Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{14 is het kleinste gemene veelvoud van 7, 7 en 2} \\ \begin{align} & \frac{x}{7}+\frac{2}{7}& = & \frac{1}{2}x-5 \\\Leftrightarrow & \color{blue}{14.} (\frac{2x}{ \color{blue}{14} }+ \frac{ 4 }{ \color{blue}{14} })& = & (\frac{7}{ \color{blue}{14} }x-\frac{70}{ \color{blue}{14} }) \color{blue}{.14} \\\Leftrightarrow & 2x+4& = & 7x-70 \\\Leftrightarrow & 2x \color{red}{+4} \color{blue}{-4} \color{blue}{-7x} & = & \color{red}{7x} -70 \color{blue}{-7x} \color{blue}{-4} \\\Leftrightarrow & -5x& = & -74 \\\Leftrightarrow & \frac{-5x}{ \color{red}{-5} }& = & \frac{-74}{-5} \\\Leftrightarrow & x = \frac{74}{5} & & \\ & V = \left\{ \frac{74}{5} \right\} & \\\end{align}\)
2. \(\text{252 is het kleinste gemene veelvoud van 7, 9 en 4} \\ \begin{align} & \frac{x}{7}+\frac{2}{9}& = & \frac{5}{4}x-8 \\\Leftrightarrow & \color{blue}{252.} (\frac{36x}{ \color{blue}{252} }+ \frac{ 56 }{ \color{blue}{252} })& = & (\frac{315}{ \color{blue}{252} }x-\frac{2016}{ \color{blue}{252} }) \color{blue}{.252} \\\Leftrightarrow & 36x+56& = & 315x-2016 \\\Leftrightarrow & 36x \color{red}{+56} \color{blue}{-56} \color{blue}{-315x} & = & \color{red}{315x} -2016 \color{blue}{-315x} \color{blue}{-56} \\\Leftrightarrow & -279x& = & -2072 \\\Leftrightarrow & \frac{-279x}{ \color{red}{-279} }& = & \frac{-2072}{-279} \\\Leftrightarrow & x = \frac{2072}{279} & & \\ & V = \left\{ \frac{2072}{279} \right\} & \\\end{align}\)
3. \(\text{21 is het kleinste gemene veelvoud van 7, 7 en 3} \\ \begin{align} & \frac{x}{7}+\frac{4}{7}& = & \frac{-8}{3}x-8 \\\Leftrightarrow & \color{blue}{21.} (\frac{3x}{ \color{blue}{21} }+ \frac{ 12 }{ \color{blue}{21} })& = & (\frac{-56}{ \color{blue}{21} }x-\frac{168}{ \color{blue}{21} }) \color{blue}{.21} \\\Leftrightarrow & 3x+12& = & -56x-168 \\\Leftrightarrow & 3x \color{red}{+12} \color{blue}{-12} \color{blue}{+56x} & = & \color{red}{-56x} -168 \color{blue}{+56x} \color{blue}{-12} \\\Leftrightarrow & 59x& = & -180 \\\Leftrightarrow & \frac{59x}{ \color{red}{59} }& = & \frac{-180}{59} \\\Leftrightarrow & x = \frac{-180}{59} & & \\ & V = \left\{ \frac{-180}{59} \right\} & \\\end{align}\)
4. \(\text{12 is het kleinste gemene veelvoud van 3, 6 en 4} \\ \begin{align} & \frac{x}{3}+\frac{5}{6}& = & \frac{1}{4}x-6 \\\Leftrightarrow & \color{blue}{12.} (\frac{4x}{ \color{blue}{12} }+ \frac{ 10 }{ \color{blue}{12} })& = & (\frac{3}{ \color{blue}{12} }x-\frac{72}{ \color{blue}{12} }) \color{blue}{.12} \\\Leftrightarrow & 4x+10& = & 3x-72 \\\Leftrightarrow & 4x \color{red}{+10} \color{blue}{-10} \color{blue}{-3x} & = & \color{red}{3x} -72 \color{blue}{-3x} \color{blue}{-10} \\\Leftrightarrow & x& = & -82 \\ & V = \left\{ -82 \right\} & \\\end{align}\)
