Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{66 is het kleinste gemene veelvoud van 2, 11 en 3} \\ \begin{align} & \frac{x}{2}+\frac{2}{11}& = & \frac{-2}{3}x-7 \\\Leftrightarrow & \color{blue}{66.} (\frac{33x}{ \color{blue}{66} }+ \frac{ 12 }{ \color{blue}{66} })& = & (\frac{-44}{ \color{blue}{66} }x-\frac{462}{ \color{blue}{66} }) \color{blue}{.66} \\\Leftrightarrow & 33x+12& = & -44x-462 \\\Leftrightarrow & 33x \color{red}{+12} \color{blue}{-12} \color{blue}{+44x} & = & \color{red}{-44x} -462 \color{blue}{+44x} \color{blue}{-12} \\\Leftrightarrow & 77x& = & -474 \\\Leftrightarrow & \frac{77x}{ \color{red}{77} }& = & \frac{-474}{77} \\\Leftrightarrow & x = \frac{-474}{77} & & \\ & V = \left\{ \frac{-474}{77} \right\} & \\\end{align}\)
2. \(\text{231 is het kleinste gemene veelvoud van 7, 11 en 3} \\ \begin{align} & \frac{x}{7}+\frac{2}{11}& = & \frac{7}{3}x+4 \\\Leftrightarrow & \color{blue}{231.} (\frac{33x}{ \color{blue}{231} }+ \frac{ 42 }{ \color{blue}{231} })& = & (\frac{539}{ \color{blue}{231} }x+\frac{924}{ \color{blue}{231} }) \color{blue}{.231} \\\Leftrightarrow & 33x+42& = & 539x+924 \\\Leftrightarrow & 33x \color{red}{+42} \color{blue}{-42} \color{blue}{-539x} & = & \color{red}{539x} +924 \color{blue}{-539x} \color{blue}{-42} \\\Leftrightarrow & -506x& = & 882 \\\Leftrightarrow & \frac{-506x}{ \color{red}{-506} }& = & \frac{882}{-506} \\\Leftrightarrow & x = \frac{-441}{253} & & \\ & V = \left\{ \frac{-441}{253} \right\} & \\\end{align}\)
3. \(\text{35 is het kleinste gemene veelvoud van 7, 7 en 5} \\ \begin{align} & \frac{x}{7}-\frac{4}{7}& = & \frac{-4}{5}x-3 \\\Leftrightarrow & \color{blue}{35.} (\frac{5x}{ \color{blue}{35} }- \frac{ 20 }{ \color{blue}{35} })& = & (\frac{-28}{ \color{blue}{35} }x-\frac{105}{ \color{blue}{35} }) \color{blue}{.35} \\\Leftrightarrow & 5x-20& = & -28x-105 \\\Leftrightarrow & 5x \color{red}{-20} \color{blue}{+20} \color{blue}{+28x} & = & \color{red}{-28x} -105 \color{blue}{+28x} \color{blue}{+20} \\\Leftrightarrow & 33x& = & -85 \\\Leftrightarrow & \frac{33x}{ \color{red}{33} }& = & \frac{-85}{33} \\\Leftrightarrow & x = \frac{-85}{33} & & \\ & V = \left\{ \frac{-85}{33} \right\} & \\\end{align}\)
4. \(\text{24 is het kleinste gemene veelvoud van 2, 8 en 3} \\ \begin{align} & \frac{x}{2}-\frac{5}{8}& = & \frac{-5}{3}x+5 \\\Leftrightarrow & \color{blue}{24.} (\frac{12x}{ \color{blue}{24} }- \frac{ 15 }{ \color{blue}{24} })& = & (\frac{-40}{ \color{blue}{24} }x+\frac{120}{ \color{blue}{24} }) \color{blue}{.24} \\\Leftrightarrow & 12x-15& = & -40x+120 \\\Leftrightarrow & 12x \color{red}{-15} \color{blue}{+15} \color{blue}{+40x} & = & \color{red}{-40x} +120 \color{blue}{+40x} \color{blue}{+15} \\\Leftrightarrow & 52x& = & 135 \\\Leftrightarrow & \frac{52x}{ \color{red}{52} }& = & \frac{135}{52} \\\Leftrightarrow & x = \frac{135}{52} & & \\ & V = \left\{ \frac{135}{52} \right\} & \\\end{align}\)
5. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}-\frac{2}{7}& = & \frac{1}{5}x+5 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }- \frac{ 60 }{ \color{blue}{210} })& = & (\frac{42}{ \color{blue}{210} }x+\frac{1050}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x-60& = & 42x+1050 \\\Leftrightarrow & 35x \color{red}{-60} \color{blue}{+60} \color{blue}{-42x} & = & \color{red}{42x} +1050 \color{blue}{-42x} \color{blue}{+60} \\\Leftrightarrow & -7x& = & 1110 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{1110}{-7} \\\Leftrightarrow & x = \frac{-1110}{7} & & \\ & V = \left\{ \frac{-1110}{7} \right\} & \\\end{align}\)
6. \(\text{30 is het kleinste gemene veelvoud van 2, 6 en 5} \\ \begin{align} & \frac{x}{2}+\frac{5}{6}& = & \frac{-2}{5}x+4 \\\Leftrightarrow & \color{blue}{30.} (\frac{15x}{ \color{blue}{30} }+ \frac{ 25 }{ \color{blue}{30} })& = & (\frac{-12}{ \color{blue}{30} }x+\frac{120}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 15x+25& = & -12x+120 \\\Leftrightarrow & 15x \color{red}{+25} \color{blue}{-25} \color{blue}{+12x} & = & \color{red}{-12x} +120 \color{blue}{+12x} \color{blue}{-25} \\\Leftrightarrow & 27x& = & 95 \\\Leftrightarrow & \frac{27x}{ \color{red}{27} }& = & \frac{95}{27} \\\Leftrightarrow & x = \frac{95}{27} & & \\ & V = \left\{ \frac{95}{27} \right\} & \\\end{align}\)
7. \(\text{70 is het kleinste gemene veelvoud van 2, 7 en 5} \\ \begin{align} & \frac{x}{2}+\frac{4}{7}& = & \frac{7}{5}x+1 \\\Leftrightarrow & \color{blue}{70.} (\frac{35x}{ \color{blue}{70} }+ \frac{ 40 }{ \color{blue}{70} })& = & (\frac{98}{ \color{blue}{70} }x+\frac{70}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 35x+40& = & 98x+70 \\\Leftrightarrow & 35x \color{red}{+40} \color{blue}{-40} \color{blue}{-98x} & = & \color{red}{98x} +70 \color{blue}{-98x} \color{blue}{-40} \\\Leftrightarrow & -63x& = & 30 \\\Leftrightarrow & \frac{-63x}{ \color{red}{-63} }& = & \frac{30}{-63} \\\Leftrightarrow & x = \frac{-10}{21} & & \\ & V = \left\{ \frac{-10}{21} \right\} & \\\end{align}\)
8. \(\text{80 is het kleinste gemene veelvoud van 5, 16 en 2} \\ \begin{align} & \frac{x}{5}+\frac{5}{16}& = & \frac{3}{2}x+6 \\\Leftrightarrow & \color{blue}{80.} (\frac{16x}{ \color{blue}{80} }+ \frac{ 25 }{ \color{blue}{80} })& = & (\frac{120}{ \color{blue}{80} }x+\frac{480}{ \color{blue}{80} }) \color{blue}{.80} \\\Leftrightarrow & 16x+25& = & 120x+480 \\\Leftrightarrow & 16x \color{red}{+25} \color{blue}{-25} \color{blue}{-120x} & = & \color{red}{120x} +480 \color{blue}{-120x} \color{blue}{-25} \\\Leftrightarrow & -104x& = & 455 \\\Leftrightarrow & \frac{-104x}{ \color{red}{-104} }& = & \frac{455}{-104} \\\Leftrightarrow & x = \frac{-35}{8} & & \\ & V = \left\{ \frac{-35}{8} \right\} & \\\end{align}\)
