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

---

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

Verbetersleutel

1. \(\text{20 is het kleinste gemene veelvoud van 4, 10 en 5} \\ \begin{align} & \frac{x}{4}+\frac{3}{10}& = & \frac{4}{5}x+7 \\\Leftrightarrow & \color{blue}{20.} (\frac{5x}{ \color{blue}{20} }+ \frac{ 6 }{ \color{blue}{20} })& = & (\frac{16}{ \color{blue}{20} }x+\frac{140}{ \color{blue}{20} }) \color{blue}{.20} \\\Leftrightarrow & 5x+6& = & 16x+140 \\\Leftrightarrow & 5x \color{red}{+6} \color{blue}{-6} \color{blue}{-16x} & = & \color{red}{16x} +140 \color{blue}{-16x} \color{blue}{-6} \\\Leftrightarrow & -11x& = & 134 \\\Leftrightarrow & \frac{-11x}{ \color{red}{-11} }& = & \frac{134}{-11} \\\Leftrightarrow & x = \frac{-134}{11} & & \\ & V = \left\{ \frac{-134}{11} \right\} & \\\end{align}\)
2. \(\text{42 is het kleinste gemene veelvoud van 3, 7 en 2} \\ \begin{align} & \frac{x}{3}+\frac{4}{7}& = & \frac{1}{2}x-7 \\\Leftrightarrow & \color{blue}{42.} (\frac{14x}{ \color{blue}{42} }+ \frac{ 24 }{ \color{blue}{42} })& = & (\frac{21}{ \color{blue}{42} }x-\frac{294}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 14x+24& = & 21x-294 \\\Leftrightarrow & 14x \color{red}{+24} \color{blue}{-24} \color{blue}{-21x} & = & \color{red}{21x} -294 \color{blue}{-21x} \color{blue}{-24} \\\Leftrightarrow & -7x& = & -318 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{-318}{-7} \\\Leftrightarrow & x = \frac{318}{7} & & \\ & V = \left\{ \frac{318}{7} \right\} & \\\end{align}\)
3. \(\text{12 is het kleinste gemene veelvoud van 4, 6 en 3} \\ \begin{align} & \frac{x}{4}-\frac{5}{6}& = & \frac{4}{3}x-5 \\\Leftrightarrow & \color{blue}{12.} (\frac{3x}{ \color{blue}{12} }- \frac{ 10 }{ \color{blue}{12} })& = & (\frac{16}{ \color{blue}{12} }x-\frac{60}{ \color{blue}{12} }) \color{blue}{.12} \\\Leftrightarrow & 3x-10& = & 16x-60 \\\Leftrightarrow & 3x \color{red}{-10} \color{blue}{+10} \color{blue}{-16x} & = & \color{red}{16x} -60 \color{blue}{-16x} \color{blue}{+10} \\\Leftrightarrow & -13x& = & -50 \\\Leftrightarrow & \frac{-13x}{ \color{red}{-13} }& = & \frac{-50}{-13} \\\Leftrightarrow & x = \frac{50}{13} & & \\ & V = \left\{ \frac{50}{13} \right\} & \\\end{align}\)
4. \(\text{220 is het kleinste gemene veelvoud van 4, 11 en 5} \\ \begin{align} & \frac{x}{4}-\frac{2}{11}& = & \frac{1}{5}x-2 \\\Leftrightarrow & \color{blue}{220.} (\frac{55x}{ \color{blue}{220} }- \frac{ 40 }{ \color{blue}{220} })& = & (\frac{44}{ \color{blue}{220} }x-\frac{440}{ \color{blue}{220} }) \color{blue}{.220} \\\Leftrightarrow & 55x-40& = & 44x-440 \\\Leftrightarrow & 55x \color{red}{-40} \color{blue}{+40} \color{blue}{-44x} & = & \color{red}{44x} -440 \color{blue}{-44x} \color{blue}{+40} \\\Leftrightarrow & 11x& = & -400 \\\Leftrightarrow & \frac{11x}{ \color{red}{11} }& = & \frac{-400}{11} \\\Leftrightarrow & x = \frac{-400}{11} & & \\ & V = \left\{ \frac{-400}{11} \right\} & \\\end{align}\)
