Vgln. eerste graad (reeks 4)

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1. \(\frac{x}{5}-\frac{3}{14}=\frac{-5}{6}x+7\)
2. \(\frac{x}{3}-\frac{2}{11}=\frac{3}{4}x-4\)
3. \(\frac{x}{7}+\frac{5}{14}=\frac{3}{2}x-2\)
4. \(\frac{x}{2}+\frac{4}{7}=\frac{-2}{3}x+1\)
5. \(\frac{x}{4}-\frac{3}{13}=\frac{6}{5}x-8\)
6. \(\frac{x}{7}-\frac{2}{11}=\frac{1}{2}x-6\)
7. \(\frac{x}{3}+\frac{3}{13}=\frac{6}{5}x-2\)
8. \(\frac{x}{3}+\frac{5}{16}=\frac{-3}{4}x+5\)
9. \(\frac{x}{6}+\frac{2}{7}=\frac{1}{5}x+5\)
10. \(\frac{x}{2}+\frac{5}{16}=\frac{-5}{3}x+5\)
11. \(\frac{x}{2}-\frac{5}{13}=\frac{-5}{3}x-6\)
12. \(\frac{x}{3}-\frac{4}{9}=\frac{1}{2}x+7\)

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

Verbetersleutel

1. \(\text{210 is het kleinste gemene veelvoud van 5, 14 en 6} \\ \begin{align} & \frac{x}{5}-\frac{3}{14}& = & \frac{-5}{6}x+7 \\\Leftrightarrow & \color{blue}{210.} (\frac{42x}{ \color{blue}{210} }- \frac{ 45 }{ \color{blue}{210} })& = & (\frac{-175}{ \color{blue}{210} }x+\frac{1470}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 42x-45& = & -175x+1470 \\\Leftrightarrow & 42x \color{red}{-45} \color{blue}{+45} \color{blue}{+175x} & = & \color{red}{-175x} +1470 \color{blue}{+175x} \color{blue}{+45} \\\Leftrightarrow & 217x& = & 1515 \\\Leftrightarrow & \frac{217x}{ \color{red}{217} }& = & \frac{1515}{217} \\\Leftrightarrow & x = \frac{1515}{217} & & \\ & V = \left\{ \frac{1515}{217} \right\} & \\\end{align}\)
2. \(\text{132 is het kleinste gemene veelvoud van 3, 11 en 4} \\ \begin{align} & \frac{x}{3}-\frac{2}{11}& = & \frac{3}{4}x-4 \\\Leftrightarrow & \color{blue}{132.} (\frac{44x}{ \color{blue}{132} }- \frac{ 24 }{ \color{blue}{132} })& = & (\frac{99}{ \color{blue}{132} }x-\frac{528}{ \color{blue}{132} }) \color{blue}{.132} \\\Leftrightarrow & 44x-24& = & 99x-528 \\\Leftrightarrow & 44x \color{red}{-24} \color{blue}{+24} \color{blue}{-99x} & = & \color{red}{99x} -528 \color{blue}{-99x} \color{blue}{+24} \\\Leftrightarrow & -55x& = & -504 \\\Leftrightarrow & \frac{-55x}{ \color{red}{-55} }& = & \frac{-504}{-55} \\\Leftrightarrow & x = \frac{504}{55} & & \\ & V = \left\{ \frac{504}{55} \right\} & \\\end{align}\)
3. \(\text{14 is het kleinste gemene veelvoud van 7, 14 en 2} \\ \begin{align} & \frac{x}{7}+\frac{5}{14}& = & \frac{3}{2}x-2 \\\Leftrightarrow & \color{blue}{14.} (\frac{2x}{ \color{blue}{14} }+ \frac{ 5 }{ \color{blue}{14} })& = & (\frac{21}{ \color{blue}{14} }x-\frac{28}{ \color{blue}{14} }) \color{blue}{.14} \\\Leftrightarrow & 2x+5& = & 21x-28 \\\Leftrightarrow & 2x \color{red}{+5} \color{blue}{-5} \color{blue}{-21x} & = & \color{red}{21x} -28 \color{blue}{-21x} \color{blue}{-5} \\\Leftrightarrow & -19x& = & -33 \\\Leftrightarrow & \frac{-19x}{ \color{red}{-19} }& = & \frac{-33}{-19} \\\Leftrightarrow & x = \frac{33}{19} & & \\ & V = \left\{ \frac{33}{19} \right\} & \\\end{align}\)
