Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{560 is het kleinste gemene veelvoud van 7, 16 en 5} \\ \begin{align} & \frac{x}{7}-\frac{5}{16}& = & \frac{1}{5}x-3 \\\Leftrightarrow & \color{blue}{560.} (\frac{80x}{ \color{blue}{560} }- \frac{ 175 }{ \color{blue}{560} })& = & (\frac{112}{ \color{blue}{560} }x-\frac{1680}{ \color{blue}{560} }) \color{blue}{.560} \\\Leftrightarrow & 80x-175& = & 112x-1680 \\\Leftrightarrow & 80x \color{red}{-175} \color{blue}{+175} \color{blue}{-112x} & = & \color{red}{112x} -1680 \color{blue}{-112x} \color{blue}{+175} \\\Leftrightarrow & -32x& = & -1505 \\\Leftrightarrow & \frac{-32x}{ \color{red}{-32} }& = & \frac{-1505}{-32} \\\Leftrightarrow & x = \frac{1505}{32} & & \\ & V = \left\{ \frac{1505}{32} \right\} & \\\end{align}\)
2. \(\text{390 is het kleinste gemene veelvoud van 6, 13 en 5} \\ \begin{align} & \frac{x}{6}-\frac{2}{13}& = & \frac{-7}{5}x-6 \\\Leftrightarrow & \color{blue}{390.} (\frac{65x}{ \color{blue}{390} }- \frac{ 60 }{ \color{blue}{390} })& = & (\frac{-546}{ \color{blue}{390} }x-\frac{2340}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 65x-60& = & -546x-2340 \\\Leftrightarrow & 65x \color{red}{-60} \color{blue}{+60} \color{blue}{+546x} & = & \color{red}{-546x} -2340 \color{blue}{+546x} \color{blue}{+60} \\\Leftrightarrow & 611x& = & -2280 \\\Leftrightarrow & \frac{611x}{ \color{red}{611} }& = & \frac{-2280}{611} \\\Leftrightarrow & x = \frac{-2280}{611} & & \\ & V = \left\{ \frac{-2280}{611} \right\} & \\\end{align}\)
3. \(\text{42 is het kleinste gemene veelvoud van 3, 7 en 2} \\ \begin{align} & \frac{x}{3}+\frac{2}{7}& = & \frac{1}{2}x+2 \\\Leftrightarrow & \color{blue}{42.} (\frac{14x}{ \color{blue}{42} }+ \frac{ 12 }{ \color{blue}{42} })& = & (\frac{21}{ \color{blue}{42} }x+\frac{84}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 14x+12& = & 21x+84 \\\Leftrightarrow & 14x \color{red}{+12} \color{blue}{-12} \color{blue}{-21x} & = & \color{red}{21x} +84 \color{blue}{-21x} \color{blue}{-12} \\\Leftrightarrow & -7x& = & 72 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{72}{-7} \\\Leftrightarrow & x = \frac{-72}{7} & & \\ & V = \left\{ \frac{-72}{7} \right\} & \\\end{align}\)
4. \(\text{30 is het kleinste gemene veelvoud van 6, 10 en 5} \\ \begin{align} & \frac{x}{6}+\frac{3}{10}& = & \frac{-4}{5}x-3 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }+ \frac{ 9 }{ \color{blue}{30} })& = & (\frac{-24}{ \color{blue}{30} }x-\frac{90}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x+9& = & -24x-90 \\\Leftrightarrow & 5x \color{red}{+9} \color{blue}{-9} \color{blue}{+24x} & = & \color{red}{-24x} -90 \color{blue}{+24x} \color{blue}{-9} \\\Leftrightarrow & 29x& = & -99 \\\Leftrightarrow & \frac{29x}{ \color{red}{29} }& = & \frac{-99}{29} \\\Leftrightarrow & x = \frac{-99}{29} & & \\ & V = \left\{ \frac{-99}{29} \right\} & \\\end{align}\)