5. \(\text{42 is het kleinste gemene veelvoud van 3, 7 en 2} \\ \begin{align} & \frac{x}{3}-\frac{2}{7}& = & \frac{1}{2}x-7 \\\Leftrightarrow & \color{blue}{42.} (\frac{14x}{ \color{blue}{42} }- \frac{ 12 }{ \color{blue}{42} })& = & (\frac{21}{ \color{blue}{42} }x-\frac{294}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 14x-12& = & 21x-294 \\\Leftrightarrow & 14x \color{red}{-12} \color{blue}{+12} \color{blue}{-21x} & = & \color{red}{21x} -294 \color{blue}{-21x} \color{blue}{+12} \\\Leftrightarrow & -7x& = & -282 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{-282}{-7} \\\Leftrightarrow & x = \frac{282}{7} & & \\ & V = \left\{ \frac{282}{7} \right\} & \\\end{align}\)
6. \(\text{180 is het kleinste gemene veelvoud van 4, 9 en 5} \\ \begin{align} & \frac{x}{4}-\frac{5}{9}& = & \frac{1}{5}x-7 \\\Leftrightarrow & \color{blue}{180.} (\frac{45x}{ \color{blue}{180} }- \frac{ 100 }{ \color{blue}{180} })& = & (\frac{36}{ \color{blue}{180} }x-\frac{1260}{ \color{blue}{180} }) \color{blue}{.180} \\\Leftrightarrow & 45x-100& = & 36x-1260 \\\Leftrightarrow & 45x \color{red}{-100} \color{blue}{+100} \color{blue}{-36x} & = & \color{red}{36x} -1260 \color{blue}{-36x} \color{blue}{+100} \\\Leftrightarrow & 9x& = & -1160 \\\Leftrightarrow & \frac{9x}{ \color{red}{9} }& = & \frac{-1160}{9} \\\Leftrightarrow & x = \frac{-1160}{9} & & \\ & V = \left\{ \frac{-1160}{9} \right\} & \\\end{align}\)
7. \(\text{42 is het kleinste gemene veelvoud van 3, 7 en 2} \\ \begin{align} & \frac{x}{3}-\frac{3}{7}& = & \frac{3}{2}x+5 \\\Leftrightarrow & \color{blue}{42.} (\frac{14x}{ \color{blue}{42} }- \frac{ 18 }{ \color{blue}{42} })& = & (\frac{63}{ \color{blue}{42} }x+\frac{210}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 14x-18& = & 63x+210 \\\Leftrightarrow & 14x \color{red}{-18} \color{blue}{+18} \color{blue}{-63x} & = & \color{red}{63x} +210 \color{blue}{-63x} \color{blue}{+18} \\\Leftrightarrow & -49x& = & 228 \\\Leftrightarrow & \frac{-49x}{ \color{red}{-49} }& = & \frac{228}{-49} \\\Leftrightarrow & x = \frac{-228}{49} & & \\ & V = \left\{ \frac{-228}{49} \right\} & \\\end{align}\)
8. \(\text{80 is het kleinste gemene veelvoud van 5, 16 en 4} \\ \begin{align} & \frac{x}{5}-\frac{5}{16}& = & \frac{-7}{4}x-2 \\\Leftrightarrow & \color{blue}{80.} (\frac{16x}{ \color{blue}{80} }- \frac{ 25 }{ \color{blue}{80} })& = & (\frac{-140}{ \color{blue}{80} }x-\frac{160}{ \color{blue}{80} }) \color{blue}{.80} \\\Leftrightarrow & 16x-25& = & -140x-160 \\\Leftrightarrow & 16x \color{red}{-25} \color{blue}{+25} \color{blue}{+140x} & = & \color{red}{-140x} -160 \color{blue}{+140x} \color{blue}{+25} \\\Leftrightarrow & 156x& = & -135 \\\Leftrightarrow & \frac{156x}{ \color{red}{156} }& = & \frac{-135}{156} \\\Leftrightarrow & x = \frac{-45}{52} & & \\ & V = \left\{ \frac{-45}{52} \right\} & \\\end{align}\)
