Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{30 is het kleinste gemene veelvoud van 6, 15 en 5} \\ \begin{align} & \frac{x}{6}+\frac{2}{15}& = & \frac{-7}{5}x-3 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }+ \frac{ 4 }{ \color{blue}{30} })& = & (\frac{-42}{ \color{blue}{30} }x-\frac{90}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x+4& = & -42x-90 \\\Leftrightarrow & 5x \color{red}{+4} \color{blue}{-4} \color{blue}{+42x} & = & \color{red}{-42x} -90 \color{blue}{+42x} \color{blue}{-4} \\\Leftrightarrow & 47x& = & -94 \\\Leftrightarrow & \frac{47x}{ \color{red}{47} }& = & \frac{-94}{47} \\\Leftrightarrow & x = -2 & & \\ & V = \left\{ -2 \right\} & \\\end{align}\)
2. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}+\frac{4}{7}& = & \frac{6}{5}x-1 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }+ \frac{ 120 }{ \color{blue}{210} })& = & (\frac{252}{ \color{blue}{210} }x-\frac{210}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x+120& = & 252x-210 \\\Leftrightarrow & 35x \color{red}{+120} \color{blue}{-120} \color{blue}{-252x} & = & \color{red}{252x} -210 \color{blue}{-252x} \color{blue}{-120} \\\Leftrightarrow & -217x& = & -330 \\\Leftrightarrow & \frac{-217x}{ \color{red}{-217} }& = & \frac{-330}{-217} \\\Leftrightarrow & x = \frac{330}{217} & & \\ & V = \left\{ \frac{330}{217} \right\} & \\\end{align}\)
3. \(\text{12 is het kleinste gemene veelvoud van 4, 12 en 3} \\ \begin{align} & \frac{x}{4}-\frac{5}{12}& = & \frac{1}{3}x-6 \\\Leftrightarrow & \color{blue}{12.} (\frac{3x}{ \color{blue}{12} }- \frac{ 5 }{ \color{blue}{12} })& = & (\frac{4}{ \color{blue}{12} }x-\frac{72}{ \color{blue}{12} }) \color{blue}{.12} \\\Leftrightarrow & 3x-5& = & 4x-72 \\\Leftrightarrow & 3x \color{red}{-5} \color{blue}{+5} \color{blue}{-4x} & = & \color{red}{4x} -72 \color{blue}{-4x} \color{blue}{+5} \\\Leftrightarrow & -x& = & -67 \\\Leftrightarrow & \frac{-x}{ \color{red}{-1} }& = & \frac{-67}{-1} \\\Leftrightarrow & x = 67 & & \\ & V = \left\{ 67 \right\} & \\\end{align}\)
4. \(\text{42 is het kleinste gemene veelvoud van 2, 7 en 3} \\ \begin{align} & \frac{x}{2}-\frac{3}{7}& = & \frac{-8}{3}x+7 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }- \frac{ 18 }{ \color{blue}{42} })& = & (\frac{-112}{ \color{blue}{42} }x+\frac{294}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x-18& = & -112x+294 \\\Leftrightarrow & 21x \color{red}{-18} \color{blue}{+18} \color{blue}{+112x} & = & \color{red}{-112x} +294 \color{blue}{+112x} \color{blue}{+18} \\\Leftrightarrow & 133x& = & 312 \\\Leftrightarrow & \frac{133x}{ \color{red}{133} }& = & \frac{312}{133} \\\Leftrightarrow & x = \frac{312}{133} & & \\ & V = \left\{ \frac{312}{133} \right\} & \\\end{align}\)
5. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}-\frac{2}{7}& = & \frac{1}{5}x-7 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }- \frac{ 60 }{ \color{blue}{210} })& = & (\frac{42}{ \color{blue}{210} }x-\frac{1470}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x-60& = & 42x-1470 \\\Leftrightarrow & 35x \color{red}{-60} \color{blue}{+60} \color{blue}{-42x} & = & \color{red}{42x} -1470 \color{blue}{-42x} \color{blue}{+60} \\\Leftrightarrow & -7x& = & -1410 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{-1410}{-7} \\\Leftrightarrow & x = \frac{1410}{7} & & \\ & V = \left\{ \frac{1410}{7} \right\} & \\\end{align}\)
6. \(\text{30 is het kleinste gemene veelvoud van 5, 15 en 6} \\ \begin{align} & \frac{x}{5}-\frac{4}{15}& = & \frac{1}{6}x-3 \\\Leftrightarrow & \color{blue}{30.} (\frac{6x}{ \color{blue}{30} }- \frac{ 8 }{ \color{blue}{30} })& = & (\frac{5}{ \color{blue}{30} }x-\frac{90}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 6x-8& = & 5x-90 \\\Leftrightarrow & 6x \color{red}{-8} \color{blue}{+8} \color{blue}{-5x} & = & \color{red}{5x} -90 \color{blue}{-5x} \color{blue}{+8} \\\Leftrightarrow & x& = & -82 \\ & V = \left\{ -82 \right\} & \\\end{align}\)
