Vgln. eerste graad (reeks 4)

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1. \(\frac{x}{7}+\frac{3}{7}=\frac{-7}{4}x+8\)
2. \(\frac{x}{7}-\frac{3}{10}=\frac{-4}{5}x+8\)
3. \(\frac{x}{2}+\frac{4}{13}=\frac{3}{5}x+7\)
4. \(\frac{x}{3}+\frac{2}{9}=\frac{3}{2}x+6\)
5. \(\frac{x}{4}+\frac{5}{8}=\frac{7}{5}x+1\)
6. \(\frac{x}{2}+\frac{3}{13}=\frac{-5}{3}x+5\)
7. \(\frac{x}{7}+\frac{4}{15}=\frac{-7}{4}x-3\)
8. \(\frac{x}{3}-\frac{5}{14}=\frac{7}{2}x+2\)
9. \(\frac{x}{4}-\frac{3}{11}=\frac{-7}{5}x-3\)
10. \(\frac{x}{5}+\frac{4}{9}=\frac{5}{6}x+2\)
11. \(\frac{x}{6}-\frac{3}{13}=\frac{-2}{5}x+7\)
12. \(\frac{x}{3}+\frac{2}{13}=\frac{1}{2}x-4\)

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

Verbetersleutel

1. \(\text{28 is het kleinste gemene veelvoud van 7, 7 en 4} \\ \begin{align} & \frac{x}{7}+\frac{3}{7}& = & \frac{-7}{4}x+8 \\\Leftrightarrow & \color{blue}{28.} (\frac{4x}{ \color{blue}{28} }+ \frac{ 12 }{ \color{blue}{28} })& = & (\frac{-49}{ \color{blue}{28} }x+\frac{224}{ \color{blue}{28} }) \color{blue}{.28} \\\Leftrightarrow & 4x+12& = & -49x+224 \\\Leftrightarrow & 4x \color{red}{+12} \color{blue}{-12} \color{blue}{+49x} & = & \color{red}{-49x} +224 \color{blue}{+49x} \color{blue}{-12} \\\Leftrightarrow & 53x& = & 212 \\\Leftrightarrow & \frac{53x}{ \color{red}{53} }& = & \frac{212}{53} \\\Leftrightarrow & x = 4 & & \\ & V = \left\{ 4 \right\} & \\\end{align}\)
2. \(\text{70 is het kleinste gemene veelvoud van 7, 10 en 5} \\ \begin{align} & \frac{x}{7}-\frac{3}{10}& = & \frac{-4}{5}x+8 \\\Leftrightarrow & \color{blue}{70.} (\frac{10x}{ \color{blue}{70} }- \frac{ 21 }{ \color{blue}{70} })& = & (\frac{-56}{ \color{blue}{70} }x+\frac{560}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 10x-21& = & -56x+560 \\\Leftrightarrow & 10x \color{red}{-21} \color{blue}{+21} \color{blue}{+56x} & = & \color{red}{-56x} +560 \color{blue}{+56x} \color{blue}{+21} \\\Leftrightarrow & 66x& = & 581 \\\Leftrightarrow & \frac{66x}{ \color{red}{66} }& = & \frac{581}{66} \\\Leftrightarrow & x = \frac{581}{66} & & \\ & V = \left\{ \frac{581}{66} \right\} & \\\end{align}\)
3. \(\text{130 is het kleinste gemene veelvoud van 2, 13 en 5} \\ \begin{align} & \frac{x}{2}+\frac{4}{13}& = & \frac{3}{5}x+7 \\\Leftrightarrow & \color{blue}{130.} (\frac{65x}{ \color{blue}{130} }+ \frac{ 40 }{ \color{blue}{130} })& = & (\frac{78}{ \color{blue}{130} }x+\frac{910}{ \color{blue}{130} }) \color{blue}{.130} \\\Leftrightarrow & 65x+40& = & 78x+910 \\\Leftrightarrow & 65x \color{red}{+40} \color{blue}{-40} \color{blue}{-78x} & = & \color{red}{78x} +910 \color{blue}{-78x} \color{blue}{-40} \\\Leftrightarrow & -13x& = & 870 \\\Leftrightarrow & \frac{-13x}{ \color{red}{-13} }& = & \frac{870}{-13} \\\Leftrightarrow & x = \frac{-870}{13} & & \\ & V = \left\{ \frac{-870}{13} \right\} & \\\end{align}\)
