Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{105 is het kleinste gemene veelvoud van 5, 7 en 3} \\ \begin{align} & \frac{x}{5}-\frac{4}{7}& = & \frac{-5}{3}x-6 \\\Leftrightarrow & \color{blue}{105.} (\frac{21x}{ \color{blue}{105} }- \frac{ 60 }{ \color{blue}{105} })& = & (\frac{-175}{ \color{blue}{105} }x-\frac{630}{ \color{blue}{105} }) \color{blue}{.105} \\\Leftrightarrow & 21x-60& = & -175x-630 \\\Leftrightarrow & 21x \color{red}{-60} \color{blue}{+60} \color{blue}{+175x} & = & \color{red}{-175x} -630 \color{blue}{+175x} \color{blue}{+60} \\\Leftrightarrow & 196x& = & -570 \\\Leftrightarrow & \frac{196x}{ \color{red}{196} }& = & \frac{-570}{196} \\\Leftrightarrow & x = \frac{-285}{98} & & \\ & V = \left\{ \frac{-285}{98} \right\} & \\\end{align}\)
2. \(\text{84 is het kleinste gemene veelvoud van 3, 7 en 4} \\ \begin{align} & \frac{x}{3}+\frac{3}{7}& = & \frac{-3}{4}x-5 \\\Leftrightarrow & \color{blue}{84.} (\frac{28x}{ \color{blue}{84} }+ \frac{ 36 }{ \color{blue}{84} })& = & (\frac{-63}{ \color{blue}{84} }x-\frac{420}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 28x+36& = & -63x-420 \\\Leftrightarrow & 28x \color{red}{+36} \color{blue}{-36} \color{blue}{+63x} & = & \color{red}{-63x} -420 \color{blue}{+63x} \color{blue}{-36} \\\Leftrightarrow & 91x& = & -456 \\\Leftrightarrow & \frac{91x}{ \color{red}{91} }& = & \frac{-456}{91} \\\Leftrightarrow & x = \frac{-456}{91} & & \\ & V = \left\{ \frac{-456}{91} \right\} & \\\end{align}\)
3. \(\text{90 is het kleinste gemene veelvoud van 5, 9 en 2} \\ \begin{align} & \frac{x}{5}-\frac{4}{9}& = & \frac{5}{2}x-1 \\\Leftrightarrow & \color{blue}{90.} (\frac{18x}{ \color{blue}{90} }- \frac{ 40 }{ \color{blue}{90} })& = & (\frac{225}{ \color{blue}{90} }x-\frac{90}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 18x-40& = & 225x-90 \\\Leftrightarrow & 18x \color{red}{-40} \color{blue}{+40} \color{blue}{-225x} & = & \color{red}{225x} -90 \color{blue}{-225x} \color{blue}{+40} \\\Leftrightarrow & -207x& = & -50 \\\Leftrightarrow & \frac{-207x}{ \color{red}{-207} }& = & \frac{-50}{-207} \\\Leftrightarrow & x = \frac{50}{207} & & \\ & V = \left\{ \frac{50}{207} \right\} & \\\end{align}\)
4. \(\text{30 is het kleinste gemene veelvoud van 6, 15 en 5} \\ \begin{align} & \frac{x}{6}+\frac{4}{15}& = & \frac{3}{5}x-2 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }+ \frac{ 8 }{ \color{blue}{30} })& = & (\frac{18}{ \color{blue}{30} }x-\frac{60}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x+8& = & 18x-60 \\\Leftrightarrow & 5x \color{red}{+8} \color{blue}{-8} \color{blue}{-18x} & = & \color{red}{18x} -60 \color{blue}{-18x} \color{blue}{-8} \\\Leftrightarrow & -13x& = & -68 \\\Leftrightarrow & \frac{-13x}{ \color{red}{-13} }& = & \frac{-68}{-13} \\\Leftrightarrow & x = \frac{68}{13} & & \\ & V = \left\{ \frac{68}{13} \right\} & \\\end{align}\)
