Leftrightarrow & 14x \color{red}{+24} \color{blue}{-24} \color{blue}{-21x} & = & \color{red}{21x} -336 \color{blue}{-21x} \color{blue}{-24} \\\Leftrightarrow & -7x& = & -360 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{-360}{-7} \\\Leftrightarrow & x = \frac{360}{7} & & \\ & V = \left\{ \frac{360}{7} \right\} & \\\end{align}\)
11. \(\text{42 is het kleinste gemene veelvoud van 2, 7 en 3} \\ \begin{align} & \frac{x}{2}-\frac{2}{7}& = & \frac{1}{3}x+1 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }- \frac{ 12 }{ \color{blue}{42} })& = & (\frac{14}{ \color{blue}{42} }x+\frac{42}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x-12& = & 14x+42 \\\Leftrightarrow & 21x \color{red}{-12} \color{blue}{+12} \color{blue}{-14x} & = & \color{red}{14x} +42 \color{blue}{-14x} \color{blue}{+12} \\\Leftrightarrow & 7x& = & 54 \\\Leftrightarrow & \frac{7x}{ \color{red}{7} }& = & \frac{54}{7} \\\Leftrightarrow & x = \frac{54}{7} & & \\ & V = \left\{ \frac{54}{7} \right\} & \\\end{align}\)
12. \(\text{210 is het kleinste gemene veelvoud van 6, 14 en 5} \\ \begin{align} & \frac{x}{6}-\frac{3}{14}& = & \frac{4}{5}x+3 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }- \frac{ 45 }{ \color{blue}{210} })& = & (\frac{168}{ \color{blue}{210} }x+\frac{630}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x-45& = & 168x+630 \\\Leftrightarrow & 35x \color{red}{-45} \color{blue}{+45} \color{blue}{-168x} & = & \color{red}{168x} +630 \color{blue}{-168x} \color{blue}{+45} \\\Leftrightarrow & -133x& = & 675 \\\Leftrightarrow & \frac{-133x}{ \color{red}{-133} }& = & \frac{675}{-133} \\\Leftrightarrow & x = \frac{-675}{133} & & \\ & V = \left\{ \frac{-675}{133} \right\} & \\\end{align}\)