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

---

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

Verbetersleutel

1. \(\text{42 is het kleinste gemene veelvoud van 3, 7 en 2} \\ \begin{align} & \frac{x}{3}+\frac{4}{7}& = & \frac{7}{2}x-2 \\\Leftrightarrow & \color{blue}{42.} (\frac{14x}{ \color{blue}{42} }+ \frac{ 24 }{ \color{blue}{42} })& = & (\frac{147}{ \color{blue}{42} }x-\frac{84}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 14x+24& = & 147x-84 \\\Leftrightarrow & 14x \color{red}{+24} \color{blue}{-24} \color{blue}{-147x} & = & \color{red}{147x} -84 \color{blue}{-147x} \color{blue}{-24} \\\Leftrightarrow & -133x& = & -108 \\\Leftrightarrow & \frac{-133x}{ \color{red}{-133} }& = & \frac{-108}{-133} \\\Leftrightarrow & x = \frac{108}{133} & & \\ & V = \left\{ \frac{108}{133} \right\} & \\\end{align}\)
2. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}-\frac{4}{7}& = & \frac{-8}{3}x-2 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }- \frac{ 48 }{ \color{blue}{84} })& = & (\frac{-224}{ \color{blue}{84} }x-\frac{168}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x-48& = & -224x-168 \\\Leftrightarrow & 21x \color{red}{-48} \color{blue}{+48} \color{blue}{+224x} & = & \color{red}{-224x} -168 \color{blue}{+224x} \color{blue}{+48} \\\Leftrightarrow & 245x& = & -120 \\\Leftrightarrow & \frac{245x}{ \color{red}{245} }& = & \frac{-120}{245} \\\Leftrightarrow & x = \frac{-24}{49} & & \\ & V = \left\{ \frac{-24}{49} \right\} & \\\end{align}\)
3. \(\text{315 is het kleinste gemene veelvoud van 7, 9 en 5} \\ \begin{align} & \frac{x}{7}-\frac{2}{9}& = & \frac{-4}{5}x+2 \\\Leftrightarrow & \color{blue}{315.} (\frac{45x}{ \color{blue}{315} }- \frac{ 70 }{ \color{blue}{315} })& = & (\frac{-252}{ \color{blue}{315} }x+\frac{630}{ \color{blue}{315} }) \color{blue}{.315} \\\Leftrightarrow & 45x-70& = & -252x+630 \\\Leftrightarrow & 45x \color{red}{-70} \color{blue}{+70} \color{blue}{+252x} & = & \color{red}{-252x} +630 \color{blue}{+252x} \color{blue}{+70} \\\Leftrightarrow & 297x& = & 700 \\\Leftrightarrow & \frac{297x}{ \color{red}{297} }& = & \frac{700}{297} \\\Leftrightarrow & x = \frac{700}{297} & & \\ & V = \left\{ \frac{700}{297} \right\} & \\\end{align}\)
4. \(\text{330 is het kleinste gemene veelvoud van 6, 11 en 5} \\ \begin{align} & \frac{x}{6}+\frac{5}{11}& = & \frac{-4}{5}x-3 \\\Leftrightarrow & \color{blue}{330.} (\frac{55x}{ \color{blue}{330} }+ \frac{ 150 }{ \color{blue}{330} })& = & (\frac{-264}{ \color{blue}{330} }x-\frac{990}{ \color{blue}{330} }) \color{blue}{.330} \\\Leftrightarrow & 55x+150& = & -264x-990 \\\Leftrightarrow & 55x \color{red}{+150} \color{blue}{-150} \color{blue}{+264x} & = & \color{red}{-264x} -990 \color{blue}{+264x} \color{blue}{-150} \\\Leftrightarrow & 319x& = & -1140 \\\Leftrightarrow & \frac{319x}{ \color{red}{319} }& = & \frac{-1140}{319} \\\Leftrightarrow & x = \frac{-1140}{319} & & \\ & V = \left\{ \frac{-1140}{319} \right\} & \\\end{align}\)
