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

---

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

Verbetersleutel

1. \(\text{240 is het kleinste gemene veelvoud van 5, 16 en 6} \\ \begin{align} & \frac{x}{5}+\frac{3}{16}& = & \frac{1}{6}x+5 \\\Leftrightarrow & \color{blue}{240.} (\frac{48x}{ \color{blue}{240} }+ \frac{ 45 }{ \color{blue}{240} })& = & (\frac{40}{ \color{blue}{240} }x+\frac{1200}{ \color{blue}{240} }) \color{blue}{.240} \\\Leftrightarrow & 48x+45& = & 40x+1200 \\\Leftrightarrow & 48x \color{red}{+45} \color{blue}{-45} \color{blue}{-40x} & = & \color{red}{40x} +1200 \color{blue}{-40x} \color{blue}{-45} \\\Leftrightarrow & 8x& = & 1155 \\\Leftrightarrow & \frac{8x}{ \color{red}{8} }& = & \frac{1155}{8} \\\Leftrightarrow & x = \frac{1155}{8} & & \\ & V = \left\{ \frac{1155}{8} \right\} & \\\end{align}\)
2. \(\text{130 is het kleinste gemene veelvoud van 2, 13 en 5} \\ \begin{align} & \frac{x}{2}+\frac{4}{13}& = & \frac{-4}{5}x+5 \\\Leftrightarrow & \color{blue}{130.} (\frac{65x}{ \color{blue}{130} }+ \frac{ 40 }{ \color{blue}{130} })& = & (\frac{-104}{ \color{blue}{130} }x+\frac{650}{ \color{blue}{130} }) \color{blue}{.130} \\\Leftrightarrow & 65x+40& = & -104x+650 \\\Leftrightarrow & 65x \color{red}{+40} \color{blue}{-40} \color{blue}{+104x} & = & \color{red}{-104x} +650 \color{blue}{+104x} \color{blue}{-40} \\\Leftrightarrow & 169x& = & 610 \\\Leftrightarrow & \frac{169x}{ \color{red}{169} }& = & \frac{610}{169} \\\Leftrightarrow & x = \frac{610}{169} & & \\ & V = \left\{ \frac{610}{169} \right\} & \\\end{align}\)
3. \(\text{42 is het kleinste gemene veelvoud van 2, 7 en 3} \\ \begin{align} & \frac{x}{2}-\frac{3}{7}& = & \frac{7}{3}x-2 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }- \frac{ 18 }{ \color{blue}{42} })& = & (\frac{98}{ \color{blue}{42} }x-\frac{84}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x-18& = & 98x-84 \\\Leftrightarrow & 21x \color{red}{-18} \color{blue}{+18} \color{blue}{-98x} & = & \color{red}{98x} -84 \color{blue}{-98x} \color{blue}{+18} \\\Leftrightarrow & -77x& = & -66 \\\Leftrightarrow & \frac{-77x}{ \color{red}{-77} }& = & \frac{-66}{-77} \\\Leftrightarrow & x = \frac{6}{7} & & \\ & V = \left\{ \frac{6}{7} \right\} & \\\end{align}\)
4. \(\text{110 is het kleinste gemene veelvoud van 5, 11 en 2} \\ \begin{align} & \frac{x}{5}-\frac{4}{11}& = & \frac{1}{2}x+4 \\\Leftrightarrow & \color{blue}{110.} (\frac{22x}{ \color{blue}{110} }- \frac{ 40 }{ \color{blue}{110} })& = & (\frac{55}{ \color{blue}{110} }x+\frac{440}{ \color{blue}{110} }) \color{blue}{.110} \\\Leftrightarrow & 22x-40& = & 55x+440 \\\Leftrightarrow & 22x \color{red}{-40} \color{blue}{+40} \color{blue}{-55x} & = & \color{red}{55x} +440 \color{blue}{-55x} \color{blue}{+40} \\\Leftrightarrow & -33x& = & 480 \\\Leftrightarrow & \frac{-33x}{ \color{red}{-33} }& = & \frac{480}{-33} \\\Leftrightarrow & x = \frac{-160}{11} & & \\ & V = \left\{ \frac{-160}{11} \right\} & \\\end{align}\)
