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

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

Verbetersleutel

1. \(\text{70 is het kleinste gemene veelvoud van 2, 14 en 5} \\ \begin{align} & \color{blue}{70.} (\frac{x}{2}-\frac{3}{14})& = & (\frac{8}{5}x+8) \color{blue}{.70} \\\Leftrightarrow & 35x-15& = & 112x+560 \\\Leftrightarrow & 35x \color{red}{-15} \color{blue}{+15} \color{blue}{-112x} & = & \color{red}{112x} +560 \color{blue}{-112x} \color{blue}{+15} \\\Leftrightarrow & -77x& = & 575 \\\Leftrightarrow & \frac{-77x}{ \color{red}{-77} }& = & \frac{575}{-77} \\\Leftrightarrow & x = \frac{-575}{77} & & \\ & V = \left\{ \frac{-575}{77} \right\} & \\\end{align}\)
2. \(\text{10 is het kleinste gemene veelvoud van 5, 10 en 2} \\ \begin{align} & \color{blue}{10.} (\frac{x}{5}+\frac{3}{10})& = & (\frac{5}{2}x-8) \color{blue}{.10} \\\Leftrightarrow & 2x+3& = & 25x-80 \\\Leftrightarrow & 2x \color{red}{+3} \color{blue}{-3} \color{blue}{-25x} & = & \color{red}{25x} -80 \color{blue}{-25x} \color{blue}{-3} \\\Leftrightarrow & -23x& = & -83 \\\Leftrightarrow & \frac{-23x}{ \color{red}{-23} }& = & \frac{-83}{-23} \\\Leftrightarrow & x = \frac{83}{23} & & \\ & V = \left\{ \frac{83}{23} \right\} & \\\end{align}\)
3. \(\text{110 is het kleinste gemene veelvoud van 2, 11 en 5} \\ \begin{align} & \color{blue}{110.} (\frac{x}{2}-\frac{5}{11})& = & (\frac{7}{5}x-8) \color{blue}{.110} \\\Leftrightarrow & 55x-50& = & 154x-880 \\\Leftrightarrow & 55x \color{red}{-50} \color{blue}{+50} \color{blue}{-154x} & = & \color{red}{154x} -880 \color{blue}{-154x} \color{blue}{+50} \\\Leftrightarrow & -99x& = & -830 \\\Leftrightarrow & \frac{-99x}{ \color{red}{-99} }& = & \frac{-830}{-99} \\\Leftrightarrow & x = \frac{830}{99} & & \\ & V = \left\{ \frac{830}{99} \right\} & \\\end{align}\)
4. \(\text{308 is het kleinste gemene veelvoud van 4, 11 en 7} \\ \begin{align} & \color{blue}{308.} (\frac{x}{4}+\frac{4}{11})& = & (\frac{5}{7}x-3) \color{blue}{.308} \\\Leftrightarrow & 77x+112& = & 220x-924 \\\Leftrightarrow & 77x \color{red}{+112} \color{blue}{-112} \color{blue}{-220x} & = & \color{red}{220x} -924 \color{blue}{-220x} \color{blue}{-112} \\\Leftrightarrow & -143x& = & -1036 \\\Leftrightarrow & \frac{-143x}{ \color{red}{-143} }& = & \frac{-1036}{-143} \\\Leftrightarrow & x = \frac{1036}{143} & & \\ & V = \left\{ \frac{1036}{143} \right\} & \\\end{align}\)
5. \(\text{364 is het kleinste gemene veelvoud van 4, 13 en 7} \\ \begin{align} & \color{blue}{364.} (\frac{x}{4}-\frac{4}{13})& = & (\frac{3}{7}x+3) \color{blue}{.364} \\\Leftrightarrow & 91x-112& = & 156x+1092 \\\Leftrightarrow & 91x \color{red}{-112} \color{blue}{+112} \color{blue}{-156x} & = & \color{red}{156x} +1092 \color{blue}{-156x} \color{blue}{+112} \\\Leftrightarrow & -65x& = & 1204 \\\Leftrightarrow & \frac{-65x}{ \color{red}{-65} }& = & \frac{1204}{-65} \\\Leftrightarrow & x = \frac{-1204}{65} & & \\ & V = \left\{ \frac{-1204}{65} \right\} & \\\end{align}\)
6. \(\text{30 is het kleinste gemene veelvoud van 5, 15 en 6} \\ \begin{align} & \color{blue}{30.} (\frac{x}{5}+\frac{4}{15})& = & (\frac{7}{6}x+3) \color{blue}{.30} \\\Leftrightarrow & 6x+8& = & 35x+90 \\\Leftrightarrow & 6x \color{red}{+8} \color{blue}{-8} \color{blue}{-35x} & = & \color{red}{35x} +90 \color{blue}{-35x} \color{blue}{-8} \\\Leftrightarrow & -29x& = & 82 \\\Leftrightarrow & \frac{-29x}{ \color{red}{-29} }& = & \frac{82}{-29} \\\Leftrightarrow & x = \frac{-82}{29} & & \\ & V = \left\{ \frac{-82}{29} \right\} & \\\end{align}\)