5. \(\text{36 is het kleinste gemene veelvoud van 4, 9 en 3} \\ \begin{align} & \frac{x}{4}-\frac{5}{9}& = & \frac{7}{3}x-5 \\\Leftrightarrow & \color{blue}{36.} (\frac{9x}{ \color{blue}{36} }- \frac{ 20 }{ \color{blue}{36} })& = & (\frac{84}{ \color{blue}{36} }x-\frac{180}{ \color{blue}{36} }) \color{blue}{.36} \\\Leftrightarrow & 9x-20& = & 84x-180 \\\Leftrightarrow & 9x \color{red}{-20} \color{blue}{+20} \color{blue}{-84x} & = & \color{red}{84x} -180 \color{blue}{-84x} \color{blue}{+20} \\\Leftrightarrow & -75x& = & -160 \\\Leftrightarrow & \frac{-75x}{ \color{red}{-75} }& = & \frac{-160}{-75} \\\Leftrightarrow & x = \frac{32}{15} & & \\ & V = \left\{ \frac{32}{15} \right\} & \\\end{align}\)
6. \(\text{40 is het kleinste gemene veelvoud van 2, 8 en 5} \\ \begin{align} & \frac{x}{2}+\frac{3}{8}& = & \frac{3}{5}x+7 \\\Leftrightarrow & \color{blue}{40.} (\frac{20x}{ \color{blue}{40} }+ \frac{ 15 }{ \color{blue}{40} })& = & (\frac{24}{ \color{blue}{40} }x+\frac{280}{ \color{blue}{40} }) \color{blue}{.40} \\\Leftrightarrow & 20x+15& = & 24x+280 \\\Leftrightarrow & 20x \color{red}{+15} \color{blue}{-15} \color{blue}{-24x} & = & \color{red}{24x} +280 \color{blue}{-24x} \color{blue}{-15} \\\Leftrightarrow & -4x& = & 265 \\\Leftrightarrow & \frac{-4x}{ \color{red}{-4} }& = & \frac{265}{-4} \\\Leftrightarrow & x = \frac{-265}{4} & & \\ & V = \left\{ \frac{-265}{4} \right\} & \\\end{align}\)
7. \(\text{240 is het kleinste gemene veelvoud van 6, 16 en 5} \\ \begin{align} & \frac{x}{6}+\frac{3}{16}& = & \frac{6}{5}x-1 \\\Leftrightarrow & \color{blue}{240.} (\frac{40x}{ \color{blue}{240} }+ \frac{ 45 }{ \color{blue}{240} })& = & (\frac{288}{ \color{blue}{240} }x-\frac{240}{ \color{blue}{240} }) \color{blue}{.240} \\\Leftrightarrow & 40x+45& = & 288x-240 \\\Leftrightarrow & 40x \color{red}{+45} \color{blue}{-45} \color{blue}{-288x} & = & \color{red}{288x} -240 \color{blue}{-288x} \color{blue}{-45} \\\Leftrightarrow & -248x& = & -285 \\\Leftrightarrow & \frac{-248x}{ \color{red}{-248} }& = & \frac{-285}{-248} \\\Leftrightarrow & x = \frac{285}{248} & & \\ & V = \left\{ \frac{285}{248} \right\} & \\\end{align}\)
8. \(\text{90 is het kleinste gemene veelvoud van 6, 9 en 5} \\ \begin{align} & \frac{x}{6}-\frac{4}{9}& = & \frac{-7}{5}x+3 \\\Leftrightarrow & \color{blue}{90.} (\frac{15x}{ \color{blue}{90} }- \frac{ 40 }{ \color{blue}{90} })& = & (\frac{-126}{ \color{blue}{90} }x+\frac{270}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 15x-40& = & -126x+270 \\\Leftrightarrow & 15x \color{red}{-40} \color{blue}{+40} \color{blue}{+126x} & = & \color{red}{-126x} +270 \color{blue}{+126x} \color{blue}{+40} \\\Leftrightarrow & 141x& = & 310 \\\Leftrightarrow & \frac{141x}{ \color{red}{141} }& = & \frac{310}{141} \\\Leftrightarrow & x = \frac{310}{141} & & \\ & V = \left\{ \frac{310}{141} \right\} & \\\end{align}\)
