Vgln. eerste graad (reeks 4)

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

1. \(\frac{x}{4}-\frac{4}{15}=\frac{-2}{5}x-3\)
2. \(\frac{x}{7}-\frac{2}{15}=\frac{5}{6}x+3\)
3. \(\frac{x}{4}-\frac{2}{9}=\frac{1}{3}x-4\)
4. \(\frac{x}{2}-\frac{5}{7}=\frac{-2}{5}x-2\)
5. \(\frac{x}{2}-\frac{3}{14}=\frac{4}{5}x+4\)
6. \(\frac{x}{4}-\frac{5}{12}=\frac{-7}{5}x+2\)
7. \(\frac{x}{3}+\frac{4}{17}=\frac{7}{5}x-3\)
8. \(\frac{x}{3}+\frac{3}{10}=\frac{5}{2}x+4\)
9. \(\frac{x}{7}-\frac{5}{11}=\frac{3}{2}x+3\)
10. \(\frac{x}{7}-\frac{2}{13}=\frac{-3}{4}x+3\)
11. \(\frac{x}{4}-\frac{2}{13}=\frac{7}{3}x-1\)
12. \(\frac{x}{4}+\frac{2}{17}=\frac{-2}{7}x-6\)

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

Verbetersleutel

1. \(\text{60 is het kleinste gemene veelvoud van 4, 15 en 5} \\ \begin{align} & \color{blue}{60.} (\frac{x}{4}-\frac{4}{15})& = & (\frac{-2}{5}x-3) \color{blue}{.60} \\\Leftrightarrow & 15x-16& = & -24x-180 \\\Leftrightarrow & 15x \color{red}{-16} \color{blue}{+16} \color{blue}{+24x} & = & \color{red}{-24x} -180 \color{blue}{+24x} \color{blue}{+16} \\\Leftrightarrow & 39x& = & -164 \\\Leftrightarrow & \frac{39x}{ \color{red}{39} }& = & \frac{-164}{39} \\\Leftrightarrow & x = \frac{-164}{39} & & \\ & V = \left\{ \frac{-164}{39} \right\} & \\\end{align}\)
2. \(\text{210 is het kleinste gemene veelvoud van 7, 15 en 6} \\ \begin{align} & \color{blue}{210.} (\frac{x}{7}-\frac{2}{15})& = & (\frac{5}{6}x+3) \color{blue}{.210} \\\Leftrightarrow & 30x-28& = & 175x+630 \\\Leftrightarrow & 30x \color{red}{-28} \color{blue}{+28} \color{blue}{-175x} & = & \color{red}{175x} +630 \color{blue}{-175x} \color{blue}{+28} \\\Leftrightarrow & -145x& = & 658 \\\Leftrightarrow & \frac{-145x}{ \color{red}{-145} }& = & \frac{658}{-145} \\\Leftrightarrow & x = \frac{-658}{145} & & \\ & V = \left\{ \frac{-658}{145} \right\} & \\\end{align}\)
3. \(\text{36 is het kleinste gemene veelvoud van 4, 9 en 3} \\ \begin{align} & \color{blue}{36.} (\frac{x}{4}-\frac{2}{9})& = & (\frac{1}{3}x-4) \color{blue}{.36} \\\Leftrightarrow & 9x-8& = & 12x-144 \\\Leftrightarrow & 9x \color{red}{-8} \color{blue}{+8} \color{blue}{-12x} & = & \color{red}{12x} -144 \color{blue}{-12x} \color{blue}{+8} \\\Leftrightarrow & -3x& = & -136 \\\Leftrightarrow & \frac{-3x}{ \color{red}{-3} }& = & \frac{-136}{-3} \\\Leftrightarrow & x = \frac{136}{3} & & \\ & V = \left\{ \frac{136}{3} \right\} & \\\end{align}\)
4. \(\text{70 is het kleinste gemene veelvoud van 2, 7 en 5} \\ \begin{align} & \color{blue}{70.} (\frac{x}{2}-\frac{5}{7})& = & (\frac{-2}{5}x-2) \color{blue}{.70} \\\Leftrightarrow & 35x-50& = & -28x-140 \\\Leftrightarrow & 35x \color{red}{-50} \color{blue}{+50} \color{blue}{+28x} & = & \color{red}{-28x} -140 \color{blue}{+28x} \color{blue}{+50} \\\Leftrightarrow & 63x& = & -90 \\\Leftrightarrow & \frac{63x}{ \color{red}{63} }& = & \frac{-90}{63} \\\Leftrightarrow & x = \frac{-10}{7} & & \\ & V = \left\{ \frac{-10}{7} \right\} & \\\end{align}\)
5. \(\text{70 is het kleinste gemene veelvoud van 2, 14 en 5} \\ \begin{align} & \color{blue}{70.} (\frac{x}{2}-\frac{3}{14})& = & (\frac{4}{5}x+4) \color{blue}{.70} \\\Leftrightarrow & 35x-15& = & 56x+280 \\\Leftrightarrow & 35x \color{red}{-15} \color{blue}{+15} \color{blue}{-56x} & = & \color{red}{56x} +280 \color{blue}{-56x} \color{blue}{+15} \\\Leftrightarrow & -21x& = & 295 \\\Leftrightarrow & \frac{-21x}{ \color{red}{-21} }& = & \frac{295}{-21} \\\Leftrightarrow & x = \frac{-295}{21} & & \\ & V = \left\{ \frac{-295}{21} \right\} & \\\end{align}\)
