Vgln. eerste graad (reeks 4)

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1. \(\frac{x}{7}+\frac{2}{11}=\frac{5}{2}x-4\)
2. \(\frac{x}{7}-\frac{3}{14}=\frac{-2}{5}x-8\)
3. \(\frac{x}{4}-\frac{4}{11}=\frac{-5}{3}x-5\)
4. \(\frac{x}{2}-\frac{3}{8}=\frac{7}{3}x+3\)
5. \(\frac{x}{5}-\frac{5}{16}=\frac{1}{4}x+5\)
6. \(\frac{x}{7}+\frac{2}{7}=\frac{1}{2}x-8\)
7. \(\frac{x}{5}-\frac{5}{11}=\frac{1}{2}x+2\)
8. \(\frac{x}{6}-\frac{4}{9}=\frac{-2}{5}x-6\)
9. \(\frac{x}{3}-\frac{5}{9}=\frac{7}{2}x-6\)
10. \(\frac{x}{3}-\frac{3}{16}=\frac{5}{2}x+6\)
11. \(\frac{x}{7}-\frac{4}{7}=\frac{1}{2}x-5\)
12. \(\frac{x}{6}+\frac{2}{13}=\frac{2}{5}x-7\)

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

Verbetersleutel

1. \(\text{154 is het kleinste gemene veelvoud van 7, 11 en 2} \\ \begin{align} & \frac{x}{7}+\frac{2}{11}& = & \frac{5}{2}x-4 \\\Leftrightarrow & \color{blue}{154.} (\frac{22x}{ \color{blue}{154} }+ \frac{ 28 }{ \color{blue}{154} })& = & (\frac{385}{ \color{blue}{154} }x-\frac{616}{ \color{blue}{154} }) \color{blue}{.154} \\\Leftrightarrow & 22x+28& = & 385x-616 \\\Leftrightarrow & 22x \color{red}{+28} \color{blue}{-28} \color{blue}{-385x} & = & \color{red}{385x} -616 \color{blue}{-385x} \color{blue}{-28} \\\Leftrightarrow & -363x& = & -644 \\\Leftrightarrow & \frac{-363x}{ \color{red}{-363} }& = & \frac{-644}{-363} \\\Leftrightarrow & x = \frac{644}{363} & & \\ & V = \left\{ \frac{644}{363} \right\} & \\\end{align}\)
2. \(\text{70 is het kleinste gemene veelvoud van 7, 14 en 5} \\ \begin{align} & \frac{x}{7}-\frac{3}{14}& = & \frac{-2}{5}x-8 \\\Leftrightarrow & \color{blue}{70.} (\frac{10x}{ \color{blue}{70} }- \frac{ 15 }{ \color{blue}{70} })& = & (\frac{-28}{ \color{blue}{70} }x-\frac{560}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 10x-15& = & -28x-560 \\\Leftrightarrow & 10x \color{red}{-15} \color{blue}{+15} \color{blue}{+28x} & = & \color{red}{-28x} -560 \color{blue}{+28x} \color{blue}{+15} \\\Leftrightarrow & 38x& = & -545 \\\Leftrightarrow & \frac{38x}{ \color{red}{38} }& = & \frac{-545}{38} \\\Leftrightarrow & x = \frac{-545}{38} & & \\ & V = \left\{ \frac{-545}{38} \right\} & \\\end{align}\)
3. \(\text{132 is het kleinste gemene veelvoud van 4, 11 en 3} \\ \begin{align} & \frac{x}{4}-\frac{4}{11}& = & \frac{-5}{3}x-5 \\\Leftrightarrow & \color{blue}{132.} (\frac{33x}{ \color{blue}{132} }- \frac{ 48 }{ \color{blue}{132} })& = & (\frac{-220}{ \color{blue}{132} }x-\frac{660}{ \color{blue}{132} }) \color{blue}{.132} \\\Leftrightarrow & 33x-48& = & -220x-660 \\\Leftrightarrow & 33x \color{red}{-48} \color{blue}{+48} \color{blue}{+220x} & = & \color{red}{-220x} -660 \color{blue}{+220x} \color{blue}{+48} \\\Leftrightarrow & 253x& = & -612 \\\Leftrightarrow & \frac{253x}{ \color{red}{253} }& = & \frac{-612}{253} \\\Leftrightarrow & x = \frac{-612}{253} & & \\ & V = \left\{ \frac{-612}{253} \right\} & \\\end{align}\)
