Vgln. eerste graad (reeks 4)

Hoofdmenu Eentje per keer  🖨

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

1. \(\frac{x}{3}+\frac{3}{16}=\frac{1}{2}x+7\)
2. \(\frac{x}{6}+\frac{5}{16}=\frac{2}{5}x+5\)
3. \(\frac{x}{3}-\frac{2}{13}=\frac{7}{4}x+6\)
4. \(\frac{x}{5}+\frac{5}{14}=\frac{1}{2}x+6\)
5. \(\frac{x}{3}+\frac{4}{7}=\frac{3}{4}x-6\)
6. \(\frac{x}{7}-\frac{3}{14}=\frac{5}{6}x+4\)
7. \(\frac{x}{7}-\frac{4}{9}=\frac{1}{6}x-1\)
8. \(\frac{x}{5}-\frac{5}{6}=\frac{5}{6}x-8\)
9. \(\frac{x}{6}-\frac{5}{11}=\frac{7}{5}x+1\)
10. \(\frac{x}{2}-\frac{4}{9}=\frac{-7}{5}x+3\)
11. \(\frac{x}{4}+\frac{5}{14}=\frac{1}{5}x+2\)
12. \(\frac{x}{4}+\frac{4}{13}=\frac{-8}{3}x+7\)

---

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

Verbetersleutel

1. \(\text{48 is het kleinste gemene veelvoud van 3, 16 en 2} \\ \begin{align} & \frac{x}{3}+\frac{3}{16}& = & \frac{1}{2}x+7 \\\Leftrightarrow & \color{blue}{48.} (\frac{16x}{ \color{blue}{48} }+ \frac{ 9 }{ \color{blue}{48} })& = & (\frac{24}{ \color{blue}{48} }x+\frac{336}{ \color{blue}{48} }) \color{blue}{.48} \\\Leftrightarrow & 16x+9& = & 24x+336 \\\Leftrightarrow & 16x \color{red}{+9} \color{blue}{-9} \color{blue}{-24x} & = & \color{red}{24x} +336 \color{blue}{-24x} \color{blue}{-9} \\\Leftrightarrow & -8x& = & 327 \\\Leftrightarrow & \frac{-8x}{ \color{red}{-8} }& = & \frac{327}{-8} \\\Leftrightarrow & x = \frac{-327}{8} & & \\ & V = \left\{ \frac{-327}{8} \right\} & \\\end{align}\)
2. \(\text{240 is het kleinste gemene veelvoud van 6, 16 en 5} \\ \begin{align} & \frac{x}{6}+\frac{5}{16}& = & \frac{2}{5}x+5 \\\Leftrightarrow & \color{blue}{240.} (\frac{40x}{ \color{blue}{240} }+ \frac{ 75 }{ \color{blue}{240} })& = & (\frac{96}{ \color{blue}{240} }x+\frac{1200}{ \color{blue}{240} }) \color{blue}{.240} \\\Leftrightarrow & 40x+75& = & 96x+1200 \\\Leftrightarrow & 40x \color{red}{+75} \color{blue}{-75} \color{blue}{-96x} & = & \color{red}{96x} +1200 \color{blue}{-96x} \color{blue}{-75} \\\Leftrightarrow & -56x& = & 1125 \\\Leftrightarrow & \frac{-56x}{ \color{red}{-56} }& = & \frac{1125}{-56} \\\Leftrightarrow & x = \frac{-1125}{56} & & \\ & V = \left\{ \frac{-1125}{56} \right\} & \\\end{align}\)
3. \(\text{156 is het kleinste gemene veelvoud van 3, 13 en 4} \\ \begin{align} & \frac{x}{3}-\frac{2}{13}& = & \frac{7}{4}x+6 \\\Leftrightarrow & \color{blue}{156.} (\frac{52x}{ \color{blue}{156} }- \frac{ 24 }{ \color{blue}{156} })& = & (\frac{273}{ \color{blue}{156} }x+\frac{936}{ \color{blue}{156} }) \color{blue}{.156} \\\Leftrightarrow & 52x-24& = & 273x+936 \\\Leftrightarrow & 52x \color{red}{-24} \color{blue}{+24} \color{blue}{-273x} & = & \color{red}{273x} +936 \color{blue}{-273x} \color{blue}{+24} \\\Leftrightarrow & -221x& = & 960 \\\Leftrightarrow & \frac{-221x}{ \color{red}{-221} }& = & \frac{960}{-221} \\\Leftrightarrow & x = \frac{-960}{221} & & \\ & V = \left\{ \frac{-960}{221} \right\} & \\\end{align}\)