4. \(\text{18 is het kleinste gemene veelvoud van 3, 9 en 2} \\ \begin{align} & \frac{x}{3}+\frac{2}{9}& = & \frac{3}{2}x+6 \\\Leftrightarrow & \color{blue}{18.} (\frac{6x}{ \color{blue}{18} }+ \frac{ 4 }{ \color{blue}{18} })& = & (\frac{27}{ \color{blue}{18} }x+\frac{108}{ \color{blue}{18} }) \color{blue}{.18} \\\Leftrightarrow & 6x+4& = & 27x+108 \\\Leftrightarrow & 6x \color{red}{+4} \color{blue}{-4} \color{blue}{-27x} & = & \color{red}{27x} +108 \color{blue}{-27x} \color{blue}{-4} \\\Leftrightarrow & -21x& = & 104 \\\Leftrightarrow & \frac{-21x}{ \color{red}{-21} }& = & \frac{104}{-21} \\\Leftrightarrow & x = \frac{-104}{21} & & \\ & V = \left\{ \frac{-104}{21} \right\} & \\\end{align}\)
5. \(\text{40 is het kleinste gemene veelvoud van 4, 8 en 5} \\ \begin{align} & \frac{x}{4}+\frac{5}{8}& = & \frac{7}{5}x+1 \\\Leftrightarrow & \color{blue}{40.} (\frac{10x}{ \color{blue}{40} }+ \frac{ 25 }{ \color{blue}{40} })& = & (\frac{56}{ \color{blue}{40} }x+\frac{40}{ \color{blue}{40} }) \color{blue}{.40} \\\Leftrightarrow & 10x+25& = & 56x+40 \\\Leftrightarrow & 10x \color{red}{+25} \color{blue}{-25} \color{blue}{-56x} & = & \color{red}{56x} +40 \color{blue}{-56x} \color{blue}{-25} \\\Leftrightarrow & -46x& = & 15 \\\Leftrightarrow & \frac{-46x}{ \color{red}{-46} }& = & \frac{15}{-46} \\\Leftrightarrow & x = \frac{-15}{46} & & \\ & V = \left\{ \frac{-15}{46} \right\} & \\\end{align}\)
6. \(\text{78 is het kleinste gemene veelvoud van 2, 13 en 3} \\ \begin{align} & \frac{x}{2}+\frac{3}{13}& = & \frac{-5}{3}x+5 \\\Leftrightarrow & \color{blue}{78.} (\frac{39x}{ \color{blue}{78} }+ \frac{ 18 }{ \color{blue}{78} })& = & (\frac{-130}{ \color{blue}{78} }x+\frac{390}{ \color{blue}{78} }) \color{blue}{.78} \\\Leftrightarrow & 39x+18& = & -130x+390 \\\Leftrightarrow & 39x \color{red}{+18} \color{blue}{-18} \color{blue}{+130x} & = & \color{red}{-130x} +390 \color{blue}{+130x} \color{blue}{-18} \\\Leftrightarrow & 169x& = & 372 \\\Leftrightarrow & \frac{169x}{ \color{red}{169} }& = & \frac{372}{169} \\\Leftrightarrow & x = \frac{372}{169} & & \\ & V = \left\{ \frac{372}{169} \right\} & \\\end{align}\)
7. \(\text{420 is het kleinste gemene veelvoud van 7, 15 en 4} \\ \begin{align} & \frac{x}{7}+\frac{4}{15}& = & \frac{-7}{4}x-3 \\\Leftrightarrow & \color{blue}{420.} (\frac{60x}{ \color{blue}{420} }+ \frac{ 112 }{ \color{blue}{420} })& = & (\frac{-735}{ \color{blue}{420} }x-\frac{1260}{ \color{blue}{420} }) \color{blue}{.420} \\\Leftrightarrow & 60x+112& = & -735x-1260 \\\Leftrightarrow & 60x \color{red}{+112} \color{blue}{-112} \color{blue}{+735x} & = & \color{red}{-735x} -1260 \color{blue}{+735x} \color{blue}{-112} \\\Leftrightarrow & 795x& = & -1372 \\\Leftrightarrow & \frac{795x}{ \color{red}{795} }& = & \frac{-1372}{795} \\\Leftrightarrow & x = \frac{-1372}{795} & & \\ & V = \left\{ \frac{-1372}{795} \right\} & \\\end{align}\)
8. \(\text{42 is het kleinste gemene veelvoud van 3, 14 en 2} \\ \begin{align} & \frac{x}{3}-\frac{5}{14}& = & \frac{7}{2}x+2 \\\Leftrightarrow & \color{blue}{42.} (\frac{14x}{ \color{blue}{42} }- \frac{ 15 }{ \color{blue}{42} })& = & (\frac{147}{ \color{blue}{42} }x+\frac{84}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 14x-15& = & 147x+84 \\\Leftrightarrow & 14x \color{red}{-15} \color{blue}{+15} \color{blue}{-147x} & = & \color{red}{147x} +84 \color{blue}{-147x} \color{blue}{+15} \\\Leftrightarrow & -133x& = & 99 \\\Leftrightarrow & \frac{-133x}{ \color{red}{-133} }& = & \frac{99}{-133} \\\Leftrightarrow & x = \frac{-99}{133} & & \\ & V = \left\{ \frac{-99}{133} \right\} & \\\end{align}\)