4. \(\text{24 is het kleinste gemene veelvoud van 2, 8 en 3} \\ \begin{align} & \frac{x}{2}-\frac{3}{8}& = & \frac{7}{3}x+3 \\\Leftrightarrow & \color{blue}{24.} (\frac{12x}{ \color{blue}{24} }- \frac{ 9 }{ \color{blue}{24} })& = & (\frac{56}{ \color{blue}{24} }x+\frac{72}{ \color{blue}{24} }) \color{blue}{.24} \\\Leftrightarrow & 12x-9& = & 56x+72 \\\Leftrightarrow & 12x \color{red}{-9} \color{blue}{+9} \color{blue}{-56x} & = & \color{red}{56x} +72 \color{blue}{-56x} \color{blue}{+9} \\\Leftrightarrow & -44x& = & 81 \\\Leftrightarrow & \frac{-44x}{ \color{red}{-44} }& = & \frac{81}{-44} \\\Leftrightarrow & x = \frac{-81}{44} & & \\ & V = \left\{ \frac{-81}{44} \right\} & \\\end{align}\)
5. \(\text{80 is het kleinste gemene veelvoud van 5, 16 en 4} \\ \begin{align} & \frac{x}{5}-\frac{5}{16}& = & \frac{1}{4}x+5 \\\Leftrightarrow & \color{blue}{80.} (\frac{16x}{ \color{blue}{80} }- \frac{ 25 }{ \color{blue}{80} })& = & (\frac{20}{ \color{blue}{80} }x+\frac{400}{ \color{blue}{80} }) \color{blue}{.80} \\\Leftrightarrow & 16x-25& = & 20x+400 \\\Leftrightarrow & 16x \color{red}{-25} \color{blue}{+25} \color{blue}{-20x} & = & \color{red}{20x} +400 \color{blue}{-20x} \color{blue}{+25} \\\Leftrightarrow & -4x& = & 425 \\\Leftrightarrow & \frac{-4x}{ \color{red}{-4} }& = & \frac{425}{-4} \\\Leftrightarrow & x = \frac{-425}{4} & & \\ & V = \left\{ \frac{-425}{4} \right\} & \\\end{align}\)
6. \(\text{14 is het kleinste gemene veelvoud van 7, 7 en 2} \\ \begin{align} & \frac{x}{7}+\frac{2}{7}& = & \frac{1}{2}x-8 \\\Leftrightarrow & \color{blue}{14.} (\frac{2x}{ \color{blue}{14} }+ \frac{ 4 }{ \color{blue}{14} })& = & (\frac{7}{ \color{blue}{14} }x-\frac{112}{ \color{blue}{14} }) \color{blue}{.14} \\\Leftrightarrow & 2x+4& = & 7x-112 \\\Leftrightarrow & 2x \color{red}{+4} \color{blue}{-4} \color{blue}{-7x} & = & \color{red}{7x} -112 \color{blue}{-7x} \color{blue}{-4} \\\Leftrightarrow & -5x& = & -116 \\\Leftrightarrow & \frac{-5x}{ \color{red}{-5} }& = & \frac{-116}{-5} \\\Leftrightarrow & x = \frac{116}{5} & & \\ & V = \left\{ \frac{116}{5} \right\} & \\\end{align}\)
7. \(\text{110 is het kleinste gemene veelvoud van 5, 11 en 2} \\ \begin{align} & \frac{x}{5}-\frac{5}{11}& = & \frac{1}{2}x+2 \\\Leftrightarrow & \color{blue}{110.} (\frac{22x}{ \color{blue}{110} }- \frac{ 50 }{ \color{blue}{110} })& = & (\frac{55}{ \color{blue}{110} }x+\frac{220}{ \color{blue}{110} }) \color{blue}{.110} \\\Leftrightarrow & 22x-50& = & 55x+220 \\\Leftrightarrow & 22x \color{red}{-50} \color{blue}{+50} \color{blue}{-55x} & = & \color{red}{55x} +220 \color{blue}{-55x} \color{blue}{+50} \\\Leftrightarrow & -33x& = & 270 \\\Leftrightarrow & \frac{-33x}{ \color{red}{-33} }& = & \frac{270}{-33} \\\Leftrightarrow & x = \frac{-90}{11} & & \\ & V = \left\{ \frac{-90}{11} \right\} & \\\end{align}\)
8. \(\text{90 is het kleinste gemene veelvoud van 6, 9 en 5} \\ \begin{align} & \frac{x}{6}-\frac{4}{9}& = & \frac{-2}{5}x-6 \\\Leftrightarrow & \color{blue}{90.} (\frac{15x}{ \color{blue}{90} }- \frac{ 40 }{ \color{blue}{90} })& = & (\frac{-36}{ \color{blue}{90} }x-\frac{540}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 15x-40& = & -36x-540 \\\Leftrightarrow & 15x \color{red}{-40} \color{blue}{+40} \color{blue}{+36x} & = & \color{red}{-36x} -540 \color{blue}{+36x} \color{blue}{+40} \\\Leftrightarrow & 51x& = & -500 \\\Leftrightarrow & \frac{51x}{ \color{red}{51} }& = & \frac{-500}{51} \\\Leftrightarrow & x = \frac{-500}{51} & & \\ & V = \left\{ \frac{-500}{51} \right\} & \\\end{align}\)