7. \(\text{63 is het kleinste gemene veelvoud van 7, 9 en 3} \\ \begin{align} & \frac{x}{7}+\frac{2}{9}& = & \frac{2}{3}x+3 \\\Leftrightarrow & \color{blue}{63.} (\frac{9x}{ \color{blue}{63} }+ \frac{ 14 }{ \color{blue}{63} })& = & (\frac{42}{ \color{blue}{63} }x+\frac{189}{ \color{blue}{63} }) \color{blue}{.63} \\\Leftrightarrow & 9x+14& = & 42x+189 \\\Leftrightarrow & 9x \color{red}{+14} \color{blue}{-14} \color{blue}{-42x} & = & \color{red}{42x} +189 \color{blue}{-42x} \color{blue}{-14} \\\Leftrightarrow & -33x& = & 175 \\\Leftrightarrow & \frac{-33x}{ \color{red}{-33} }& = & \frac{175}{-33} \\\Leftrightarrow & x = \frac{-175}{33} & & \\ & V = \left\{ \frac{-175}{33} \right\} & \\\end{align}\)
8. \(\text{80 is het kleinste gemene veelvoud van 2, 16 en 5} \\ \begin{align} & \frac{x}{2}+\frac{3}{16}& = & \frac{-4}{5}x-5 \\\Leftrightarrow & \color{blue}{80.} (\frac{40x}{ \color{blue}{80} }+ \frac{ 15 }{ \color{blue}{80} })& = & (\frac{-64}{ \color{blue}{80} }x-\frac{400}{ \color{blue}{80} }) \color{blue}{.80} \\\Leftrightarrow & 40x+15& = & -64x-400 \\\Leftrightarrow & 40x \color{red}{+15} \color{blue}{-15} \color{blue}{+64x} & = & \color{red}{-64x} -400 \color{blue}{+64x} \color{blue}{-15} \\\Leftrightarrow & 104x& = & -415 \\\Leftrightarrow & \frac{104x}{ \color{red}{104} }& = & \frac{-415}{104} \\\Leftrightarrow & x = \frac{-415}{104} & & \\ & V = \left\{ \frac{-415}{104} \right\} & \\\end{align}\)
9. \(\text{90 is het kleinste gemene veelvoud van 2, 9 en 5} \\ \begin{align} & \frac{x}{2}+\frac{2}{9}& = & \frac{6}{5}x-8 \\\Leftrightarrow & \color{blue}{90.} (\frac{45x}{ \color{blue}{90} }+ \frac{ 20 }{ \color{blue}{90} })& = & (\frac{108}{ \color{blue}{90} }x-\frac{720}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 45x+20& = & 108x-720 \\\Leftrightarrow & 45x \color{red}{+20} \color{blue}{-20} \color{blue}{-108x} & = & \color{red}{108x} -720 \color{blue}{-108x} \color{blue}{-20} \\\Leftrightarrow & -63x& = & -740 \\\Leftrightarrow & \frac{-63x}{ \color{red}{-63} }& = & \frac{-740}{-63} \\\Leftrightarrow & x = \frac{740}{63} & & \\ & V = \left\{ \frac{740}{63} \right\} & \\\end{align}\)
10. \(\text{42 is het kleinste gemene veelvoud van 2, 7 en 3} \\ \begin{align} & \frac{x}{2}+\frac{4}{7}& = & \frac{4}{3}x-1 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }+ \frac{ 24 }{ \color{blue}{42} })& = & (\frac{56}{ \color{blue}{42} }x-\frac{42}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x+24& = & 56x-42 \\\Leftrightarrow & 21x \color{red}{+24} \color{blue}{-24} \color{blue}{-56x} & = & \color{red}{56x} -42 \color{blue}{-56x} \color{blue}{-24} \\\Leftrightarrow & -35x& = & -66 \\\Leftrightarrow & \frac{-35x}{ \color{red}{-35} }& = & \frac{-66}{-35} \\\Leftrightarrow & x = \frac{66}{35} & & \\ & V = \left\{ \frac{66}{35} \right\} & \\\end{align}\)
11. \(\text{220 is het kleinste gemene veelvoud van 4, 11 en 5} \\ \begin{align} & \frac{x}{4}-\frac{5}{11}& = & \frac{7}{5}x+3 \\\Leftrightarrow & \color{blue}{220.} (\frac{55x}{ \color{blue}{220} }- \frac{ 100 }{ \color{blue}{220} })& = & (\frac{308}{ \color{blue}{220} }x+\frac{660}{ \color{blue}{220} }) \color{blue}{.220} \\\Leftrightarrow & 55x-100& = & 308x+660 \\\Leftrightarrow & 55x \color{red}{-100} \color{blue}{+100} \color{blue}{-308x} & = & \color{red}{308x} +660 \color{blue}{-308x} \color{blue}{+100} \\\Leftrightarrow & -253x& = & 760 \\\Leftrightarrow & \frac{-253x}{ \color{red}{-253} }& = & \frac{760}{-253} \\\Leftrightarrow & x = \frac{-760}{253} & & \\ & V = \left\{ \frac{-760}{253} \right\} & \\\end{align}\)
12. \(\text{48 is het kleinste gemene veelvoud van 4, 16 en 3} \\ \begin{align} & \frac{x}{4}+\frac{5}{16}& = & \frac{1}{3}x+5 \\\Leftrightarrow & \color{blue}{48.} (\frac{12x}{ \color{blue}{48} }+ \frac{ 15 }{ \color{blue}{48} })& = & (\frac{16}{ \color{blue}{48} }x+\frac{240}{ \color{blue}{48} }) \color{blue}{.48} \\\Leftrightarrow & 12x+15& = & 16x+240 \\\Leftrightarrow & 12x \color{red}{+15} \color{blue}{-15} \color{blue}{-16x} & = & \color{red}{16x} +240 \color{blue}{-16x} \color{blue}{-15} \\\Leftrightarrow & -4x& = & 225 \\\Leftrightarrow & \frac{-4x}{ \color{red}{-4} }& = & \frac{225}{-4} \\\Leftrightarrow & x = \frac{-225}{4} & & \\ & V = \left\{ \frac{-225}{4} \right\} & \\\end{align}\)