Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{36 is het kleinste gemene veelvoud van 3, 9 en 4} \\ \begin{align} & \frac{x}{3}+\frac{4}{9}& = & \frac{-3}{4}x+7 \\\Leftrightarrow & \color{blue}{36.} (\frac{12x}{ \color{blue}{36} }+ \frac{ 16 }{ \color{blue}{36} })& = & (\frac{-27}{ \color{blue}{36} }x+\frac{252}{ \color{blue}{36} }) \color{blue}{.36} \\\Leftrightarrow & 12x+16& = & -27x+252 \\\Leftrightarrow & 12x \color{red}{+16} \color{blue}{-16} \color{blue}{+27x} & = & \color{red}{-27x} +252 \color{blue}{+27x} \color{blue}{-16} \\\Leftrightarrow & 39x& = & 236 \\\Leftrightarrow & \frac{39x}{ \color{red}{39} }& = & \frac{236}{39} \\\Leftrightarrow & x = \frac{236}{39} & & \\ & V = \left\{ \frac{236}{39} \right\} & \\\end{align}\)
2. \(\text{20 is het kleinste gemene veelvoud van 4, 10 en 5} \\ \begin{align} & \frac{x}{4}+\frac{3}{10}& = & \frac{-4}{5}x-4 \\\Leftrightarrow & \color{blue}{20.} (\frac{5x}{ \color{blue}{20} }+ \frac{ 6 }{ \color{blue}{20} })& = & (\frac{-16}{ \color{blue}{20} }x-\frac{80}{ \color{blue}{20} }) \color{blue}{.20} \\\Leftrightarrow & 5x+6& = & -16x-80 \\\Leftrightarrow & 5x \color{red}{+6} \color{blue}{-6} \color{blue}{+16x} & = & \color{red}{-16x} -80 \color{blue}{+16x} \color{blue}{-6} \\\Leftrightarrow & 21x& = & -86 \\\Leftrightarrow & \frac{21x}{ \color{red}{21} }& = & \frac{-86}{21} \\\Leftrightarrow & x = \frac{-86}{21} & & \\ & V = \left\{ \frac{-86}{21} \right\} & \\\end{align}\)
3. \(\text{390 is het kleinste gemene veelvoud van 5, 13 en 6} \\ \begin{align} & \frac{x}{5}-\frac{2}{13}& = & \frac{5}{6}x-3 \\\Leftrightarrow & \color{blue}{390.} (\frac{78x}{ \color{blue}{390} }- \frac{ 60 }{ \color{blue}{390} })& = & (\frac{325}{ \color{blue}{390} }x-\frac{1170}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 78x-60& = & 325x-1170 \\\Leftrightarrow & 78x \color{red}{-60} \color{blue}{+60} \color{blue}{-325x} & = & \color{red}{325x} -1170 \color{blue}{-325x} \color{blue}{+60} \\\Leftrightarrow & -247x& = & -1110 \\\Leftrightarrow & \frac{-247x}{ \color{red}{-247} }& = & \frac{-1110}{-247} \\\Leftrightarrow & x = \frac{1110}{247} & & \\ & V = \left\{ \frac{1110}{247} \right\} & \\\end{align}\)
4. \(\text{120 is het kleinste gemene veelvoud van 6, 8 en 5} \\ \begin{align} & \frac{x}{6}+\frac{5}{8}& = & \frac{2}{5}x+3 \\\Leftrightarrow & \color{blue}{120.} (\frac{20x}{ \color{blue}{120} }+ \frac{ 75 }{ \color{blue}{120} })& = & (\frac{48}{ \color{blue}{120} }x+\frac{360}{ \color{blue}{120} }) \color{blue}{.120} \\\Leftrightarrow & 20x+75& = & 48x+360 \\\Leftrightarrow & 20x \color{red}{+75} \color{blue}{-75} \color{blue}{-48x} & = & \color{red}{48x} +360 \color{blue}{-48x} \color{blue}{-75} \\\Leftrightarrow & -28x& = & 285 \\\Leftrightarrow & \frac{-28x}{ \color{red}{-28} }& = & \frac{285}{-28} \\\Leftrightarrow & x = \frac{-285}{28} & & \\ & V = \left\{ \frac{-285}{28} \right\} & \\\end{align}\)