4. \(\text{42 is het kleinste gemene veelvoud van 2, 7 en 3} \\ \begin{align} & \frac{x}{2}+\frac{4}{7}& = & \frac{-2}{3}x+1 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }+ \frac{ 24 }{ \color{blue}{42} })& = & (\frac{-28}{ \color{blue}{42} }x+\frac{42}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x+24& = & -28x+42 \\\Leftrightarrow & 21x \color{red}{+24} \color{blue}{-24} \color{blue}{+28x} & = & \color{red}{-28x} +42 \color{blue}{+28x} \color{blue}{-24} \\\Leftrightarrow & 49x& = & 18 \\\Leftrightarrow & \frac{49x}{ \color{red}{49} }& = & \frac{18}{49} \\\Leftrightarrow & x = \frac{18}{49} & & \\ & V = \left\{ \frac{18}{49} \right\} & \\\end{align}\)
5. \(\text{260 is het kleinste gemene veelvoud van 4, 13 en 5} \\ \begin{align} & \frac{x}{4}-\frac{3}{13}& = & \frac{6}{5}x-8 \\\Leftrightarrow & \color{blue}{260.} (\frac{65x}{ \color{blue}{260} }- \frac{ 60 }{ \color{blue}{260} })& = & (\frac{312}{ \color{blue}{260} }x-\frac{2080}{ \color{blue}{260} }) \color{blue}{.260} \\\Leftrightarrow & 65x-60& = & 312x-2080 \\\Leftrightarrow & 65x \color{red}{-60} \color{blue}{+60} \color{blue}{-312x} & = & \color{red}{312x} -2080 \color{blue}{-312x} \color{blue}{+60} \\\Leftrightarrow & -247x& = & -2020 \\\Leftrightarrow & \frac{-247x}{ \color{red}{-247} }& = & \frac{-2020}{-247} \\\Leftrightarrow & x = \frac{2020}{247} & & \\ & V = \left\{ \frac{2020}{247} \right\} & \\\end{align}\)
6. \(\text{154 is het kleinste gemene veelvoud van 7, 11 en 2} \\ \begin{align} & \frac{x}{7}-\frac{2}{11}& = & \frac{1}{2}x-6 \\\Leftrightarrow & \color{blue}{154.} (\frac{22x}{ \color{blue}{154} }- \frac{ 28 }{ \color{blue}{154} })& = & (\frac{77}{ \color{blue}{154} }x-\frac{924}{ \color{blue}{154} }) \color{blue}{.154} \\\Leftrightarrow & 22x-28& = & 77x-924 \\\Leftrightarrow & 22x \color{red}{-28} \color{blue}{+28} \color{blue}{-77x} & = & \color{red}{77x} -924 \color{blue}{-77x} \color{blue}{+28} \\\Leftrightarrow & -55x& = & -896 \\\Leftrightarrow & \frac{-55x}{ \color{red}{-55} }& = & \frac{-896}{-55} \\\Leftrightarrow & x = \frac{896}{55} & & \\ & V = \left\{ \frac{896}{55} \right\} & \\\end{align}\)
7. \(\text{195 is het kleinste gemene veelvoud van 3, 13 en 5} \\ \begin{align} & \frac{x}{3}+\frac{3}{13}& = & \frac{6}{5}x-2 \\\Leftrightarrow & \color{blue}{195.} (\frac{65x}{ \color{blue}{195} }+ \frac{ 45 }{ \color{blue}{195} })& = & (\frac{234}{ \color{blue}{195} }x-\frac{390}{ \color{blue}{195} }) \color{blue}{.195} \\\Leftrightarrow & 65x+45& = & 234x-390 \\\Leftrightarrow & 65x \color{red}{+45} \color{blue}{-45} \color{blue}{-234x} & = & \color{red}{234x} -390 \color{blue}{-234x} \color{blue}{-45} \\\Leftrightarrow & -169x& = & -435 \\\Leftrightarrow & \frac{-169x}{ \color{red}{-169} }& = & \frac{-435}{-169} \\\Leftrightarrow & x = \frac{435}{169} & & \\ & V = \left\{ \frac{435}{169} \right\} & \\\end{align}\)
8. \(\text{48 is het kleinste gemene veelvoud van 3, 16 en 4} \\ \begin{align} & \frac{x}{3}+\frac{5}{16}& = & \frac{-3}{4}x+5 \\\Leftrightarrow & \color{blue}{48.} (\frac{16x}{ \color{blue}{48} }+ \frac{ 15 }{ \color{blue}{48} })& = & (\frac{-36}{ \color{blue}{48} }x+\frac{240}{ \color{blue}{48} }) \color{blue}{.48} \\\Leftrightarrow & 16x+15& = & -36x+240 \\\Leftrightarrow & 16x \color{red}{+15} \color{blue}{-15} \color{blue}{+36x} & = & \color{red}{-36x} +240 \color{blue}{+36x} \color{blue}{-15} \\\Leftrightarrow & 52x& = & 225 \\\Leftrightarrow & \frac{52x}{ \color{red}{52} }& = & \frac{225}{52} \\\Leftrightarrow & x = \frac{225}{52} & & \\ & V = \left\{ \frac{225}{52} \right\} & \\\end{align}\)
9. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}+\frac{2}{7}& = & \frac{1}{5}x+5 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }+ \frac{ 60 }{ \color{blue}{210} })& = & (\frac{42}{ \color{blue}{210} }x+\frac{1050}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x+60& = & 42x+1050 \\\Leftrightarrow & 35x \color{red}{+60} \color{blue}{-60} \color{blue}{-42x} & = & \color{red}{42x} +1050 \color{blue}{-42x} \color{blue}{-60} \\\Leftrightarrow & -7x& = & 990 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{990}{-7} \\\Leftrightarrow & x = \frac{-990}{7} & & \\ & V = \left\{ \frac{-990}{7} \right\} & \\\end{align}\)
10. \(\text{48 is het kleinste gemene veelvoud van 2, 16 en 3} \\ \begin{align} & \frac{x}{2}+\frac{5}{16}& = & \frac{-5}{3}x+5 \\\Leftrightarrow & \color{blue}{48.} (\frac{24x}{ \color{blue}{48} }+ \frac{ 15 }{ \color{blue}{48} })& = & (\frac{-80}{ \color{blue}{48} }x+\frac{240}{ \color{blue}{48} }) \color{blue}{.48} \\\Leftrightarrow & 24x+15& = & -80x+240 \\\Leftrightarrow & 24x \color{red}{+15} \color{blue}{-15} \color{blue}{+80x} & = & \color{red}{-80x} +240 \color{blue}{+80x} \color{blue}{-15} \\\Leftrightarrow & 104x& = & 225 \\\Leftrightarrow & \frac{104x}{ \color{red}{104} }& = & \frac{225}{104} \\\Leftrightarrow & x = \frac{225}{104} & & \\ & V = \left\{ \frac{225}{104} \right\} & \\\end{align}\)
11. \(\text{78 is het kleinste gemene veelvoud van 2, 13 en 3} \\ \begin{align} & \frac{x}{2}-\frac{5}{13}& = & \frac{-5}{3}x-6 \\\Leftrightarrow & \color{blue}{78.} (\frac{39x}{ \color{blue}{78} }- \frac{ 30 }{ \color{blue}{78} })& = & (\frac{-130}{ \color{blue}{78} }x-\frac{468}{ \color{blue}{78} }) \color{blue}{.78} \\\Leftrightarrow & 39x-30& = & -130x-468 \\\Leftrightarrow & 39x \color{red}{-30} \color{blue}{+30} \color{blue}{+130x} & = & \color{red}{-130x} -468 \color{blue}{+130x} \color{blue}{+30} \\\Leftrightarrow & 169x& = & -438 \\\Leftrightarrow & \frac{169x}{ \color{red}{169} }& = & \frac{-438}{169} \\\Leftrightarrow & x = \frac{-438}{169} & & \\ & V = \left\{ \frac{-438}{169} \right\} & \\\end{align}\)
12. \(\text{18 is het kleinste gemene veelvoud van 3, 9 en 2} \\ \begin{align} & \frac{x}{3}-\frac{4}{9}& = & \frac{1}{2}x+7 \\\Leftrightarrow & \color{blue}{18.} (\frac{6x}{ \color{blue}{18} }- \frac{ 8 }{ \color{blue}{18} })& = & (\frac{9}{ \color{blue}{18} }x+\frac{126}{ \color{blue}{18} }) \color{blue}{.18} \\\Leftrightarrow & 6x-8& = & 9x+126 \\\Leftrightarrow & 6x \color{red}{-8} \color{blue}{+8} \color{blue}{-9x} & = & \color{red}{9x} +126 \color{blue}{-9x} \color{blue}{+8} \\\Leftrightarrow & -3x& = & 134 \\\Leftrightarrow & \frac{-3x}{ \color{red}{-3} }& = & \frac{134}{-3} \\\Leftrightarrow & x = \frac{-134}{3} & & \\ & V = \left\{ \frac{-134}{3} \right\} & \\\end{align}\)