9. \(\text{18 is het kleinste gemene veelvoud van 3, 9 en 2} \\ \begin{align} & \frac{x}{3}-\frac{5}{9}& = & \frac{7}{2}x-6 \\\Leftrightarrow & \color{blue}{18.} (\frac{6x}{ \color{blue}{18} }- \frac{ 10 }{ \color{blue}{18} })& = & (\frac{63}{ \color{blue}{18} }x-\frac{108}{ \color{blue}{18} }) \color{blue}{.18} \\\Leftrightarrow & 6x-10& = & 63x-108 \\\Leftrightarrow & 6x \color{red}{-10} \color{blue}{+10} \color{blue}{-63x} & = & \color{red}{63x} -108 \color{blue}{-63x} \color{blue}{+10} \\\Leftrightarrow & -57x& = & -98 \\\Leftrightarrow & \frac{-57x}{ \color{red}{-57} }& = & \frac{-98}{-57} \\\Leftrightarrow & x = \frac{98}{57} & & \\ & V = \left\{ \frac{98}{57} \right\} & \\\end{align}\)
10. \(\text{48 is het kleinste gemene veelvoud van 3, 16 en 2} \\ \begin{align} & \frac{x}{3}-\frac{3}{16}& = & \frac{5}{2}x+6 \\\Leftrightarrow & \color{blue}{48.} (\frac{16x}{ \color{blue}{48} }- \frac{ 9 }{ \color{blue}{48} })& = & (\frac{120}{ \color{blue}{48} }x+\frac{288}{ \color{blue}{48} }) \color{blue}{.48} \\\Leftrightarrow & 16x-9& = & 120x+288 \\\Leftrightarrow & 16x \color{red}{-9} \color{blue}{+9} \color{blue}{-120x} & = & \color{red}{120x} +288 \color{blue}{-120x} \color{blue}{+9} \\\Leftrightarrow & -104x& = & 297 \\\Leftrightarrow & \frac{-104x}{ \color{red}{-104} }& = & \frac{297}{-104} \\\Leftrightarrow & x = \frac{-297}{104} & & \\ & V = \left\{ \frac{-297}{104} \right\} & \\\end{align}\)
11. \(\text{14 is het kleinste gemene veelvoud van 7, 7 en 2} \\ \begin{align} & \frac{x}{7}-\frac{4}{7}& = & \frac{1}{2}x-5 \\\Leftrightarrow & \color{blue}{14.} (\frac{2x}{ \color{blue}{14} }- \frac{ 8 }{ \color{blue}{14} })& = & (\frac{7}{ \color{blue}{14} }x-\frac{70}{ \color{blue}{14} }) \color{blue}{.14} \\\Leftrightarrow & 2x-8& = & 7x-70 \\\Leftrightarrow & 2x \color{red}{-8} \color{blue}{+8} \color{blue}{-7x} & = & \color{red}{7x} -70 \color{blue}{-7x} \color{blue}{+8} \\\Leftrightarrow & -5x& = & -62 \\\Leftrightarrow & \frac{-5x}{ \color{red}{-5} }& = & \frac{-62}{-5} \\\Leftrightarrow & x = \frac{62}{5} & & \\ & V = \left\{ \frac{62}{5} \right\} & \\\end{align}\)
12. \(\text{390 is het kleinste gemene veelvoud van 6, 13 en 5} \\ \begin{align} & \frac{x}{6}+\frac{2}{13}& = & \frac{2}{5}x-7 \\\Leftrightarrow & \color{blue}{390.} (\frac{65x}{ \color{blue}{390} }+ \frac{ 60 }{ \color{blue}{390} })& = & (\frac{156}{ \color{blue}{390} }x-\frac{2730}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 65x+60& = & 156x-2730 \\\Leftrightarrow & 65x \color{red}{+60} \color{blue}{-60} \color{blue}{-156x} & = & \color{red}{156x} -2730 \color{blue}{-156x} \color{blue}{-60} \\\Leftrightarrow & -91x& = & -2790 \\\Leftrightarrow & \frac{-91x}{ \color{red}{-91} }& = & \frac{-2790}{-91} \\\Leftrightarrow & x = \frac{2790}{91} & & \\ & V = \left\{ \frac{2790}{91} \right\} & \\\end{align}\)