9. \(\text{280 is het kleinste gemene veelvoud van 7, 8 en 5} \\ \begin{align} & \frac{x}{7}+\frac{3}{8}& = & \frac{-2}{5}x-5 \\\Leftrightarrow & \color{blue}{280.} (\frac{40x}{ \color{blue}{280} }+ \frac{ 105 }{ \color{blue}{280} })& = & (\frac{-112}{ \color{blue}{280} }x-\frac{1400}{ \color{blue}{280} }) \color{blue}{.280} \\\Leftrightarrow & 40x+105& = & -112x-1400 \\\Leftrightarrow & 40x \color{red}{+105} \color{blue}{-105} \color{blue}{+112x} & = & \color{red}{-112x} -1400 \color{blue}{+112x} \color{blue}{-105} \\\Leftrightarrow & 152x& = & -1505 \\\Leftrightarrow & \frac{152x}{ \color{red}{152} }& = & \frac{-1505}{152} \\\Leftrightarrow & x = \frac{-1505}{152} & & \\ & V = \left\{ \frac{-1505}{152} \right\} & \\\end{align}\)
10. \(\text{220 is het kleinste gemene veelvoud van 4, 11 en 5} \\ \begin{align} & \frac{x}{4}+\frac{2}{11}& = & \frac{3}{5}x+8 \\\Leftrightarrow & \color{blue}{220.} (\frac{55x}{ \color{blue}{220} }+ \frac{ 40 }{ \color{blue}{220} })& = & (\frac{132}{ \color{blue}{220} }x+\frac{1760}{ \color{blue}{220} }) \color{blue}{.220} \\\Leftrightarrow & 55x+40& = & 132x+1760 \\\Leftrightarrow & 55x \color{red}{+40} \color{blue}{-40} \color{blue}{-132x} & = & \color{red}{132x} +1760 \color{blue}{-132x} \color{blue}{-40} \\\Leftrightarrow & -77x& = & 1720 \\\Leftrightarrow & \frac{-77x}{ \color{red}{-77} }& = & \frac{1720}{-77} \\\Leftrightarrow & x = \frac{-1720}{77} & & \\ & V = \left\{ \frac{-1720}{77} \right\} & \\\end{align}\)
11. \(\text{240 is het kleinste gemene veelvoud van 5, 16 en 3} \\ \begin{align} & \frac{x}{5}-\frac{3}{16}& = & \frac{-2}{3}x+8 \\\Leftrightarrow & \color{blue}{240.} (\frac{48x}{ \color{blue}{240} }- \frac{ 45 }{ \color{blue}{240} })& = & (\frac{-160}{ \color{blue}{240} }x+\frac{1920}{ \color{blue}{240} }) \color{blue}{.240} \\\Leftrightarrow & 48x-45& = & -160x+1920 \\\Leftrightarrow & 48x \color{red}{-45} \color{blue}{+45} \color{blue}{+160x} & = & \color{red}{-160x} +1920 \color{blue}{+160x} \color{blue}{+45} \\\Leftrightarrow & 208x& = & 1965 \\\Leftrightarrow & \frac{208x}{ \color{red}{208} }& = & \frac{1965}{208} \\\Leftrightarrow & x = \frac{1965}{208} & & \\ & V = \left\{ \frac{1965}{208} \right\} & \\\end{align}\)
12. \(\text{390 is het kleinste gemene veelvoud van 6, 13 en 5} \\ \begin{align} & \frac{x}{6}-\frac{4}{13}& = & \frac{1}{5}x-7 \\\Leftrightarrow & \color{blue}{390.} (\frac{65x}{ \color{blue}{390} }- \frac{ 120 }{ \color{blue}{390} })& = & (\frac{78}{ \color{blue}{390} }x-\frac{2730}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 65x-120& = & 78x-2730 \\\Leftrightarrow & 65x \color{red}{-120} \color{blue}{+120} \color{blue}{-78x} & = & \color{red}{78x} -2730 \color{blue}{-78x} \color{blue}{+120} \\\Leftrightarrow & -13x& = & -2610 \\\Leftrightarrow & \frac{-13x}{ \color{red}{-13} }& = & \frac{-2610}{-13} \\\Leftrightarrow & x = \frac{2610}{13} & & \\ & V = \left\{ \frac{2610}{13} \right\} & \\\end{align}\)