5. \(\text{195 is het kleinste gemene veelvoud van 3, 13 en 5} \\ \begin{align} & \frac{x}{3}+\frac{3}{13}& = & \frac{4}{5}x+5 \\\Leftrightarrow & \color{blue}{195.} (\frac{65x}{ \color{blue}{195} }+ \frac{ 45 }{ \color{blue}{195} })& = & (\frac{156}{ \color{blue}{195} }x+\frac{975}{ \color{blue}{195} }) \color{blue}{.195} \\\Leftrightarrow & 65x+45& = & 156x+975 \\\Leftrightarrow & 65x \color{red}{+45} \color{blue}{-45} \color{blue}{-156x} & = & \color{red}{156x} +975 \color{blue}{-156x} \color{blue}{-45} \\\Leftrightarrow & -91x& = & 930 \\\Leftrightarrow & \frac{-91x}{ \color{red}{-91} }& = & \frac{930}{-91} \\\Leftrightarrow & x = \frac{-930}{91} & & \\ & V = \left\{ \frac{-930}{91} \right\} & \\\end{align}\)
6. \(\text{42 is het kleinste gemene veelvoud van 7, 7 en 6} \\ \begin{align} & \frac{x}{7}-\frac{4}{7}& = & \frac{1}{6}x+8 \\\Leftrightarrow & \color{blue}{42.} (\frac{6x}{ \color{blue}{42} }- \frac{ 24 }{ \color{blue}{42} })& = & (\frac{7}{ \color{blue}{42} }x+\frac{336}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 6x-24& = & 7x+336 \\\Leftrightarrow & 6x \color{red}{-24} \color{blue}{+24} \color{blue}{-7x} & = & \color{red}{7x} +336 \color{blue}{-7x} \color{blue}{+24} \\\Leftrightarrow & -x& = & 360 \\\Leftrightarrow & \frac{-x}{ \color{red}{-1} }& = & \frac{360}{-1} \\\Leftrightarrow & x = -360 & & \\ & V = \left\{ -360 \right\} & \\\end{align}\)
7. \(\text{182 is het kleinste gemene veelvoud van 7, 13 en 2} \\ \begin{align} & \frac{x}{7}-\frac{3}{13}& = & \frac{1}{2}x+2 \\\Leftrightarrow & \color{blue}{182.} (\frac{26x}{ \color{blue}{182} }- \frac{ 42 }{ \color{blue}{182} })& = & (\frac{91}{ \color{blue}{182} }x+\frac{364}{ \color{blue}{182} }) \color{blue}{.182} \\\Leftrightarrow & 26x-42& = & 91x+364 \\\Leftrightarrow & 26x \color{red}{-42} \color{blue}{+42} \color{blue}{-91x} & = & \color{red}{91x} +364 \color{blue}{-91x} \color{blue}{+42} \\\Leftrightarrow & -65x& = & 406 \\\Leftrightarrow & \frac{-65x}{ \color{red}{-65} }& = & \frac{406}{-65} \\\Leftrightarrow & x = \frac{-406}{65} & & \\ & V = \left\{ \frac{-406}{65} \right\} & \\\end{align}\)
8. \(\text{132 is het kleinste gemene veelvoud van 4, 11 en 3} \\ \begin{align} & \frac{x}{4}+\frac{4}{11}& = & \frac{7}{3}x+5 \\\Leftrightarrow & \color{blue}{132.} (\frac{33x}{ \color{blue}{132} }+ \frac{ 48 }{ \color{blue}{132} })& = & (\frac{308}{ \color{blue}{132} }x+\frac{660}{ \color{blue}{132} }) \color{blue}{.132} \\\Leftrightarrow & 33x+48& = & 308x+660 \\\Leftrightarrow & 33x \color{red}{+48} \color{blue}{-48} \color{blue}{-308x} & = & \color{red}{308x} +660 \color{blue}{-308x} \color{blue}{-48} \\\Leftrightarrow & -275x& = & 612 \\\Leftrightarrow & \frac{-275x}{ \color{red}{-275} }& = & \frac{612}{-275} \\\Leftrightarrow & x = \frac{-612}{275} & & \\ & V = \left\{ \frac{-612}{275} \right\} & \\\end{align}\)
