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

---

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

Verbetersleutel

1. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}+\frac{4}{7}& = & \frac{1}{5}x+5 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }+ \frac{ 120 }{ \color{blue}{210} })& = & (\frac{42}{ \color{blue}{210} }x+\frac{1050}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x+120& = & 42x+1050 \\\Leftrightarrow & 35x \color{red}{+120} \color{blue}{-120} \color{blue}{-42x} & = & \color{red}{42x} +1050 \color{blue}{-42x} \color{blue}{-120} \\\Leftrightarrow & -7x& = & 930 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{930}{-7} \\\Leftrightarrow & x = \frac{-930}{7} & & \\ & V = \left\{ \frac{-930}{7} \right\} & \\\end{align}\)
2. \(\text{66 is het kleinste gemene veelvoud van 3, 11 en 2} \\ \begin{align} & \frac{x}{3}-\frac{2}{11}& = & \frac{1}{2}x+3 \\\Leftrightarrow & \color{blue}{66.} (\frac{22x}{ \color{blue}{66} }- \frac{ 12 }{ \color{blue}{66} })& = & (\frac{33}{ \color{blue}{66} }x+\frac{198}{ \color{blue}{66} }) \color{blue}{.66} \\\Leftrightarrow & 22x-12& = & 33x+198 \\\Leftrightarrow & 22x \color{red}{-12} \color{blue}{+12} \color{blue}{-33x} & = & \color{red}{33x} +198 \color{blue}{-33x} \color{blue}{+12} \\\Leftrightarrow & -11x& = & 210 \\\Leftrightarrow & \frac{-11x}{ \color{red}{-11} }& = & \frac{210}{-11} \\\Leftrightarrow & x = \frac{-210}{11} & & \\ & V = \left\{ \frac{-210}{11} \right\} & \\\end{align}\)
3. \(\text{390 is het kleinste gemene veelvoud van 5, 13 en 6} \\ \begin{align} & \frac{x}{5}-\frac{2}{13}& = & \frac{5}{6}x-5 \\\Leftrightarrow & \color{blue}{390.} (\frac{78x}{ \color{blue}{390} }- \frac{ 60 }{ \color{blue}{390} })& = & (\frac{325}{ \color{blue}{390} }x-\frac{1950}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 78x-60& = & 325x-1950 \\\Leftrightarrow & 78x \color{red}{-60} \color{blue}{+60} \color{blue}{-325x} & = & \color{red}{325x} -1950 \color{blue}{-325x} \color{blue}{+60} \\\Leftrightarrow & -247x& = & -1890 \\\Leftrightarrow & \frac{-247x}{ \color{red}{-247} }& = & \frac{-1890}{-247} \\\Leftrightarrow & x = \frac{1890}{247} & & \\ & V = \left\{ \frac{1890}{247} \right\} & \\\end{align}\)
4. \(\text{56 is het kleinste gemene veelvoud van 7, 8 en 2} \\ \begin{align} & \frac{x}{7}+\frac{3}{8}& = & \frac{1}{2}x+5 \\\Leftrightarrow & \color{blue}{56.} (\frac{8x}{ \color{blue}{56} }+ \frac{ 21 }{ \color{blue}{56} })& = & (\frac{28}{ \color{blue}{56} }x+\frac{280}{ \color{blue}{56} }) \color{blue}{.56} \\\Leftrightarrow & 8x+21& = & 28x+280 \\\Leftrightarrow & 8x \color{red}{+21} \color{blue}{-21} \color{blue}{-28x} & = & \color{red}{28x} +280 \color{blue}{-28x} \color{blue}{-21} \\\Leftrightarrow & -20x& = & 259 \\\Leftrightarrow & \frac{-20x}{ \color{red}{-20} }& = & \frac{259}{-20} \\\Leftrightarrow & x = \frac{-259}{20} & & \\ & V = \left\{ \frac{-259}{20} \right\} & \\\end{align}\)
5. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}+\frac{2}{7}& = & \frac{-2}{5}x+2 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }+ \frac{ 60 }{ \color{blue}{210} })& = & (\frac{-84}{ \color{blue}{210} }x+\frac{420}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x+60& = & -84x+420 \\\Leftrightarrow & 35x \color{red}{+60} \color{blue}{-60} \color{blue}{+84x} & = & \color{red}{-84x} +420 \color{blue}{+84x} \color{blue}{-60} \\\Leftrightarrow & 119x& = & 360 \\\Leftrightarrow & \frac{119x}{ \color{red}{119} }& = & \frac{360}{119} \\\Leftrightarrow & x = \frac{360}{119} & & \\ & V = \left\{ \frac{360}{119} \right\} & \\\end{align}\)
6. \(\text{195 is het kleinste gemene veelvoud van 3, 13 en 5} \\ \begin{align} & \frac{x}{3}-\frac{4}{13}& = & \frac{7}{5}x+3 \\\Leftrightarrow & \color{blue}{195.} (\frac{65x}{ \color{blue}{195} }- \frac{ 60 }{ \color{blue}{195} })& = & (\frac{273}{ \color{blue}{195} }x+\frac{585}{ \color{blue}{195} }) \color{blue}{.195} \\\Leftrightarrow & 65x-60& = & 273x+585 \\\Leftrightarrow & 65x \color{red}{-60} \color{blue}{+60} \color{blue}{-273x} & = & \color{red}{273x} +585 \color{blue}{-273x} \color{blue}{+60} \\\Leftrightarrow & -208x& = & 645 \\\Leftrightarrow & \frac{-208x}{ \color{red}{-208} }& = & \frac{645}{-208} \\\Leftrightarrow & x = \frac{-645}{208} & & \\ & V = \left\{ \frac{-645}{208} \right\} & \\\end{align}\)
7. \(\text{30 is het kleinste gemene veelvoud van 3, 10 en 2} \\ \begin{align} & \frac{x}{3}+\frac{3}{10}& = & \frac{1}{2}x+7 \\\Leftrightarrow & \color{blue}{30.} (\frac{10x}{ \color{blue}{30} }+ \frac{ 9 }{ \color{blue}{30} })& = & (\frac{15}{ \color{blue}{30} }x+\frac{210}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 10x+9& = & 15x+210 \\\Leftrightarrow & 10x \color{red}{+9} \color{blue}{-9} \color{blue}{-15x} & = & \color{red}{15x} +210 \color{blue}{-15x} \color{blue}{-9} \\\Leftrightarrow & -5x& = & 201 \\\Leftrightarrow & \frac{-5x}{ \color{red}{-5} }& = & \frac{201}{-5} \\\Leftrightarrow & x = \frac{-201}{5} & & \\ & V = \left\{ \frac{-201}{5} \right\} & \\\end{align}\)
8. \(\text{182 is het kleinste gemene veelvoud van 7, 13 en 2} \\ \begin{align} & \frac{x}{7}-\frac{3}{13}& = & \frac{3}{2}x+7 \\\Leftrightarrow & \color{blue}{182.} (\frac{26x}{ \color{blue}{182} }- \frac{ 42 }{ \color{blue}{182} })& = & (\frac{273}{ \color{blue}{182} }x+\frac{1274}{ \color{blue}{182} }) \color{blue}{.182} \\\Leftrightarrow & 26x-42& = & 273x+1274 \\\Leftrightarrow & 26x \color{red}{-42} \color{blue}{+42} \color{blue}{-273x} & = & \color{red}{273x} +1274 \color{blue}{-273x} \color{blue}{+42} \\\Leftrightarrow & -247x& = & 1316 \\\Leftrightarrow & \frac{-247x}{ \color{red}{-247} }& = & \frac{1316}{-247} \\\Leftrightarrow & x = \frac{-1316}{247} & & \\ & V = \left\{ \frac{-1316}{247} \right\} & \\\end{align}\)
