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. \(\frac{x}{2}+\frac{4}{7}=\frac{-7}{5}x+3\)
2. \(\frac{x}{6}+\frac{2}{15}=\frac{6}{5}x+3\)
3. \(\frac{x}{3}+\frac{5}{7}=\frac{1}{2}x+3\)
4. \(\frac{x}{2}-\frac{3}{13}=\frac{-7}{5}x-7\)
5. \(\frac{x}{6}+\frac{4}{13}=\frac{-4}{5}x-6\)
6. \(\frac{x}{6}-\frac{2}{13}=\frac{2}{5}x-1\)
7. \(\frac{x}{6}-\frac{3}{13}=\frac{-7}{5}x-5\)
8. \(\frac{x}{4}+\frac{5}{8}=\frac{4}{5}x+5\)
9. \(\frac{x}{4}-\frac{4}{13}=\frac{-2}{5}x-3\)
10. \(\frac{x}{2}+\frac{2}{13}=\frac{1}{3}x-2\)
11. \(\frac{x}{4}-\frac{3}{16}=\frac{5}{3}x+6\)
12. \(\frac{x}{7}+\frac{5}{6}=\frac{7}{2}x-7\)

---

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

Verbetersleutel

1. \(\text{70 is het kleinste gemene veelvoud van 2, 7 en 5} \\ \begin{align} & \frac{x}{2}+\frac{4}{7}& = & \frac{-7}{5}x+3 \\\Leftrightarrow & \color{blue}{70.} (\frac{35x}{ \color{blue}{70} }+ \frac{ 40 }{ \color{blue}{70} })& = & (\frac{-98}{ \color{blue}{70} }x+\frac{210}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 35x+40& = & -98x+210 \\\Leftrightarrow & 35x \color{red}{+40} \color{blue}{-40} \color{blue}{+98x} & = & \color{red}{-98x} +210 \color{blue}{+98x} \color{blue}{-40} \\\Leftrightarrow & 133x& = & 170 \\\Leftrightarrow & \frac{133x}{ \color{red}{133} }& = & \frac{170}{133} \\\Leftrightarrow & x = \frac{170}{133} & & \\ & V = \left\{ \frac{170}{133} \right\} & \\\end{align}\)
2. \(\text{30 is het kleinste gemene veelvoud van 6, 15 en 5} \\ \begin{align} & \frac{x}{6}+\frac{2}{15}& = & \frac{6}{5}x+3 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }+ \frac{ 4 }{ \color{blue}{30} })& = & (\frac{36}{ \color{blue}{30} }x+\frac{90}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x+4& = & 36x+90 \\\Leftrightarrow & 5x \color{red}{+4} \color{blue}{-4} \color{blue}{-36x} & = & \color{red}{36x} +90 \color{blue}{-36x} \color{blue}{-4} \\\Leftrightarrow & -31x& = & 86 \\\Leftrightarrow & \frac{-31x}{ \color{red}{-31} }& = & \frac{86}{-31} \\\Leftrightarrow & x = \frac{-86}{31} & & \\ & V = \left\{ \frac{-86}{31} \right\} & \\\end{align}\)
3. \(\text{42 is het kleinste gemene veelvoud van 3, 7 en 2} \\ \begin{align} & \frac{x}{3}+\frac{5}{7}& = & \frac{1}{2}x+3 \\\Leftrightarrow & \color{blue}{42.} (\frac{14x}{ \color{blue}{42} }+ \frac{ 30 }{ \color{blue}{42} })& = & (\frac{21}{ \color{blue}{42} }x+\frac{126}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 14x+30& = & 21x+126 \\\Leftrightarrow & 14x \color{red}{+30} \color{blue}{-30} \color{blue}{-21x} & = & \color{red}{21x} +126 \color{blue}{-21x} \color{blue}{-30} \\\Leftrightarrow & -7x& = & 96 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{96}{-7} \\\Leftrightarrow & x = \frac{-96}{7} & & \\ & V = \left\{ \frac{-96}{7} \right\} & \\\end{align}\)
