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

---

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

Verbetersleutel

1. \(\text{126 is het kleinste gemene veelvoud van 7, 9 en 2} \\ \begin{align} & \frac{x}{7}+\frac{2}{9}& = & \frac{7}{2}x+8 \\\Leftrightarrow & \color{blue}{126.} (\frac{18x}{ \color{blue}{126} }+ \frac{ 28 }{ \color{blue}{126} })& = & (\frac{441}{ \color{blue}{126} }x+\frac{1008}{ \color{blue}{126} }) \color{blue}{.126} \\\Leftrightarrow & 18x+28& = & 441x+1008 \\\Leftrightarrow & 18x \color{red}{+28} \color{blue}{-28} \color{blue}{-441x} & = & \color{red}{441x} +1008 \color{blue}{-441x} \color{blue}{-28} \\\Leftrightarrow & -423x& = & 980 \\\Leftrightarrow & \frac{-423x}{ \color{red}{-423} }& = & \frac{980}{-423} \\\Leftrightarrow & x = \frac{-980}{423} & & \\ & V = \left\{ \frac{-980}{423} \right\} & \\\end{align}\)
2. \(\text{220 is het kleinste gemene veelvoud van 5, 11 en 4} \\ \begin{align} & \frac{x}{5}+\frac{4}{11}& = & \frac{5}{4}x-4 \\\Leftrightarrow & \color{blue}{220.} (\frac{44x}{ \color{blue}{220} }+ \frac{ 80 }{ \color{blue}{220} })& = & (\frac{275}{ \color{blue}{220} }x-\frac{880}{ \color{blue}{220} }) \color{blue}{.220} \\\Leftrightarrow & 44x+80& = & 275x-880 \\\Leftrightarrow & 44x \color{red}{+80} \color{blue}{-80} \color{blue}{-275x} & = & \color{red}{275x} -880 \color{blue}{-275x} \color{blue}{-80} \\\Leftrightarrow & -231x& = & -960 \\\Leftrightarrow & \frac{-231x}{ \color{red}{-231} }& = & \frac{-960}{-231} \\\Leftrightarrow & x = \frac{320}{77} & & \\ & V = \left\{ \frac{320}{77} \right\} & \\\end{align}\)
3. \(\text{12 is het kleinste gemene veelvoud van 4, 12 en 3} \\ \begin{align} & \frac{x}{4}-\frac{5}{12}& = & \frac{7}{3}x+8 \\\Leftrightarrow & \color{blue}{12.} (\frac{3x}{ \color{blue}{12} }- \frac{ 5 }{ \color{blue}{12} })& = & (\frac{28}{ \color{blue}{12} }x+\frac{96}{ \color{blue}{12} }) \color{blue}{.12} \\\Leftrightarrow & 3x-5& = & 28x+96 \\\Leftrightarrow & 3x \color{red}{-5} \color{blue}{+5} \color{blue}{-28x} & = & \color{red}{28x} +96 \color{blue}{-28x} \color{blue}{+5} \\\Leftrightarrow & -25x& = & 101 \\\Leftrightarrow & \frac{-25x}{ \color{red}{-25} }& = & \frac{101}{-25} \\\Leftrightarrow & x = \frac{-101}{25} & & \\ & V = \left\{ \frac{-101}{25} \right\} & \\\end{align}\)
4. \(\text{70 is het kleinste gemene veelvoud van 5, 7 en 2} \\ \begin{align} & \frac{x}{5}-\frac{2}{7}& = & \frac{1}{2}x-4 \\\Leftrightarrow & \color{blue}{70.} (\frac{14x}{ \color{blue}{70} }- \frac{ 20 }{ \color{blue}{70} })& = & (\frac{35}{ \color{blue}{70} }x-\frac{280}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 14x-20& = & 35x-280 \\\Leftrightarrow & 14x \color{red}{-20} \color{blue}{+20} \color{blue}{-35x} & = & \color{red}{35x} -280 \color{blue}{-35x} \color{blue}{+20} \\\Leftrightarrow & -21x& = & -260 \\\Leftrightarrow & \frac{-21x}{ \color{red}{-21} }& = & \frac{-260}{-21} \\\Leftrightarrow & x = \frac{260}{21} & & \\ & V = \left\{ \frac{260}{21} \right\} & \\\end{align}\)