9. \(\text{42 is het kleinste gemene veelvoud van 7, 7 en 6} \\ \begin{align} & \frac{x}{7}+\frac{2}{7}& = & \frac{5}{6}x-5 \\\Leftrightarrow & \color{blue}{42.} (\frac{6x}{ \color{blue}{42} }+ \frac{ 12 }{ \color{blue}{42} })& = & (\frac{35}{ \color{blue}{42} }x-\frac{210}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 6x+12& = & 35x-210 \\\Leftrightarrow & 6x \color{red}{+12} \color{blue}{-12} \color{blue}{-35x} & = & \color{red}{35x} -210 \color{blue}{-35x} \color{blue}{-12} \\\Leftrightarrow & -29x& = & -222 \\\Leftrightarrow & \frac{-29x}{ \color{red}{-29} }& = & \frac{-222}{-29} \\\Leftrightarrow & x = \frac{222}{29} & & \\ & V = \left\{ \frac{222}{29} \right\} & \\\end{align}\)
10. \(\text{56 is het kleinste gemene veelvoud van 7, 8 en 2} \\ \begin{align} & \frac{x}{7}-\frac{5}{8}& = & \frac{7}{2}x+7 \\\Leftrightarrow & \color{blue}{56.} (\frac{8x}{ \color{blue}{56} }- \frac{ 35 }{ \color{blue}{56} })& = & (\frac{196}{ \color{blue}{56} }x+\frac{392}{ \color{blue}{56} }) \color{blue}{.56} \\\Leftrightarrow & 8x-35& = & 196x+392 \\\Leftrightarrow & 8x \color{red}{-35} \color{blue}{+35} \color{blue}{-196x} & = & \color{red}{196x} +392 \color{blue}{-196x} \color{blue}{+35} \\\Leftrightarrow & -188x& = & 427 \\\Leftrightarrow & \frac{-188x}{ \color{red}{-188} }& = & \frac{427}{-188} \\\Leftrightarrow & x = \frac{-427}{188} & & \\ & V = \left\{ \frac{-427}{188} \right\} & \\\end{align}\)
11. \(\text{12 is het kleinste gemene veelvoud van 4, 12 en 3} \\ \begin{align} & \frac{x}{4}-\frac{5}{12}& = & \frac{7}{3}x-3 \\\Leftrightarrow & \color{blue}{12.} (\frac{3x}{ \color{blue}{12} }- \frac{ 5 }{ \color{blue}{12} })& = & (\frac{28}{ \color{blue}{12} }x-\frac{36}{ \color{blue}{12} }) \color{blue}{.12} \\\Leftrightarrow & 3x-5& = & 28x-36 \\\Leftrightarrow & 3x \color{red}{-5} \color{blue}{+5} \color{blue}{-28x} & = & \color{red}{28x} -36 \color{blue}{-28x} \color{blue}{+5} \\\Leftrightarrow & -25x& = & -31 \\\Leftrightarrow & \frac{-25x}{ \color{red}{-25} }& = & \frac{-31}{-25} \\\Leftrightarrow & x = \frac{31}{25} & & \\ & V = \left\{ \frac{31}{25} \right\} & \\\end{align}\)
12. \(\text{70 is het kleinste gemene veelvoud van 5, 7 en 2} \\ \begin{align} & \frac{x}{5}+\frac{2}{7}& = & \frac{1}{2}x-6 \\\Leftrightarrow & \color{blue}{70.} (\frac{14x}{ \color{blue}{70} }+ \frac{ 20 }{ \color{blue}{70} })& = & (\frac{35}{ \color{blue}{70} }x-\frac{420}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 14x+20& = & 35x-420 \\\Leftrightarrow & 14x \color{red}{+20} \color{blue}{-20} \color{blue}{-35x} & = & \color{red}{35x} -420 \color{blue}{-35x} \color{blue}{-20} \\\Leftrightarrow & -21x& = & -440 \\\Leftrightarrow & \frac{-21x}{ \color{red}{-21} }& = & \frac{-440}{-21} \\\Leftrightarrow & x = \frac{440}{21} & & \\ & V = \left\{ \frac{440}{21} \right\} & \\\end{align}\)