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

---

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

Verbetersleutel

1. \(\text{66 is het kleinste gemene veelvoud van 2, 11 en 3} \\ \begin{align} & \frac{x}{2}+\frac{3}{11}& = & \frac{-2}{3}x-6 \\\Leftrightarrow & \color{blue}{66.} (\frac{33x}{ \color{blue}{66} }+ \frac{ 18 }{ \color{blue}{66} })& = & (\frac{-44}{ \color{blue}{66} }x-\frac{396}{ \color{blue}{66} }) \color{blue}{.66} \\\Leftrightarrow & 33x+18& = & -44x-396 \\\Leftrightarrow & 33x \color{red}{+18} \color{blue}{-18} \color{blue}{+44x} & = & \color{red}{-44x} -396 \color{blue}{+44x} \color{blue}{-18} \\\Leftrightarrow & 77x& = & -414 \\\Leftrightarrow & \frac{77x}{ \color{red}{77} }& = & \frac{-414}{77} \\\Leftrightarrow & x = \frac{-414}{77} & & \\ & V = \left\{ \frac{-414}{77} \right\} & \\\end{align}\)
2. \(\text{195 is het kleinste gemene veelvoud van 5, 13 en 3} \\ \begin{align} & \frac{x}{5}-\frac{2}{13}& = & \frac{-2}{3}x+7 \\\Leftrightarrow & \color{blue}{195.} (\frac{39x}{ \color{blue}{195} }- \frac{ 30 }{ \color{blue}{195} })& = & (\frac{-130}{ \color{blue}{195} }x+\frac{1365}{ \color{blue}{195} }) \color{blue}{.195} \\\Leftrightarrow & 39x-30& = & -130x+1365 \\\Leftrightarrow & 39x \color{red}{-30} \color{blue}{+30} \color{blue}{+130x} & = & \color{red}{-130x} +1365 \color{blue}{+130x} \color{blue}{+30} \\\Leftrightarrow & 169x& = & 1395 \\\Leftrightarrow & \frac{169x}{ \color{red}{169} }& = & \frac{1395}{169} \\\Leftrightarrow & x = \frac{1395}{169} & & \\ & V = \left\{ \frac{1395}{169} \right\} & \\\end{align}\)
3. \(\text{10 is het kleinste gemene veelvoud van 2, 10 en 5} \\ \begin{align} & \frac{x}{2}-\frac{3}{10}& = & \frac{3}{5}x-6 \\\Leftrightarrow & \color{blue}{10.} (\frac{5x}{ \color{blue}{10} }- \frac{ 3 }{ \color{blue}{10} })& = & (\frac{6}{ \color{blue}{10} }x-\frac{60}{ \color{blue}{10} }) \color{blue}{.10} \\\Leftrightarrow & 5x-3& = & 6x-60 \\\Leftrightarrow & 5x \color{red}{-3} \color{blue}{+3} \color{blue}{-6x} & = & \color{red}{6x} -60 \color{blue}{-6x} \color{blue}{+3} \\\Leftrightarrow & -x& = & -57 \\\Leftrightarrow & \frac{-x}{ \color{red}{-1} }& = & \frac{-57}{-1} \\\Leftrightarrow & x = 57 & & \\ & V = \left\{ 57 \right\} & \\\end{align}\)
4. \(\text{66 is het kleinste gemene veelvoud van 3, 11 en 2} \\ \begin{align} & \frac{x}{3}+\frac{5}{11}& = & \frac{1}{2}x+3 \\\Leftrightarrow & \color{blue}{66.} (\frac{22x}{ \color{blue}{66} }+ \frac{ 30 }{ \color{blue}{66} })& = & (\frac{33}{ \color{blue}{66} }x+\frac{198}{ \color{blue}{66} }) \color{blue}{.66} \\\Leftrightarrow & 22x+30& = & 33x+198 \\\Leftrightarrow & 22x \color{red}{+30} \color{blue}{-30} \color{blue}{-33x} & = & \color{red}{33x} +198 \color{blue}{-33x} \color{blue}{-30} \\\Leftrightarrow & -11x& = & 168 \\\Leftrightarrow & \frac{-11x}{ \color{red}{-11} }& = & \frac{168}{-11} \\\Leftrightarrow & x = \frac{-168}{11} & & \\ & V = \left\{ \frac{-168}{11} \right\} & \\\end{align}\)