4. \(\text{70 is het kleinste gemene veelvoud van 5, 14 en 2} \\ \begin{align} & \frac{x}{5}+\frac{5}{14}& = & \frac{1}{2}x+6 \\\Leftrightarrow & \color{blue}{70.} (\frac{14x}{ \color{blue}{70} }+ \frac{ 25 }{ \color{blue}{70} })& = & (\frac{35}{ \color{blue}{70} }x+\frac{420}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 14x+25& = & 35x+420 \\\Leftrightarrow & 14x \color{red}{+25} \color{blue}{-25} \color{blue}{-35x} & = & \color{red}{35x} +420 \color{blue}{-35x} \color{blue}{-25} \\\Leftrightarrow & -21x& = & 395 \\\Leftrightarrow & \frac{-21x}{ \color{red}{-21} }& = & \frac{395}{-21} \\\Leftrightarrow & x = \frac{-395}{21} & & \\ & V = \left\{ \frac{-395}{21} \right\} & \\\end{align}\)
5. \(\text{84 is het kleinste gemene veelvoud van 3, 7 en 4} \\ \begin{align} & \frac{x}{3}+\frac{4}{7}& = & \frac{3}{4}x-6 \\\Leftrightarrow & \color{blue}{84.} (\frac{28x}{ \color{blue}{84} }+ \frac{ 48 }{ \color{blue}{84} })& = & (\frac{63}{ \color{blue}{84} }x-\frac{504}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 28x+48& = & 63x-504 \\\Leftrightarrow & 28x \color{red}{+48} \color{blue}{-48} \color{blue}{-63x} & = & \color{red}{63x} -504 \color{blue}{-63x} \color{blue}{-48} \\\Leftrightarrow & -35x& = & -552 \\\Leftrightarrow & \frac{-35x}{ \color{red}{-35} }& = & \frac{-552}{-35} \\\Leftrightarrow & x = \frac{552}{35} & & \\ & V = \left\{ \frac{552}{35} \right\} & \\\end{align}\)
6. \(\text{42 is het kleinste gemene veelvoud van 7, 14 en 6} \\ \begin{align} & \frac{x}{7}-\frac{3}{14}& = & \frac{5}{6}x+4 \\\Leftrightarrow & \color{blue}{42.} (\frac{6x}{ \color{blue}{42} }- \frac{ 9 }{ \color{blue}{42} })& = & (\frac{35}{ \color{blue}{42} }x+\frac{168}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 6x-9& = & 35x+168 \\\Leftrightarrow & 6x \color{red}{-9} \color{blue}{+9} \color{blue}{-35x} & = & \color{red}{35x} +168 \color{blue}{-35x} \color{blue}{+9} \\\Leftrightarrow & -29x& = & 177 \\\Leftrightarrow & \frac{-29x}{ \color{red}{-29} }& = & \frac{177}{-29} \\\Leftrightarrow & x = \frac{-177}{29} & & \\ & V = \left\{ \frac{-177}{29} \right\} & \\\end{align}\)
7. \(\text{126 is het kleinste gemene veelvoud van 7, 9 en 6} \\ \begin{align} & \frac{x}{7}-\frac{4}{9}& = & \frac{1}{6}x-1 \\\Leftrightarrow & \color{blue}{126.} (\frac{18x}{ \color{blue}{126} }- \frac{ 56 }{ \color{blue}{126} })& = & (\frac{21}{ \color{blue}{126} }x-\frac{126}{ \color{blue}{126} }) \color{blue}{.126} \\\Leftrightarrow & 18x-56& = & 21x-126 \\\Leftrightarrow & 18x \color{red}{-56} \color{blue}{+56} \color{blue}{-21x} & = & \color{red}{21x} -126 \color{blue}{-21x} \color{blue}{+56} \\\Leftrightarrow & -3x& = & -70 \\\Leftrightarrow & \frac{-3x}{ \color{red}{-3} }& = & \frac{-70}{-3} \\\Leftrightarrow & x = \frac{70}{3} & & \\ & V = \left\{ \frac{70}{3} \right\} & \\\end{align}\)
8. \(\text{30 is het kleinste gemene veelvoud van 5, 6 en 6} \\ \begin{align} & \frac{x}{5}-\frac{5}{6}& = & \frac{5}{6}x-8 \\\Leftrightarrow & \color{blue}{30.} (\frac{6x}{ \color{blue}{30} }- \frac{ 25 }{ \color{blue}{30} })& = & (\frac{25}{ \color{blue}{30} }x-\frac{240}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 6x-25& = & 25x-240 \\\Leftrightarrow & 6x \color{red}{-25} \color{blue}{+25} \color{blue}{-25x} & = & \color{red}{25x} -240 \color{blue}{-25x} \color{blue}{+25} \\\Leftrightarrow & -19x& = & -215 \\\Leftrightarrow & \frac{-19x}{ \color{red}{-19} }& = & \frac{-215}{-19} \\\Leftrightarrow & x = \frac{215}{19} & & \\ & V = \left\{ \frac{215}{19} \right\} & \\\end{align}\)