4. \(\text{130 is het kleinste gemene veelvoud van 2, 13 en 5} \\ \begin{align} & \frac{x}{2}-\frac{3}{13}& = & \frac{-7}{5}x-7 \\\Leftrightarrow & \color{blue}{130.} (\frac{65x}{ \color{blue}{130} }- \frac{ 30 }{ \color{blue}{130} })& = & (\frac{-182}{ \color{blue}{130} }x-\frac{910}{ \color{blue}{130} }) \color{blue}{.130} \\\Leftrightarrow & 65x-30& = & -182x-910 \\\Leftrightarrow & 65x \color{red}{-30} \color{blue}{+30} \color{blue}{+182x} & = & \color{red}{-182x} -910 \color{blue}{+182x} \color{blue}{+30} \\\Leftrightarrow & 247x& = & -880 \\\Leftrightarrow & \frac{247x}{ \color{red}{247} }& = & \frac{-880}{247} \\\Leftrightarrow & x = \frac{-880}{247} & & \\ & V = \left\{ \frac{-880}{247} \right\} & \\\end{align}\)
5. \(\text{390 is het kleinste gemene veelvoud van 6, 13 en 5} \\ \begin{align} & \frac{x}{6}+\frac{4}{13}& = & \frac{-4}{5}x-6 \\\Leftrightarrow & \color{blue}{390.} (\frac{65x}{ \color{blue}{390} }+ \frac{ 120 }{ \color{blue}{390} })& = & (\frac{-312}{ \color{blue}{390} }x-\frac{2340}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 65x+120& = & -312x-2340 \\\Leftrightarrow & 65x \color{red}{+120} \color{blue}{-120} \color{blue}{+312x} & = & \color{red}{-312x} -2340 \color{blue}{+312x} \color{blue}{-120} \\\Leftrightarrow & 377x& = & -2460 \\\Leftrightarrow & \frac{377x}{ \color{red}{377} }& = & \frac{-2460}{377} \\\Leftrightarrow & x = \frac{-2460}{377} & & \\ & V = \left\{ \frac{-2460}{377} \right\} & \\\end{align}\)
6. \(\text{390 is het kleinste gemene veelvoud van 6, 13 en 5} \\ \begin{align} & \frac{x}{6}-\frac{2}{13}& = & \frac{2}{5}x-1 \\\Leftrightarrow & \color{blue}{390.} (\frac{65x}{ \color{blue}{390} }- \frac{ 60 }{ \color{blue}{390} })& = & (\frac{156}{ \color{blue}{390} }x-\frac{390}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 65x-60& = & 156x-390 \\\Leftrightarrow & 65x \color{red}{-60} \color{blue}{+60} \color{blue}{-156x} & = & \color{red}{156x} -390 \color{blue}{-156x} \color{blue}{+60} \\\Leftrightarrow & -91x& = & -330 \\\Leftrightarrow & \frac{-91x}{ \color{red}{-91} }& = & \frac{-330}{-91} \\\Leftrightarrow & x = \frac{330}{91} & & \\ & V = \left\{ \frac{330}{91} \right\} & \\\end{align}\)
7. \(\text{390 is het kleinste gemene veelvoud van 6, 13 en 5} \\ \begin{align} & \frac{x}{6}-\frac{3}{13}& = & \frac{-7}{5}x-5 \\\Leftrightarrow & \color{blue}{390.} (\frac{65x}{ \color{blue}{390} }- \frac{ 90 }{ \color{blue}{390} })& = & (\frac{-546}{ \color{blue}{390} }x-\frac{1950}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 65x-90& = & -546x-1950 \\\Leftrightarrow & 65x \color{red}{-90} \color{blue}{+90} \color{blue}{+546x} & = & \color{red}{-546x} -1950 \color{blue}{+546x} \color{blue}{+90} \\\Leftrightarrow & 611x& = & -1860 \\\Leftrightarrow & \frac{611x}{ \color{red}{611} }& = & \frac{-1860}{611} \\\Leftrightarrow & x = \frac{-1860}{611} & & \\ & V = \left\{ \frac{-1860}{611} \right\} & \\\end{align}\)
8. \(\text{40 is het kleinste gemene veelvoud van 4, 8 en 5} \\ \begin{align} & \frac{x}{4}+\frac{5}{8}& = & \frac{4}{5}x+5 \\\Leftrightarrow & \color{blue}{40.} (\frac{10x}{ \color{blue}{40} }+ \frac{ 25 }{ \color{blue}{40} })& = & (\frac{32}{ \color{blue}{40} }x+\frac{200}{ \color{blue}{40} }) \color{blue}{.40} \\\Leftrightarrow & 10x+25& = & 32x+200 \\\Leftrightarrow & 10x \color{red}{+25} \color{blue}{-25} \color{blue}{-32x} & = & \color{red}{32x} +200 \color{blue}{-32x} \color{blue}{-25} \\\Leftrightarrow & -22x& = & 175 \\\Leftrightarrow & \frac{-22x}{ \color{red}{-22} }& = & \frac{175}{-22} \\\Leftrightarrow & x = \frac{-175}{22} & & \\ & V = \left\{ \frac{-175}{22} \right\} & \\\end{align}\)