5. \(\text{18 is het kleinste gemene veelvoud van 3, 9 en 2} \\ \begin{align} & \frac{x}{3}+\frac{4}{9}& = & \frac{7}{2}x+4 \\\Leftrightarrow & \color{blue}{18.} (\frac{6x}{ \color{blue}{18} }+ \frac{ 8 }{ \color{blue}{18} })& = & (\frac{63}{ \color{blue}{18} }x+\frac{72}{ \color{blue}{18} }) \color{blue}{.18} \\\Leftrightarrow & 6x+8& = & 63x+72 \\\Leftrightarrow & 6x \color{red}{+8} \color{blue}{-8} \color{blue}{-63x} & = & \color{red}{63x} +72 \color{blue}{-63x} \color{blue}{-8} \\\Leftrightarrow & -57x& = & 64 \\\Leftrightarrow & \frac{-57x}{ \color{red}{-57} }& = & \frac{64}{-57} \\\Leftrightarrow & x = \frac{-64}{57} & & \\ & V = \left\{ \frac{-64}{57} \right\} & \\\end{align}\)
6. \(\text{30 is het kleinste gemene veelvoud van 5, 6 en 3} \\ \begin{align} & \frac{x}{5}+\frac{5}{6}& = & \frac{7}{3}x-5 \\\Leftrightarrow & \color{blue}{30.} (\frac{6x}{ \color{blue}{30} }+ \frac{ 25 }{ \color{blue}{30} })& = & (\frac{70}{ \color{blue}{30} }x-\frac{150}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 6x+25& = & 70x-150 \\\Leftrightarrow & 6x \color{red}{+25} \color{blue}{-25} \color{blue}{-70x} & = & \color{red}{70x} -150 \color{blue}{-70x} \color{blue}{-25} \\\Leftrightarrow & -64x& = & -175 \\\Leftrightarrow & \frac{-64x}{ \color{red}{-64} }& = & \frac{-175}{-64} \\\Leftrightarrow & x = \frac{175}{64} & & \\ & V = \left\{ \frac{175}{64} \right\} & \\\end{align}\)
7. \(\text{24 is het kleinste gemene veelvoud van 2, 8 en 3} \\ \begin{align} & \frac{x}{2}-\frac{5}{8}& = & \frac{-5}{3}x+8 \\\Leftrightarrow & \color{blue}{24.} (\frac{12x}{ \color{blue}{24} }- \frac{ 15 }{ \color{blue}{24} })& = & (\frac{-40}{ \color{blue}{24} }x+\frac{192}{ \color{blue}{24} }) \color{blue}{.24} \\\Leftrightarrow & 12x-15& = & -40x+192 \\\Leftrightarrow & 12x \color{red}{-15} \color{blue}{+15} \color{blue}{+40x} & = & \color{red}{-40x} +192 \color{blue}{+40x} \color{blue}{+15} \\\Leftrightarrow & 52x& = & 207 \\\Leftrightarrow & \frac{52x}{ \color{red}{52} }& = & \frac{207}{52} \\\Leftrightarrow & x = \frac{207}{52} & & \\ & V = \left\{ \frac{207}{52} \right\} & \\\end{align}\)
8. \(\text{182 is het kleinste gemene veelvoud van 7, 13 en 2} \\ \begin{align} & \frac{x}{7}+\frac{2}{13}& = & \frac{1}{2}x+2 \\\Leftrightarrow & \color{blue}{182.} (\frac{26x}{ \color{blue}{182} }+ \frac{ 28 }{ \color{blue}{182} })& = & (\frac{91}{ \color{blue}{182} }x+\frac{364}{ \color{blue}{182} }) \color{blue}{.182} \\\Leftrightarrow & 26x+28& = & 91x+364 \\\Leftrightarrow & 26x \color{red}{+28} \color{blue}{-28} \color{blue}{-91x} & = & \color{red}{91x} +364 \color{blue}{-91x} \color{blue}{-28} \\\Leftrightarrow & -65x& = & 336 \\\Leftrightarrow & \frac{-65x}{ \color{red}{-65} }& = & \frac{336}{-65} \\\Leftrightarrow & x = \frac{-336}{65} & & \\ & V = \left\{ \frac{-336}{65} \right\} & \\\end{align}\)