5. \(\text{156 is het kleinste gemene veelvoud van 4, 13 en 3} \\ \begin{align} & \frac{x}{4}-\frac{3}{13}& = & \frac{-2}{3}x-2 \\\Leftrightarrow & \color{blue}{156.} (\frac{39x}{ \color{blue}{156} }- \frac{ 36 }{ \color{blue}{156} })& = & (\frac{-104}{ \color{blue}{156} }x-\frac{312}{ \color{blue}{156} }) \color{blue}{.156} \\\Leftrightarrow & 39x-36& = & -104x-312 \\\Leftrightarrow & 39x \color{red}{-36} \color{blue}{+36} \color{blue}{+104x} & = & \color{red}{-104x} -312 \color{blue}{+104x} \color{blue}{+36} \\\Leftrightarrow & 143x& = & -276 \\\Leftrightarrow & \frac{143x}{ \color{red}{143} }& = & \frac{-276}{143} \\\Leftrightarrow & x = \frac{-276}{143} & & \\ & V = \left\{ \frac{-276}{143} \right\} & \\\end{align}\)
6. \(\text{24 is het kleinste gemene veelvoud van 2, 8 en 3} \\ \begin{align} & \frac{x}{2}-\frac{5}{8}& = & \frac{-8}{3}x+2 \\\Leftrightarrow & \color{blue}{24.} (\frac{12x}{ \color{blue}{24} }- \frac{ 15 }{ \color{blue}{24} })& = & (\frac{-64}{ \color{blue}{24} }x+\frac{48}{ \color{blue}{24} }) \color{blue}{.24} \\\Leftrightarrow & 12x-15& = & -64x+48 \\\Leftrightarrow & 12x \color{red}{-15} \color{blue}{+15} \color{blue}{+64x} & = & \color{red}{-64x} +48 \color{blue}{+64x} \color{blue}{+15} \\\Leftrightarrow & 76x& = & 63 \\\Leftrightarrow & \frac{76x}{ \color{red}{76} }& = & \frac{63}{76} \\\Leftrightarrow & x = \frac{63}{76} & & \\ & V = \left\{ \frac{63}{76} \right\} & \\\end{align}\)
7. \(\text{78 is het kleinste gemene veelvoud van 2, 13 en 3} \\ \begin{align} & \frac{x}{2}-\frac{5}{13}& = & \frac{7}{3}x-7 \\\Leftrightarrow & \color{blue}{78.} (\frac{39x}{ \color{blue}{78} }- \frac{ 30 }{ \color{blue}{78} })& = & (\frac{182}{ \color{blue}{78} }x-\frac{546}{ \color{blue}{78} }) \color{blue}{.78} \\\Leftrightarrow & 39x-30& = & 182x-546 \\\Leftrightarrow & 39x \color{red}{-30} \color{blue}{+30} \color{blue}{-182x} & = & \color{red}{182x} -546 \color{blue}{-182x} \color{blue}{+30} \\\Leftrightarrow & -143x& = & -516 \\\Leftrightarrow & \frac{-143x}{ \color{red}{-143} }& = & \frac{-516}{-143} \\\Leftrightarrow & x = \frac{516}{143} & & \\ & V = \left\{ \frac{516}{143} \right\} & \\\end{align}\)
8. \(\text{63 is het kleinste gemene veelvoud van 7, 9 en 3} \\ \begin{align} & \frac{x}{7}+\frac{5}{9}& = & \frac{4}{3}x+1 \\\Leftrightarrow & \color{blue}{63.} (\frac{9x}{ \color{blue}{63} }+ \frac{ 35 }{ \color{blue}{63} })& = & (\frac{84}{ \color{blue}{63} }x+\frac{63}{ \color{blue}{63} }) \color{blue}{.63} \\\Leftrightarrow & 9x+35& = & 84x+63 \\\Leftrightarrow & 9x \color{red}{+35} \color{blue}{-35} \color{blue}{-84x} & = & \color{red}{84x} +63 \color{blue}{-84x} \color{blue}{-35} \\\Leftrightarrow & -75x& = & 28 \\\Leftrightarrow & \frac{-75x}{ \color{red}{-75} }& = & \frac{28}{-75} \\\Leftrightarrow & x = \frac{-28}{75} & & \\ & V = \left\{ \frac{-28}{75} \right\} & \\\end{align}\)
9. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}-\frac{2}{7}& = & \frac{4}{3}x+1 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }- \frac{ 24 }{ \color{blue}{84} })& = & (\frac{112}{ \color{blue}{84} }x+\frac{84}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x-24& = & 112x+84 \\\Leftrightarrow & 21x \color{red}{-24} \color{blue}{+24} \color{blue}{-112x} & = & \color{red}{112x} +84 \color{blue}{-112x} \color{blue}{+24} \\\Leftrightarrow & -91x& = & 108 \\\Leftrightarrow & \frac{-91x}{ \color{red}{-91} }& = & \frac{108}{-91} \\\Leftrightarrow & x = \frac{-108}{91} & & \\ & V = \left\{ \frac{-108}{91} \right\} & \\\end{align}\)
10. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}-\frac{4}{7}& = & \frac{1}{3}x+8 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }- \frac{ 48 }{ \color{blue}{84} })& = & (\frac{28}{ \color{blue}{84} }x+\frac{672}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x-48& = & 28x+672 \\\Leftrightarrow & 21x \color{red}{-48} \color{blue}{+48} \color{blue}{-28x} & = & \color{red}{28x} +672 \color{blue}{-28x} \color{blue}{+48} \\\Leftrightarrow & -7x& = & 720 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{720}{-7} \\\Leftrightarrow & x = \frac{-720}{7} & & \\ & V = \left\{ \frac{-720}{7} \right\} & \\\end{align}\)
11. \(\text{6 is het kleinste gemene veelvoud van 3, 6 en 2} \\ \begin{align} & \frac{x}{3}-\frac{5}{6}& = & \frac{1}{2}x+3 \\\Leftrightarrow & \color{blue}{6.} (\frac{2x}{ \color{blue}{6} }- \frac{ 5 }{ \color{blue}{6} })& = & (\frac{3}{ \color{blue}{6} }x+\frac{18}{ \color{blue}{6} }) \color{blue}{.6} \\\Leftrightarrow & 2x-5& = & 3x+18 \\\Leftrightarrow & 2x \color{red}{-5} \color{blue}{+5} \color{blue}{-3x} & = & \color{red}{3x} +18 \color{blue}{-3x} \color{blue}{+5} \\\Leftrightarrow & -x& = & 23 \\\Leftrightarrow & \frac{-x}{ \color{red}{-1} }& = & \frac{23}{-1} \\\Leftrightarrow & x = -23 & & \\ & V = \left\{ -23 \right\} & \\\end{align}\)
12. \(\text{105 is het kleinste gemene veelvoud van 5, 7 en 3} \\ \begin{align} & \frac{x}{5}+\frac{2}{7}& = & \frac{5}{3}x+1 \\\Leftrightarrow & \color{blue}{105.} (\frac{21x}{ \color{blue}{105} }+ \frac{ 30 }{ \color{blue}{105} })& = & (\frac{175}{ \color{blue}{105} }x+\frac{105}{ \color{blue}{105} }) \color{blue}{.105} \\\Leftrightarrow & 21x+30& = & 175x+105 \\\Leftrightarrow & 21x \color{red}{+30} \color{blue}{-30} \color{blue}{-175x} & = & \color{red}{175x} +105 \color{blue}{-175x} \color{blue}{-30} \\\Leftrightarrow & -154x& = & 75 \\\Leftrightarrow & \frac{-154x}{ \color{red}{-154} }& = & \frac{75}{-154} \\\Leftrightarrow & x = \frac{-75}{154} & & \\ & V = \left\{ \frac{-75}{154} \right\} & \\\end{align}\)