5. \(\text{180 is het kleinste gemene veelvoud van 4, 9 en 5} \\ \begin{align} & \frac{x}{4}-\frac{4}{9}& = & \frac{1}{5}x+8 \\\Leftrightarrow & \color{blue}{180.} (\frac{45x}{ \color{blue}{180} }- \frac{ 80 }{ \color{blue}{180} })& = & (\frac{36}{ \color{blue}{180} }x+\frac{1440}{ \color{blue}{180} }) \color{blue}{.180} \\\Leftrightarrow & 45x-80& = & 36x+1440 \\\Leftrightarrow & 45x \color{red}{-80} \color{blue}{+80} \color{blue}{-36x} & = & \color{red}{36x} +1440 \color{blue}{-36x} \color{blue}{+80} \\\Leftrightarrow & 9x& = & 1520 \\\Leftrightarrow & \frac{9x}{ \color{red}{9} }& = & \frac{1520}{9} \\\Leftrightarrow & x = \frac{1520}{9} & & \\ & V = \left\{ \frac{1520}{9} \right\} & \\\end{align}\)
6. \(\text{182 is het kleinste gemene veelvoud van 7, 13 en 2} \\ \begin{align} & \frac{x}{7}-\frac{4}{13}& = & \frac{1}{2}x+7 \\\Leftrightarrow & \color{blue}{182.} (\frac{26x}{ \color{blue}{182} }- \frac{ 56 }{ \color{blue}{182} })& = & (\frac{91}{ \color{blue}{182} }x+\frac{1274}{ \color{blue}{182} }) \color{blue}{.182} \\\Leftrightarrow & 26x-56& = & 91x+1274 \\\Leftrightarrow & 26x \color{red}{-56} \color{blue}{+56} \color{blue}{-91x} & = & \color{red}{91x} +1274 \color{blue}{-91x} \color{blue}{+56} \\\Leftrightarrow & -65x& = & 1330 \\\Leftrightarrow & \frac{-65x}{ \color{red}{-65} }& = & \frac{1330}{-65} \\\Leftrightarrow & x = \frac{-266}{13} & & \\ & V = \left\{ \frac{-266}{13} \right\} & \\\end{align}\)
7. \(\text{84 is het kleinste gemene veelvoud van 7, 6 en 4} \\ \begin{align} & \frac{x}{7}+\frac{5}{6}& = & \frac{5}{4}x-8 \\\Leftrightarrow & \color{blue}{84.} (\frac{12x}{ \color{blue}{84} }+ \frac{ 70 }{ \color{blue}{84} })& = & (\frac{105}{ \color{blue}{84} }x-\frac{672}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 12x+70& = & 105x-672 \\\Leftrightarrow & 12x \color{red}{+70} \color{blue}{-70} \color{blue}{-105x} & = & \color{red}{105x} -672 \color{blue}{-105x} \color{blue}{-70} \\\Leftrightarrow & -93x& = & -742 \\\Leftrightarrow & \frac{-93x}{ \color{red}{-93} }& = & \frac{-742}{-93} \\\Leftrightarrow & x = \frac{742}{93} & & \\ & V = \left\{ \frac{742}{93} \right\} & \\\end{align}\)
8. \(\text{140 is het kleinste gemene veelvoud van 4, 7 en 5} \\ \begin{align} & \frac{x}{4}-\frac{4}{7}& = & \frac{-7}{5}x+1 \\\Leftrightarrow & \color{blue}{140.} (\frac{35x}{ \color{blue}{140} }- \frac{ 80 }{ \color{blue}{140} })& = & (\frac{-196}{ \color{blue}{140} }x+\frac{140}{ \color{blue}{140} }) \color{blue}{.140} \\\Leftrightarrow & 35x-80& = & -196x+140 \\\Leftrightarrow & 35x \color{red}{-80} \color{blue}{+80} \color{blue}{+196x} & = & \color{red}{-196x} +140 \color{blue}{+196x} \color{blue}{+80} \\\Leftrightarrow & 231x& = & 220 \\\Leftrightarrow & \frac{231x}{ \color{red}{231} }& = & \frac{220}{231} \\\Leftrightarrow & x = \frac{20}{21} & & \\ & V = \left\{ \frac{20}{21} \right\} & \\\end{align}\)