5. \(\text{14 is het kleinste gemene veelvoud van 7, 7 en 2} \\ \begin{align} & \frac{x}{7}+\frac{4}{7}& = & \frac{1}{2}x-6 \\\Leftrightarrow & \color{blue}{14.} (\frac{2x}{ \color{blue}{14} }+ \frac{ 8 }{ \color{blue}{14} })& = & (\frac{7}{ \color{blue}{14} }x-\frac{84}{ \color{blue}{14} }) \color{blue}{.14} \\\Leftrightarrow & 2x+8& = & 7x-84 \\\Leftrightarrow & 2x \color{red}{+8} \color{blue}{-8} \color{blue}{-7x} & = & \color{red}{7x} -84 \color{blue}{-7x} \color{blue}{-8} \\\Leftrightarrow & -5x& = & -92 \\\Leftrightarrow & \frac{-5x}{ \color{red}{-5} }& = & \frac{-92}{-5} \\\Leftrightarrow & x = \frac{92}{5} & & \\ & V = \left\{ \frac{92}{5} \right\} & \\\end{align}\)
6. \(\text{42 is het kleinste gemene veelvoud van 7, 6 en 3} \\ \begin{align} & \frac{x}{7}-\frac{5}{6}& = & \frac{4}{3}x+4 \\\Leftrightarrow & \color{blue}{42.} (\frac{6x}{ \color{blue}{42} }- \frac{ 35 }{ \color{blue}{42} })& = & (\frac{56}{ \color{blue}{42} }x+\frac{168}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 6x-35& = & 56x+168 \\\Leftrightarrow & 6x \color{red}{-35} \color{blue}{+35} \color{blue}{-56x} & = & \color{red}{56x} +168 \color{blue}{-56x} \color{blue}{+35} \\\Leftrightarrow & -50x& = & 203 \\\Leftrightarrow & \frac{-50x}{ \color{red}{-50} }& = & \frac{203}{-50} \\\Leftrightarrow & x = \frac{-203}{50} & & \\ & V = \left\{ \frac{-203}{50} \right\} & \\\end{align}\)
7. \(\text{78 is het kleinste gemene veelvoud van 2, 13 en 3} \\ \begin{align} & \frac{x}{2}+\frac{2}{13}& = & \frac{-8}{3}x+4 \\\Leftrightarrow & \color{blue}{78.} (\frac{39x}{ \color{blue}{78} }+ \frac{ 12 }{ \color{blue}{78} })& = & (\frac{-208}{ \color{blue}{78} }x+\frac{312}{ \color{blue}{78} }) \color{blue}{.78} \\\Leftrightarrow & 39x+12& = & -208x+312 \\\Leftrightarrow & 39x \color{red}{+12} \color{blue}{-12} \color{blue}{+208x} & = & \color{red}{-208x} +312 \color{blue}{+208x} \color{blue}{-12} \\\Leftrightarrow & 247x& = & 300 \\\Leftrightarrow & \frac{247x}{ \color{red}{247} }& = & \frac{300}{247} \\\Leftrightarrow & x = \frac{300}{247} & & \\ & V = \left\{ \frac{300}{247} \right\} & \\\end{align}\)
8. \(\text{18 is het kleinste gemene veelvoud van 2, 9 en 3} \\ \begin{align} & \frac{x}{2}+\frac{2}{9}& = & \frac{4}{3}x+1 \\\Leftrightarrow & \color{blue}{18.} (\frac{9x}{ \color{blue}{18} }+ \frac{ 4 }{ \color{blue}{18} })& = & (\frac{24}{ \color{blue}{18} }x+\frac{18}{ \color{blue}{18} }) \color{blue}{.18} \\\Leftrightarrow & 9x+4& = & 24x+18 \\\Leftrightarrow & 9x \color{red}{+4} \color{blue}{-4} \color{blue}{-24x} & = & \color{red}{24x} +18 \color{blue}{-24x} \color{blue}{-4} \\\Leftrightarrow & -15x& = & 14 \\\Leftrightarrow & \frac{-15x}{ \color{red}{-15} }& = & \frac{14}{-15} \\\Leftrightarrow & x = \frac{-14}{15} & & \\ & V = \left\{ \frac{-14}{15} \right\} & \\\end{align}\)
9. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}+\frac{4}{7}& = & \frac{-8}{3}x+3 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }+ \frac{ 48 }{ \color{blue}{84} })& = & (\frac{-224}{ \color{blue}{84} }x+\frac{252}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x+48& = & -224x+252 \\\Leftrightarrow & 21x \color{red}{+48} \color{blue}{-48} \color{blue}{+224x} & = & \color{red}{-224x} +252 \color{blue}{+224x} \color{blue}{-48} \\\Leftrightarrow & 245x& = & 204 \\\Leftrightarrow & \frac{245x}{ \color{red}{245} }& = & \frac{204}{245} \\\Leftrightarrow & x = \frac{204}{245} & & \\ & V = \left\{ \frac{204}{245} \right\} & \\\end{align}\)
10. \(\text{42 is het kleinste gemene veelvoud van 7, 7 en 6} \\ \begin{align} & \frac{x}{7}+\frac{2}{7}& = & \frac{1}{6}x-8 \\\Leftrightarrow & \color{blue}{42.} (\frac{6x}{ \color{blue}{42} }+ \frac{ 12 }{ \color{blue}{42} })& = & (\frac{7}{ \color{blue}{42} }x-\frac{336}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 6x+12& = & 7x-336 \\\Leftrightarrow & 6x \color{red}{+12} \color{blue}{-12} \color{blue}{-7x} & = & \color{red}{7x} -336 \color{blue}{-7x} \color{blue}{-12} \\\Leftrightarrow & -x& = & -348 \\\Leftrightarrow & \frac{-x}{ \color{red}{-1} }& = & \frac{-348}{-1} \\\Leftrightarrow & x = 348 & & \\ & V = \left\{ 348 \right\} & \\\end{align}\)
11. \(\text{120 is het kleinste gemene veelvoud van 6, 8 en 5} \\ \begin{align} & \frac{x}{6}+\frac{5}{8}& = & \frac{7}{5}x-8 \\\Leftrightarrow & \color{blue}{120.} (\frac{20x}{ \color{blue}{120} }+ \frac{ 75 }{ \color{blue}{120} })& = & (\frac{168}{ \color{blue}{120} }x-\frac{960}{ \color{blue}{120} }) \color{blue}{.120} \\\Leftrightarrow & 20x+75& = & 168x-960 \\\Leftrightarrow & 20x \color{red}{+75} \color{blue}{-75} \color{blue}{-168x} & = & \color{red}{168x} -960 \color{blue}{-168x} \color{blue}{-75} \\\Leftrightarrow & -148x& = & -1035 \\\Leftrightarrow & \frac{-148x}{ \color{red}{-148} }& = & \frac{-1035}{-148} \\\Leftrightarrow & x = \frac{1035}{148} & & \\ & V = \left\{ \frac{1035}{148} \right\} & \\\end{align}\)
12. \(\text{12 is het kleinste gemene veelvoud van 4, 6 en 3} \\ \begin{align} & \frac{x}{4}+\frac{5}{6}& = & \frac{2}{3}x+5 \\\Leftrightarrow & \color{blue}{12.} (\frac{3x}{ \color{blue}{12} }+ \frac{ 10 }{ \color{blue}{12} })& = & (\frac{8}{ \color{blue}{12} }x+\frac{60}{ \color{blue}{12} }) \color{blue}{.12} \\\Leftrightarrow & 3x+10& = & 8x+60 \\\Leftrightarrow & 3x \color{red}{+10} \color{blue}{-10} \color{blue}{-8x} & = & \color{red}{8x} +60 \color{blue}{-8x} \color{blue}{-10} \\\Leftrightarrow & -5x& = & 50 \\\Leftrightarrow & \frac{-5x}{ \color{red}{-5} }& = & \frac{50}{-5} \\\Leftrightarrow & x = -10 & & \\ & V = \left\{ -10 \right\} & \\\end{align}\)