9. \(\text{35 is het kleinste gemene veelvoud van 7, 7 en 5} \\ \begin{align} & \frac{x}{7}-\frac{4}{7}& = & \frac{3}{5}x+5 \\\Leftrightarrow & \color{blue}{35.} (\frac{5x}{ \color{blue}{35} }- \frac{ 20 }{ \color{blue}{35} })& = & (\frac{21}{ \color{blue}{35} }x+\frac{175}{ \color{blue}{35} }) \color{blue}{.35} \\\Leftrightarrow & 5x-20& = & 21x+175 \\\Leftrightarrow & 5x \color{red}{-20} \color{blue}{+20} \color{blue}{-21x} & = & \color{red}{21x} +175 \color{blue}{-21x} \color{blue}{+20} \\\Leftrightarrow & -16x& = & 195 \\\Leftrightarrow & \frac{-16x}{ \color{red}{-16} }& = & \frac{195}{-16} \\\Leftrightarrow & x = \frac{-195}{16} & & \\ & V = \left\{ \frac{-195}{16} \right\} & \\\end{align}\)
10. \(\text{260 is het kleinste gemene veelvoud van 4, 13 en 5} \\ \begin{align} & \frac{x}{4}+\frac{4}{13}& = & \frac{7}{5}x+7 \\\Leftrightarrow & \color{blue}{260.} (\frac{65x}{ \color{blue}{260} }+ \frac{ 80 }{ \color{blue}{260} })& = & (\frac{364}{ \color{blue}{260} }x+\frac{1820}{ \color{blue}{260} }) \color{blue}{.260} \\\Leftrightarrow & 65x+80& = & 364x+1820 \\\Leftrightarrow & 65x \color{red}{+80} \color{blue}{-80} \color{blue}{-364x} & = & \color{red}{364x} +1820 \color{blue}{-364x} \color{blue}{-80} \\\Leftrightarrow & -299x& = & 1740 \\\Leftrightarrow & \frac{-299x}{ \color{red}{-299} }& = & \frac{1740}{-299} \\\Leftrightarrow & x = \frac{-1740}{299} & & \\ & V = \left\{ \frac{-1740}{299} \right\} & \\\end{align}\)
11. \(\text{84 is het kleinste gemene veelvoud van 7, 6 en 4} \\ \begin{align} & \frac{x}{7}+\frac{5}{6}& = & \frac{-3}{4}x-6 \\\Leftrightarrow & \color{blue}{84.} (\frac{12x}{ \color{blue}{84} }+ \frac{ 70 }{ \color{blue}{84} })& = & (\frac{-63}{ \color{blue}{84} }x-\frac{504}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 12x+70& = & -63x-504 \\\Leftrightarrow & 12x \color{red}{+70} \color{blue}{-70} \color{blue}{+63x} & = & \color{red}{-63x} -504 \color{blue}{+63x} \color{blue}{-70} \\\Leftrightarrow & 75x& = & -574 \\\Leftrightarrow & \frac{75x}{ \color{red}{75} }& = & \frac{-574}{75} \\\Leftrightarrow & x = \frac{-574}{75} & & \\ & V = \left\{ \frac{-574}{75} \right\} & \\\end{align}\)
12. \(\text{120 is het kleinste gemene veelvoud van 5, 8 en 6} \\ \begin{align} & \frac{x}{5}+\frac{5}{8}& = & \frac{1}{6}x-1 \\\Leftrightarrow & \color{blue}{120.} (\frac{24x}{ \color{blue}{120} }+ \frac{ 75 }{ \color{blue}{120} })& = & (\frac{20}{ \color{blue}{120} }x-\frac{120}{ \color{blue}{120} }) \color{blue}{.120} \\\Leftrightarrow & 24x+75& = & 20x-120 \\\Leftrightarrow & 24x \color{red}{+75} \color{blue}{-75} \color{blue}{-20x} & = & \color{red}{20x} -120 \color{blue}{-20x} \color{blue}{-75} \\\Leftrightarrow & 4x& = & -195 \\\Leftrightarrow & \frac{4x}{ \color{red}{4} }& = & \frac{-195}{4} \\\Leftrightarrow & x = \frac{-195}{4} & & \\ & V = \left\{ \frac{-195}{4} \right\} & \\\end{align}\)