5. \(\text{70 is het kleinste gemene veelvoud van 2, 14 en 5} \\ \begin{align} & \frac{x}{2}-\frac{3}{14}& = & \frac{6}{5}x-3 \\\Leftrightarrow & \color{blue}{70.} (\frac{35x}{ \color{blue}{70} }- \frac{ 15 }{ \color{blue}{70} })& = & (\frac{84}{ \color{blue}{70} }x-\frac{210}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 35x-15& = & 84x-210 \\\Leftrightarrow & 35x \color{red}{-15} \color{blue}{+15} \color{blue}{-84x} & = & \color{red}{84x} -210 \color{blue}{-84x} \color{blue}{+15} \\\Leftrightarrow & -49x& = & -195 \\\Leftrightarrow & \frac{-49x}{ \color{red}{-49} }& = & \frac{-195}{-49} \\\Leftrightarrow & x = \frac{195}{49} & & \\ & V = \left\{ \frac{195}{49} \right\} & \\\end{align}\)
6. \(\text{70 is het kleinste gemene veelvoud van 5, 7 en 2} \\ \begin{align} & \frac{x}{5}+\frac{3}{7}& = & \frac{1}{2}x+5 \\\Leftrightarrow & \color{blue}{70.} (\frac{14x}{ \color{blue}{70} }+ \frac{ 30 }{ \color{blue}{70} })& = & (\frac{35}{ \color{blue}{70} }x+\frac{350}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 14x+30& = & 35x+350 \\\Leftrightarrow & 14x \color{red}{+30} \color{blue}{-30} \color{blue}{-35x} & = & \color{red}{35x} +350 \color{blue}{-35x} \color{blue}{-30} \\\Leftrightarrow & -21x& = & 320 \\\Leftrightarrow & \frac{-21x}{ \color{red}{-21} }& = & \frac{320}{-21} \\\Leftrightarrow & x = \frac{-320}{21} & & \\ & V = \left\{ \frac{-320}{21} \right\} & \\\end{align}\)
7. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}+\frac{3}{7}& = & \frac{1}{3}x-4 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }+ \frac{ 36 }{ \color{blue}{84} })& = & (\frac{28}{ \color{blue}{84} }x-\frac{336}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x+36& = & 28x-336 \\\Leftrightarrow & 21x \color{red}{+36} \color{blue}{-36} \color{blue}{-28x} & = & \color{red}{28x} -336 \color{blue}{-28x} \color{blue}{-36} \\\Leftrightarrow & -7x& = & -372 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{-372}{-7} \\\Leftrightarrow & x = \frac{372}{7} & & \\ & V = \left\{ \frac{372}{7} \right\} & \\\end{align}\)
8. \(\text{30 is het kleinste gemene veelvoud van 5, 6 en 3} \\ \begin{align} & \frac{x}{5}+\frac{5}{6}& = & \frac{-2}{3}x-4 \\\Leftrightarrow & \color{blue}{30.} (\frac{6x}{ \color{blue}{30} }+ \frac{ 25 }{ \color{blue}{30} })& = & (\frac{-20}{ \color{blue}{30} }x-\frac{120}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 6x+25& = & -20x-120 \\\Leftrightarrow & 6x \color{red}{+25} \color{blue}{-25} \color{blue}{+20x} & = & \color{red}{-20x} -120 \color{blue}{+20x} \color{blue}{-25} \\\Leftrightarrow & 26x& = & -145 \\\Leftrightarrow & \frac{26x}{ \color{red}{26} }& = & \frac{-145}{26} \\\Leftrightarrow & x = \frac{-145}{26} & & \\ & V = \left\{ \frac{-145}{26} \right\} & \\\end{align}\)