5. \(\text{42 is het kleinste gemene veelvoud van 3, 7 en 2} \\ \begin{align} & \frac{x}{3}+\frac{3}{7}& = & \frac{1}{2}x+7 \\\Leftrightarrow & \color{blue}{42.} (\frac{14x}{ \color{blue}{42} }+ \frac{ 18 }{ \color{blue}{42} })& = & (\frac{21}{ \color{blue}{42} }x+\frac{294}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 14x+18& = & 21x+294 \\\Leftrightarrow & 14x \color{red}{+18} \color{blue}{-18} \color{blue}{-21x} & = & \color{red}{21x} +294 \color{blue}{-21x} \color{blue}{-18} \\\Leftrightarrow & -7x& = & 276 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{276}{-7} \\\Leftrightarrow & x = \frac{-276}{7} & & \\ & V = \left\{ \frac{-276}{7} \right\} & \\\end{align}\)
6. \(\text{132 is het kleinste gemene veelvoud van 3, 11 en 4} \\ \begin{align} & \frac{x}{3}-\frac{5}{11}& = & \frac{1}{4}x-5 \\\Leftrightarrow & \color{blue}{132.} (\frac{44x}{ \color{blue}{132} }- \frac{ 60 }{ \color{blue}{132} })& = & (\frac{33}{ \color{blue}{132} }x-\frac{660}{ \color{blue}{132} }) \color{blue}{.132} \\\Leftrightarrow & 44x-60& = & 33x-660 \\\Leftrightarrow & 44x \color{red}{-60} \color{blue}{+60} \color{blue}{-33x} & = & \color{red}{33x} -660 \color{blue}{-33x} \color{blue}{+60} \\\Leftrightarrow & 11x& = & -600 \\\Leftrightarrow & \frac{11x}{ \color{red}{11} }& = & \frac{-600}{11} \\\Leftrightarrow & x = \frac{-600}{11} & & \\ & V = \left\{ \frac{-600}{11} \right\} & \\\end{align}\)
7. \(\text{14 is het kleinste gemene veelvoud van 7, 7 en 2} \\ \begin{align} & \frac{x}{7}+\frac{3}{7}& = & \frac{5}{2}x-5 \\\Leftrightarrow & \color{blue}{14.} (\frac{2x}{ \color{blue}{14} }+ \frac{ 6 }{ \color{blue}{14} })& = & (\frac{35}{ \color{blue}{14} }x-\frac{70}{ \color{blue}{14} }) \color{blue}{.14} \\\Leftrightarrow & 2x+6& = & 35x-70 \\\Leftrightarrow & 2x \color{red}{+6} \color{blue}{-6} \color{blue}{-35x} & = & \color{red}{35x} -70 \color{blue}{-35x} \color{blue}{-6} \\\Leftrightarrow & -33x& = & -76 \\\Leftrightarrow & \frac{-33x}{ \color{red}{-33} }& = & \frac{-76}{-33} \\\Leftrightarrow & x = \frac{76}{33} & & \\ & V = \left\{ \frac{76}{33} \right\} & \\\end{align}\)
8. \(\text{84 is het kleinste gemene veelvoud van 4, 14 en 3} \\ \begin{align} & \frac{x}{4}-\frac{5}{14}& = & \frac{-2}{3}x-6 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }- \frac{ 30 }{ \color{blue}{84} })& = & (\frac{-56}{ \color{blue}{84} }x-\frac{504}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x-30& = & -56x-504 \\\Leftrightarrow & 21x \color{red}{-30} \color{blue}{+30} \color{blue}{+56x} & = & \color{red}{-56x} -504 \color{blue}{+56x} \color{blue}{+30} \\\Leftrightarrow & 77x& = & -474 \\\Leftrightarrow & \frac{77x}{ \color{red}{77} }& = & \frac{-474}{77} \\\Leftrightarrow & x = \frac{-474}{77} & & \\ & V = \left\{ \frac{-474}{77} \right\} & \\\end{align}\)