9. \(\text{70 is het kleinste gemene veelvoud van 5, 7 en 2} \\ \begin{align} & \frac{x}{5}+\frac{2}{7}& = & \frac{3}{2}x+8 \\\Leftrightarrow & \color{blue}{70.} (\frac{14x}{ \color{blue}{70} }+ \frac{ 20 }{ \color{blue}{70} })& = & (\frac{105}{ \color{blue}{70} }x+\frac{560}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 14x+20& = & 105x+560 \\\Leftrightarrow & 14x \color{red}{+20} \color{blue}{-20} \color{blue}{-105x} & = & \color{red}{105x} +560 \color{blue}{-105x} \color{blue}{-20} \\\Leftrightarrow & -91x& = & 540 \\\Leftrightarrow & \frac{-91x}{ \color{red}{-91} }& = & \frac{540}{-91} \\\Leftrightarrow & x = \frac{-540}{91} & & \\ & V = \left\{ \frac{-540}{91} \right\} & \\\end{align}\)
10. \(\text{42 is het kleinste gemene veelvoud van 7, 7 en 6} \\ \begin{align} & \frac{x}{7}-\frac{4}{7}& = & \frac{1}{6}x+6 \\\Leftrightarrow & \color{blue}{42.} (\frac{6x}{ \color{blue}{42} }- \frac{ 24 }{ \color{blue}{42} })& = & (\frac{7}{ \color{blue}{42} }x+\frac{252}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 6x-24& = & 7x+252 \\\Leftrightarrow & 6x \color{red}{-24} \color{blue}{+24} \color{blue}{-7x} & = & \color{red}{7x} +252 \color{blue}{-7x} \color{blue}{+24} \\\Leftrightarrow & -x& = & 276 \\\Leftrightarrow & \frac{-x}{ \color{red}{-1} }& = & \frac{276}{-1} \\\Leftrightarrow & x = -276 & & \\ & V = \left\{ -276 \right\} & \\\end{align}\)
11. \(\text{105 is het kleinste gemene veelvoud van 7, 15 en 5} \\ \begin{align} & \frac{x}{7}-\frac{2}{15}& = & \frac{-7}{5}x+5 \\\Leftrightarrow & \color{blue}{105.} (\frac{15x}{ \color{blue}{105} }- \frac{ 14 }{ \color{blue}{105} })& = & (\frac{-147}{ \color{blue}{105} }x+\frac{525}{ \color{blue}{105} }) \color{blue}{.105} \\\Leftrightarrow & 15x-14& = & -147x+525 \\\Leftrightarrow & 15x \color{red}{-14} \color{blue}{+14} \color{blue}{+147x} & = & \color{red}{-147x} +525 \color{blue}{+147x} \color{blue}{+14} \\\Leftrightarrow & 162x& = & 539 \\\Leftrightarrow & \frac{162x}{ \color{red}{162} }& = & \frac{539}{162} \\\Leftrightarrow & x = \frac{539}{162} & & \\ & V = \left\{ \frac{539}{162} \right\} & \\\end{align}\)
12. \(\text{42 is het kleinste gemene veelvoud van 2, 7 en 3} \\ \begin{align} & \frac{x}{2}+\frac{3}{7}& = & \frac{1}{3}x-1 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }+ \frac{ 18 }{ \color{blue}{42} })& = & (\frac{14}{ \color{blue}{42} }x-\frac{42}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x+18& = & 14x-42 \\\Leftrightarrow & 21x \color{red}{+18} \color{blue}{-18} \color{blue}{-14x} & = & \color{red}{14x} -42 \color{blue}{-14x} \color{blue}{-18} \\\Leftrightarrow & 7x& = & -60 \\\Leftrightarrow & \frac{7x}{ \color{red}{7} }& = & \frac{-60}{7} \\\Leftrightarrow & x = \frac{-60}{7} & & \\ & V = \left\{ \frac{-60}{7} \right\} & \\\end{align}\)