9. \(\text{220 is het kleinste gemene veelvoud van 4, 11 en 5} \\ \begin{align} & \frac{x}{4}-\frac{3}{11}& = & \frac{-7}{5}x-3 \\\Leftrightarrow & \color{blue}{220.} (\frac{55x}{ \color{blue}{220} }- \frac{ 60 }{ \color{blue}{220} })& = & (\frac{-308}{ \color{blue}{220} }x-\frac{660}{ \color{blue}{220} }) \color{blue}{.220} \\\Leftrightarrow & 55x-60& = & -308x-660 \\\Leftrightarrow & 55x \color{red}{-60} \color{blue}{+60} \color{blue}{+308x} & = & \color{red}{-308x} -660 \color{blue}{+308x} \color{blue}{+60} \\\Leftrightarrow & 363x& = & -600 \\\Leftrightarrow & \frac{363x}{ \color{red}{363} }& = & \frac{-600}{363} \\\Leftrightarrow & x = \frac{-200}{121} & & \\ & V = \left\{ \frac{-200}{121} \right\} & \\\end{align}\)
10. \(\text{90 is het kleinste gemene veelvoud van 5, 9 en 6} \\ \begin{align} & \frac{x}{5}+\frac{4}{9}& = & \frac{5}{6}x+2 \\\Leftrightarrow & \color{blue}{90.} (\frac{18x}{ \color{blue}{90} }+ \frac{ 40 }{ \color{blue}{90} })& = & (\frac{75}{ \color{blue}{90} }x+\frac{180}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 18x+40& = & 75x+180 \\\Leftrightarrow & 18x \color{red}{+40} \color{blue}{-40} \color{blue}{-75x} & = & \color{red}{75x} +180 \color{blue}{-75x} \color{blue}{-40} \\\Leftrightarrow & -57x& = & 140 \\\Leftrightarrow & \frac{-57x}{ \color{red}{-57} }& = & \frac{140}{-57} \\\Leftrightarrow & x = \frac{-140}{57} & & \\ & V = \left\{ \frac{-140}{57} \right\} & \\\end{align}\)
11. \(\text{390 is het kleinste gemene veelvoud van 6, 13 en 5} \\ \begin{align} & \frac{x}{6}-\frac{3}{13}& = & \frac{-2}{5}x+7 \\\Leftrightarrow & \color{blue}{390.} (\frac{65x}{ \color{blue}{390} }- \frac{ 90 }{ \color{blue}{390} })& = & (\frac{-156}{ \color{blue}{390} }x+\frac{2730}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 65x-90& = & -156x+2730 \\\Leftrightarrow & 65x \color{red}{-90} \color{blue}{+90} \color{blue}{+156x} & = & \color{red}{-156x} +2730 \color{blue}{+156x} \color{blue}{+90} \\\Leftrightarrow & 221x& = & 2820 \\\Leftrightarrow & \frac{221x}{ \color{red}{221} }& = & \frac{2820}{221} \\\Leftrightarrow & x = \frac{2820}{221} & & \\ & V = \left\{ \frac{2820}{221} \right\} & \\\end{align}\)
12. \(\text{78 is het kleinste gemene veelvoud van 3, 13 en 2} \\ \begin{align} & \frac{x}{3}+\frac{2}{13}& = & \frac{1}{2}x-4 \\\Leftrightarrow & \color{blue}{78.} (\frac{26x}{ \color{blue}{78} }+ \frac{ 12 }{ \color{blue}{78} })& = & (\frac{39}{ \color{blue}{78} }x-\frac{312}{ \color{blue}{78} }) \color{blue}{.78} \\\Leftrightarrow & 26x+12& = & 39x-312 \\\Leftrightarrow & 26x \color{red}{+12} \color{blue}{-12} \color{blue}{-39x} & = & \color{red}{39x} -312 \color{blue}{-39x} \color{blue}{-12} \\\Leftrightarrow & -13x& = & -324 \\\Leftrightarrow & \frac{-13x}{ \color{red}{-13} }& = & \frac{-324}{-13} \\\Leftrightarrow & x = \frac{324}{13} & & \\ & V = \left\{ \frac{324}{13} \right\} & \\\end{align}\)