9. \(\text{14 is het kleinste gemene veelvoud van 7, 14 en 2} \\ \begin{align} & \frac{x}{7}+\frac{5}{14}& = & \frac{1}{2}x-8 \\\Leftrightarrow & \color{blue}{14.} (\frac{2x}{ \color{blue}{14} }+ \frac{ 5 }{ \color{blue}{14} })& = & (\frac{7}{ \color{blue}{14} }x-\frac{112}{ \color{blue}{14} }) \color{blue}{.14} \\\Leftrightarrow & 2x+5& = & 7x-112 \\\Leftrightarrow & 2x \color{red}{+5} \color{blue}{-5} \color{blue}{-7x} & = & \color{red}{7x} -112 \color{blue}{-7x} \color{blue}{-5} \\\Leftrightarrow & -5x& = & -117 \\\Leftrightarrow & \frac{-5x}{ \color{red}{-5} }& = & \frac{-117}{-5} \\\Leftrightarrow & x = \frac{117}{5} & & \\ & V = \left\{ \frac{117}{5} \right\} & \\\end{align}\)
10. \(\text{330 is het kleinste gemene veelvoud van 6, 11 en 5} \\ \begin{align} & \frac{x}{6}-\frac{3}{11}& = & \frac{6}{5}x+8 \\\Leftrightarrow & \color{blue}{330.} (\frac{55x}{ \color{blue}{330} }- \frac{ 90 }{ \color{blue}{330} })& = & (\frac{396}{ \color{blue}{330} }x+\frac{2640}{ \color{blue}{330} }) \color{blue}{.330} \\\Leftrightarrow & 55x-90& = & 396x+2640 \\\Leftrightarrow & 55x \color{red}{-90} \color{blue}{+90} \color{blue}{-396x} & = & \color{red}{396x} +2640 \color{blue}{-396x} \color{blue}{+90} \\\Leftrightarrow & -341x& = & 2730 \\\Leftrightarrow & \frac{-341x}{ \color{red}{-341} }& = & \frac{2730}{-341} \\\Leftrightarrow & x = \frac{-2730}{341} & & \\ & V = \left\{ \frac{-2730}{341} \right\} & \\\end{align}\)
11. \(\text{30 is het kleinste gemene veelvoud van 2, 15 en 5} \\ \begin{align} & \frac{x}{2}+\frac{2}{15}& = & \frac{-4}{5}x+8 \\\Leftrightarrow & \color{blue}{30.} (\frac{15x}{ \color{blue}{30} }+ \frac{ 4 }{ \color{blue}{30} })& = & (\frac{-24}{ \color{blue}{30} }x+\frac{240}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 15x+4& = & -24x+240 \\\Leftrightarrow & 15x \color{red}{+4} \color{blue}{-4} \color{blue}{+24x} & = & \color{red}{-24x} +240 \color{blue}{+24x} \color{blue}{-4} \\\Leftrightarrow & 39x& = & 236 \\\Leftrightarrow & \frac{39x}{ \color{red}{39} }& = & \frac{236}{39} \\\Leftrightarrow & x = \frac{236}{39} & & \\ & V = \left\{ \frac{236}{39} \right\} & \\\end{align}\)
12. \(\text{140 is het kleinste gemene veelvoud van 4, 7 en 5} \\ \begin{align} & \frac{x}{4}-\frac{2}{7}& = & \frac{3}{5}x+3 \\\Leftrightarrow & \color{blue}{140.} (\frac{35x}{ \color{blue}{140} }- \frac{ 40 }{ \color{blue}{140} })& = & (\frac{84}{ \color{blue}{140} }x+\frac{420}{ \color{blue}{140} }) \color{blue}{.140} \\\Leftrightarrow & 35x-40& = & 84x+420 \\\Leftrightarrow & 35x \color{red}{-40} \color{blue}{+40} \color{blue}{-84x} & = & \color{red}{84x} +420 \color{blue}{-84x} \color{blue}{+40} \\\Leftrightarrow & -49x& = & 460 \\\Leftrightarrow & \frac{-49x}{ \color{red}{-49} }& = & \frac{460}{-49} \\\Leftrightarrow & x = \frac{-460}{49} & & \\ & V = \left\{ \frac{-460}{49} \right\} & \\\end{align}\)