9. \(\text{330 is het kleinste gemene veelvoud van 6, 11 en 5} \\ \begin{align} & \frac{x}{6}-\frac{5}{11}& = & \frac{7}{5}x+1 \\\Leftrightarrow & \color{blue}{330.} (\frac{55x}{ \color{blue}{330} }- \frac{ 150 }{ \color{blue}{330} })& = & (\frac{462}{ \color{blue}{330} }x+\frac{330}{ \color{blue}{330} }) \color{blue}{.330} \\\Leftrightarrow & 55x-150& = & 462x+330 \\\Leftrightarrow & 55x \color{red}{-150} \color{blue}{+150} \color{blue}{-462x} & = & \color{red}{462x} +330 \color{blue}{-462x} \color{blue}{+150} \\\Leftrightarrow & -407x& = & 480 \\\Leftrightarrow & \frac{-407x}{ \color{red}{-407} }& = & \frac{480}{-407} \\\Leftrightarrow & x = \frac{-480}{407} & & \\ & V = \left\{ \frac{-480}{407} \right\} & \\\end{align}\)
10. \(\text{90 is het kleinste gemene veelvoud van 2, 9 en 5} \\ \begin{align} & \frac{x}{2}-\frac{4}{9}& = & \frac{-7}{5}x+3 \\\Leftrightarrow & \color{blue}{90.} (\frac{45x}{ \color{blue}{90} }- \frac{ 40 }{ \color{blue}{90} })& = & (\frac{-126}{ \color{blue}{90} }x+\frac{270}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 45x-40& = & -126x+270 \\\Leftrightarrow & 45x \color{red}{-40} \color{blue}{+40} \color{blue}{+126x} & = & \color{red}{-126x} +270 \color{blue}{+126x} \color{blue}{+40} \\\Leftrightarrow & 171x& = & 310 \\\Leftrightarrow & \frac{171x}{ \color{red}{171} }& = & \frac{310}{171} \\\Leftrightarrow & x = \frac{310}{171} & & \\ & V = \left\{ \frac{310}{171} \right\} & \\\end{align}\)
11. \(\text{140 is het kleinste gemene veelvoud van 4, 14 en 5} \\ \begin{align} & \frac{x}{4}+\frac{5}{14}& = & \frac{1}{5}x+2 \\\Leftrightarrow & \color{blue}{140.} (\frac{35x}{ \color{blue}{140} }+ \frac{ 50 }{ \color{blue}{140} })& = & (\frac{28}{ \color{blue}{140} }x+\frac{280}{ \color{blue}{140} }) \color{blue}{.140} \\\Leftrightarrow & 35x+50& = & 28x+280 \\\Leftrightarrow & 35x \color{red}{+50} \color{blue}{-50} \color{blue}{-28x} & = & \color{red}{28x} +280 \color{blue}{-28x} \color{blue}{-50} \\\Leftrightarrow & 7x& = & 230 \\\Leftrightarrow & \frac{7x}{ \color{red}{7} }& = & \frac{230}{7} \\\Leftrightarrow & x = \frac{230}{7} & & \\ & V = \left\{ \frac{230}{7} \right\} & \\\end{align}\)
12. \(\text{156 is het kleinste gemene veelvoud van 4, 13 en 3} \\ \begin{align} & \frac{x}{4}+\frac{4}{13}& = & \frac{-8}{3}x+7 \\\Leftrightarrow & \color{blue}{156.} (\frac{39x}{ \color{blue}{156} }+ \frac{ 48 }{ \color{blue}{156} })& = & (\frac{-416}{ \color{blue}{156} }x+\frac{1092}{ \color{blue}{156} }) \color{blue}{.156} \\\Leftrightarrow & 39x+48& = & -416x+1092 \\\Leftrightarrow & 39x \color{red}{+48} \color{blue}{-48} \color{blue}{+416x} & = & \color{red}{-416x} +1092 \color{blue}{+416x} \color{blue}{-48} \\\Leftrightarrow & 455x& = & 1044 \\\Leftrightarrow & \frac{455x}{ \color{red}{455} }& = & \frac{1044}{455} \\\Leftrightarrow & x = \frac{1044}{455} & & \\ & V = \left\{ \frac{1044}{455} \right\} & \\\end{align}\)