9. \(\text{260 is het kleinste gemene veelvoud van 4, 13 en 5} \\ \begin{align} & \frac{x}{4}-\frac{4}{13}& = & \frac{-2}{5}x-3 \\\Leftrightarrow & \color{blue}{260.} (\frac{65x}{ \color{blue}{260} }- \frac{ 80 }{ \color{blue}{260} })& = & (\frac{-104}{ \color{blue}{260} }x-\frac{780}{ \color{blue}{260} }) \color{blue}{.260} \\\Leftrightarrow & 65x-80& = & -104x-780 \\\Leftrightarrow & 65x \color{red}{-80} \color{blue}{+80} \color{blue}{+104x} & = & \color{red}{-104x} -780 \color{blue}{+104x} \color{blue}{+80} \\\Leftrightarrow & 169x& = & -700 \\\Leftrightarrow & \frac{169x}{ \color{red}{169} }& = & \frac{-700}{169} \\\Leftrightarrow & x = \frac{-700}{169} & & \\ & V = \left\{ \frac{-700}{169} \right\} & \\\end{align}\)
10. \(\text{78 is het kleinste gemene veelvoud van 2, 13 en 3} \\ \begin{align} & \frac{x}{2}+\frac{2}{13}& = & \frac{1}{3}x-2 \\\Leftrightarrow & \color{blue}{78.} (\frac{39x}{ \color{blue}{78} }+ \frac{ 12 }{ \color{blue}{78} })& = & (\frac{26}{ \color{blue}{78} }x-\frac{156}{ \color{blue}{78} }) \color{blue}{.78} \\\Leftrightarrow & 39x+12& = & 26x-156 \\\Leftrightarrow & 39x \color{red}{+12} \color{blue}{-12} \color{blue}{-26x} & = & \color{red}{26x} -156 \color{blue}{-26x} \color{blue}{-12} \\\Leftrightarrow & 13x& = & -168 \\\Leftrightarrow & \frac{13x}{ \color{red}{13} }& = & \frac{-168}{13} \\\Leftrightarrow & x = \frac{-168}{13} & & \\ & V = \left\{ \frac{-168}{13} \right\} & \\\end{align}\)
11. \(\text{48 is het kleinste gemene veelvoud van 4, 16 en 3} \\ \begin{align} & \frac{x}{4}-\frac{3}{16}& = & \frac{5}{3}x+6 \\\Leftrightarrow & \color{blue}{48.} (\frac{12x}{ \color{blue}{48} }- \frac{ 9 }{ \color{blue}{48} })& = & (\frac{80}{ \color{blue}{48} }x+\frac{288}{ \color{blue}{48} }) \color{blue}{.48} \\\Leftrightarrow & 12x-9& = & 80x+288 \\\Leftrightarrow & 12x \color{red}{-9} \color{blue}{+9} \color{blue}{-80x} & = & \color{red}{80x} +288 \color{blue}{-80x} \color{blue}{+9} \\\Leftrightarrow & -68x& = & 297 \\\Leftrightarrow & \frac{-68x}{ \color{red}{-68} }& = & \frac{297}{-68} \\\Leftrightarrow & x = \frac{-297}{68} & & \\ & V = \left\{ \frac{-297}{68} \right\} & \\\end{align}\)
12. \(\text{42 is het kleinste gemene veelvoud van 7, 6 en 2} \\ \begin{align} & \frac{x}{7}+\frac{5}{6}& = & \frac{7}{2}x-7 \\\Leftrightarrow & \color{blue}{42.} (\frac{6x}{ \color{blue}{42} }+ \frac{ 35 }{ \color{blue}{42} })& = & (\frac{147}{ \color{blue}{42} }x-\frac{294}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 6x+35& = & 147x-294 \\\Leftrightarrow & 6x \color{red}{+35} \color{blue}{-35} \color{blue}{-147x} & = & \color{red}{147x} -294 \color{blue}{-147x} \color{blue}{-35} \\\Leftrightarrow & -141x& = & -329 \\\Leftrightarrow & \frac{-141x}{ \color{red}{-141} }& = & \frac{-329}{-141} \\\Leftrightarrow & x = \frac{7}{3} & & \\ & V = \left\{ \frac{7}{3} \right\} & \\\end{align}\)