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

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

Verbetersleutel

1. \(\text{12 is het kleinste gemene veelvoud van 2, 12 en 3} \\ \begin{align} & \color{blue}{12.} (\frac{x}{2}+\frac{5}{12})& = & (\frac{2}{3}x+6) \color{blue}{.12} \\\Leftrightarrow & 6x+5& = & 8x+72 \\\Leftrightarrow & 6x \color{red}{+5} \color{blue}{-5} \color{blue}{-8x} & = & \color{red}{8x} +72 \color{blue}{-8x} \color{blue}{-5} \\\Leftrightarrow & -2x& = & 67 \\\Leftrightarrow & \frac{-2x}{ \color{red}{-2} }& = & \frac{67}{-2} \\\Leftrightarrow & x = \frac{-67}{2} & & \\ & V = \left\{ \frac{-67}{2} \right\} & \\\end{align}\)
2. \(\text{455 is het kleinste gemene veelvoud van 5, 13 en 7} \\ \begin{align} & \color{blue}{455.} (\frac{x}{5}-\frac{3}{13})& = & (\frac{6}{7}x-3) \color{blue}{.455} \\\Leftrightarrow & 91x-105& = & 390x-1365 \\\Leftrightarrow & 91x \color{red}{-105} \color{blue}{+105} \color{blue}{-390x} & = & \color{red}{390x} -1365 \color{blue}{-390x} \color{blue}{+105} \\\Leftrightarrow & -299x& = & -1260 \\\Leftrightarrow & \frac{-299x}{ \color{red}{-299} }& = & \frac{-1260}{-299} \\\Leftrightarrow & x = \frac{1260}{299} & & \\ & V = \left\{ \frac{1260}{299} \right\} & \\\end{align}\)
3. \(\text{180 is het kleinste gemene veelvoud van 4, 9 en 5} \\ \begin{align} & \color{blue}{180.} (\frac{x}{4}+\frac{2}{9})& = & (\frac{6}{5}x-2) \color{blue}{.180} \\\Leftrightarrow & 45x+40& = & 216x-360 \\\Leftrightarrow & 45x \color{red}{+40} \color{blue}{-40} \color{blue}{-216x} & = & \color{red}{216x} -360 \color{blue}{-216x} \color{blue}{-40} \\\Leftrightarrow & -171x& = & -400 \\\Leftrightarrow & \frac{-171x}{ \color{red}{-171} }& = & \frac{-400}{-171} \\\Leftrightarrow & x = \frac{400}{171} & & \\ & V = \left\{ \frac{400}{171} \right\} & \\\end{align}\)
4. \(\text{45 is het kleinste gemene veelvoud van 5, 9 en 3} \\ \begin{align} & \color{blue}{45.} (\frac{x}{5}-\frac{4}{9})& = & (\frac{4}{3}x-4) \color{blue}{.45} \\\Leftrightarrow & 9x-20& = & 60x-180 \\\Leftrightarrow & 9x \color{red}{-20} \color{blue}{+20} \color{blue}{-60x} & = & \color{red}{60x} -180 \color{blue}{-60x} \color{blue}{+20} \\\Leftrightarrow & -51x& = & -160 \\\Leftrightarrow & \frac{-51x}{ \color{red}{-51} }& = & \frac{-160}{-51} \\\Leftrightarrow & x = \frac{160}{51} & & \\ & V = \left\{ \frac{160}{51} \right\} & \\\end{align}\)
5. \(\text{78 is het kleinste gemene veelvoud van 3, 13 en 2} \\ \begin{align} & \color{blue}{78.} (\frac{x}{3}-\frac{2}{13})& = & (\frac{1}{2}x+6) \color{blue}{.78} \\\Leftrightarrow & 26x-12& = & 39x+468 \\\Leftrightarrow & 26x \color{red}{-12} \color{blue}{+12} \color{blue}{-39x} & = & \color{red}{39x} +468 \color{blue}{-39x} \color{blue}{+12} \\\Leftrightarrow & -13x& = & 480 \\\Leftrightarrow & \frac{-13x}{ \color{red}{-13} }& = & \frac{480}{-13} \\\Leftrightarrow & x = \frac{-480}{13} & & \\ & V = \left\{ \frac{-480}{13} \right\} & \\\end{align}\)
6. \(\text{154 is het kleinste gemene veelvoud van 2, 11 en 7} \\ \begin{align} & \color{blue}{154.} (\frac{x}{2}-\frac{4}{11})& = & (\frac{3}{7}x-1) \color{blue}{.154} \\\Leftrightarrow & 77x-56& = & 66x-154 \\\Leftrightarrow & 77x \color{red}{-56} \color{blue}{+56} \color{blue}{-66x} & = & \color{red}{66x} -154 \color{blue}{-66x} \color{blue}{+56} \\\Leftrightarrow & 11x& = & -98 \\\Leftrightarrow & \frac{11x}{ \color{red}{11} }& = & \frac{-98}{11} \\\Leftrightarrow & x = \frac{-98}{11} & & \\ & V = \left\{ \frac{-98}{11} \right\} & \\\end{align}\)