9. \(\text{220 is het kleinste gemene veelvoud van 4, 11 en 5} \\ \begin{align} & \frac{x}{4}+\frac{2}{11}& = & \frac{-2}{5}x-8 \\\Leftrightarrow & \color{blue}{220.} (\frac{55x}{ \color{blue}{220} }+ \frac{ 40 }{ \color{blue}{220} })& = & (\frac{-88}{ \color{blue}{220} }x-\frac{1760}{ \color{blue}{220} }) \color{blue}{.220} \\\Leftrightarrow & 55x+40& = & -88x-1760 \\\Leftrightarrow & 55x \color{red}{+40} \color{blue}{-40} \color{blue}{+88x} & = & \color{red}{-88x} -1760 \color{blue}{+88x} \color{blue}{-40} \\\Leftrightarrow & 143x& = & -1800 \\\Leftrightarrow & \frac{143x}{ \color{red}{143} }& = & \frac{-1800}{143} \\\Leftrightarrow & x = \frac{-1800}{143} & & \\ & V = \left\{ \frac{-1800}{143} \right\} & \\\end{align}\)
10. \(\text{90 is het kleinste gemene veelvoud van 6, 9 en 5} \\ \begin{align} & \frac{x}{6}+\frac{4}{9}& = & \frac{7}{5}x+6 \\\Leftrightarrow & \color{blue}{90.} (\frac{15x}{ \color{blue}{90} }+ \frac{ 40 }{ \color{blue}{90} })& = & (\frac{126}{ \color{blue}{90} }x+\frac{540}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 15x+40& = & 126x+540 \\\Leftrightarrow & 15x \color{red}{+40} \color{blue}{-40} \color{blue}{-126x} & = & \color{red}{126x} +540 \color{blue}{-126x} \color{blue}{-40} \\\Leftrightarrow & -111x& = & 500 \\\Leftrightarrow & \frac{-111x}{ \color{red}{-111} }& = & \frac{500}{-111} \\\Leftrightarrow & x = \frac{-500}{111} & & \\ & V = \left\{ \frac{-500}{111} \right\} & \\\end{align}\)
11. \(\text{70 is het kleinste gemene veelvoud van 5, 7 en 2} \\ \begin{align} & \frac{x}{5}-\frac{4}{7}& = & \frac{5}{2}x-5 \\\Leftrightarrow & \color{blue}{70.} (\frac{14x}{ \color{blue}{70} }- \frac{ 40 }{ \color{blue}{70} })& = & (\frac{175}{ \color{blue}{70} }x-\frac{350}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 14x-40& = & 175x-350 \\\Leftrightarrow & 14x \color{red}{-40} \color{blue}{+40} \color{blue}{-175x} & = & \color{red}{175x} -350 \color{blue}{-175x} \color{blue}{+40} \\\Leftrightarrow & -161x& = & -310 \\\Leftrightarrow & \frac{-161x}{ \color{red}{-161} }& = & \frac{-310}{-161} \\\Leftrightarrow & x = \frac{310}{161} & & \\ & V = \left\{ \frac{310}{161} \right\} & \\\end{align}\)
12. \(\text{14 is het kleinste gemene veelvoud van 7, 7 en 2} \\ \begin{align} & \frac{x}{7}+\frac{5}{7}& = & \frac{5}{2}x-5 \\\Leftrightarrow & \color{blue}{14.} (\frac{2x}{ \color{blue}{14} }+ \frac{ 10 }{ \color{blue}{14} })& = & (\frac{35}{ \color{blue}{14} }x-\frac{70}{ \color{blue}{14} }) \color{blue}{.14} \\\Leftrightarrow & 2x+10& = & 35x-70 \\\Leftrightarrow & 2x \color{red}{+10} \color{blue}{-10} \color{blue}{-35x} & = & \color{red}{35x} -70 \color{blue}{-35x} \color{blue}{-10} \\\Leftrightarrow & -33x& = & -80 \\\Leftrightarrow & \frac{-33x}{ \color{red}{-33} }& = & \frac{-80}{-33} \\\Leftrightarrow & x = \frac{80}{33} & & \\ & V = \left\{ \frac{80}{33} \right\} & \\\end{align}\)