6. \(\text{60 is het kleinste gemene veelvoud van 4, 12 en 5} \\ \begin{align} & \color{blue}{60.} (\frac{x}{4}-\frac{5}{12})& = & (\frac{-7}{5}x+2) \color{blue}{.60} \\\Leftrightarrow & 15x-25& = & -84x+120 \\\Leftrightarrow & 15x \color{red}{-25} \color{blue}{+25} \color{blue}{+84x} & = & \color{red}{-84x} +120 \color{blue}{+84x} \color{blue}{+25} \\\Leftrightarrow & 99x& = & 145 \\\Leftrightarrow & \frac{99x}{ \color{red}{99} }& = & \frac{145}{99} \\\Leftrightarrow & x = \frac{145}{99} & & \\ & V = \left\{ \frac{145}{99} \right\} & \\\end{align}\)
7. \(\text{255 is het kleinste gemene veelvoud van 3, 17 en 5} \\ \begin{align} & \color{blue}{255.} (\frac{x}{3}+\frac{4}{17})& = & (\frac{7}{5}x-3) \color{blue}{.255} \\\Leftrightarrow & 85x+60& = & 357x-765 \\\Leftrightarrow & 85x \color{red}{+60} \color{blue}{-60} \color{blue}{-357x} & = & \color{red}{357x} -765 \color{blue}{-357x} \color{blue}{-60} \\\Leftrightarrow & -272x& = & -825 \\\Leftrightarrow & \frac{-272x}{ \color{red}{-272} }& = & \frac{-825}{-272} \\\Leftrightarrow & x = \frac{825}{272} & & \\ & V = \left\{ \frac{825}{272} \right\} & \\\end{align}\)
8. \(\text{30 is het kleinste gemene veelvoud van 3, 10 en 2} \\ \begin{align} & \color{blue}{30.} (\frac{x}{3}+\frac{3}{10})& = & (\frac{5}{2}x+4) \color{blue}{.30} \\\Leftrightarrow & 10x+9& = & 75x+120 \\\Leftrightarrow & 10x \color{red}{+9} \color{blue}{-9} \color{blue}{-75x} & = & \color{red}{75x} +120 \color{blue}{-75x} \color{blue}{-9} \\\Leftrightarrow & -65x& = & 111 \\\Leftrightarrow & \frac{-65x}{ \color{red}{-65} }& = & \frac{111}{-65} \\\Leftrightarrow & x = \frac{-111}{65} & & \\ & V = \left\{ \frac{-111}{65} \right\} & \\\end{align}\)
9. \(\text{154 is het kleinste gemene veelvoud van 7, 11 en 2} \\ \begin{align} & \color{blue}{154.} (\frac{x}{7}-\frac{5}{11})& = & (\frac{3}{2}x+3) \color{blue}{.154} \\\Leftrightarrow & 22x-70& = & 231x+462 \\\Leftrightarrow & 22x \color{red}{-70} \color{blue}{+70} \color{blue}{-231x} & = & \color{red}{231x} +462 \color{blue}{-231x} \color{blue}{+70} \\\Leftrightarrow & -209x& = & 532 \\\Leftrightarrow & \frac{-209x}{ \color{red}{-209} }& = & \frac{532}{-209} \\\Leftrightarrow & x = \frac{-28}{11} & & \\ & V = \left\{ \frac{-28}{11} \right\} & \\\end{align}\)
10. \(\text{364 is het kleinste gemene veelvoud van 7, 13 en 4} \\ \begin{align} & \color{blue}{364.} (\frac{x}{7}-\frac{2}{13})& = & (\frac{-3}{4}x+3) \color{blue}{.364} \\\Leftrightarrow & 52x-56& = & -273x+1092 \\\Leftrightarrow & 52x \color{red}{-56} \color{blue}{+56} \color{blue}{+273x} & = & \color{red}{-273x} +1092 \color{blue}{+273x} \color{blue}{+56} \\\Leftrightarrow & 325x& = & 1148 \\\Leftrightarrow & \frac{325x}{ \color{red}{325} }& = & \frac{1148}{325} \\\Leftrightarrow & x = \frac{1148}{325} & & \\ & V = \left\{ \frac{1148}{325} \right\} & \\\end{align}\)
11. \(\text{156 is het kleinste gemene veelvoud van 4, 13 en 3} \\ \begin{align} & \color{blue}{156.} (\frac{x}{4}-\frac{2}{13})& = & (\frac{7}{3}x-1) \color{blue}{.156} \\\Leftrightarrow & 39x-24& = & 364x-156 \\\Leftrightarrow & 39x \color{red}{-24} \color{blue}{+24} \color{blue}{-364x} & = & \color{red}{364x} -156 \color{blue}{-364x} \color{blue}{+24} \\\Leftrightarrow & -325x& = & -132 \\\Leftrightarrow & \frac{-325x}{ \color{red}{-325} }& = & \frac{-132}{-325} \\\Leftrightarrow & x = \frac{132}{325} & & \\ & V = \left\{ \frac{132}{325} \right\} & \\\end{align}\)
12. \(\text{476 is het kleinste gemene veelvoud van 4, 17 en 7} \\ \begin{align} & \color{blue}{476.} (\frac{x}{4}+\frac{2}{17})& = & (\frac{-2}{7}x-6) \color{blue}{.476} \\\Leftrightarrow & 119x+56& = & -136x-2856 \\\Leftrightarrow & 119x \color{red}{+56} \color{blue}{-56} \color{blue}{+136x} & = & \color{red}{-136x} -2856 \color{blue}{+136x} \color{blue}{-56} \\\Leftrightarrow & 255x& = & -2912 \\\Leftrightarrow & \frac{255x}{ \color{red}{255} }& = & \frac{-2912}{255} \\\Leftrightarrow & x = \frac{-2912}{255} & & \\ & V = \left\{ \frac{-2912}{255} \right\} & \\\end{align}\)