7. \(\text{165 is het kleinste gemene veelvoud van 3, 11 en 5} \\ \begin{align} & \color{blue}{165.} (\frac{x}{3}-\frac{4}{11})& = & (\frac{2}{5}x+4) \color{blue}{.165} \\\Leftrightarrow & 55x-60& = & 66x+660 \\\Leftrightarrow & 55x \color{red}{-60} \color{blue}{+60} \color{blue}{-66x} & = & \color{red}{66x} +660 \color{blue}{-66x} \color{blue}{+60} \\\Leftrightarrow & -11x& = & 720 \\\Leftrightarrow & \frac{-11x}{ \color{red}{-11} }& = & \frac{720}{-11} \\\Leftrightarrow & x = \frac{-720}{11} & & \\ & V = \left\{ \frac{-720}{11} \right\} & \\\end{align}\)
8. \(\text{18 is het kleinste gemene veelvoud van 3, 9 en 2} \\ \begin{align} & \color{blue}{18.} (\frac{x}{3}-\frac{4}{9})& = & (\frac{5}{2}x-8) \color{blue}{.18} \\\Leftrightarrow & 6x-8& = & 45x-144 \\\Leftrightarrow & 6x \color{red}{-8} \color{blue}{+8} \color{blue}{-45x} & = & \color{red}{45x} -144 \color{blue}{-45x} \color{blue}{+8} \\\Leftrightarrow & -39x& = & -136 \\\Leftrightarrow & \frac{-39x}{ \color{red}{-39} }& = & \frac{-136}{-39} \\\Leftrightarrow & x = \frac{136}{39} & & \\ & V = \left\{ \frac{136}{39} \right\} & \\\end{align}\)
9. \(\text{273 is het kleinste gemene veelvoud van 7, 13 en 3} \\ \begin{align} & \color{blue}{273.} (\frac{x}{7}-\frac{4}{13})& = & (\frac{4}{3}x+3) \color{blue}{.273} \\\Leftrightarrow & 39x-84& = & 364x+819 \\\Leftrightarrow & 39x \color{red}{-84} \color{blue}{+84} \color{blue}{-364x} & = & \color{red}{364x} +819 \color{blue}{-364x} \color{blue}{+84} \\\Leftrightarrow & -325x& = & 903 \\\Leftrightarrow & \frac{-325x}{ \color{red}{-325} }& = & \frac{903}{-325} \\\Leftrightarrow & x = \frac{-903}{325} & & \\ & V = \left\{ \frac{-903}{325} \right\} & \\\end{align}\)
10. \(\text{315 is het kleinste gemene veelvoud van 7, 9 en 5} \\ \begin{align} & \color{blue}{315.} (\frac{x}{7}-\frac{2}{9})& = & (\frac{-4}{5}x-2) \color{blue}{.315} \\\Leftrightarrow & 45x-70& = & -252x-630 \\\Leftrightarrow & 45x \color{red}{-70} \color{blue}{+70} \color{blue}{+252x} & = & \color{red}{-252x} -630 \color{blue}{+252x} \color{blue}{+70} \\\Leftrightarrow & 297x& = & -560 \\\Leftrightarrow & \frac{297x}{ \color{red}{297} }& = & \frac{-560}{297} \\\Leftrightarrow & x = \frac{-560}{297} & & \\ & V = \left\{ \frac{-560}{297} \right\} & \\\end{align}\)
11. \(\text{140 is het kleinste gemene veelvoud van 5, 7 en 4} \\ \begin{align} & \color{blue}{140.} (\frac{x}{5}-\frac{5}{7})& = & (\frac{5}{4}x-5) \color{blue}{.140} \\\Leftrightarrow & 28x-100& = & 175x-700 \\\Leftrightarrow & 28x \color{red}{-100} \color{blue}{+100} \color{blue}{-175x} & = & \color{red}{175x} -700 \color{blue}{-175x} \color{blue}{+100} \\\Leftrightarrow & -147x& = & -600 \\\Leftrightarrow & \frac{-147x}{ \color{red}{-147} }& = & \frac{-600}{-147} \\\Leftrightarrow & x = \frac{200}{49} & & \\ & V = \left\{ \frac{200}{49} \right\} & \\\end{align}\)
12. \(\text{80 is het kleinste gemene veelvoud van 5, 16 en 4} \\ \begin{align} & \color{blue}{80.} (\frac{x}{5}-\frac{3}{16})& = & (\frac{5}{4}x-7) \color{blue}{.80} \\\Leftrightarrow & 16x-15& = & 100x-560 \\\Leftrightarrow & 16x \color{red}{-15} \color{blue}{+15} \color{blue}{-100x} & = & \color{red}{100x} -560 \color{blue}{-100x} \color{blue}{+15} \\\Leftrightarrow & -84x& = & -545 \\\Leftrightarrow & \frac{-84x}{ \color{red}{-84} }& = & \frac{-545}{-84} \\\Leftrightarrow & x = \frac{545}{84} & & \\ & V = \left\{ \frac{545}{84} \right\} & \\\end{align}\)