9. \(\text{210 is het kleinste gemene veelvoud van 5, 7 en 6} \\ \begin{align} & \frac{x}{5}+\frac{4}{7}& = & \frac{5}{6}x+7 \\\Leftrightarrow & \color{blue}{210.} (\frac{42x}{ \color{blue}{210} }+ \frac{ 120 }{ \color{blue}{210} })& = & (\frac{175}{ \color{blue}{210} }x+\frac{1470}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 42x+120& = & 175x+1470 \\\Leftrightarrow & 42x \color{red}{+120} \color{blue}{-120} \color{blue}{-175x} & = & \color{red}{175x} +1470 \color{blue}{-175x} \color{blue}{-120} \\\Leftrightarrow & -133x& = & 1350 \\\Leftrightarrow & \frac{-133x}{ \color{red}{-133} }& = & \frac{1350}{-133} \\\Leftrightarrow & x = \frac{-1350}{133} & & \\ & V = \left\{ \frac{-1350}{133} \right\} & \\\end{align}\)
10. \(\text{165 is het kleinste gemene veelvoud van 3, 11 en 5} \\ \begin{align} & \frac{x}{3}-\frac{5}{11}& = & \frac{1}{5}x-7 \\\Leftrightarrow & \color{blue}{165.} (\frac{55x}{ \color{blue}{165} }- \frac{ 75 }{ \color{blue}{165} })& = & (\frac{33}{ \color{blue}{165} }x-\frac{1155}{ \color{blue}{165} }) \color{blue}{.165} \\\Leftrightarrow & 55x-75& = & 33x-1155 \\\Leftrightarrow & 55x \color{red}{-75} \color{blue}{+75} \color{blue}{-33x} & = & \color{red}{33x} -1155 \color{blue}{-33x} \color{blue}{+75} \\\Leftrightarrow & 22x& = & -1080 \\\Leftrightarrow & \frac{22x}{ \color{red}{22} }& = & \frac{-1080}{22} \\\Leftrightarrow & x = \frac{-540}{11} & & \\ & V = \left\{ \frac{-540}{11} \right\} & \\\end{align}\)
11. \(\text{105 is het kleinste gemene veelvoud van 7, 15 en 5} \\ \begin{align} & \frac{x}{7}+\frac{4}{15}& = & \frac{-4}{5}x+4 \\\Leftrightarrow & \color{blue}{105.} (\frac{15x}{ \color{blue}{105} }+ \frac{ 28 }{ \color{blue}{105} })& = & (\frac{-84}{ \color{blue}{105} }x+\frac{420}{ \color{blue}{105} }) \color{blue}{.105} \\\Leftrightarrow & 15x+28& = & -84x+420 \\\Leftrightarrow & 15x \color{red}{+28} \color{blue}{-28} \color{blue}{+84x} & = & \color{red}{-84x} +420 \color{blue}{+84x} \color{blue}{-28} \\\Leftrightarrow & 99x& = & 392 \\\Leftrightarrow & \frac{99x}{ \color{red}{99} }& = & \frac{392}{99} \\\Leftrightarrow & x = \frac{392}{99} & & \\ & V = \left\{ \frac{392}{99} \right\} & \\\end{align}\)
12. \(\text{390 is het kleinste gemene veelvoud van 6, 13 en 5} \\ \begin{align} & \frac{x}{6}-\frac{2}{13}& = & \frac{1}{5}x+7 \\\Leftrightarrow & \color{blue}{390.} (\frac{65x}{ \color{blue}{390} }- \frac{ 60 }{ \color{blue}{390} })& = & (\frac{78}{ \color{blue}{390} }x+\frac{2730}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 65x-60& = & 78x+2730 \\\Leftrightarrow & 65x \color{red}{-60} \color{blue}{+60} \color{blue}{-78x} & = & \color{red}{78x} +2730 \color{blue}{-78x} \color{blue}{+60} \\\Leftrightarrow & -13x& = & 2790 \\\Leftrightarrow & \frac{-13x}{ \color{red}{-13} }& = & \frac{2790}{-13} \\\Leftrightarrow & x = \frac{-2790}{13} & & \\ & V = \left\{ \frac{-2790}{13} \right\} & \\\end{align}\)