9. \(\text{42 is het kleinste gemene veelvoud van 2, 7 en 3} \\ \begin{align} & \frac{x}{2}-\frac{5}{7}& = & \frac{-2}{3}x+3 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }- \frac{ 30 }{ \color{blue}{42} })& = & (\frac{-28}{ \color{blue}{42} }x+\frac{126}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x-30& = & -28x+126 \\\Leftrightarrow & 21x \color{red}{-30} \color{blue}{+30} \color{blue}{+28x} & = & \color{red}{-28x} +126 \color{blue}{+28x} \color{blue}{+30} \\\Leftrightarrow & 49x& = & 156 \\\Leftrightarrow & \frac{49x}{ \color{red}{49} }& = & \frac{156}{49} \\\Leftrightarrow & x = \frac{156}{49} & & \\ & V = \left\{ \frac{156}{49} \right\} & \\\end{align}\)
10. \(\text{140 is het kleinste gemene veelvoud van 7, 10 en 4} \\ \begin{align} & \frac{x}{7}-\frac{3}{10}& = & \frac{1}{4}x-6 \\\Leftrightarrow & \color{blue}{140.} (\frac{20x}{ \color{blue}{140} }- \frac{ 42 }{ \color{blue}{140} })& = & (\frac{35}{ \color{blue}{140} }x-\frac{840}{ \color{blue}{140} }) \color{blue}{.140} \\\Leftrightarrow & 20x-42& = & 35x-840 \\\Leftrightarrow & 20x \color{red}{-42} \color{blue}{+42} \color{blue}{-35x} & = & \color{red}{35x} -840 \color{blue}{-35x} \color{blue}{+42} \\\Leftrightarrow & -15x& = & -798 \\\Leftrightarrow & \frac{-15x}{ \color{red}{-15} }& = & \frac{-798}{-15} \\\Leftrightarrow & x = \frac{266}{5} & & \\ & V = \left\{ \frac{266}{5} \right\} & \\\end{align}\)
11. \(\text{273 is het kleinste gemene veelvoud van 7, 13 en 3} \\ \begin{align} & \frac{x}{7}-\frac{2}{13}& = & \frac{4}{3}x+7 \\\Leftrightarrow & \color{blue}{273.} (\frac{39x}{ \color{blue}{273} }- \frac{ 42 }{ \color{blue}{273} })& = & (\frac{364}{ \color{blue}{273} }x+\frac{1911}{ \color{blue}{273} }) \color{blue}{.273} \\\Leftrightarrow & 39x-42& = & 364x+1911 \\\Leftrightarrow & 39x \color{red}{-42} \color{blue}{+42} \color{blue}{-364x} & = & \color{red}{364x} +1911 \color{blue}{-364x} \color{blue}{+42} \\\Leftrightarrow & -325x& = & 1953 \\\Leftrightarrow & \frac{-325x}{ \color{red}{-325} }& = & \frac{1953}{-325} \\\Leftrightarrow & x = \frac{-1953}{325} & & \\ & V = \left\{ \frac{-1953}{325} \right\} & \\\end{align}\)
12. \(\text{80 is het kleinste gemene veelvoud van 4, 16 en 5} \\ \begin{align} & \frac{x}{4}-\frac{5}{16}& = & \frac{-2}{5}x+2 \\\Leftrightarrow & \color{blue}{80.} (\frac{20x}{ \color{blue}{80} }- \frac{ 25 }{ \color{blue}{80} })& = & (\frac{-32}{ \color{blue}{80} }x+\frac{160}{ \color{blue}{80} }) \color{blue}{.80} \\\Leftrightarrow & 20x-25& = & -32x+160 \\\Leftrightarrow & 20x \color{red}{-25} \color{blue}{+25} \color{blue}{+32x} & = & \color{red}{-32x} +160 \color{blue}{+32x} \color{blue}{+25} \\\Leftrightarrow & 52x& = & 185 \\\Leftrightarrow & \frac{52x}{ \color{red}{52} }& = & \frac{185}{52} \\\Leftrightarrow & x = \frac{185}{52} & & \\ & V = \left\{ \frac{185}{52} \right\} & \\\end{align}\)