9. \(\text{84 is het kleinste gemene veelvoud van 3, 7 en 4} \\ \begin{align} & \frac{x}{3}-\frac{4}{7}& = & \frac{-7}{4}x+7 \\\Leftrightarrow & \color{blue}{84.} (\frac{28x}{ \color{blue}{84} }- \frac{ 48 }{ \color{blue}{84} })& = & (\frac{-147}{ \color{blue}{84} }x+\frac{588}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 28x-48& = & -147x+588 \\\Leftrightarrow & 28x \color{red}{-48} \color{blue}{+48} \color{blue}{+147x} & = & \color{red}{-147x} +588 \color{blue}{+147x} \color{blue}{+48} \\\Leftrightarrow & 175x& = & 636 \\\Leftrightarrow & \frac{175x}{ \color{red}{175} }& = & \frac{636}{175} \\\Leftrightarrow & x = \frac{636}{175} & & \\ & V = \left\{ \frac{636}{175} \right\} & \\\end{align}\)
10. \(\text{14 is het kleinste gemene veelvoud van 7, 7 en 2} \\ \begin{align} & \frac{x}{7}-\frac{4}{7}& = & \frac{7}{2}x+7 \\\Leftrightarrow & \color{blue}{14.} (\frac{2x}{ \color{blue}{14} }- \frac{ 8 }{ \color{blue}{14} })& = & (\frac{49}{ \color{blue}{14} }x+\frac{98}{ \color{blue}{14} }) \color{blue}{.14} \\\Leftrightarrow & 2x-8& = & 49x+98 \\\Leftrightarrow & 2x \color{red}{-8} \color{blue}{+8} \color{blue}{-49x} & = & \color{red}{49x} +98 \color{blue}{-49x} \color{blue}{+8} \\\Leftrightarrow & -47x& = & 106 \\\Leftrightarrow & \frac{-47x}{ \color{red}{-47} }& = & \frac{106}{-47} \\\Leftrightarrow & x = \frac{-106}{47} & & \\ & V = \left\{ \frac{-106}{47} \right\} & \\\end{align}\)
11. \(\text{70 is het kleinste gemene veelvoud van 2, 7 en 5} \\ \begin{align} & \frac{x}{2}+\frac{3}{7}& = & \frac{1}{5}x-7 \\\Leftrightarrow & \color{blue}{70.} (\frac{35x}{ \color{blue}{70} }+ \frac{ 30 }{ \color{blue}{70} })& = & (\frac{14}{ \color{blue}{70} }x-\frac{490}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 35x+30& = & 14x-490 \\\Leftrightarrow & 35x \color{red}{+30} \color{blue}{-30} \color{blue}{-14x} & = & \color{red}{14x} -490 \color{blue}{-14x} \color{blue}{-30} \\\Leftrightarrow & 21x& = & -520 \\\Leftrightarrow & \frac{21x}{ \color{red}{21} }& = & \frac{-520}{21} \\\Leftrightarrow & x = \frac{-520}{21} & & \\ & V = \left\{ \frac{-520}{21} \right\} & \\\end{align}\)
12. \(\text{24 is het kleinste gemene veelvoud van 2, 8 en 3} \\ \begin{align} & \frac{x}{2}-\frac{5}{8}& = & \frac{-2}{3}x-2 \\\Leftrightarrow & \color{blue}{24.} (\frac{12x}{ \color{blue}{24} }- \frac{ 15 }{ \color{blue}{24} })& = & (\frac{-16}{ \color{blue}{24} }x-\frac{48}{ \color{blue}{24} }) \color{blue}{.24} \\\Leftrightarrow & 12x-15& = & -16x-48 \\\Leftrightarrow & 12x \color{red}{-15} \color{blue}{+15} \color{blue}{+16x} & = & \color{red}{-16x} -48 \color{blue}{+16x} \color{blue}{+15} \\\Leftrightarrow & 28x& = & -33 \\\Leftrightarrow & \frac{28x}{ \color{red}{28} }& = & \frac{-33}{28} \\\Leftrightarrow & x = \frac{-33}{28} & & \\ & V = \left\{ \frac{-33}{28} \right\} & \\\end{align}\)