7. \(\text{182 is het kleinste gemene veelvoud van 2, 13 en 7} \\ \begin{align} & \color{blue}{182.} (\frac{x}{2}+\frac{4}{13})& = & (\frac{1}{7}x+2) \color{blue}{.182} \\\Leftrightarrow & 91x+56& = & 26x+364 \\\Leftrightarrow & 91x \color{red}{+56} \color{blue}{-56} \color{blue}{-26x} & = & \color{red}{26x} +364 \color{blue}{-26x} \color{blue}{-56} \\\Leftrightarrow & 65x& = & 308 \\\Leftrightarrow & \frac{65x}{ \color{red}{65} }& = & \frac{308}{65} \\\Leftrightarrow & x = \frac{308}{65} & & \\ & V = \left\{ \frac{308}{65} \right\} & \\\end{align}\)
8. \(\text{14 is het kleinste gemene veelvoud van 7, 7 en 2} \\ \begin{align} & \color{blue}{14.} (\frac{x}{7}+\frac{4}{7})& = & (\frac{1}{2}x-6) \color{blue}{.14} \\\Leftrightarrow & 2x+8& = & 7x-84 \\\Leftrightarrow & 2x \color{red}{+8} \color{blue}{-8} \color{blue}{-7x} & = & \color{red}{7x} -84 \color{blue}{-7x} \color{blue}{-8} \\\Leftrightarrow & -5x& = & -92 \\\Leftrightarrow & \frac{-5x}{ \color{red}{-5} }& = & \frac{-92}{-5} \\\Leftrightarrow & x = \frac{92}{5} & & \\ & V = \left\{ \frac{92}{5} \right\} & \\\end{align}\)
9. \(\text{30 is het kleinste gemene veelvoud van 6, 15 en 5} \\ \begin{align} & \color{blue}{30.} (\frac{x}{6}+\frac{2}{15})& = & (\frac{8}{5}x-1) \color{blue}{.30} \\\Leftrightarrow & 5x+4& = & 48x-30 \\\Leftrightarrow & 5x \color{red}{+4} \color{blue}{-4} \color{blue}{-48x} & = & \color{red}{48x} -30 \color{blue}{-48x} \color{blue}{-4} \\\Leftrightarrow & -43x& = & -34 \\\Leftrightarrow & \frac{-43x}{ \color{red}{-43} }& = & \frac{-34}{-43} \\\Leftrightarrow & x = \frac{34}{43} & & \\ & V = \left\{ \frac{34}{43} \right\} & \\\end{align}\)
10. \(\text{252 is het kleinste gemene veelvoud van 4, 9 en 7} \\ \begin{align} & \color{blue}{252.} (\frac{x}{4}+\frac{5}{9})& = & (\frac{-6}{7}x+7) \color{blue}{.252} \\\Leftrightarrow & 63x+140& = & -216x+1764 \\\Leftrightarrow & 63x \color{red}{+140} \color{blue}{-140} \color{blue}{+216x} & = & \color{red}{-216x} +1764 \color{blue}{+216x} \color{blue}{-140} \\\Leftrightarrow & 279x& = & 1624 \\\Leftrightarrow & \frac{279x}{ \color{red}{279} }& = & \frac{1624}{279} \\\Leftrightarrow & x = \frac{1624}{279} & & \\ & V = \left\{ \frac{1624}{279} \right\} & \\\end{align}\)
11. \(\text{546 is het kleinste gemene veelvoud van 6, 13 en 7} \\ \begin{align} & \color{blue}{546.} (\frac{x}{6}-\frac{3}{13})& = & (\frac{6}{7}x+7) \color{blue}{.546} \\\Leftrightarrow & 91x-126& = & 468x+3822 \\\Leftrightarrow & 91x \color{red}{-126} \color{blue}{+126} \color{blue}{-468x} & = & \color{red}{468x} +3822 \color{blue}{-468x} \color{blue}{+126} \\\Leftrightarrow & -377x& = & 3948 \\\Leftrightarrow & \frac{-377x}{ \color{red}{-377} }& = & \frac{3948}{-377} \\\Leftrightarrow & x = \frac{-3948}{377} & & \\ & V = \left\{ \frac{-3948}{377} \right\} & \\\end{align}\)
12. \(\text{42 is het kleinste gemene veelvoud van 6, 14 en 7} \\ \begin{align} & \color{blue}{42.} (\frac{x}{6}-\frac{3}{14})& = & (\frac{8}{7}x+4) \color{blue}{.42} \\\Leftrightarrow & 7x-9& = & 48x+168 \\\Leftrightarrow & 7x \color{red}{-9} \color{blue}{+9} \color{blue}{-48x} & = & \color{red}{48x} +168 \color{blue}{-48x} \color{blue}{+9} \\\Leftrightarrow & -41x& = & 177 \\\Leftrightarrow & \frac{-41x}{ \color{red}{-41} }& = & \frac{177}{-41} \\\Leftrightarrow & x = \frac{-177}{41} & & \\ & V = \left\{ \frac{-177}{41} \right\} & \\\end{align}\)